Reduced Density Matrix Functional Theory at Finite ...

Reduced Density Matrix Functional Theoryat Finite Temperature

Dissertation

zu Erlangung des akademischen Grades

Doctor rerum naturalium

vorgelegt von

Tim Baldsiefen

Institut fur Theoretische Physik

Freie Universitat Berlin

Oktober 2012

Erstgutacher Prof Dr E K U Gross

Zweitgutachter Prof Dr K Schotte

Datum der Disputation 31102012

Eigenstandigkeitserklarung

Hiermit versichere ich dass ich die vorliegende Dissertationsschrift mit dem Titel ldquoReduced Den-

sity Matrix Functional Theory at Finite Temperaturerdquo selbstandig und ohne die Benutzung andererals der angegeben Hilfsmittel angefertigt habe Die Stellen der Arbeit die dem Wortlaut oder demSinn nach anderen Werken entnommen sind wurden unter Angabe der Quelle kenntlich gemacht

Berlin den 10102012

Abstract

Density functional theory (DFT) is highly successful in many fields of research There are how-ever areas in which its performance is rather limited An important example is the description ofthermodynamical variables of a quantum system in thermodynamical equilibrium Although thefinite-temperature version of DFT (FT-DFT) rests on a firm theoretical basis and is only one yearyounger than its brother groundstate DFT it has been successfully applied to only a few prob-lems Because FT-DFT like DFT is in principle exact these shortcomings can be attributed to thedifficulties of deriving valuable functionals for FT-DFT

In this thesis we are going to present an alternative theoretical description of quantum systems inthermal equilibrium It is based on the 1-reduced density matrix (1RDM) of the system rather thanon its density and will rather cumbersomly be called finite-temperature reduced density matrix func-tional theory (FT-RDMFT) Its zero-temperature counterpart (RDMFT) proved to be successful inseveral fields formerly difficult to address via DFT These fields include for example the calculationof dissociation energies or the calculation of the fundamental gap also for Mott insulators

This success is mainly due to the fact that the 1RDM carries more directly accessible ldquomany-bodyrdquo information than the density alone leading for example to an exact description of the kineticenergy functional This sparks the hope that a description of thermodynamical systems employingthe 1RDM via FT-RDMFT can yield an improvement over FT-DFT

Giving a short review of RDMFT and pointing out difficulties when describing spin-polarized sys-tems initiates our work We will then lay the theoretical framework for FT-RDMFT by proving therequired Hohenberg-Kohn-like theorems investigating and determining the domain of FT-RDMFTfunctionals and by deriving several properties of the exact functional Subsequently we will presenta perturbative method to iteratively construct approximate functionals for FT-RDMFT The min-imization of the corresponding first-order functional is shown to be equivalent to a solution of thefinite-temperature Hartree-Fock (FT-HF) equations We will then present a self-consistent mini-mization scheme much like the Kohn-Sham minimization scheme in DFT and show that it can alsobe employed to effectively and efficiently minimize functionals from RDMFT

Finally we will investigate the temperature-dependent homogeneous electron gas (HEG) employ-ing various techniques which include finite-temperature many-body perturbation theory (FT-MBPT)and FT-RDMFT We will focus on the description of the magnetic phase diagram and the temperature-dependent quasi-particle spectrum for collinear as well as chiral spin configurations

Table of Contents

List of Figures v

List of Tables vii

List of Abbreviations ix

1 Introduction 1

2 Mathematical prerequisites 521 Hamiltonian and spin notation 522 Statistical density operator (SDO) 523 1-reduced density matrix (1RDM) 624 Homogeneous electron gas (HEG) 8

3 Zero-temperature RDMFT 1131 Theoretical foundations 11

311 Exchange-correlation functionals 1332 The problem of the construction of an RDMFT-LDA 1433 BOW functional 1634 The problem of the description of magnetic systems 1935 Summary and outlook 22

4 Finite-temperature RDMFT (FT-RDMFT) 2341 Physical situation and standard quantum mechanical treatment 24

411 Properties of thermodynamic variables 2542 Mathematical foundation of FT-RDMFT 27

421 Existence of grand potential functional Ω[γ] 28422 Lieb formulation of the universal functional 29423 Kohn-Sham system for FT-RDMFT 31424 Functionals in FT-RDMFT 31425 Finite-temperature Hartree-Fock (FT-HF) 32

43 Properties of the universal functional 33431 Existence of minimum 33432 Lower semicontinuity 35433 Convexity 36

44 Eq-V-representability 3645 Relation to microcanonical and canonical formulations 3846 Summary and outlook 40

5 Properties of the exact FT-RDMFT functionals 4351 Negativity of the correlation functionals ΩcWc Sc 4352 Adiabatic connection formula 4453 Uniform coordiante scaling 4554 Temperature and interaction scaling 4855 Summary and outlook 49

i

TABLE OF CONTENTS

6 Constructing approximate functionals 5161 Why not use standard FT-MBPT 5162 Methodology of modified perturbation theory 53

621 Elimination of Veff 5463 Importance of eq-V-representability 5864 Scaling behaviour 5965 Summary and outlook 60

7 Numerical treatment 6171 Key idea of self-consistent minimization 6172 Effective Hamiltonian 6373 Temperature tensor 6474 Small step investigation 66

741 Occupation number (ON) contribution 68742 Natural orbital (NO) contribution 68

75 Convergence measures 6976 Sample calculations 70

761 Occupation number (ON) minimization 70762 Full minimization 72763 Summary and outlook 74

8 Applications 7581 Magnetic phase transitions in the HEG 7682 Homogeneous electron gas with contact interaction 7983 FT-MBPT 80

831 Exchange-only 81832 Exchange + RPA 83833 Exchange + RPA + second-order Born 86

84 FT-RDMFT 88841 Collinear spin configuration 89842 Planar spin spirals (PSS) 93

85 FT-HF dispersion relations 9886 Correlation in FT-RDMFT 100

861 DFT-LSDA correlation 100862 FT-DFT-RPA correlation 102863 Zero-temperature RDMFT correlation 103864 FT-RDMFT perturbation expansion 106

87 Summary and outlook 110

A APPENDIX 113A1 Probabilistic interpretation of the ONs of the 1RDM 113A2 Equilibrium ONs in general systems 114A3 Zero-temperature mapping between potentials and wavefunctions 115A4 Noninteracting grand canonical equilibrium 116A5 Feynman rules 118A6 Second-order Born diagram 119

B Bibliography 121

C Deutsche Kurzfassung 131

D Publications 133

ii

TABLE OF CONTENTS

E Acknowledgements 135

iii

List of Figures

21 Sketch of the spaces of ensemble-N-representable SDOs and 1RDMs and their embed-ment in a Banach space 6

31 Correlation energy of the 3D-HEG for various RDMFT functionals 1632 Occupation number dependence function f for the BOW and α functionals 1733 Momentum distributions for the 3D-HEG exact and approximate 1834 Correlation energy of the 2D-HEG for various RDMFT functionals 1835 Inner- and inter-spin-channel correlation in the 3D-HEG 2041 Physical situation referring to a grand canonical ensemble 2442 Illustration of the concepts of convexity and lower semicontinuity 2543 Merminrsquos theorem justifies FT-RDMFT 2844 Sketch of all relevant spaces of SDOs and 1RDMs and their embedment in a Banach

space 4045 Logical outline of Section 4 4151 Scaled hydrogen wavefunctions and densities 4552 Constraints for the correlation grand potential functional Ωc[γ] from uniform coordi-

nate scaling 4861 Goerling-Levy type perturbation in FT-RDMFT 5262 Second-order contributions to the interacting Greenrsquos function G(x ν xprime νprime) 5563 Importance of eq-V-representability in FT-RDMFT 5871 Iterative minimization of Ω[γ] by employing an effective noninteracting system 6272 Projected grand potential surfaces for a model system with constant temperature 6573 Projected grand potential surfaces for a model system with temperature tensor 6674 Self-consistent minimization scheme in FT-RDMFT 6775 ON-minimization of the α functional applied to LiH 7176 Full minimization scheme 7277 NO-minimization of the α functional applied to LiH 7378 NO-convergence for different effective temperatures 7481 Determination of eq-polarization 7782 Sketch of the expected qualitative behaviour of the phase diagram of the HEG in

collinear configuration 7883 HEG phase diagram from Stoner model 8084 Exchange and second-order Born universal functions 8285 HEG phase diagram from FT-MBPT exchange-only 8386 RPA free energy contributions 8487 Accuracy of extrapolated zero-temperature results 8588 HEG phase diagram from FT-MBPT exchange + RPA 8589 HEG phase diagram from FT-MBPT exchange + RPA + second-order Born 87810 Comparison of different free energy contributions from FT-MBPT at rs = 7au 88811 Temperature and density dependence of eq-polarization of HEG 90812 Temperature and polarization dependence of free energy contributions from first-order

FT-RDMFT 90813 Comparison of temperature and polarization dependence of free energy contributions

in FT-MBPT and FT-RDMFT 91814 HEG phase diagram from first-order FT-RDMFT collinear phases 92815 Sketch of q-dependence of Fermi surface of a fully polarized PSS state 95816 Temperature and wavevector dependence of free energy contributions of the PSS phase

in first-order FT-RDMFT 95817 Temperature-induced first-order phase transition of PSS phase in first-order FT-RDMFT 96818 Temperature and density dependence of eq-PSS amplitudes 97819 HEG phase diagram from first-order FT-RDMFT collinear and PSS phases 98

v

LIST OF FIGURES

820 Hartree-Fock dispersion relation for the collinear HEG in three and two dimensions 99821 Hartree-Fock dispersion relation for the PSS phase in the HEG 100822 HEG phase diagram from FT-RDMFT with DFT-LSDA correlation 102823 HEG phase diagram from FT-RDMFT with FT-DFT-RPA correlation 102824 Temperature dependence of momentum distributions 104825 HEG phase diagram from FT-RDMFT with BOW-TIE correlation 105826 Correlation-energy of the three-dimensional electron gas for the KAPPA-functional 109827 Momentum distributions from the KAPPA-functional 109828 HEG phase diagram from FT-RDMFT with KAPPA-TIE correlation 110

vi

List of Tables

31 Best parameters to describe the correlation energy of a HEG in three and two dimen-sions for the α and BOW functionals 21

32 Optimal parameters for the BOW-TIE functional 2261 Feynman graphs in FT-RDMFT 5481 General assumptions about behaviour of the phase diagram of the HEG in collinear

configuration 7882 Critical temperatures of the PSS phase 9783 Dependence of critical temperatures and Wigner-Seitz radii of the collinear phase on

the parameter in the α functional 10484 Dependence of critical temperatures of the PSS phase for the α functional 10485 Best parameters to describe the correlation-energy of a HEG with the KAPPA- func-

tional 10986 Best parameters to describe the correlation-energy of a HEG with the KAPPA-TIE-

functional 11087 Agreement of phase diagrams for different approximations with ldquoexactrdquo properties 112

vii

List of Abbreviations

1RDM 1-reduced density matrix


BB Buijse-BaerendsMuller functional

BBCN Corrections to the Buijse-BaerendsMuller functional

BOW Baldsiefen and Gross 2012 functional

BOW-TIE Trans-channel interaction energy version of the BOW functional

CDFT Current density functional theory

CGA xc functional by Csanyi Goedecker and Arias

COHSEX Coulomb hole plus screened exchange approximation

DFT Density functional theory

EPX Effective potential expansion method

EQ Equilibrium

FM Ferromagnetic

FP-LAPW Full potential linearily augmented plane waves

FT-DFT Finite-temperature density functional theory

FT-HF Finite-temperature Hartree-Fock

FT-MBPT Finite-temperature many-body perturbation theory

FT-RDMFT Finite-temperature reduced matrix functional theory

GGA Generalized gradient approximation

GS Ground state

GZ Gori-Giorgi and Ziesche

HEG Homogeneous electron gas

HF Hartree-Fock

HS Hilbert-Schmidt

KAPPA Baldsiefen 2010 FT-RDMFT correlation functional

KAPPA-TIE Trans-channel interaction energy version of the KAPPA functional

LDA Local density approximation

LRDMA Local 1RDM approximation

LSDA Local spin density approximation

MBPT Many-body perturbation theory

ix


MC Monte-Carlo

MLML-SIC Marques-Lathiotakis empirical functional

NO Natural orbitals

ON Occupation numbers

PBE Perdew-Burke-Ernzerhof functional

PM Paramagnetic

PNOF0 Piris natural orbital functionals

PP Partially polarized

PSS Planar spin spirals

PWCA Perdew and Wang parametrization of Monte-Carlo results for the HEG byCeperley and Alder

RDMFT Reduced density matrix functional theory

RPA Random phase approximation

SC Self-consistent

SC-DFT Superconductiong density functional theory

SDO Statistical density operator

SDW Spin density waves

SW Spin waves

TD-DFT Time-dependent density functional theory

XC Exchange-correlation

x

1 Introduction

Already 86 years have past since Schrodinger formulated the fundamental equation of quantum me-chanics [1] Although this equation is in principle capable of describing all nonrelativistic quantumsystems completely its solution is practically impossible for most many-particle systems The wave-function keeps track of all coordinates of all particles making it such a tremendeously complicatedobject that an actual numerical determination is out of reach of classical computation techniques atpresent and presumably will be so in the forseeable future

From a practical point of view it certainly appears inefficient to retain all information contained inthe wavefunction if in the end one is interested in averaged quantities like the energy for example Ittherefore suggests the search for objects other than the wavefunction and methods other than solvingthe Schrodinger equation to determine the groundstate (gs) properties of a quantum mechanicalsystem The most important relation utilized for this purpose is the variational principle singlingout the groundstate as the state which minimizes the energy of the system If one could describe theenergy of a state by an object of lower dimensionality than the wavefunction one could utilize thevariational principle to find the groundstate energy There is however one important nicety in thisprocedure In the minimization process one has to know which objects are physical ie stem from awavefunction of the correct symmetry This problem prevents one from using the variational principlein combination with the Greenrsquos function or the 2-reduced density matrix [2] which otherwise wouldbe able to yield the exact gs energy

On the other hand the conditions for the density to be physical are known and rather sim-ple namely the requirements of semipositivity and normalizability It is by virtue of the work ofHohenberg and Kohn [3] that we know that the density also contains all the information needed todetermine the groundstate energy The formulation of a density functional theory (DFT) is thereforewell founded

The cost one has to pay for the lower dimensionality of the density is that the functional de-pendence on the density of the observables is generally not known and has to be approximated Amajor improvement in the context of DFT was the introduction of a Kohn-Sham system [4] ie anoninteracting system having the same groundstate density as the interacting one The functionalfor the energy of the interacting system then consists of four parts These are the functional forthe noninteracting kinetic energy of the Kohn-Sham system the Hartree energy functional describ-ing the classical part of the interaction energy the exchange energy functional paying tribute toPaulirsquos exclusion principle and finally the remaining part the correlation energy functional Findingapproximations for the correlation part is the main obstacle in DFT Two of the most prominentapproximations the local density approximation (LDA) [3] and the generalized gradient approxi-mation (GGA) [5] show remarkable success in many different applications eg in determining gsenergies and structure constants Both approximations are based on Monte-Carlo (MC) results forthe homogeneous electron gas (HEG) which underlines the importance of this simple model system

Because of the success of the original formulation of DFT a wide range of extensions has emergedin the past decades expanding the realm of possible applications They can be cast roughly into twogroups The first one represents different varieties of Hamiltonians leading to eg spin-DFT [6 7]which is capable of describing spin-polarized systems current density functional theory (CDFT)[8] describing systems in external magnetic fields or superconducting DFT (SC-DFT) [9] treatingsuperconducting states The second group deals with problems which go beyond groundstate cal-culations like time-dependent DFT (TD-DFT) [10] describing the time evolution of a system orfinite-temperature DFT (FT-DFT) [11] trying to determine thermodynamic properties of quantumsystems in thermodynamic equilibrium

All these extensions of DFT are theoretically well founded but are not necessarily equally suc-cessful in their respective fields of application Two prominent examples for physical problemswhere DFT struggles are the calculation of dissociation energies [12 13] and the determination ofthe fundamental gap of Mott insulators Futhermore FT-DFT in general fails to achieve accura-cies comparable to those of gs-DFT This is unfortunate in view of the wide variety of possible

1135

1 INTRODUCTION

applications These include eg the description of phase transitions either magnetic [14 15] orsuperconducting [16 17] the description of warm matter [18] and hot plasmas [19 20 21] femto-chemistry at surfaces of solids [22] the description of shock waves in hot dense gases [23 24] or thecomposition of different phases of water in the interior of giant planets [25]

With regard to the zero-temperature case there are several approaches to remedy existing prob-lems One promising approach is reduced density matrix functional theory (RDMFT) which employsthe 1-reduced density matrix (1RDM) rather than the density as central variable In contrast toDFT this theory based on a theorem by Gilbert [26] is capable of describing systems subject toa nonlocal external potential Such potentials are sometimes used as mathematical tools [27 28]and are therefore an important consideration The theory of RDMFT is successful in calculatingdissociation energy curves [29 30 31 32] and predicting fundamental gaps for atoms and molecules[31 33] as well as for Mott insulators [34]

The main concern of this work motivated by the success of RDMFT at zero temperature is to laythe theoretical framework of a finite-temperature version of RDMFT (FT-RDMFT) and investigateits properties and performance

We have divided the thesis in the following wayAfter presenting a selection of mathematical prerequisites in Section 2 we shall give a short

review of RDMFT in Section 3 From an investigation of several existing approximations we willbe able to define a new exchange-correlation functional which is capable of describing the correlationenergy of a homogeneous electron gas to unprecedented accuracy It furthermore recovers importantqualitative features of the momentum distribution a capability most existing functionals are missingThen by using only Monte-Carlo results for the groundstate energy of a HEG we will be able todemonstrate that a certain class of correlation functionals in RDMFT is bound to fail when appliedto the spin-polarized HEG The investigation of this problem will give some valuable insight intothe relation of correlation contributions which stem from interactions of electrons of the same spincompared to electrons with different spin

In Section 4 we will lay the mathematical framework of FT-RDMFT This will include provinga Hohenberg-Kohn-like theorem as well as investigating the domain of functionals Special attentionis given to the question of equilibrium-V-representability This is how 1RDMs which come froman equilibrium state corresponding to a Hamiltonian with fixed two-particle interaction and someexternal potential can be characterized As we will point out repeatedly in this work this question isof more than just mathematical interest which also explains why its zero-temperature counterpartin DFT received considerable attention during the course of the theoretical development of DFT[35 36 37 38]

Also in FT-RDMFT some part of the free energy functional has to be approximated Accordinglythe success of FT-RDMFT depends strongly on the quality of these approximations To guide thedevelopment of correlation functionals in Section 5 we will derive various properties of the exactfree energy functional One major tool utilized for this is the concept of uniform coordinate scalingwhich proved to be very useful also in the context of DFT [39 40 41 42] and FT-DFT [43]

Utilizing the fact that in FT-RDMFT there is a Kohn-Sham system in Section 6 we will establisha methodology to construct FT-RDMFT correlation functionals using the powerful methods of finite-temperature many-body perturbation theory (FT-MBPT) By construction these functionals satisfymost exact properties derived in the previous section

Having laid a solid theoretical foundation in Section 7 we will then turn to the problem ofnumerical minimization of FT-RDMFT functionals We will develop a self-consistent minimizationscheme which will be able to effectively and efficiently minimize functionals from FT-RDMFT aswell as RDMFT

In the final Section 8 we will investigate the effect of several FT-RDMFT functionals on themagnetic phase diagram of the HEG The results will be compared to results from Monte-Carlo cal-culations as well as FT-MBPT approximations We will furthermore be able to employ FT-RDMFTto calculate the finite-temperature Hartree-Fock dispersion relations for both a collinear as well as a

2135

1 INTRODUCTION

chiral spin configuration

We have chosen to make each chapter of this thesis relatively self contained We hope that thiswill increase its usefulness as a reference work and we are gladly willing to pay the price of a higherdegree of redundancy

Throughout this thesis unless mentioned otherwise we will use atomic units (e = me = ~ = 1)

3135

4135

2 Mathematical prerequisites

The theory of RDMFT both at zero as well as finite temperature to a large extent relies on thesame theoretical concepts Accordingly before proceeding with the investigation of RDMFT we aregoing to introduce some important mathematical requisites in the following

Section Description

21 We will define the Hamiltonian mostly used in this thesis especially focussing on spinnotation

2223

We will then define the statistical density operator (SDO) and 1RDM of a quantumsystem and investigate some of their properties

24 Finally we will briefly introduce the concept of the HEG It will become an importanttesting system for RDMFT at both zero as well as finite temperature

Related publications [44]

21 Hamiltonian and spin notation

In the course of this work we will focus mainly on systems subject to a two-particle interactionW and a generally nonlocal external potential Vext In combination with the kinetic energy partdescribed by T this yields the following Hamiltonian

H = T + W + Vext (21)

A subject of major interest will be temperature-driven magnetic phase transitions We will thereforehave to keep track of the spin quantum number Not to overload notation we will combine the spinindex σ with the spacial index r into the joint variable x = (σ r) Integrals over x then refer toa spin summation and a spacial integration

intdx =

sumσ

intd3r The second-quantized forms of the

operators introduced before then become

T =

intdx lim

xprimerarrx

(minusnabla

2

2

)Ψ+(xprime)Ψ(x) (22)

Vext =

intdxdxprimevext(x x

prime)Ψ+(xprime)Ψ(x) (23)

W =

intdxdxprimew(x xprime)Ψ+(xprime)Ψ+(x)Ψ(x)Ψ(xprime) (24)

where Ψ are the common fermionic field operators The most important type of interaction in thiswork is the Coulomb interaction whose kernel w(x xprime) is given by

w(x xprime) =1

|r minus rprime| (25)

22 Statistical density operator (SDO)

A quantum mechanical system is generally described by a SDO ie a weighted sum of projectionoperators on the underlying Hilbert space H

D =sum

wi|ψi〉〈ψi| 0 le wi le 1sum

i

wi = 1 (26)

where ψi constitutes a basis of H The set of all possible SDOs yielding a particle number N willbe denoted by DN

DN =D | D as defined in Eq (26) with TrDN = N

(27)

5135

Prerequisites

T

T1

DN

21a Hierarchy of spaces as definedin Eqs (29)(211) and (27)

T

TradicN

ΓN

21b Hierarchy of spaces as definedin Eqs (29)(223) and (222)

Figure 21 Sketch of the spaces of ensemble-N-representable SDOs and 1RDMs and their embedmentin a Banach space

where the particle number operatore N is defined as

N =

intdxΨ(x)+Ψ(x) (28)

By definition DN sub T where T is the set of all trace-class operators ie compact operators withfinite trace

T =A | A trace-class

(29)

To define a distance between SDOs we introduce the Hilbert-Schmidt (HS) norm middot HS

AHS =

radicTrA+A (210)

T together with middot HS therefore constitutes a Banach space ie a complete normed vector spacewhich we will simply denote by T in the following From Eq (26) we see that every D isin DN hasa HS-norm smaller or equal to 1 DN is therefore a subset of the norm closed ball T1 of radius 1

T1 =A | AHS le 1

(211)

It follows that DN sub T1 sub T as depicted in Figure 21a This property ie the embedment ofDN in a norm closed subset (T1) of a Banach space (T) will become crucial for the investigationof properties of the universal functional in FT-RDMFT in Section 43

23 1-reduced density matrix (1RDM)

The 1RDM γ(x xprime) of a system whose state is described by D is formally defined with the help ofthe field operators Ψ(x)

γ(x xprime) = TrDΨ+(xprime)Ψ(x) (212)

By definition γ(x xprime) is hermitean and can therefore be written in spectral representation as

γ(x xprime) =sum

i

niφlowasti (x

prime)φi(x) (213)

6135

Prerequisites

The eigenstates φi(x) of the 1RDM are called natural orbitals (NO) and the eigenvalues nioccupation numbers (ON) [45] It is sometimes desirable to treat spin and spacial variables separatelyTo this end we introduce a two-component (Pauli) spinor notation

Φi(r) =

(φi1(r)

φi2(r)

) (214)

where φiσ(r) = φi(x) = φi(σ r) (σ = 1 2) are the orbitals of Eq (213) The 1RDM can then bewritten as a matrix in spin space as

γ(r rprime) =sum

i

niΦdaggeri (r

prime)otimes Φi(r) (215)

=

(γ11(r r

prime) γ12(r rprime)

γ21(r rprime) γ22(r r

prime)

)=sum

i

ni

(φlowasti1(r

prime)φi1(r) φlowasti2(rprime)φi1(r)

φlowasti1(rprime)φi2(r) φlowasti2(r

prime)φi2(r)

)(216)

There are some cases in which one can treat different spin channels separately One of these specialcases is the particulary important situation of collinear spin configuration For these systems thenatural orbitals are so-called spin orbitals ie spinors containing only one spin component Wecharacterize these spin orbitals by an additional index

Φi1(r) =

(φi1(r)

0

)Φi2(r) =

(0

φi2(r)

)(217)

This then leads to a 1RDM where every 2x2 matrix in Eq (216) contains only one nonvanishingentry either the 11 or the 22 one Hence the complete 1RDM is diagonal wrt the spin coordinate

γσσprime(r rprime) = δσσprime

sum

i

niσφlowastiσ(r

prime)φiσ(r) (218)

where niσ are the occupation numbers of the special spinors Φiσ(r) of Eq (217) Another specialcase where this seperation is possible is the case of spin spiral states which will be dealt with inSection 842

Because we want to establish a functional theory based on a variational scheme over 1RDMs it isimportant to know which 1RDMs stem from a physical state D as defined in Eq(26) Fortunatelyas Coleman showed [46] this set of 1RDMs is simply characterized by the following constraints onthe ONs and NOs of the 1RDMs

Theorem 21 [ensemble-N-representability]Let γ(x xprime) =

sumi niφ

lowasti (x

prime)φi(x) There exists a SDO D isin DN so that TrDΨ+(xprime)Ψ(x) = γ(x xprime)if and only if

0 le ni le 1 (219)sum

i

ni = N (220)

φi forms a complete orthonormal set (221)

A 1RDM which fulfills these requirements is said to be ensemble-N-representable

The set of all ensemble-N-representable 1RDMs is denoted by ΓN

ΓN =γ | γ fulfills Eqs (219)(220) and (221)

(222)

7135

Prerequisites

In analogy to our investigation of SDOs we see that every ensemble-N-representable 1RDM γ isin ΓN

has a Hilbert-Schmidt norm smaller or equal toradicN ΓN is therefore embedded in the norm closed

ball TradicN of radius

radicN

TradicN =

A | AHS le

radicN

(223)

The hierarchy of sets is then ΓN sub TradicN sub T as depicted in Figure 21b

We show in Appendix A1 that an ON ni bears the following important information It reflectsthe contribution of its corresponding NO φi to the state of the system If one expands ψi inEq (26) in Slater determinants constructed from the NOs then niN is the relative frequency ofoccurrence of the orbital φi Eg a state consisting only of a single projection operator on a Slaterdeterminant of orbitals φi (a noninteracting nondegenerate groundstate) will yield ONs which areeither 0 or 1 On the other hand a thermodynamical equilibrium (eq) state will have contributionsfrom all possible eigenstates of the Hamiltonian Therefore every NO will contribute and all ONs willbe bigger 0 Furthermore they will be also smaller than 1 because otherwise that would imply thatstates not containing the orbital would not contribute to the equilibrium state A more detailedproof of this argument is presented in Appendix A2

Another important property of the 1RDM was shown by Lowdin in 1954 [45] If one uses the NOsas basis functions for the Slater determinants in a configuration interaction (CI) expansion then oneachieves the most rapid convergence This fact however cannot simply be exploited because thegs-1RDM is not known a priori

Finally following from Eq (212) we point out that the density n(x) is simply given as thediagonal of the 1RDM

n(x) = γ(x x) (224)

If one is interested in the spin-resolved density nσ which has to be done when considering systemssubject to magnetic fields then one can take only the spacial diagonal to get

nσ(r) = γσσ(r r) (225)

24 Homogeneous electron gas (HEG)

In the course of this work we will be concerned mostly with systems subject to Coulomb interactionsas defined in Eq (25) This interaction as well as the kinetic energy operator is spacially invariantie w(r rprime) = w(r minus rprime) If the external potential also fulfills this property then the full Hamilto-

nian commutes with the displacement operator J = eiP middotr with P being the momentum operatorConsequently all eigenstates of H can be chosen to be momentum eigenstates as well Accordinglythe Greenrsquos function as well as the 1RDM is also spacially invariant yielding a constant densityBecause of this constant density the system is usually termed homogeneous electron gas (HEG) TheHEG constitutes a central model system for the theoretical description of many particle quantumsystems and an extensive investigation of it can be found in Ref [47]

If one considers local potentials the requirement of spacial invariance can only be fulfilled bya constant potential Following from the Hohenberg-Kohn theorem [3] one can characterize theHEG completely by the density Rather than using the density itself it is common practice to usethe Wigner-Seitz radius rs instead It is given as the radius of a sphere of constant density whichcontains one electron

rs =

(3

4πn

)13

(226)

8135

Prerequisites

The Fermi wavevector kF the Fermi energy εF and the Fermi temperature TF are then defined as

kF =(3π2n

) 13 =

(9

4π

) 13

rminus1s (227)

εF =k2F2

(228)

TF =εFkB

(229)

where kB is Boltzmannrsquos constant with kB = 317 middot 10minus6(HaK) When considering spin-dependentsystems one has to use the full spin-resolved density For collinear spin configurations one can eitheruse the diagonal elements of the the spin-resolved densities nuarr and ndarr or one can use the averageddensity or Wigner-Seitz radius in combination with the polarization ξ

ξ =nuarr minus ndarrnuarr + ndarr

(230)

The Fermi variables from Eqs (227)-(229) can then also be defined for the different spin channels

nσ = n(1plusmn ξ) (231)

kFσ = kF (1plusmn ξ)1d (232)

εFσ = εF (1plusmn ξ)2d (233)

TFσ = TF (1plusmn ξ)2d (234)

where d denotes the number dimensions of the HEGThe gs-energy of the HEG has been calculated for the spin-polarized case in three dimensions

[48 49] and for the paramagnetic case in two dimensions [50 51] These results have also beenreadily parametrized [52 49] Because these results are very accurate the HEG is often used asa testing system in theoretical solid state physics Furthermore the 3D-results are employed inthe very successful local density approximation (LDA) of DFT where the correlation energy of anonuniform system Enu

c [n(r)] is calculated as the average of the correlation energy per particle ofthe HEG ec(n)

Enuc [n(r)] =

intdrn(r)ec(n(r)) (235)

The treatment of the HEG is greatly simplified by the fact that the external potential is only aconstant In the course of this work however we will also have to deal with nonlocal externalpotentials The requirement of spacial invariance then leads to the following form for Vext

Vext σσprime =

intdkvext σσprime(k)nσσprime(k) (236)

where vext σσprime(k) is the Fourier transform of vext σσprime(r minus rprime) and nσσprime(k) is the momentum densityoperator Gilbert showed [26] that for these potentials the density alone is not sufficient anymore andone has to resort to the 1RDM The ONs of a HEG describe what is usually called the momentumdistribution The determination of the gs-momentum distribution is complicated because the gs-energy is relatively insensitive to small changes in the ONs This then usually leads to big statisticalvariances in the determination of the gs-momentum distribution via Monte-Carlo calculations Itwas Takada and Kita [53] who for the case of the three-dimensional HEG derived a self-consistencyrelation between the momentum distribution and the correlation energy which allows to test theaccuracy of momentum distributions It also allows the development of an effective potential ex-pansion (EPX) method [54] to calculate the momentum distributions to high accuracy Gori-Giorgiand Ziesche [55] then derived a parametrization for the gs-momentum distribution of the HEG forrs lt 12 both checking against Monte-Carlo as well as EPX results

9135

10135

3 Zero-temperature RDMFT

DFT proved to be highly successful in the description of several properties of many-particle quantumsystems However although in principle exact it has to rely on approximations for the correlationenergy which will inevitably fail to describe all possible systems accurately An investigation of thefailures of DFT can then lead to a better understanding of the basic processes governing the quantumworld and stimulate the derivation of improved functionals and techniques

One example for a theory which rests on the same theoretical concepts as DFT is RDMFTemploying the 1RDM as central variable rather than the density The aim of this part of our workis to introduce the main concepts of RDMFT and discuss their accomplishments and shortcomingsand to propose possible improvements We will structure this section as the following

Section Description

31 We start this section by giving a short introduction to the theoretical foundations ofRDMFT reviewing several approximate functionals

32 The construction of an LDA for RDMFT is complicated by several difficulties We willreview some of these obstacles and propose improvements

33 One improvement will be the development of a novel correlation functional for RDMFTIt is shown to be able to reproduce the gs-energy of the HEG for a wide range ofdensities more accurately than most common RDMFT functionals It will furthermorebe able to reproduce the effects of depletion of low momentum states in the momentumdistribution as well as occupation of all high momentum states leading to a qualitativelybetter description of the gs-momentum distribution

34 Finally we will consider the description of magnetic systems Using only results fromMonte-Carlo calculations we will be able to show that a certain class of RDMFT func-tionals is intrinsically incapable of describing polarized systems accurately

Related publications [44 56]

31 Theoretical foundations

At zero temperature the system is from now on assumed to be in a nondegenerate groundstate sothat its SDO D is given by a single projection operator

D = |Ψgs〉〈Ψgs| (31)

The energy of the system is then given by

E = Ek[γ] + Vext[γ] +W [Γ(2)] (32)

Ek[γ] =

intdx lim

xprimerarrx

(minusnabla

2

2

)γ(xprime x) (33)

Vext[γ] =


prime)γ(xprime x) (34)

W [Γ(2)] =

intdxdxprimew(x xprime)Γ(2)(x xprimex xprime) (35)

where Γ(2) is the 2-reduced density matrix (2RDM) of the system defined as

Γ(2)(x1 x2x3 x4) =N(N minus 1)

2TrDΨ+(x4)Ψ

+(x3)Ψ+(x1)Ψ

+(x2) (36)

11135

ZT-RDMFT

The corresponding 1RDM γ(x xprime) can be calculated from Γ(2) via

γ(x xprime) =2

N minus 1

intdx2Γ

(2)(x x2xprime x2) (37)

We conclude that knowledge of Γ(2) is sufficient to calculate the corresponding energy exactly Onemight try to exploit this fact in combination with the variational principle to find the gs-energy ofthe system by a minimization of the energy functional as defined in Eq(32) wrt the 2RDM Onedifficulty arising in this procedure is the problem of ensemble-N-representability (see Section 23) ofthe 2RDM This is the question of how a 2RDM coming from a state of the correct symmetry viaEq (36) can be characterized Although the conditions are known in principle [57 58] they are toocomplicated to be used in a numerical procedure The apparent possiblity of an exact descriptionof a quantum gs led to an tremendous effort to approx-mate the exact conditions to be treated inpractical terms [59 60 61 62 63 64 2] So far however one was only successful in deriving severalsimple separate necessary or sufficient conditions but not an inclusive combination of both

On the other hand as we have seen in Section 23 the conditions for the 1RDM to be ensemble-N-representable are known exactly and rather simple (see Thm 21) Because the density is simplyrelated to the 1RDM via Eq 224 the Hohenberg-Kohn theorem justifies the incorporation of the1RDM as central variable

W [Γ(2)] =W [γ] (38)

As Gilbert showed in 1975 [26] the nonlocal nature of the 1RDM allows the treatment of nonlocalpotentials which sometimes appear in model Hamiltonians [27 28] This is a problem formerlyinaccessible via DFT

An important difference between DFT and RDMFT concerns the uniqueness of the mappingbetween external potentials and 1RDMs Assume eg the following two potentials

v(x xprime) = vHF (x xprime) (39)

vprime(x xprime) = vHF (x xprime) + αγgs(x x

prime) (310)

where vHF (x xprime) denotes the Hartree-Fock potential and γgs(x x

prime) is the corresponding gs-1RDMBoth potentials yield the same gs-determinant for systems with a discrete spectrum and sufficientlysmall α Therefore the mapping of external potentials to the corresponding gs-1RDMs is not uniqueHowever we were able to show in Appendix A3 that in general the one-to-one correspondencebetween potential and wavefunction persists if and only if there are no pinned ONs ie no ONsequal to 0 or 1 The case described above exhibits only pinned states and can therefore be be seenas an extreme case of our general result from Appendix A3 Following these considerations it isnow interesting to investigate the properties of the gs-1RDMs of an interacting system with regardto the distribution of their ONs

When dealing with Coulomb systems ie systems with a two-particle interaction with w(x xprime) =1|r minus rprime| the following important statement was shown [65]

Theorem 31 [Cusp condition]The gs-1RDM of a Coulomb system has an infinite number of occupied orbitals ie orbitals withON ni gt 0

Proof [of Theorem 31]The original proof is rather involved and we therefore give an alternative simpler version in thefollowingAs Kato showed in 1957 [66] the divergence of the Coulomb interaction for vanishing interparticledistances leads to a nucleus-electron as well as an electron-electron cusp in the gs-wavefunction Thisgs-wavefunction is an element of the general N-particle Hilbert space of the system and can therefore

12135

ZT-RDMFT

be expanded in any basis of this Hilbert space Furthermore we know that one can create an N-particle basis by constructing all possible N-particle Slater determinants of a fixed one-particle basisBecause in a Slater determinant a particular one-particle wavefunction ldquodoes not knowrdquo about thecoordinates of another the electron-electron cusp condition cannot be fulfilled by a superposition of afinite number of Slater determinants One therefore needs an infinite number of Slater determinantsto reproduce the cusp and therefore the gs-wavefunction of a Coulomb-system This argument isvalid for all one-particle bases including the set of NOs of the gs-1RDM If an infinite number ofN-particle wavefunctions contributes then the number of contributing one-particle NOs also has tobe infinite concluding the proof

We know from Sec 23 that a 1RDM with partially occupied ONs cannot be the 1RDM of anondegenerate gs Therefore in combination with Thm 31 we conclude that in RDMFT thereexists no Kohn-Sham system for Coulomb systems We whould like to point out however that adegenerate noninteracting system is capable of reproducing a 1RDM with partially occupied NOs[67] but the degree of degeneracy is equal to the number of partially occupied states which for aCoulomb system is infinite

The main task of RDMFT is now to find accurate approximations for the interaction energy interms of the 1RDM It is common practice to separate the classical Hartree energy EH and theexchange energy Ex which is induced by Paulirsquos inclusion principle The remainder will then becalled the correlation energy

W [γ] = EH [γ] + Ex[γ] + Ec[γ] (311)

EH [γ] =1

2

intdxdxprimew(x xprime)γ(x x)γ(xprime xprime) (312)

Ex[γ] = minus1

2

intdxdxprimew(x xprime)γ(x xprime)γ(xprime x) (313)

Ec[γ] =W [γ]minus EH [γ]minus Ex[γ] (314)

We see from Eq 33 that in contrast to DFT the kinetic energy functional is known exactly interms of the 1RDM Therefore the correlation energy in RDMFT does not contain any kineticcontribution This has some conceptual advantages for example the simpler behaviour of Ec[γ]under coordinate scaling (see Section 53)

Before we review several existing functionals in RDMFT we should like to mention the respectiveareas of success RDMFT is capable of describing several properties of small systems quite accuratelyThese include dissociation curves of several small open-shell molecules references of which will begiven in the next section Furthermore RDMFT was also shown to be able to predict correctlythe fundamental gap of small systems [68 69 33] as well as of strongly correlated Mott insulators[34 33] the latter being an extremely difficult task within DFT However the accurate descriptionof magnetic systems and their properties is not very satisfactory We will try to shed some light onthe intrinsic problems of an RDMFT description of these systems in Section 34 and propose someconceptual improvements

Starting with the derivation of the famous Muller functional [70] in 1984 there emerged severalfunctionals trying to incorporate the correlation energy in the framework of RDMFT The most com-mon class of functionals namely the exchange-correlation (xc) functionals Exc[γ] will be discussedin the following section

311 Exchange-correlation functionals

Most energy functionals in RDMFT try to incorporate correlation via a modification of the ex-change functional Eq (313) The resulting functionals are accordingly called exchange-correlation

13135

ZT-RDMFT

functionals and can generally be written as

Exc[γ] = Ex[γ] + Ec[γ] (315)

Exc[γ] = minus1

2

sum

ij

f(ni nj)

intdxdxprimew(x xprime)φlowasti (x

prime)φi(x)φlowastj (x)φj(x

prime) (316)

Choosing fx(ni nj) = ninj reproduces the exchange-only functional neglecting correlation com-pletely

The first approximation to f(ni nj) was done by Muller in 1984 [70] leading to fMueller(ni nj) =radicninj The same approximation was rederived almost 20 years later from a different ansatz by Buijse

and Baerends [71] and is therefore sometimes also called BB functional This Muller functional wasable to describe correctly the dissociation limit of several small dimers of open-shell atoms but itoverestimates the correlation energy quite considerably Goedecker and Umrigar [72] attributedthis fact to the inclusion of self-interaction terms in the Muller functional proposing a remedy byomitting the diagonal terms of fMueller(ni nj) They succeeded in improving correlation energiesbut failed to retain the good results for the dissociation limit [73] Inveigled by the simplicity of theform of the xc approximation and the primary success of the Muller functional Gritsenko Pernaland Baerends developed what is now known as the BBC1 BBC2 and BBC3 functionals [29] The keydifference in their approach is the different treatment of orbitals which were occupied or unoccupiedin a Hartree-Fock (HF) solution (termed strongly and weakly occupied in the following) In additionBBC2 and BBC3 effectively mix parts of the exchange and Muller functionals A similar approachwas taken by Piris et al [74] Their approach differs slightly in the distinction of strongly and weaklyoccupied orbitals removes parts of the self interaction (PNOF0) and tries to incorporate particle-hole symmetry (PNOF) These more elaborate functionals BBC123 and PNOF0 are capable ofreproducing good dissociation energies as well as correlation energies

Whereas the previous functionals are mainly derived from physical arguments Marques andLathiotakis [75] pursued a different way by proposing a two-parameter Pade form for f(ni nj)These parameters are then optimized to minimize the deviation of correlation energies when appliedto the molecules of the G2 and G2-1 sets The resulting functionals are called ML and ML-SICeither including or excluding self interaction The ML and ML-SIC functionals achieve an unprece-dented precision reaching the accuracy of second-order Moslashller-Plesset perturbation theory for thecalculation of the correlation energies An overview of several of these functionals as well as thecorresponding energies for many molecules can be found in Ref [76]

Another rather empirical functional which will become important in this work was derived bySharma et al [34] They realized that both the exchange-only functionals as well as the Mullerfunctional can be seen as instances of a more general functional namely the α or Power functionaldescribed by fα(ni nj) = (ninj)

α Because the Muller functional underestimates the correlationenergy one expects to improve the results for values of α between 05 and 1 This assumptionproved to be valid [31] leading to an accurate description of the dissociation energy curve of H2 (adiscussion of the performance of several DFT functionals for this problem can be found in [77])

So far we have only considered small finite systems Because the xc functional should in principlebe general it is now instructive to investigate how the different functionals perform for extendedsystems We therefore turn to the description of a HEG via RDMFT in the following section

32 The problem of the construction of an RDMFT-LDA

One reason for the wide acceptance of DFT as the method of choice for modern day quantumcomputations is the tremendous success of the LDA Because of this success one should like toderive a similar approximation in the framework of RDMFT But there arise some conceptualproblems

In DFT one only has to calculate the correlation-energy of the HEG as a function of the constantdensity n In RDMFT on the other hand one has to calculate it for all possible density matrices γ(rminus

14135

ZT-RDMFT

rprime) that are compatible with translational invariance ie for all possible momentum distributionsn(k) The correlation-energy per volume of the HEG in RDMFT therefore becomes a functionalεHEGc [n(k)] of n(k) rather than just a function of n Under the assumption that such a functionalwas accessible the local 1RDM approximation (LRDMA) of RDMFT is defined in the following way

ELRDMAc [γ(r rprime)] =

intd3RεHEG

c [γ(kR)] (317)

where ELRDMAc is the approximate correlation-energy of a nonuniform system and γ(kR) describes

the Wigner transform of the 1RDM of the nonuniform system under study

γ(kR) =

intd3sγ(R+ s2Rminus s2)eikmiddots (318)

This procedure is similar to the definition of the LDA in SC-DFT [78 9] While there are manyLDA constructions conceivable that correctly reduce to the homogeneous limit Eq (317) is theonly definition that correctly reproduces the correlation energy of a weakly inhomogeneous electrongas [78]

A parametrization of εHEGc [n(k)] has not been carried out so far and remains an important task

for the future An extension of the LRDMA fomalism to spin-dependent systems and systems at finitetemperature is conceptually straightforward However the explicit calculation of the temperature-dependent εHEG

c [n(k) T ] although in principle possible by means of path integral Monte-Carlotechniques [79] is rather involved due to the fermionic sign problem [80]

An alternative approach for further theoretical development in RDMFT and FT-RDMFT there-fore consists in the derivation of approximate correlation functionals for the HEG and their im-plementation in a (FT-)LRDMA We point out that a functional in FT-RDMFT should not onlyreproduce an accurate eq-free energy but also has to yield a good momentum distribution

As a first step we investigated the performance of several of the previously mentioned RDMFTfunctionals for the calculation of the correlation energy of the HEG The results are shown in Figure31 The CGA functional [81] which relies on an approximate tensor product expansion of the2RDM was derived explicitly to describe the HEG and succeeds to do so for high densities but failsfor low ones including the important range of metallic densities 1 lt rs lt 6 From the functionalsmentioned so far only the α functional is capable of describing the correlation energy over the wholerange of densities considerably accurately Worth mentioning is the fact that the energy of the HEGfor the Muller functional can be solved analytically so long as there are no fully occupied states Itcan be shown [82] that this is the case if and only if rs gt 577

To make any statements about whether or not a given functional reproduces an accurate mo-mentum distribution we will have to rely on the results by Gori-Giorgi and Ziesche [55] which weintroduced briefly in Section 24 Three important properties of the momentum distribution derivedfrom these results are listed in the following list

(1) Nonzero occupation of all high momentum states [83] which can be understood by the help ofthe electronic cusp condition [65 44]

(2) A discontinuity and rather symmetrical behaviour of the momentum distribution at the Fermilevel (as also suggested by Landau-Liquid theory)

(3) Depletion of low momentum states

All functionals mentioned so far succeed in recovering property (1) This is due to the large value

of the derivative partExc[γ]partni

for ni rarr 0 All but the MLML-SIC functionals show a divergence for thisderivative which will then lead to a partial occupation of all states The MLML-SIC functionalsexhibit only a very big but finite derivative Therefore states of very high momentum are not tobe expected to show partial occupation When it comes to the description of property (2) only

15135

ZT-RDMFT

10010101001

minus030

minus020

minus010

0

minus005

minus015

minus025

rs(au)

e c(Ha)

PWCABOW (α = 061)BBC1BBC2CGAMuller

MLα (α = 055)

Figure 31 Correlation energy of the three-dimensional electron gas The black dashed line denotesMonte-Carlo results in the PWCA [52] parametrization The other labels are explained in Sections311 and 33 For rs gt 1 which includes the metallic density range the BOW functional shows anunprecedented accuracy

the BBC and the PNOF functionals recover a discontinuity at the Fermi surface This is due tothe discontinuous change in f(ni nj) when changing from a strongly occupied state to a weaklyoccupied one We should like to point out that this discontinuous behaviour is not intrinsicallynecessary to produce a discontinuous momentum distribution We have managed to achieve the samefeature for model functionals which were partially like the exchange-only functional but shifted bya constant ie f test(ni nj) = (ninj + c) on an area around the Fermi level The discontinuitycreated by the BBC functionals shows wrong behaviour when changing rs Although expected todecrease with decreasing density it increases [84] Furthermore the behaviour around the Fermilevel is not symmetric Neither does it qualitatively resemble the Monte-Carlo results Property(3) a qualitatively correct depletion of low momentum states is not fulfilled by the investigatedfunctionals Only the BBC1 BBC2 and PNOF functionals show a small decrease of occupation butnot nearly as much as prevalent in the exact momentum distributions

We believe that the depletion of low momentum states is an important physical effect whichmust be recovered by an RDMFT functional Motivated by this credo we are going to design anappropriate functional in the following section

33 BOW functional

We wish to construct an xc functional which is capable of reproducing both the occupation ofhigh momentum states as well as the depletion of low momentum ones We will not focus on thediscontinuity at the Fermi level and propose it as a task for future studies

Considering the occupation of high momentum states we will be guided by the success of the αfunctional and include a term of (ninj)

α in our functional To achieve a depletion of low momentumstates we then require our functional to have a vanishing derivative for ni = nj = 1 In thisway it is possible to reduce the occupation number of fully occupied orbitals without changing theenergy As the exchange contribution is negative this excess charge can be used to lower the energyThe variational principle will therefore lead to a groundstate where the orbitals are never fullyoccupied A possible choice of functional with vanishing derivative for full occupation could then

16135

ZT-RDMFT

0

02

04

06

08

10

0 02 04 06 08 10

ninj

f

fHF

fBOW (05)fα(05)fBOW (07)fα(07)

Figure 32 Occupation number dependence function f for the BOW and α functionals for differentvalues of the parameter α The Hartree-Fock function is reproduced by both functionals with α = 1

be f(ni nj) = ((ninj)α minus αninj)(1 minus α) We found however that this choice underestimates the

correlation-energy considerably The partially occupied states are given too much influence on theenergy We therefore introduce a simple counter-term to decrease this effect It incorporates theinverse of the α functional leading to our final choice for the xc functional which we will call BOWfunctional in the following

EBOWxc [γα] = minus1

2

sum

ij

fBOW (ni nj α)



prime) (319)

fBOW (ni nj α) = (ninj)α minus αninj + αminus α(1minus ninj)1α (320)

As a necessary property the BOW-functional leads to the reproduction of the exchange-only func-tional for uncorrelated momentum distributions

fBOW (ni nj α)|ninj=0 = 0 (321)

fBOW (ni nj α)|ni=nj=1 = 1 (322)

Furthermore we recover the exchange-only functional by choosing α = 1 ie fBOW (ni nj 1) =fx(ni nj) We can now use the parameter α to tune the influence of correlation We showfBOW (ni nj α) and the function from the α-functional fα(ni nj α) = (ninj)

α for several val-ues of α in Figure 32 A decrease in α leads to a bow-like shape which lead to our choice of nameAs we show in Figure 31 with the right choice of α the BOW functional is capable of describingthe correlation energy for rs gt 1 very accurately exceeding the precision of all other functionals Itsperformance for higher densities becomes less superb but is still considerably accurate compared toits contestants

We can now turn to the actual problem we were concerned with in the beginning namely thedescription of the gs-momentum distribution As we can see in Figure 33 the BOW functional issuccessful in describing both occupation of high momentum states as well as depletion of low mo-mentum ones Because of the success in describing both energy quantitatively as well as momentumdistribution qualitatively for the three-dimensional (3D) HEG we also investigated the performance

17135

ZT-RDMFT

00 05 10 15 2000

02

04

06

08

10

kkF

n(kk

F)

rs = 05

rs = 1

GZBOW

GZBOW

00 05 10 15 2000

02

04

06

08

10

kkF

n(kk

F)

rs = 5

rs = 10

GZBOW

GZBOW

Figure 33 Momentum distributions of the HEG for rs isin 05 1 5 10au from the parametrizationby Gori-Giorgi and Ziesche (GZ) [55] and from the BOW functional for α = 056 Low momentumstate depletion high momentum state occupation and symmetrical behaviour around the Fermilevel are reproduced

10010101001

minus08

minus07

minus06

minus05

minus04

minus03

minus02

minus01

0

MCBOW (α = 06)BBC1BBC2CGAMuller

MLα (α = 06)

rs(au)

e c(Ha)

Figure 34 Correlation energy of the two-dimensional electron gas The black dashed line denotesMonte-Carlo results in the Attaccalite [50] parametrization The other labels are explained in Sec-tions 311 and 33

of the different functionals for the two-dimensional (2D) HEG The correlation energies of a 2D-HEGare known from Monte-Carlo calculations and are readily parametrized [50 51] We show the resultsin Figure 34 As one can see also for the 2D-HEG the BOW functional succeeds in describing thecorrelation energy over the whole range of densities

In summary the BOW functional shows remarkable success in describing the correlation energyof the HEG both in 3 and 2 dimensions over a wide range of densities (001 lt rs lt 100) includingthe range of metallic densities (1 lt rs lt 6) It for the first time manages to qualitatively correctlydescribe the depletion of low momentum states an exact property known from Monte-Carlo resultsThe reproduction of a discontinuity at the Fermi level is a task for the future To achieve thisproperty one could follow the idea of the BBC and PNOF functionals to describe states above andbelow the Fermi energy differently which will then lead to a discontinuity

18135

ZT-RDMFT

34 The problem of the description of magnetic systems

As mentioned in the previous section a formulation of an RDMFT-LDA and an RDMFT-LSDA istheoretically possible but has not been carried out so far The energy differences between differentmagnetic phases of the HEG become very close for low densities Therefore even in Monte-Carlocalculations the estimates for the critical Wigner-Seitz radius rc where a phase transition between anunpolarized and totally polarized configuration occurs vary Ceperley and Alder [48] find rc = 75auwhereas Zhong et al [85] find rs = 50au An approximate functional in RDMFT would need toexhibit a very high accuracy to be able to reproduce the phase transition in the HEG which seemsto be a hard problem to solve However because physical systems usually show phase transitions formuch lower densities one suspectshopes that a medium accuracy would suffice In the followingwe should like to elucidate why the use of a functional of the kind of Eq (316) will have severedifficulties to achieve even a mediocre accuracy in describing spin-polarized systems

When trying to describe spin-polarized systems in collinear configuration the kinetic the externalpotential and the exchange part can be separated into a sum of contributions from only spin-up andonly spin-down NOs and ONs To describe the xc functional for a spin-polarized system a commonapproach in RDMFT is to make the same ansatz of spin-channel seperability

Exc[γ] = Exc[γuarruarr γdarrdarr] = ESxc[γuarruarr] + ES

xc[γdarrdarr] (323)

where γuarruarr and γdarrdarr are the diagonal elements of the 1RDM γ from Eq (216) and ES [γ] denotes theseparated functional

We claim that such an approximation is intrinsically incapable of describing both spin-polarizedand spin-unpolarized configurations together The underlying reason for this problem is that a sepa-rable functional might describe accurately the correlation contributions which arise from interactionsof the electrons of the same spin (ie inner-spin-channel correlation) but will not be able to describethe contribution coming from the correlation of electrons of different spin (ie inter-spin-channelcorrelation) In the following we will elaborate on this problem by considering a spin-polarized HEG(see Section 24) We will again rely on the PWCA[52] parametrization of the polarization-dependentcorrelation energies

For a collinear spin configuration the fundamental quantities are the spin-up density nuarr and thespin-down density ndarr and the energy can be written as E(nuarr ndarr) If the assumption of spin-cannelseperability is valid the following two relations would hold

E(nuarr ndarr) = E(nuarr 0) + E(0 ndarr) = EF (324)

E(nuarr ndarr) =1

2(E(nuarr nuarr) + E(ndarr ndarr)) = EP (325)

In Eq (324) the partially polarized system is given as a sum of two fully polarized ie ferromagneticsystems whereas in Eq (325) one constructs the partially polarized one out of two unpolarized ieparamagnetic systems hence the notations EF and EP

We can now investigate if Eqs (324) and (325) are valid for the case of a partially polarizedHEG by calculating the differences between the exact ie Monte-Carlo results and the expectedresults EF and EP

∆F = E minus EF (326)

∆P = E minus EP (327)

The results are shown in Figure 35 As we can see in Figure 35a ∆F vanishes by constructionfor a totally polarized system (ξ = 1) Decreasing the polarization then leads to a decrease of∆F ie E lt EF Because in EF the inter-spin-channel correlation is neglected we deduce thatit has to be negative Considering the paramagnetic case in Figure 35b we see that again byconstruction ∆P vanishes now for the paramagnetic configuration (ξ = 0) and then increases with

19135

ZT-RDMFT

001 01 1 10 100minus010

minus008

minus006

minus004

minus002

0

rs(au)

∆F(Ha)

ξ00204060810

35a Energy differences ∆F from Eq (326)

001 01 1 10 100

0

002

004

006

008

010

rs(au)

∆P(Ha)

ξ00204060810

35b Energy differences ∆P from Eq (327)

0

01

02

03

04

001 01 1 10 100rs(au)

δF

00204

060810

35c Relative energy deviations δF from Eq (328)

minus08

minus06

minus04

minus02

0

001 01 1 10 100rs(au)

δP

00204

060810

35d Relative energy deviations δP from Eq (329)

Figure 35 Energy difference ∆F and ∆P and relative energy fractions δF and δP as defined in Eqs(326) - (329) with respect to rs for different values of the polarization ξ

increasing polarization This can be understood by realizing that EP basically double counts theinter-spin-channel correlation contribution leading to E gt EP

In RDMFT the only correlation contribution comes from the interaction alone We thereforeshould like to know how the inter-spin-channel correlation contribution relates to this correlationcontribution Wc We can use the PWCA parametrization of the correlation contributions to theenergy to calculate Wc = Ec minus Tc and investigate the following two fractions

δF =∆F

Wc(328)

δP =∆P

Wc (329)

The results are shown in Figure 35 We see that by assuming spin-channel seperability one will yielderrors for the correlation energy of up to 40 However a remarkable feature of Figures 35c and35d is that the relative deviations over the whole range of considered densities vary only slightlyApparently both inner- as well as inter-spin-channel correlation are affected in the same way by achange in the density leading to only a small change in their fraction This result becomes importantat the end of this section where we model the inner-spin-channel correlation by a term similar tothe inter-spin-channel contribution

The importance of the previous results lies in the following Let us assume that we founda spin-channel separable functional which described the polarized (unpolarized) HEG perfectly

20135

ZT-RDMFT

3D

ξ 00 01 02 03 04 05 06 07 08 09 10

α 056 056 056 057 057 057 058 059 061 063 066

BOW 061 061 061 062 062 063 063 064 065 067 069

2D

ξ 00 01 02 03 04 05 06 07 08 09 10

α 063 063 063 063 064 064 065 066 067 07 074

BOW 066 066 066 067 067 068 069 071 073 076 080

Table 31 Best parameters to describe the correlation energy of a HEG in three and two dimensionsin the range 1 lt rs lt 100 for the α and BOW functionals

Applying this functional now to an unpolarized (polarized) HEG will inevitably yield an error ofabout 30-40 in the correlation energy Most of the previously introduced functionals (MullerBBC123PNOF0MLML-SICCGA) are spin-channel separable and there seems to be no easyway to remedy this problem The α and BOW functionals on the other hand offer a simple way outof the seperability dilemma because they exhibit one parameter Making this parameter polarization-dependent makes the functional inseparable and offers an easy solution to the problem given thatthe functionals also describe the partially polarized systems accurately We investigated the α andBOW functionals for partially polarized systems and found a good agreement of the correlation en-ergies for the respective best parameters We show the resulting parameters in Table 31 Howeveralthough both the α as well as the BOW functional with polarization-dependent parameter describethe correlation energy qualitatively correctly they fail to predict a phase transition between param-agnetic and ferromagnetic phases This is due to the fact that a smaller coefficient in the low densitylimit in both functionals leads to a bigger contribution to the correlation energy and therefore byreproducing the energy for intermediate densities well they favour the paramagnetic phase for lowdensity where the phase transition should occur

This is a serious backlash because the phase transition is a physical property of considerable inter-est Because in Section 8 we want to study the phase diagram of the HEG at finite temperatures wewill now propose a rather phenomenological way to recover the phase transition at zero temperatureFrom the good agreement of the α as well as of the BOW functional results with the Monte-Carloresults for fully polarized configurations we deduce that one can at least to some extent describethe inner-spin-channel correlation effects by employing the exchange integral We therefore proposeto use a similar approach to include the opposite-spin-channel contributions additionally Our ex-pression for this ldquotrans-channel interaction energyrdquo (BOW-TIE) modification reads with an explicitmentioning of the spin index

EBOWminusTIExc [γ] = minus1

2

sum

ijσ

fBOW (niσ njσαP )K(i j)minus

csum

ij

((niuarrnjdarr)

αU

(1minus niuarrnjdarr)αU)K(i j) (330)

K(i j) represents the exchange integral corresponding to the NOs φi and φj and αP stand forthe best parameter for the description of the fully polarized HEG The second term in Eq 330vanishes for a spin-polarized system and contributes increasingly with decreasing polarization Thecoefficients αU and c are fitted to reproduce a critical density closer to the Monte-Carlo result whilemaintaining the good overall accuracy of the correlation-energy for different spin polarizations The

21135

ZT-RDMFT

αP αU c

BOW-TIE 070 20 019

Table 32 Optimal parameters for the TIE-version of the BOW xc functional as defined in Eq(330)

resulting parameters are shown in Table 32 and lead to an instantaneous phase transition at acritical density of rc asymp 28au

We should like to emphasize again that we did not deduce this opposite-spin-channel contributionfrom higher principles but rather postulated it to create a model functional which reproduces thecritical density of the HEG more accurately With a different choice for the inter-spin channelcorrelation energy ie one which favours partially polarized configurations more strongly one mightbe able to get rid of the instantaneous transition between unpolarized and polarized phases andreproduce a qualitatively correct continuous quantum phase transition

This functional can then later on be used as a model function to study the effect of temperatureon the magnetic phase diagram of the HEG

35 Summary and outlook

In this section we gave a short review of the theoretical foundations of RDMFT at zero temperatureWe then investigated the performance of several popular xc functionals for the HEG in three andtwo dimensions Apart from the accurate description of the gs-energy we laid our focus on thereproduction of the gs-momentum distribution We found out that most functionals used in actualcalculations fail to yield a depletion of low momentum states We therefore designed a functionalthe BOW functional which is capable of reproducing exactly this feature in a qualitatively correctmanner It also reproduces the gs-energies for 1au lt rs lt 100au to unprecedented accuracy

We then turned to the problem of the description of partially polarized systems We showedthat most of the functionals used in actual calculations are intrinsically incapable of describing boththe unpolarized as well as the polarized HEG together We attributed this problem to the propertyof spin-channel seperability Motivated by these considerations we proposed two ways to constructexplicitly non spin-channel separable functionals The first one constitutes of making the functionalparameter polarization-dependent This procedure applied to the α and BOW functional succeededin qualitatively reproducing the gs-energies of the 3D- and 2D-HEG but because of wrong low densitybehaviour failed to reproduce the magnetic phase transition between unpolarized and polarizedconfigurations The second procedure introduces an explicit coupling between the diagonal elementsof the spin-dependent 1RDM By choosing appropriate parameters for this interaction we were ableto reproduce the gs-energies accurately including a magnetic phase transition at rs asymp 40au Thisfunctional will later on be used as a testing functional for investigation of the effect of temperatureon the free energy phase diagram of the 3D-HEG (see Section 8)

There are several open questions to be dealt with in the future For example one should tryto modify the BOW functional to reproduce the discontinuity of the momentum distribution at theFermi surface Another topic which is of considerable interest is the theoretical justification ofthe postulated coupling between the diagonal elements of the spin-dependent 1RDM As this effectseems to be able to reproduce the main features of the gs-energy of a partially polarized HEG itwould be desirable to get a more physical than phenomenological understanding of it Finally onewould have to apply the ldquonot spin-channel separablerdquo functionals to real systems and investigate themagnetic properties of the respective gs-configurations

22135

4 Finite-temperature RDMFT (FT-RDMFT)

After giving a short review of RDMFT at zero temperature in Section 3 we will now allow the systemto exchange particles and energy with its surrounding We will pursue the goal of laying a solidfoundation for the description of quantum systems in grand canonical thermodynamic equilibriumusing the 1RDM as central variable The resulting theory will be called FT-RDMFT

It was shortly after the original work by Hohenberg and Kohn [3] that Mermin extended theirproofs to grand canonical ensembles He proved that no two local potentials can yield the sameeq-density and therefore paved the way for a finite-temperature version of DFT (FT-DFT) Thisoriginal formulation of FT-DFT just like its zero-temperature counterpart is only valid for densitieswhich are equilibrium-V-representable ie eq-densities corresponding to some external potential V In general it is not known how this set of densities can be characterized which leads to some math-ematical difficulties for example the problem of defining a functional derivative It was Lieb [37]who reformulated DFT using the concept of Legendre-transforms solving several of these conceptualmathematical problems The same path was followed by Eschrig [86] putting FT-DFT on a solidmathematical footing Knowing about these stumbling blocks but without the claim of absolutemathematical rigor we are going to investigate the theoretical foundations of FT-RDMFT as thor-oughly as possible in this section The outline will be as follows

Section Description

41 At first we are going to define the problem we are interested in namely the descriptionof quantum mechanical system in grand canonical equilibrium We will then repeathow standard quantum mechanics approaches this problem and investigate propertiesof functionals of the SDO We will later be able to translate most of these propertiesto the functionals in FT-RDMFT

42 We will then state a proof of a Hohenberg-Kohn theorem for quantum systems in grandcanonical equilibrium with possibly nonlocal external potential This allows the for-mulation of a functional theory employing the 1RDM We will furthermore extend thedomain of the functionals to the whole set of ensemble-N-representable 1RDMs ΓN

43 To aid further investigations we will then investigate some mathematical properties ofthe FT-RDMFT functionals focussing on convexity and lower semicontinuity

44 Subsequently we will consider the question of equilibrium-V-representability ie thequestion how 1RDMs which come from an equilibrium state can be characterizedWe will find that the set of equilibrium-V-representable 1RDMs ΓV lies dense in theset of ensemble-N-representable 1RDMs ΓN a result which will become considerablyimportant for the remainder of this work

45 Finally we will investigate how the results from our considerations of grand canonicalensembles translate to the case of microcanonical and canonical ones


23135

FT-RDMFT

41 Physical situation and standard quantum mechanical treatment

H

particle and heat bath

∆Nmicro

∆E β

Figure 41 Sketch of a grand canonical ensemble The subsystem governed by Hamiltonian Hexchanges energy E and particles N with the surrounding infinite bath The strength of the couplingsto the bath is governed by the two Lagrangian multipliers micro and β

In contrast to the zero-temperature problem we will now allow the system to exchange energy andparticles with an infinite bath The exchange of particles then requires an extension of the N-particleHilbert space to the Fock space H which is given as the direct sum of symmetrized tensor productsof the one-particle Hilbert space h

H =

infinoplus

n=0

Shotimesn (41)

Instead of the energy the main thermodynamic variable of this system is the grand potential Ω Fora given state D the grand potential is given by

Ω[D] = TrD(H minus microN + 1β ln D) (42)

The Lagrangian parameters micro (the chemical potential) and 1β = kBT (the temperature) describethe couplings to the particle and heat bath The entropy of the system is then defined as

S[D] = minusTrD ln D (43)

which from the definition of D in Eq (26) is positive definite The equilibrium state is now definedas the state which minimizes Eq (42) leading to the finite-temperature variational principle (theGibbs principle)

Ω[D] gt Ω[Deq] for all D 6= Deq (44)

It was shown in Ref [11] that the equilibrium state Deq is uniquely given by

Deq = eminusβ(HminusmicroN)Zeq (45)

Zeq = Treminusβ(HminusmicroN) (46)

24135

FT-RDMFT

F [D]

D

downwards convex

upwards convex

42a Convexity

F [D]

D

42b Lower semicontinuity

Figure 42 Illustration of the concepts of convexity and lower semicontinuity

Zeq is called the partition function and allows a simple representation of the equilibrium grand

potential Ωeq = Ω[Deq]

Ωeq = minus1β lnZeq (47)

We will now introduce the set of 1RDMs which come from an eq-state They will be calledequilibrium-V-representable and are defined as the following

Definition 41 [equilibrium-V-representability]If for a given 1RDM γ there exists a potential V so that

γ(x xprime) = Tr

eminusβ(T+V+W )

ZΨ+(xprime)Ψ(x)

(48)

then γ is called equilibrium-V-representable (eq-V-representable) The set of all eq-V-representable1RDMs is denoted by ΓV

ΓV =γ | γ eq-V-representable

(49)

The conditions for an arbitrary 1RDM to be eq-V-representable are not known and the only

knowledge we have about this set so far is that it is a subset of ΓN and therefore also of TradicN and

T (see Section 23)

ΓV sube ΓN sub TradicN sub T (410)

However in Section 44 we will be able to show that the set ΓV is dense in the set ΓN This meansthat given any γ isin ΓN there is a γ isin ΓV arbitrarily close to it (employing the Banach-space norm)This will become crucial for the investigation of properties of the exact functionals in FT-RDMFT(see Section 5) and for the development of a methodology to derive approximate functionals (seeSection 6) But before digging into the theoretical treatment of FT-RDMFT we are going to havea closer look at the properties of Ω[D] and S[D] in the following The reason for this is that by thevirtue of Theorem 48 we will be able to translate most of these exact properties to the functionalsin FT-RDMFT

411 Properties of thermodynamic variables

The two main properties are convexity and lower semicontinuity as defined in the following andillustrated in Figures 42a and 42b

25135

FT-RDMFT

Definition 42 [Convexity]A functional F [D] is called upwards (downwards) convex if for all D1 D2 and for all λ isin R with0 le λ le 1

F [λD1 + (1minus λ)D2] le (ge)λF [D1] + (1minus λ)F [D2] (411)

Definition 43 [Lower semicontinuity]Let Dk converge (weakly) to D A functional F [D] is called (weakly) lower semicontinuous if

F [D] le lim infkrarrinfin

F [Dk] (412)

Because in FT-RDMFT the equilibrium state will be found by a minimization of a grand potentialfunctional Ω[γ] the convexity property will prove to be very useful Furthermore the propertyof lower semicontinuity (lsc) will become important when dealing with the problem of eq-V-representability in Section 44 For an investigation of the convexity and continuity properties of thethermodynamic variables we are now going to introduce the concept of relative entropy [87 88] Fortwo density matrices A and B the relative entropy is defined as

S[A B] = TrA(ln Aminus ln B) (413)

S[A B] can be related to the grand potential of the system by setting A = D and B = Deq

S[D Deq] = β(Ω[D]minus Ωeq) (414)

and to the entropy by setting A = D and B = 1

S[D 1] = minusβS[D] (415)

The relative entropy S[A B] was shown to be upwards convex in both arguments [89] ie if A =λA1 + (1minus λ)A2 B = λB1 + (1minus λ)B2 and 0 le λ le 1 then

S[A B] le λS[A1 B1] + (1minus λ)S[A2 B2] (416)

By using Eqs (414) and (415) this property translates directly to Ω[D] and S[D]

Ω[λD1 + (1minus λ)D2] le λΩ[D1] + (1minus λ)Ω[D2] (417)

S[λD1 + (1minus λ)D2] ge λS[D1] + (1minus λ)S[D2] (418)

Because A and B do not necessarily commute the following representation of S[A B] [90] will beuseful

S[A B] = supλ

((1λ)

(S[λA+ (1minus λ)B]minus λS[A]minus (1minus λ)S[B]

))(419)

S[A B] was shown in [90] to be lower semicontinuous with respect to the trace norm The require-ment of norm convergence can be softened and the modified proof is stated in the following

Theorem 44 [Lower semicontinuity]Let Ak and Bk be infinite series of density operators If there are A and B so that TrP (AkminusA) rarr 0 and TrP (Bk minus B) rarr 0 for every finite-dimensional projection operator P then

S[A B] le lim infkrarrinfin

S[Ak Bk] (420)

26135

FT-RDMFT

Proof [of Theorem 44]We will use the following two relations

TrA = supP

TrP A (421)

TrP A le PTrA (422)

where the norm middot is the operator norm From the conditions TrP (Akminus A) rarr 0 and TrP (BkminusB) rarr 0 it follows that for all 0 le λ le 1

TrP (λAk + (1minus λ)Bk minus λAminus (1minus λ)B) rarr 0 (423)

Because Ak Bk A and B have only semipositive eigenvalues and because x lnx is a continuousfinite function on [0 infin) this leads to

TrP (S[λAk + (1minus λ)Bk)] rarr TrP (S[λA+ (1minus λ)B]) (424)

Realizing that P = 1 and using Eqs (419) and (424) one can conclude the proof

S[A B] = supP λ

((1λ)tr

P(S[λA+ (1minus λ)B]minus λS[A]minus (1minus λ)S[B]

))(425)

le lim infkrarrinfin

S[Ak Bk] (426)

This result can again be related to the grand potential and entropy by setting Ak = Dk and Bk = Deq

and Ak = Dk and Bk = 1 respectively

Corollary 45 Let Dk be an infinite series of density operators so that TrP (Dk minus D) rarr 0 forall finite-dimensional projection operators P Then

Ω[D] le lim infkrarrinfin

Ω[Dk] (427)

S[D] ge lim infkrarrinfin

S[Dk] (428)

After reviewing several important concepts from standard quantum mechanics we can now turn tothe formulation of a finite-temperature version of RDMFT

42 Mathematical foundation of FT-RDMFT

As mentioned in the introduction Mermin [11] showed that there is a one-to-one correspondencebetween the eq-density neq and the SDO Deq for a system with local external potential Since thediagonal of the 1RDM gives the density (n(x) = γ(x x)) it follows that for such systems there is alsoa one-to-one correspondence between the Deq and the equilibrium 1RDM γeq (see Figure 43)

In 1974 Gilbert [26] extended the zero-temperature Hohenberg-Kohn theorems to systems withnonlocal external potential In these systems the groundstate is not uniquely determined by thedensity anymore but by the 1RDM In the following we will show that for such a system at finitetemperature the 1RDM is still sufficient to describe the equilibrium properties This will be achievedby showing that the map between Deq and γeq(x x

prime) is invertible which in turn implies the existenceof a grand potential functional Ω[γ]

27135

FT-RDMFT

Deq

Deq

Deq

γeq

γeq neq

neq

neq

+ =rArr

Figure 43 The one-to-one map between D and n implies the existence of a one-to-one map betweenD and γ

421 Existence of grand potential functional Ω[γ]

We want to point out that in the subsequent discussion we explicitly consider only eq-V-representable1RDMs γ(x xprime) (see Def 41) which we do not know how to characterize This lack of knowledgeof the constraints on the domain of Ω[γ] might pose a serious problem in the course of numericalminimization Fortunately in Section 43 we will be able to extend the domain to the full set of ΓN which is easy to deal with The proof of the existence of a one-to-one mapping between the eq-SDODeq (see Eq(45)) and the eq-1RDM (see Eq(212)) can be divided in two parts First the one-to-

one mapping between Deq and the external potential minus the chemical potential (vext(x xprime)minus micro)

will be shown then the one-to-one correspondence between (vext(x xprime)minusmicro) and γeq(x xprime) is proven

bull Deq1minus1larrrarr (vext(x x

prime)minus micro) Let H and H prime be two different Hamiltonians and assume they lead to the same SDO D H prime shalldiffer from H only by a one-particle potential contribution U With Eq (45) this reads

eminusβ(HminusmicroN)Z = eminusβ(H+UminusmicroN)Z prime (429)

where Z and Z prime are the partition functions (eg Z = Treminusβ(HminusmicroN)) Solving Eq (429) for Uyields

U =

intdxdxprimeu(xprime x)ψ+(xprime)ψ(x) = minus 1

βlnZ

Z prime (430)

We now need to show that there is no one-particle potential u(x xprime) 6= 0 which fulfills this equalitythereby contradicting our initial assumption To proceed we assume three different Slater determi-nants |X1〉 = |1 0 0 〉 |X2〉 = |0 1 0 〉 and |X3〉 = |1 1 0 〉 in the basis χi The potential inthis basis is denoted by uij =

intdxdxprimeu(xprime x)χlowast

i (xprime)χj(x) Calculating the expectation value of both

sides of Eq (430) wrt these three Slater determinants we get the following system of equations

minus 1

βlnZ

Z prime = u11 = u22 = u11 + u22 (431)

which can only be fulfilled by u11 = u22 = 0 and Z = Z prime A repetition of this argument for allpossible bases then shows that only U = 0 fulfills Eq (430) which in turn proves the one-to-onecorrespondence between Deq and (vext(x x

prime) minus micro) It has to be noted that this proof intrinsicallyrelies on the fact that in the case of grand canonical ensembles we work in a Fock space ie aHilbert space with varying particle number If we had restricted ourselves to a canonical situationie a Hilbert space with fixed particle number we would have found that the external potential isuniquely determined only up to within an additional constant

28135

FT-RDMFT

bull (vext(x xprime)minus micro) 1minus1larrrarr γ(x xprime) In order to prove the one-to-one correspondence between (vext(x x

prime)minus micro) and γeq(x xprime) we assume

that H and H prime differ only in their external potentials The corresponding grand potentials are givenby

Ω[Deq] = TrDeq(H minus microN + 1β ln Deq) (432)

Ωprime[Dprimeeq] = TrDprime

eq(Hprime minus microprimeN + 1β ln Dprime

eq) (433)

where Deq and Dprimeeq are defined according to Eq (45) The variational principle (Eq (44)) then

leads to

Ω[Deq] lt Ω[Dprimeeq] (434)

= TrDprimeeq(H minus microN + 1β ln Dprime

eq) (435)

= Ωprime[Dprimeeq] + TrDprime

eq((H minus microN)minus (H prime minus microprimeN)) (436)

Now by exchanging primed and unprimed objects one obtains

Ω[Deq] lt Ωprime[Dprimeeq] +

intdxdxprime((vext(x x

prime)minus micro)minus (vprimeext(x xprime)minus microprime))γprime(xprime x) (437)

Ωprime[Dprimeeq] lt Ω[Deq] +

intdxdxprime((vprimeext(x x

prime)minus microprime)minus (vext(x xprime)minus micro))γ(xprime x) (438)

Adding these two equations leads to the following relation


prime)minus microprime)minus (vext(x xprime)minus micro))(γ(xprime x)minus γprime(xprime x)) gt 0 (439)

The existence of two different sets of external and chemical potentials yielding the same eq-1RDMlets the integral in Eq(439) vanish which leads to a contradiction Hence the initial assumption isfalsifiedThis proof of the existence of a one-to-one mapping between Deq and γeq allows us to define thegrand potential as a functional of the 1RDM

Ω[γeq] = TrD[γeq](H minus microN + 1β ln(D[γeq])) (440)

In Eq (440) the contributions from the external and the chemical potential can be separatedyielding the definition of a universal functional F [γeq] for FT-RDMFT which then reads

Ω[γeq] =

intdxdxprime(vext(x x

prime)minus micro)γeq(x xprime) + F [γeq] (441)

F [γeq] = TrD[γeq](T + W + 1β ln D[γeq]) (442)

As mentioned before this functional F [γ] is defined only on the set ΓV of 1RDMs coming from SDOsof the form of Eq (45) which we do nt know how to characterize Fortunately as we will show inthe following section we can extend the domain by using a different formulation for the universalfunctional

422 Lieb formulation of the universal functional

Analoguous to the Lieb-formulation [91 37] of DFT we start by restating the variational principle

Ωeq = minDisinDN

Ω[D] (443)

29135

FT-RDMFT

Because every D isin DN yields a 1RDM γ isin ΓN we can divide the minimization as follows

Ωeq = minγisinΓN

infDisinDNrarrγ

Ω[D[γ]] (444)

The infimum in Eq (444) appears because for a 1RDM γ isin ΓNΓV ie a 1RDM which is noteq-V-representable it is not clear so far if there exists a minimizing SDO D rarr γ We will come backto this question in Section 431 and show that there is in fact a minimizing D in Eq (444) for allγ isin ΓN Therefore in the following we replace the infimum by a minimumWe can now separate the external potential part from Eq (444) to get

Ωeq = minγisinΓN

(F[γ] +


prime)minus micro)γ(xprime x)) (445)

where the universal functional F[γ] is defined as

F[γ] = minDisinDNrarrγ

TrD(T + W + 1β ln D) (446)

The grand potential functional Ω[γ] is then given by

Ω[γ] = F[γ] +


prime)minus micro)γ(xprime x) (447)

The equilibrium grand potential is now found by a minimization of the functional Ω[γ]

Ωeq = minγisinγN

Ω[γ] (448)

(The corresonding zero-temperature formulation is also sometimes referred to as the grand canonicalensemble formulation [92])

We would also like to establish an Euler-Lagrange equation for the eq-1RDM γeq of the followingkind

δF [γ]

δγ(xprime x)

∣∣∣∣γeq

+ vext(x xprime) = micro (449)

However the question of the existence of a functional derivative of the exact F [γ] at the equilibrium1RDM is not answered so far and is subject to continued studies (also in DFT this question is stillunder discussion [93]) In an approximate treatment however one will most likely define a functionalF approx[γ] for which the functional derivative exists justifying the use of the Euler-Lagrange equationin a minimization scheme Two differences to the case of zero-temperature RDMFT have to bepointed out here Because at zero temperature the minimizing 1RDM can be on the boundary ofthe domain ΓN (ie one can have ONs equal to 0 or 1) the Euler-Lagrange equation does not takethe form of Eq (449) but has to incorporate the constraints on the eigenvalues of the 1RDM byfurther Kuhn-Tucker multipliers [94] In the case of FT-RDMFT where the eq-1RDM refers to agrand canonical ensemble we pointed out in Section 23 that there cannot be ONs on the boundaryof ΓN rendering these additional multipliers unnecessary

The second and maybe more important difference concerns the existence of a Kohn-Sham sys-tem We have seen in Section 3 that because of the cusp condition there exists no Kohn-Shamsystem in RDMFT at zero temperature (see Thm 31) In the following section we will show thatin FT-RDMFT this drawback disappears and that there exists a noninteracting system which ingrand canonical equilibrium reproduces the eq-1RDM of an interacting system We will also showhow the corresponding Kohn-Sham Hamiltonian can be constructed directly from the ONs and NOsof the interacting 1RDM This direct construction is a conceptual advantage over DFT where theKohn-Sham Hamiltonian had to be found by an inversion of the Kohn-Sham equation which is ingeneral a nontrivial task and has to be done by iterative methods [95]

30135

FT-RDMFT

423 Kohn-Sham system for FT-RDMFT

To show the existence of a Kohn-Sham system in the context of FT-RDMFT it is instructive to

consider an arbitrary noninteracting system defined by the one-particle Hamiltonian H(1)s with

eigenvalues εi and eigenfunctions φi(x)

H(1)s =

sum

i

εi|φi〉〈φi| (450)

For a grand canonical ensemble at chemical potential micro the eq-1RDM is then [96] given by

γeq(x xprime) =

sum

i

niφlowasti (x

prime)φi(x) (451)

where the ONs ni are determined completely by the eigenvalues εi and the chemical potential micro(we show the explicit derivation in Appendix A4)

ni =1

1 + eβ(εiminusmicro) (452)

This relation can be inverted to give the Kohn-Sham energies in terms of the corresponding ONsand the chemical potential

εi minus micro =1

βln

(1minus nini

) (453)

From Eq(453) it is now possible to construct the Kohn-Sham Hamiltonian from a given 1RDM Itseigenfunctions are given by the NOs of the 1RDM while the eigenvalues are defined up to a commonconstant by Eq(453) This is of course just possible if the 1RDM has no ONs equal to 0 or 1 butas we have pointed out in Section 23 and Appendix A2 this cannot be for an eq-1RDM of a grandcanonical ensembleDefining the kinetic operator in the basis of NOs (tij = 〈φi|T |φj〉) the effective one-particle potentialveff (x x

prime) can be expressed as

veff (x xprime) =

sum

ij

(δijεi minus tij)φlowasti (xprime)φj(x) (454)

which is generally nonlocal in spatial coordinates The existence of a Kohn-Sham system now suggeststhe definition of correlation functionals following the ideas of DFT

424 Functionals in FT-RDMFT

In contrast to DFT in the framework of RDMFT the functionals for the kinetic energy Ek[γ] theHartree energy EH [γ] and the exchange energy Ex[γ] of the interacting system are known exactlyFor now we postulate that the functional forms of these contributions stay the same for finite-temperature ensembles A detailed investigation of the correlation functional in Section 6 will provethis assumption to be valid Furthermore the entropy of a noninteracting system with eq-1RDM γie the Kohn-Sham entropy S0[γ] is a trivial functional of the ONs of the 1RDM (see AppendixA4) The grand potential functional is then given by

Ω[γ] = Ωk[γ] + Ωext[γ]minus microN [γ]minus 1βS0[γ] + ΩH [γ] + Ωx[γ] + Ωc[γ] (455)

31135

FT-RDMFT

where the individual contributions are defined as follows

Ωk[γ] =

intdxprime lim

xrarrxprime

(minusnabla

2

2

)γ(x xprime) (456)

N [γ] =

intdxγ(x x) (457)

Ωext[γ] =



ΩH [γ] =1

2


Ωx[γ] = minus1

2


S0[γ] = minussum

i

(ni lnni + (1minus ni) ln(1minus ni)) (461)

The universal functional F [γ] can then be written as

F [γ] = Ωk[γ]minus 1βS0[γ] + ΩH [γ] + Ωx[γ] + Ωc[γ] (462)

We can now show that neglecting correlation completely will lead to a grand potential functionalwhose minimization is equivalent to a solution of the finite-temperature Hartree-Fock (FT-HF) equa-tions

425 Finite-temperature Hartree-Fock (FT-HF)

Neglecting Ωc[γ] in Eq (455) yields the following functional

Ω[γ] =

intdxprime lim

xrarrxprime

(minusnabla

2

2

)γ(x xprime) +

intdxvext(x x

prime)γ(xprime x)minus microintdxγ(x x)+

1

2

intdxdxprimew(x xprime)γ(x x)γ(xprime xprime)minus 1

2

intdxdxprimew(x xprime)γ(x xprime)γ(xprime x)+

1βsum

i


In equilibrium at finite temperature there will be no pinned states ie no states with ONs equalto 0 or 1 Furthermore Eq (463) is an explicit functional of the 1RDM Therefore the functionalderivative wrt the 1RDM exists and at the minimum the functional fulfills the following Euler-Lagrange equation

δΩ[γ]

δγ(xprime x)= 0 (464)

We will now apply this condition to the correlation-free functional from Eq(463) and project theresult on the i-th NO This will then lead to the FT-HF equations

0 =

intdxprimeφi(x

prime)δΩ[γ]

δγ(xprime x)(465)

=

(minusnabla

2

2

)φi(x) +

intdxprimevext(x x

prime)φi(xprime)minus

intdxprimew(x xprime)γ(x xprime)φi(x

prime)+

(intdxprimew(x xprime)γ(xprime xprime)

)φi(x)minus εiφi (466)

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FT-RDMFT

where in the last term we used Eq(453) As a first valuable result from our investigation ofFT-RDMFT we therefore derived an alternative way to solve the FT-HF equations by a mini-mization rather than by iterative diagonalization We will use this fact in Section 84 to investigatethe temperature dependence of the quasi-particle dispersion relation as well as the magnetic phasediagram for collinear and chiral spin configurations in FT-HF We will now turn to a more thoroughinvestigation of the mathematical foundations of FT-RDMFT

43 Properties of the universal functional

We showed in Section 421 that a quantum mechanical system with fixed interaction and with localor nonlocal external potential which is in grand canonical equilibrium can be described solely byits 1RDM Furthermore we established a one-to-one correspondence between the external potentialminus the chemical potential and the 1RDM ((vext(x x

prime) minus micro) larrrarr γ(x xprime)) Therefore the grandpotential can be written as a functional of the 1RDM

Ω[γ] = F [γ] +

intdxdxprimeγ(xprime x)(vext(x x

prime)minus micro) (467)

431 Existence of minimum

We are now going to show that we were allowed to replace the infimum from Eq (444) by a minimumin Eq (446) This is important because it leads to a one-to-one mapping between the ensemble-N-representable 1RDMs and a subset DN

L of DN which allows one to translate several exact properties

of the SDO grand potential functional Ω[D] to the corresponding 1RDM-functional Ω[γ]The corresponding problem in DFT was dealt with for zero-temperature DFT in Refs [37 38]

with an equivalent outcome We will follow a very similar path now in the context of FT-RDMFTWe will have to deal with questions of convergence on Banach-spaces In particular we need theconcept of weak-lowast convergence and the Banach-Alaoglu theorem which we restate in the following

Definition 46 [Weak-lowast convergence]Let B be a Banach space and Blowast its dual space A series fk isin Blowast is said to be weak-lowast convergentif for all x isin B

limk(fk(x)minus f(x))rarr 0 (468)

Theorem 47 [Banach-Alaoglu theorem]Let B be a Banach space and Blowast its dual space Let Blowast

1 be a norm-closed subset of radius 1 ie

Blowast1 = f isin Blowast|fBlowast le 1 (469)

Then Blowast1 is compact with respect to the weak-lowast topology (see Def 46)

With these two tools at hand we will be able to prove the existence of a minimal SDO D in Eq444

Theorem 48 [(γ isin ΓN )1minus1larrrarr (D isin DN

L sub DN )]

For every γ isin Γ there exists a density operator D isin DN with D rarr γ so that

F [γ] = TrD(H0 + 1β ln D) (470)

ie the infimum in Eq (444) is a minimum

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FT-RDMFT

Proof of theorem 48 Let Dk be a sequence of density operators with Dk isin DN sub T so thateach Dk rarr γ and

limkrarrinfin

TrDk(H0 + 1β ln Dk) = F [γ] (471)

We will divide the proof in three parts and show

(1) There exists a D with Dk rarr D in some sensetopology

(2) This D yields γ

(3) D fulfills TrD(H0 + 1β ln D) = F [γ]

(1) There exists a D with Dk rarr D in some sensetopologyFrom Section 22 we know that for each density operator the Hilbert-Schmidt norm yields Dk le 1ie Dk isin T1 By the virtue of the Banach-Alaoglu theorem 47 we know that there is a D withD le 1 so that Dk

lowastrarr D in the weak-lowast topology (see Definition 46) By definition Dk are traceclass operators The set of trace class operators is the dual of the set of compact operators Theweak-lowast convergence therefore implies that for all compact operators A

Tr(Dk minus D)A rarr 0 (472)

This implies that each matrix element of Dk converges weakly against the corresponding one of Dwhich then implies that the eigenvalues of D are between 0 and 1(2) D yields γAlthough each Dk yields the same 1RDM we will now for didactical reasons denote them by γk Wenow want to use the weak-lowast convergence of Dk to prove the weak convergence of γk Because wechose all γk = γ this would then also imply strong convergence and therefore D rarr γ We denotethe 1RDM resulting from D by γ γ is trace class The dual of the set of trace class operators arebounded operators Therefore γk rarr γ weakly if and only if for all bounded f

intdxdxprime(γk(x x

prime)minus γ(x xprime))f(xprime x)rarr 0 (473)

To relate the 1RDMs with their corresponding SDOs we now introduce the following operator

Mf =

intdxdxprimef(xprime x)ψ+(xprime)ψ(x) (474)

With this help we can reformulate the requirement from Eq (473) as

Tr(Dk minus D)Mf rarr 0 (475)

Mf although it originates from a bounded function f(x xprime) is not bounded which is why we cannot

just use Eq (472) to proceed However we know that each eigenvalue of Mf is finite We now

introduce an arbitrary finite-dimensional projection operator P Then the product Mf P is compactagain The left hand side of Eq (475) can therefore be written as

Tr(Dk minus D)Mf P+Tr(Dk minus D)Mf (1minus P ) (476)

Because of Eq(472) we know that for any choice of compact P the first part of Eq(476) goes tozero We now just have to show that we can always choose a P so that for every ǫ the second termin Eq (476) will fall below ǫ This would show weak convergence of γk It is here that our proofdiffers from the DFT version [37] This is because γ and f are not diagonal in the same basis iethe spatial one We will proof that the second term in Eq (476) falls below ǫ by showing that both

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FT-RDMFT

terms individually do We remember that the expectation values of TrDkMf and TrDMf arefinite by construction Furthermore the eigenvalues of Dk and D are positive Considering the Dk

operators this implies that one can always find an M0 so that for all M gt M0

infinsum

i=M+1

wk i〈ψk i|Mf |ψk i〉 lt ε (477)

The same argument applies to D which then proofs that given any ε gt 0 we can find a M stthe right hand side of Eq (476) surpasses ε To elucidate this statement we go back to the def-inition of weak convergence We want to show that for all ε there is one K st for all k gt K|Tr(Dk minus D)Mf| lt ε We now choose an ε and K1 Because of Eq (477) we know that there is

an M st for all k gt K1 Tr(Dk minus D)Mf (1 minus PM ) lt ε We furthermore know from the weak-lowast

convergence of Dk that there is also a K2 st Tr(Dk minus D)MfPM lt ε for all k gt K2 Thenchoosing the bigger one of K1 and K2 as K yields the proof of weak convergence of γk rarr γ Andbecause we chose γk = γ this then finally leads to D rarr γ We have therefore shown that the weak-lowast

convergence implies weak convergence This is due to the fact that the p-1 norm or absolute valuenorm of all D isin DN is bounded (it is in fact by construction equal to 1) This is not true for allelements of T1 because of which generally weak-lowast and weak convergence differ

(3) D fulfills TrD(H0 + 1β ln D) = F [γ]As before we use that a finite-dimensional projection P is compact Replacing A in (472) with P one gets

TrP (Dk minus D) rarr 0 (478)

Using Corollary 45 and Eq (446) one gets

TrD(H0 + 1β ln D) = F [γ] (479)

where the equality sign follows from the definition of Dk as the minimizing sequence

The existence of a minimizing D in Eq (444) for all γ isin ΓN now simplifies the proof that thefunctionals F [γ] and therefore also Ω[γ] are convex and lower semicontinuous

432 Lower semicontinuity

Theorem 49 [Semicontinuity of F [γ]]Let γk γ isin ΓN with γk rarr γ weakly Then the following relation holds

F [γ] le lim infkrarrinfin

F [γk] (480)

Proof of theorem 49 Because of Theorem 48 for each γk there exists a Dk so that F [γk] =TrDk(H0+1β ln Dk) This defines a sequence Dk which by the Banach-Alaoglu theorem convergesagainst a D in the weak-lowast topology Following the same steps as in the proof of Theorem 48 onecan show D rarr γ weakly Again using Eq (427) yields

F [γ] le TrD(H0 + 1β ln D) (481)


F [γk] (482)

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FT-RDMFT

433 Convexity

Theorem 410 [Convexity of F [γ]]Let γ1 γ2 isin ΓN and 0 le λ le 1 then

F [λγ1 + (1minus λ)γ2] le λF [γ1] + (1minus λ)F [γ2] (483)

Proof of theorem 410 Because of Theorem 48 there exist D1 rarr γ1 and D2 rarr γ2 Thereforeusing Eq 417 and removing the external contribution proves the theorem

We have thus managed to derive the general properties of convexity and lower semicontinuity ofthe exact grand potential functional in FT-RDMFT Convexity is important when one considers anumerical minimization of a functional lower semicontinuity on the other hand will help us in thefollowing section to address the question of eq-V-representability in FT-RDMFT

44 Eq-V-representability

For every potential V the eq-SDO Deq can be constructed via Eq (45) This density operatorthen yields an eq-1RDM γ(x xprime) via Eq (212) Following from Definition 41 such an eq-1RDM iscalled eq-V-representable This section is now concerned with the question of how the set of all eq-V-representable 1RDMs can be classified The zero-temperature counterpart in DFT the questionof V-representability has been dealt with already at early stages of the theoretical developmentof DFT It has been proven that the set of V-representable densities coincides with the set of N-representable densities for finite-dimensional state spaces [35] for general quantum lattice systems[97] and for systems with coarse-grained densities [38] On the other hand it was shown that thesame statement is invalid for general infinite-dimensional spaces [98 37] As an illustrative exampleRef [62] demonstrates how the spaces of N- and V-representable densities differ for a simple modelinteraction

It has to be noted that the treatment of this problem is of more than just purely mathematicalinterest The assumption of eq-V-representability allows a detailed investigation of properties of theexact correlation functionals in FT-RDMFT (see Section 5) as well as allows the development of amethodology to derive correlation functionals by employing methods from many-body perturbationtheory (see Section 6)Following ideas of Liebrsquos prolific work [37]we will be able to show that the set of all eq-V-representable1RDMs ΓV lies dense in the set of all ensemble-N-representable 1RDMs ΓN Therefore for anygiven 1RDM γ there is a eq-V-representable γ arbitrarily close to it But to show this we will haveto rephrase the definition of eq-V-representability by the help of the universal functional F [γ] asdefined in Eq (446) To disencumber notation we are going to omit the notion of the two spatialcoordinates in integrals and arguments of potentials and 1RDMs in the following

Definition 411 [Eq-V-representability II]If for a given γ0 isin ΓN there exists a potential v0 st

infγisinΓN

(F [γ] +

intv0γ

)= F [γ0] +

intv0γ0 (484)

then γ0 is called eq-V-representable

At first glance it might seem that the existence of a minimum in Eq (484) could be fulfilled

easily for all γ0 isin ΓN by setting v0 = minus δF [γ]δγ0

But there is one important problem The existence

of a functional derivative of F [γ] on the whole set of ΓN is not known so far In fact the sameproblem occurs in zero-temperature DFT Initially it was claimed [99] that the DFT-universal Lieb-functional FDFT [ρ] is differentiable for all ensemble-V-representable densities and nowhere else Butquite recently new light was shed on this particular field of subject [93] and it seems that this

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FT-RDMFT

statement cannot be uphold without further constraints on the domain of FDFT [ρ] Fortunatelyas it will turn out the question of eq-V-representability in FT-RDMFT can be dealt with alsowithout the use of functional derivatives To further investigate this question we have to introducethe concept of continuous tangent functionals

Definition 412 Let F be a real functional on a subset A of a Banach space B and let γ0 isin A Alinear functional L on B is said to be a tangent functional (tf) at γ0 if and only if for all γ isin A

F [γ] ge F [γ0]minus L[γ minus γ0] (485)

If L is furthermore continuous then it is called a continuous tangent functional (ctf) We cannow prove the following important theorem

Theorem 413 The universal functional F [γ] has a unique continuous tangent functional at everyeq-V-representable 1RDM and nowhere else

Proof [of Theorem 413]bull ctf rArr eq-VThe proof is done by reductio ad absurdo Suppose F [γ] exhibits a ctf at γ0 but γ0 is not eq-V-representable Denoting the ctf as v it follows

F [γ] ge F [γ0]minusintv(γ minus γ0) (486)

F [γ] +

intvγ ge F [γ0] +

intvγ0 (487)

infγisinΓN

(F [γ] +

intvγ

)ge F [γ0] +

intvγ0 (488)

where the equality sign is fulfilled for γ0 But we assumed in the beginning that γ0 is not eq-V-representable and therefore the infimum on the left of Eq (488) should never be assumed for anyγ This leads to a contradiction and proves that the existence of a continuous tangent functional atγ0 implies the eq-V-representability of γ0

bull eq-V rArr ctfBy definition of eq-V-representability (see Eq(484)) we deduce the following relation

F [γ0] +

intv0γ0 = inf

γisinΓN

(F [γ] +

intv0γ

)(489)

le F [γ] +intv0γ for all γ (490)

which immediately proves the existence of a continuous tangent functional

F [γ] ge F [γ0]minusintv0(γ minus γ0) (491)

Uniqueness is then proven by assuming the existence of a different ctf v 6= v0 One is then again leadto Eqs (486)-(488) But because v 6= v0 Eq (488) will again be violated proving the uniquenessof the ctf

Theorem 413 transforms the question of eq-V-representability to the question of the existence ofa unique ctf We have shown already in Theorems 410 and 49 that F [γ] is convex and lowersemicontinuous and we can therefore use the following theorem

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FT-RDMFT

Theorem 414 [Bishop-Phelps theorem]Let F be a lower semicontinuous convex functional on a real Banach space B F can take the value+infin but not everywhere Suppose γ0 isin B and v0 isin Blowast with F [γ0] lt infin For every ε gt 0 thereexists γε isin B and vε isin Blowast so that

1 vε minus v0 le ε

2 vε is ctf to F at γε

3 εγε minus γ0 le F [γ0] +intv0γ0 minus infγisinBF [γ] +

intv0γ

Part 3 of theorem 414 makes an assertion about distances between elements in B We can usethis to prove the final theorem of this section

Theorem 415 For any given 1RDM γ isin ΓN there exists a sequence γk isin ΓN so that

1 γk rarr γ

2 F has a ctf at each γk ie each γk is eq-V-representable

This theorem is equivalent to the statement that ΓV is dense in ΓN

Proof [of Theorem 415]The right hand side of part 3 of Theorem 414 is finite and independent of ε We will denote it by∆0

∆0 = F [γ0] +

intv0γ0 minus inf

γisinΓN

F [γ] +

intv0γ

(492)

We now choose a set εk with εk = k∆0 From Theorem 414 we also know that for each of theseεk we can find a 1RDM γk so that

γk minus γ0 le ∆0ε = 1k (493)

Finally part 2 of Theorem 414 then ensures eq-V-representability of γk

With this final proof we have succeeded in laying a firm theoretical basis for FT-RDMFT As thederivation of our results rested mainly on a few general properties of quantum systems in (grandcanonical) equilibrium we will now investigate how these results translate if one considers differentensembles ie microcanonical or canonicl ones

45 Relation to microcanonical and canonical formulations

The main results of this section rely only on the properties of the sets of SDOs and 1RDMs consideredthe variational principle and on the convexity and lower semicontinuity of the SDO-functionalsThe fact that one works with a grand canonical ensemble is only required in the investigation of theexistence of a Kohn-Sham system All other results can be translated to the cases of microcanonicaland canonical ensembles with an appropriate replacement of the main thermodynamical variablesto energy or free energy respectively One might wonder how the claim that also for microcanonicalensembles ΓV should be dense in ΓN is compatible with the statement that there is no Kohn-Shamsystem in zero-temperature RDMFT The solution to this apparent contradiction is that our resultsdo not claim that the corresponding potential will not lead to degeneracies in the noninteractingHamiltonian If one allows the noninteracting system to be degenerate then there is a Kohn-Shamsystem in RDMFT [67]

Because the description of systems in canonical equilibrium will ultimately become importantin our investigation of phase transitions of the HEG in Section 8 we will briefly state the required

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FT-RDMFT

concepts in the following The appropriate Hilbert space for the description of canonical ensemblesof particle number N is given by the N -particle subspace HN of the Fock space H of Eq (41)

HN = ShotimesN (494)

and the SDOs are weighted sums of projection operators on HN

Dc =sum

α

wαN |ΨαN 〉〈ΨαN | wαN ge 0sum

α

wαN = 1 (495)

The variational principle which governs this situation involves the free energy F rather than thegrand potential

F [D] = TrD(H + 1β ln D) (496)

The corresponding eq-SDO is given by

Dceq =

eminusβH

TreminusβH (497)

where H is now the N -particle Hamiltonian of the system We already see from Eq (497) thatthe external potential is only defined up to a constant ie there are infinitely many potentialsall yielding the same canonical eq-1RDM The Lieb-construction now allows to define a canonicaluniversal functional Fc[γ] on the whole domain of ensemble-N-representable 1RDMs as

Fc[γ] = infDisinHNrarrγ

TrD(T + W + 1β ln D) (498)

The equilibrium of the system is then found by a minimization of the free energy functional

F [γ] = Fc[γ] + Vext[γ] (499)

So far the formulation of FT-RDMFT for canonical ensembles follows in the same steps as forgrand canonical ones and is equally well founded As pointed out before the main difference tothe grand canonical ensemble occurs when we investigate the canonical Kohn-Sham system Asin the grand canonical case the NOs of an eq-1RDM of a noninteracting system will be given bythe eigenstates of the one-particle Hamiltonian and the ONs will lie in the interior of the set ofensemble N-representable 1RDMs However because there is no simple analytic relation betweenthe eigenvalues of the one-particle Hamiltonian and the ONs as in Eq (452) we do not knowif every 1RDM with 0 lt ni lt 1 is a canonical eq-1RDM and we can only state that the set ofnoninteracting canonical eq-1RDMs lies dense in the set of ensemble-N-representable ones If weknow that a given 1RDM corresponds to a noninteracting canonical equilibrium the correspondingpotential has to be found by iterative methods similar to [95] The reason why the construction ofthe noninteracting Hamiltonian from the canonical eq-1RDM is more complicated than in the grandcanonical case lies in the fact that the exponent of the eq-SDO in a canonical ensemble contains onlyN -particle contributions which will become important in our derivation of a perturbative method toapproximate FT-RDMFT-functionals in Section 6

In general FT-RDMFT functionals describing either a canonical or a grand canonical equilibriumare different However considering the special case of the system being in the thermodynamic limitthe thermodynamic variables and therefore also the corresponding functionals of grand canonicaland canonical ensembles coincide We can therefore use a functional for the grand potential Ω[γ] tocalculate the free energy

F [γ] = Ω[γ] + microN [γ] (4100)

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FT-RDMFT

T

T1

DN

DV

DNL


T

TradicN

ΓN

ΓV


Figure 44 Sketch of all relevant spaces of SDOs and 1RDMs and their embedment in a Banachspace


We started this section by giving a definition of the physical system we are interested in ie quantummechanical systems in grand canonical equilibrium We then reviewed how standard quantum me-chanics approaches this problem focussing on the description via functionals of SDOs Subsequentlywe derived several properties of these functionals eg convexity and lower semicontinuity Becauseemploying a SDO in a variational principle is far from being applicable in calculations we then laidthe mathematical foundation for the description of grand canonical ensembles via a functional theorywrt 1RDMs The corresponding existence-theorem (see Section 421) justifies this formulation forall eq-V-representable 1RDMs implying a one-to-one correspondence between ΓV and DV the setof all eq-V-representable SDOs However because it is not known how to characterize ΓV we thenused a formulation of the FT-RDMFT functionals similar to the Lieb construction in DFT We werethen able to show that the Lieb construction leads to an extension of the one-to-one correspondenceto ΓN Ie for every γ isin ΓN there is a D isin DN so that Eq (446) attains its minimum for D Theset of all D corresponding to a γ via Eq (446) is denoted by DN

L In summary the mappings areas follows

DN many-1larrminusminusminusminusminusminusrarr ΓN (4101)

DV 1-1larrminusminusminusminusminusminusrarr ΓV (4102)

DNL

1-1larrminusminusminusminusminusminusrarr ΓN (4103)

We show a sketch of the several sets defined in this section in Figure 44 Subsequently using theprevious results we were then able to show that the set ΓV is dense in ΓN ie given a γN isin ΓN there is a γV isin ΓV arbitrarily close to it This will prove to be important in the investigation ofexact properties of the FT-RDMFT functionals in Section 5 as well as for the development of amethodology to derive approximate functionals via methods from FT-MBPT in Section 6 In Figure45 we depict the logical outline of this section backing up the importance of these mathematicalconsiderations

Finally we investigated how our results which were orginally derived for grand canonical ensem-bles translate to microcanonical and canonical ones We found that the only difference concerns theexistence ie construction of the appropriate Kohn-Sham systems and that the general formulationof a canonical version of FT-RDMFT is theoretically well founded

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FT-RDMFT

Goal Describe eq properties of QM system in grand canon-ical equilibrium (Figure 41)

Hohenberg-Kohn + Gilbert theorem (Sec 421)Ensures existence of functional Ω[γ]

Domain of Ω[γ] is ΓV which is not known

Domain of Ω[γ] extended to all ΓN via Lieb construction(Eq(446))

Is infimum in Lieb construction Eq (444) also minimum

Theorem 48 Yes leading to DNL

1minus1larrrarr ΓN

Convexity and lower semicontinuity of Ω[γ] proven in Thms49 and 410

ΓV dense in ΓN (Theorem 415)

Derivation of exact properties (Sec 5)Construction of functionals via MBPT (Sec 6)

solved by

leads to

solved by

leads to

solved by

enables

used for

allows

Figure 45 Logical outline of Section 4

41135

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5 Properties of the exact FT-RDMFT functionals

In the previous section we laid the foundations for the description of grand canonical ensembles viaFT-RDMFT What remains to be done is the development of correlation functionals Rather thanstumbling through the dark it will be helpful to know properties of the exact correlation functionalwhich may guide the development of approximations As a motivating example from DFT we shouldlike to mention the PBE functional [5] which incorporates exact coordinate scaling relations leadingto an increase in accuracy over a wide range of applications We are going to structure this sectionas folows

Section Description

51 Using the variational principle we will show that the correlation functionals of the grandpotential Ωc[γ] as well as separately the interaction Wc[γ] and entropy Sc[γ] are alwaysnegative

52 We will then express the correlation grand potential functional Ωc[γ] through the cor-relation interaction functional Wc[γ] via the method of adiabatic connection

53 Introducing the concept of uniform coordinate scaling we can then derive several exactrelations between the different correlation contributions

54 Finally we will be able to derive properties of the correlation functionals linking thebehaviour at different temperatures and interaction strengths

Related publications [100]It is worth pointing out that in the derivation of all these exact properties the concept of eq-V-representability (see Section 44) plays a central role We always assume that a given 1RDM canbe seen as an eq-1RDM of a noninteracting as well as of an interacting system To help the readerdistinguish between results and derivations we chose to put the exact relations in frames in thissection Furthermore we will use a temperature variable τ measured in units of kB τ = 1β

51 Negativity of the correlation functionals ΩcWc Sc

As mentioned before we assume eq-V-representability Therefore a given 1RDM can be seen asthe eq-1RDM of either an interacting or a noninteracting system with eq-SDOs Dw and D0 Thevariational principle Eq (44) then yields

Ωw[Dw] lt Ωw[D0] (51)

Ω0[D0] lt Ω0[Dw] (52)

where ΩwΩ0 denote the grand potentials of the interacting and noninteracting system respectivelyIt is now in order to define the correlation contributions to interaction energy Wc[γ] entropy Sc[γ]and grand potential Ωc[γ] as

Wc[γ] = Tr(Dw[γ]minus D0[γ])W (53)

Sc[γ] = minusTrDw[γ] ln Dw[γ]+TrD0[γ] ln D0[γ] (54)

Ωc[γ] =Wc[γ]minus τSc[γ] (55)

Because Dw[γ] and D0[γ] by construction both yield the same 1RDM the respective expectationvalues of one-particle operators will yield the same result Therefore Eqs (51) and (52) reduce to

TrDw[γ](W + τ ln Dw[γ]) lt TrD0[γ](W + τ ln D0[γ]) (56)

TrD0[γ] ln D0[γ] lt TrDw[γ] ln Dw[γ] (57)

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Exact properties

This can be used to get our first exact relations for the correlation energy entropy and grandpotential

Sc[γ] lt 0

Wc[γ] lt τSc[γ] lt 0

Ωc[γ] lt 0

(58)

(59)

(510)

We will now proceed by deriving an adiabatic connection formula between Ωc[γ] and Wc[γ]

52 Adiabatic connection formula

We consider a parameter-dependent Hamiltonian of the form

Hλ = T + V0 + λW (511)

For this Hamiltonian the grand potential functional as defined in Eq 446 becomes

Ωλ[γ] = minDrarrγ

TrD(Hλ minus microN + τ ln D

)(512)

= minDrarrγ

(TrD(Hλ + V λ[γ]minus microN + τ ln D

)minus Tr

DV λ[γ]

) (513)

V λ[γ] is chosen so that the eq-1RDM of Hλ + V λ[γ] is γ for each λ Because V λ[γ] is a one-particleoperator the last term in Eq 514 can be taken out of the minimization The minimizing SDO

is then given by the equilibrium operator Dλ[γ] = eminusβ(Hλ+V λ[γ]minusmicroN)Zλ and the grand potentialfunctional becomes

Ωλ[γ] = TrDλ[γ]

(Hλ + V λ[γ]minus microN + τ ln Dλ[γ]

)minus Tr

D[γ]V λ[γ]

(514)

D[γ] can be chosen to be any SDO yielding γ ie also a λ-independent one For each λ the firstterm on the right hand side of Eq (514) describes a system in equilibrium with 1RDM γ Takingthe derivative with respect to λ therefore does not yield a contribution from the derivatives of theSDO Dλ[γ] and the total derivative of the grand potential functional Ωλ[γ] wrt λ becomes

dΩλ[γ]

dλ= Tr

Dλ[γ]

(W +

d

dλV λ[γ]

)minus d

dλTrD[γ]V λ[γ]

(515)

Again using the one-particle character of V λ[γ] and integrating Eq (515) yields

Ω[γ] = Ω0[γ] +

int 1

0

dλTrDλ[γ]W

(516)

The Hartree and exchange functionals (Eqs (459) and (460)) separately fulfill this equation andare substracted This then leads to

Ωc[γ] =

int 1

0

dλ

λWλ

c [γ] (517)

whereWλc [γ] = Tr(Dλ[γ]minusD0[γ])λW The correlation grand potential therefore can be interpreted

as a coupling constant average over the correlation interaction energy ie over the interaction energydifference between a noninteracting and an interacting ensemble It is interesting to note a similaritybetween DFT and FT-RDMFT at this point In DFT the method of adiabatic connection expresses

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Exact properties

minus4 minus2 2 4

minus02

minus01

01

02

z

Ψ210(0 0 z)

51a Wavefunction

minus4 minus2 2 4

001

002

003

004

0z

ρ210(0 0 z)

51b Density

Figure 51 Hydrogen wavefunctions Ψnlm(x y z) and densities ρnlm(x y z) for n = 2 l = 1m = 0on the z-axis The blue dashed graphs denote the unscaled functions whereas the red graphs followfrom a a uniform coordinate scaling by a factor of 2 according to Eq (519)

the correlation energy Ec[n] throughWc[n] The kinetic correlation contribution Tc[n] = T 1[n]minusT 0[n]is therefore completely taken care of by the coupling constant integration In FT-RDMFT wherethere is no kinetic correlation contribution the coupling constant integration manages to take careof the entropic correlation contribution Sc[γ] = S1[γ]minus S0[γ] It seems surprising that both kineticas well as entropic contributions which are naıvely expected to be very different in nature are dealtwith equivalently in the different frameworks of DFT and FT-RDMFT

Employing the method of uniform coordinate scaling we can now proceed to derive several furtherexact properties of the different contributions to the exact functional in FT-RDMFT

53 Uniform coordiante scaling

The method of uniform coordinate scaling proved to be a valuable tool in the derivation of exactproperties of the functionals in zero-temperature DFT as well as FT-DFT In DFT it was suc-cessfully used to derive several scaling inequalities of the correlation functionals [40] as well as fordetermining bounds for the xc functional [41] Some of the exact uniform scaling relations were laterincorporated into the famous PBE functional [5] Furthermore as we will demonstrate in Section54 one can make a connection between coordinate scaling and a combined coupling constant andtemperature scaling The TD-DFT equivalent was exploited in Ref [42] to determine the correlationenergy of the HEG from the frequency-dependent exchange-correlation kernel to assess the quality ofseveral approximations for the kernel The method of uniform coordinate scaling is not restriced to aspecific form of the SDO describing the quantum mechanical state and can therefore also be used toderive exact properties of the exact functionals in FT-DFT [43] This situation however comparedto FT-RDMFT is complicated by the fact that the correlation functional in FT-DFT consists ofthree different contributions kinetic interaction and entropic whereas in FT-RDMFT the kineticpart is treated exactly We might therefore claim that in this particular aspect FT-RDMFT is ad-vantageous compared to FT-DFT because it allows a more detailed investigation of exact propertiesvia uniform coordinate scaling

To introduce the concept of coordinate scaling we consider an arbitrary element |ψ〉 of the Hilbertspace under consideration Its spatial representation is given by ψ(r1 rN ) = 〈r1 rN |ψ〉 Onenow introduces a transformation U(λ) so that

ψλ(r1 rN ) = 〈r1 rN |U(λ)|ψ〉 (518)

= λ3N2ψ(λr1 λrN ) (519)

We illustrate the effect of scaling in Figure 51 It then seems natural to define a ldquoscaledrdquo operator

45135

Exact properties

Oλ = U(λ)OU(1λ) For this operator it follows that

〈ψλ|Oλ|ψλ〉 = 〈ψ|O|ψ〉 (520)

The scaled operators for kinetic energy particle interaction and particle number can be simplyrelated to their unscaled counterparts

Tλ =1

λ2T (521)

Wλ =1

λW (522)

Nλ = N (523)

We would now like to know how the exact 1RDM functionals in FT-RDMFT behave under scalingof the 1RDM From the explicit forms of Ωk[γ]ΩH [γ]Ωx[γ] and S0[γ] (Eqs (456) - (461)) onecan immediately deduce

T [γλ] = λ2Ωk[γ] (524)

ΩH [γλ] = λΩH [γ] (525)

Ωx[γλ] = λΩx[γ] (526)

S0[γλ] = S0[γ] (527)

Wc[γ] Sc[γ] and Ωc[γ] are not known explicitly but we can derive exact relations for their behaviourunder coordinate scaling We will use the fact that a scaled SDO Dλ[γ] leads to a scaled 1RDM γλThis can be recast in the more convenient form

D 1λ[γλ]rarr γ (528)

D 1λ[γλ] now decribes a system with interaction W 1

λ We can again use the variational principle to

get

TrD[γ](W + τ ln D[γ]) le TrD 1λ[γλ](W + τ ln D 1

λ[γλ]) (529)

TrD 1λ[γλ](W 1

λ+ τ ln D 1

λ[γλ]) le TrD[γ](W 1

λ+ τ ln D[γ]) (530)

With the help of Eqs (522) and using the scaling behaviour of ΩH Ωx and S0 this can be writtenas

Wc[γ]minus τSc[γ] le1

λWc[γλ]minus τSc[γλ] (531)

Wc[γλ]minus τSc[γλ] le λWc[γ]minus τSc[γ] (532)

which directly yields the following relation

λ(τSc[γλ]minus τSc[γ]) lt Wc[γλ]minus λWc[γ] lt τSc[γλ]minus τSc[γ] (533)

Eqs(531)(532) and (533) now allow us to find exact relations for the correlation functionals

(λminus 1)τSc[γλ] le Ωc[γλ]minus λΩc[γ] le (λminus 1)τSc[γ] (534)

(1minus λ)(Wc[γλ]minus λWc[γ]) ge 0

(1minus λ)(Sc[γλ]minus Sc[γ]) ge 0

(535)

(536)

46135

Exact properties

From Eq (535) it follows that Wc[γλ] gt λWc[γ] for λ lt 1 Because Wc lt τSc (Eq (59)) thisimplies

Wc[γλ]λrarr0minusrarr 0

Sc[γλ]λrarr0minusrarr 0

Ωc[γλ]λrarr0minusrarr 0

(537)

(538)

(539)

The correlation contributions vanish in the limit λ rarr 0 We can now derive differential equationsrelating the different correlation functionals It will be instructive to use the variational principle ina different form

0 =d

dλ

(Ω[D 1

λ[γλ]]

)∣∣∣λ=1

(540)

As usual getting rid of the one-particle operators we obtain

0 =d

dλTr

D 1

λ[γλ]

(1

λW 1

λ+ τ ln(D 1

λ[γλ]

)∣∣∣∣λ=1

(541)

which gives the following differential equation

W [γ] =d

dλ(W [γλ]minus τS[γλ])|λ=1 (542)

ΩH Ωx and S0 fulfill this equation separately and can be substracted Applying the remainingcorrelation functionals to γν and consecutively renaming ν rarr λ yields

Wc[γλ] = λd

dλΩc[γλ] (543)

Because Ωc[γλ] vanishes in the limit λrarr 0 (Eq(539)) the solution of Eq(543) is given by

Ωc[γλ] =

int λ

0

dmicro

microWc[γmicro] (544)

From Eq (55) we can furthermore derive

Sc[γλ] = λ2(part

partλ

1

λΩc[γλ]

)(545)

which can be used to finally relate Wc[γλ] and Sc[γλ]

d

dλτSc[γλ] = λ

d

dλ

(1

λWc[γλ]

) (546)

From Eq (543) we see that Ωc[γ] is monotonically decreasing with λ We can show that this isalso true for Wc[γ] and Sc[γ] For this purpose we expand the functionals in Eqs (535) and (536)around λ = 1 We then replace γ rarr γmicro and subsequently substitute λmicrorarr λ

d

dλΩc[γλ] =

1

λWc[γλ] lt 0

d

dλWc[γλ] lt

d

dλΩc[γλ] lt 0

d

dλSc[γλ] lt 0

(547)

(548)

(549)

47135

Exact properties

Ωc[γλ]

λ1

minusτSc[γ]

0

Ωc[γ] λΩc[γ] + (λminus 1)τSc[γ]

Ωc[γλ]

λΩc[γ] + (λminus 1)τSc[γλ]

Figure 52 Given an approximation for Ωc[γ] one can test if Ωc[γλ] fulfills the exact conditionsderived so far Both its first and second derivatives wrt λ have to be negative (Eqs (547) and(551)) Furthermore it has to lie in the allowed (grey shaded) area defined by the relations in Eq(534) and the negativity constraint (510) Sc[γλ] is to be derived from Eq (545)

Finally we can also prove negativity of the second derivative of Ωc[γλ] with respect to λ by differ-entiating Eq (547) and using Eq (548)

d2

dλ2Ωc[γλ] = minus

1

λ2Wc[γλ] +

1

λ

d

dλWc[γλ] lt minus

1

λ2Wc[γλ] +

1

λ

d

dλΩc[γλ] (550)

d2

dλ2Ωc[γλ] lt 0 (551)

To give a more descriptive representation of the relations derived so far we sketch the behaviour ofa model-functional in Figure 52

We will now have a closer look at Eq (544) It looks very similar to our adiabatic connectionformula in Eq (517) To further investigate this similarity we will show in the following sectionthat a coordinate scaling is equivalent to an appropriate scaling of the temperature as well as of theinteraction strength of the system

54 Temperature and interaction scaling

Because we will now work with systems at different temperatures and interaction strengths we willinclude these parameters as arguments The scaled equilibrium SDO for a system at temperatureτ = 1β and interaction strength w is then given by

Dλ(τ w)[γ] = eminusβ(T λ2+Vλ[γ]+WλminusmicroN)Zλ (552)

One knows that Dλ[γ] yields γλ and can therefore deduce

Dλ(τ w)[γ] = D(λ2τ λw)[γλ] (553)

48135

Exact properties

With this relation and the scaling relations from Eqs (521)(522) and (523) we can show thatall contributions to the grand potential functional show the same behaviour under scaling Theyacquire a prefactor of λ2 while the temperature is scaled by 1λ2 and the interaction strength by1λ This reads for the correlation functionals as

Wc(τ w)[γλ] = λ2Wc(τλ2 wλ)[γ]

τSc(τ w)[γλ] = λ2τ

λ2Sc(τλ

2 wλ)[γ]

Ωc(τ w)[γλ] = λ2Ωc(τλ2 wλ)[γ]

(554)

(555)

(556)

We can now rewrite Eq(544) for λ = 1 and compare it with Eq(517)

Ωc(τ w)[γ] =

int 1

0

dλ

λWc(τ λw)[γ] (557)

Ωc(τ w)[γ] =

int 1

0

dλλWc(τλ2 wλ)[γ] (558)

Eqs (557) and (558) accomplish to relate the temperature and interaction strength dependence ofthe correlation functionals They can be used to test if an approximate Ωapprox

c (τ w)[γ] shows aphysical temperature and interaction strength dependence ie if it respects Eqs (557) and (558)


In this section we focussed on the task of deriving properties of the exact FT-RDMFT functionalsUnder the assumption of eq-V-representability we could employ the variational principle to provethe negativity of the separate correlation contributions Wc Sc and Ωc The variational principlefurthermore allowed us to express the correlation contribution to the grand potantial Ωc via acoupling constant integration over the correlation contribution to the interaction Wc which is verysimilar to the method of adiabatic connection in DFT We then employed the method of uniformcoordinate scaling This allowed us on one hand to derive differential equations relating thedifferent correlation functionals and on the other hand led to the derivation of bounds for thescaled correlation contributions Finally we were able to relate uniform coordinate scaling to ascaling of temperature and interaction

The properties derived in this section can be used to test existing approximate functionals inFT-RDMFT or guide the development of new ones Further progress in the determination of exactproperties could be achieved by considering exact properties known from zero-temperature RDMFTlike particle-hole symmetry or size consistency Another useful investigation could be concerned withthe derivation of exact bounds on the correlation grand potential eg in the spirit of the Lieb-Oxfordbound in DFT [101]

We present in the next section a rigorous method to develop approximate grand potential func-tionals by employing methods of FT-MBPT As a first application of the results derived in thepresent section we will show that these functionals fulfill all exact relations derived so far under afew conditions

49135

50135

6 Constructing approximate functionals

After deriving the mathematical foundations of FT-RDMFT in Section 4 we now turn to the prob-lem of deriving approximate functionals for the grand potential functional Ω[γ] One key differenceto RDMFT is the fact that at finite temperature there exists a Kohn-Sham system (see Section423) We shall employ this fact in the development of a methodology to construct functionals usingthe powerful methods of FT-MBPT The unperturbed system will then be the Kohn-Sham systemrather than just the original system stripped from its interaction part This section will be structuredas follows

Section Description

61 We will start by pointing out the conceptual problems of employing FT-MBPT to derivecorrelation functionals in FT-RDMFT

62 We will then show how these problems can be solved by introducing a modified pertur-bation consisting of the original two-particle interaction together with an appropriateone-particle correction

63 The method introduced will have one major conceptual uncertainty namely the as-sumption of eq-V-representability of the 1RDMs under consideration We will pointout why the results from Section 44 prevent this uncertainty from having notworthyimpact on our method therefore both substantiating our perturbative scheme as wellas justifying our efforts from Section 44

64 Finally we will investigate how arbitrary diagrams in our method will behave underuniform coordinate scaling We will find that they naturally fulfill most of the exactconditions derived in Section 5


61 Why not use standard FT-MBPT

To better understand the conceptual differences between a standard FT-MBPT treatment and ourapproach of FT-RDMFT we will give a short review of FT-MBPT in the followingLet the system under consideration be governed by a Hamiltonian H = H0 + W where H0 is aone-particle operator and W represents a two-particle interaction (see Eqs (24)) The questionFT-MBPT tries to answer can be phrased as follows

ldquoGiven a Hamiltonian H a temperature τ and a chemical potential micro What are the thermody-namic equilibrium properties of the systemrdquo

The central assumption of FT-MBPT is then that the two-particle interaction W can be treatedas a perturbation In grand canonical equilibrium the noninteracting system governed by h0(x x

prime) =sumi εiφ

lowasti (x

prime)φi(x) exhibits the following finite-temperature or Matsubara Greenrsquos function G0(x ν xprime νprime)[102]

G0(x ν xprime νprime) =sum

i

φi(x)φlowasti (x

prime)eminus(εiminusmicro)(νminusνprime)

ni ν gt νprime

ni minus 1 ν gt νprime(61)

The eq-1RDM is then given as the temperature diagonal of G0(x ν xprime νprime)

γ0(x xprime) = G0(x ν xprime ν+) =sum

i

niφlowasti (x

prime)φi(x) (62)

51135

Approximate functionals

W + Veff

G0(x 0 xprime 0+) = γ(x xprime) = G(x 0 xprime 0+)

Figure 61 Perturbation in FT-RDMFT It consists of a two-particle interaction (arrows) and anadditional nonlocal one-particle potential (wavy lines) designed to leave the 1RDM invariant

where the eigenvalues εi and occupation numbers ni are related via the Fermi distribution (see Eqs(452) and (453))

ni =1

1 + eβ(εiminusmicro)(63)

εi =1

βln

(1minus nini

)+ micro (64)

We see that knowledge of the noninteracting 1RDM determines the noninteracting Greenrsquos functioncompletely By the virtue of the finite-temperature version of Wickrsquos theorem [102] one can nowrelate the interacting and noninteracting systems by using the method of Feynman diagrams Inthis procedure the chemical potential (and the temperature) is held constant But this means thatin general the interacting system has a different particle number from the noninteracting one Ifthe particle numbers are not the same then surely also the 1RDMs are different In FT-RDMFThowever the question one is asking differs from the one in FT-MBPT It reads

ldquoGiven a Hamiltonian H and a temperature τ what are the thermodynamic properties of the in-teracting system with eq-1RDM γrdquo

Therefore one explicitly has the interacting 1RDM at hand If one should now resort to just usinga perturbative expansion from FT-MBPT one would encounter the problem that the noninteracting1RDMs and therefore also the noninteracting Greenrsquos function would not be known Therefore theFeynman diagrams appearing in a perturbative expansion of the interacting Greenrsquos function cannotbe evaluated Our approach to solve this conceptual problem is the introduction of an additional ingeneral nonlocal one-particle potential which lets the 1RDM stay invariant under this new pertur-bation We depict this idea in Figure 61 Accordingly the interacting and noninteracting 1RDMswould be the same and one could easily calculate the noninteracting Greenrsquos function We shouldlike to acknowledge that a similar approach was pursued in the course of DFT [103] There inaccordance with the nature of DFT the additional one-particle potential was local and designed tokeep the density constant

52135


62 Methodology of modified perturbation theory

After these preliminary considerations we can proceed to derive Feynman rules to construct approx-imate functionals for Ω[γ] We start from the adiabatic connection formula Eq (516) which we willrestate in the following

Ω[γ] = Ω0[γ] +

int 1

0

dλTrDλ[γ]W

(65)

The λ-dependent SDO is given by

Dλ[γ] = eminusβ(T+V λ[γ]+λWminusmicroN)Z (66)

where V λ[γ] is chosen so that γ(x xprime) = TrDλ[γ]Ψ+(xprime)Ψ(x) stays invariant with respect to λIt is now known (eg[104] p230) that 〈λW 〉λ = TrDλ[γ]λW can be calculated from the Greenrsquosfunction of the system under consideration

〈λW 〉λ =1

2

intdxdxprime lim

νprimerarrν+

(minusδ(xminus xprime) part

partνminus k0(λ)(x xprime)

)Gλ(x ν xprime νprime) (67)

where β = 1τ and k0(λ) is defined by K0(λ) = T + V0 minus microN + V λ[γ] As explained beforewe can now relate the interacting system with its Kohn-Sham system which allows to express theresulting Feynman diagrams in terms of the ONs and NOs of the 1RDM The Hamiltonian of thisnoninteracting system is given by K0(0) Eq (67) therefore is rewritten as

〈λW 〉λ =1

2

intdxdxprime lim

νprimerarrν+

(minus δ(xminus xprime) part

partνminus k0(0)(x xprime)

)Gλ(x ν xprime νprime)+

1

2

intd3xd3xprime

(v0[γ](xprime x)minus vλ[γ](xprime x)

)γ(x xprime) (68)

Now methods of FT-MBPT can be applied The unperturbed Hamiltonian is K0(0) whereas theinteracting one reads K(λ) = K0(λ) + λW This defines the perturbation as (λW + V λ minus V 0) Theone-particle part will be denoted by vλeff

vλeff = vλ minus v0 (69)

The perturbation now consists of a two-particle interaction λw(r rprime) and a possibly nonlocal one-particle interaction vλeff (r r

prime) The proof of Wickrsquos theorem is still applicable for this kind ofperturbation and the same Feynman graphs appear (see Table 61)

In the special situation of a temperature-independent Hamiltonian and a spatially uniform systemEq (68) can be written in a compact form

〈λW 〉unifλ =1

2

Σlowast minus

(610)

where Σlowast denotes the irreducible self energy In practice one has to select a particular set ofFeynman diagrams We will demonstrate in the following section that the requirement that bothinteracting and noninteracting systems exhibit the same eq-1RDM is sufficient for an elimination ofthe Kohn-Sham potential contributions We will be left with diagrams containing only two-particleinteractions and noninteracting Greenrsquos functions which can be calculated using the Feynman rulesof Appendix A5

53135


λw(r rprime)

vλ(x xprime)

Gλ(0)(x τ xprime τ prime)


Table 61 Feynman graph contributions for the construction of correlation functionals inFT-RDMFT

621 Elimination of Veff

In Figure 62 we show all contributions to the interacting Greenrsquos function G(x ν xprime νprime) up to second-order in w and veff We now expand veff in orders of w as

veff = v(1)eff + v

(2)eff + (611)

It is now possible to iteratively solve for the different contributions for v(n)eff from the requirement

G(x 0 xprime 0+) = G0(x 0 xprime 0+) It can be recast in terms of Feynman diagrams as

0

0+

=

0

0+

+

0

0+

Σlowast (612)

The interacting and noninteracting 1RDMs are required to be equal Therefore we are led to theconditional equation for veff

0 =

0

0+

Σlowast (613)

54135


+ + + +

+ + + + + +

+ + + +

+ + + + + +

Figure 62 Contributions to the interacting Greenrsquos function G(x ν xprime νprime) up to second-order in Wand Veff

We can now utilize Eq (613) to solve for v(1)eff by only considering first-order contributions We get

(1)

0

0+

= minus

0

0+

minus

0+

0

(614)

55135


intdydyprime

int β

0

dνdνprimev(1)eff (y yprime)δ(ν minus νprime)G0(x 0 y ν)G0(yprime νprime xprime 0+)

= minusintdydyprime

int β

0

dνdνprimew(y yprime)δ(ν minus νprime)G0(x 0 y ν)G0(y ν yprime νprime)G0(yprime νprime xprime 0+)minusintdydyprime

int β

0

dνdνprimew(y yprime)δ(ν minus νprime)G0(x 0 y ν)G0(y ν xprime 0+)G0(y ν yprime νprime) (615)

We can now replace the noninteracting Greenrsquos function with expression Eq (61) It is exactly herethat the nonlocality of the 1RDM pays off because we can use the orthonormality and completenessof φ and the fact that 0 lt τ lt β to get

v(1)eff (x x

prime) = minusδ(xminus xprime)intdyw(x y)γ(xprime xprime)minus w(x xprime)γ(x xprime) (616)

or in terms of Feynman diagrams

= minus minus (617)

This result bears the following important information Up to first order in the interaction the Kohn-Sham potential Veff cancels completely the interaction contributions coming from W implying thatthe Greenrsquos functions of the noninteracting and interacting system are equal This represents theresult we obtained in Section 425 namely that the first-order functional in FT-RDMFT is equivalentto finite-temperature Hartree-Fock theory which is effectively a noninteracting theory

The next step is now to replace all first-order contributions to veff by Eq (617) There willbe many cancellations of diagrams and we get for the second-order contribution to the Kohn-Sham

potential v(2)eff

(2)

0

0+

= minus

0+

0

minus

0+

0

(618)

The solution of this equation is rather more complicated than the corresponding one from first-orderdiagrams The reason for this is that the in and outgoing Greenrsquos functions will have differenttemperature arguments As a simplification we write

(2)

0

0+

= minus

0

0+

M (619)

Written in spatial representation this becomes

intdydyprime

int β

0

dτdτ primev(1)eff (y yprime)δ(τ minus τ prime)G0(x 0 y τ)G0(yprime τ prime xprime 0+)

= minusintdydyprime

int β

0

dτdτ primeM(y τ yprime τ prime)G0(x 0 y τ)G0(yprime τ prime xprime 0+) (620)

56135


which will be solved by

v(2)eff =

sumij φi(x)φ

lowastj (x

prime)int β

0dνdνprimeMij(ν ν

prime)eν(εiminusmicro)eminusνprime(εjminusmicro)

int β

0eν

prime(εiminusεj) (621)

where

Mij(ν νprime) =

intdydyprimeM(x ν xprime νprime)φi(x

prime)φlowastj (x) (622)

An evaluation of Eq (621) with the appropriate M(x ν xprime νprime) then yields the second-order con-

tribution to veff Exactly in the same fashion as before we could now replace all appearing v(2)eff

and solve for the third-order term v(3)eff We would be led to an equation just like Eq (619) with

different M(x ν xprime νprime) and therefore to an Eq (619) which would then yield the third-order con-tribution Applying this method iteratively therefore determines veff We would now like to pointout a small sublety It seems that by the arguments above one could solve for veff for any given1RDM γ isin ΓN This would lead to the conclusion that all ΓN = ΓV The flaw in this argumentis the assumption that the perturbation expansion of G converges This question of convergence ofperturbation expansions is a very complicated and so far generally unsolved problem in FT-MBPTTherefore we cannot conclude ΓN = ΓV However given any FT-MBPT approximation for G wecan still calculate the corresponding veff

It is now possible to express veff in terms of Feynman diagrams by the introduction of anadditional graphical contribution

M =

sumij φi(x)φ

lowastj (x

prime)int β



int β

0eν


Combining our first-order and second-order results we get the following expression for the interactingGreenrsquos function

= + minus + minus (624)

Any choice of diagrams leads to an approximation for G(x ν xprime νprime) from which we can derive Ω[γ]via Eqs (516) and (67)

Considering now only the first-order contribution to the grand potential functional we can justifyour claim from Section 424 that the form of the Hartree and exchange functionals from zero-

temperature RDMFT carry over to the finite-temperature case The first-order functional Ω(1)xc is

given in terms of Feynman diagrams as

Ω(1)xc [γ] = + (625)

which translates to Eqs (459) and (460)

Ωxc[γ] =1

2


2


57135


In the case of a translationally invariant Hamiltonian Eq (610) allows a simple representationof the grand potential functional up to second order

Ω(2)xc [γ] = + + + (627)

We want to point out that an inclusion of higher-order diagrams will lead to contributions to Ωc

including the new Feynman graphs introduced in Eq (623) Up to second order they are cancelledby the effective potential contribution in Eq 610

Applied to the HEG the first and third diagrams in Eq 627 are diverging The first one isthen cancelled by the positive background charge and the third one is usually subjected to an RPAscreening rendering it finite We will investigate the behaviour of the different contributions to thexc functional as described in Eq 627 in Section 84

Having derived a methodology to iteratively construct functionals for FT-RDMFT we will nowcome back to the problem already mentioned in the introduction to Section 6 namely the questionof eq-V-representability of the 1RDMs under consideration

63 Importance of eq-V-representability

Starting from a noninteracting 1RDM γ isin ΓN we explicitly assumed that it was also eq-V-representablefor interaction W But this might contain the following conceptual problem Generally speaking theperturbative procedure is only valid on ΓV Let us now assume that all the assumptions of FT-MBPTare well fulfilled and that we derived a good functional Ω[γ] Good means that a minimization ofΩ[γ] on ΓV yields a minimum close to the exact one Applying this functional on ΓNΓV it is notat all justified to assume that the functional retains its good behaviour ie that one is not led to acompletely different minimum on ΓNΓV We illustrate this problem in Figure 63 In summary theproblem is that one minimizes a functional which is well behaved on one set on a bigger set wherethere is nothing known about its behaviour It is now that Theorem 415 unveils its importance Itstates that ΓV is dense in ΓN which ensures that given any 1RDM γN isin ΓN there is another 1RDMγV isin ΓV arbitrarily close to it Because the functionals derived from FT-MBPT will be considerablysmooth this means that Ω[γN ] will be arbitrarily close to Ω[γV ] Therefore a minimization of Ω[γ]over the full set ΓN will lead to a minimum arbitrarily close to the minimum over ΓV (see Figure63)

Ω[γ]

γΓV ΓNΓV ΓV

(γV EV

) (γN EN

)

63a ΓV not being dense in ΓN Minima on dif-ferent sets can be very different

Ω[γ]

γΓN ΓV

(γV EV

) (γN EN

)

63b ΓV dense in ΓN Minima on different setsare infinitesimal close

Figure 63 Illustration of the importance of eq-V-representability in the context of functional con-struction via FT-MBPT The grey regions in both figures denote the set ΓNΓV In Figure 63bthis region is enlarged for graphical purposes

58135


64 Scaling behaviour

We can now investigate how the single diagrams appearing in a certain approximation to Ωc[γ]behave under scaling of the 1RDM temperature and interaction strength We will consider a scalingof the 1RDM first From Eqs (61) and (64) we see that with k0(x x

prime) = h0(x xprime)minus micro we get

k0[γλ](x xprime) = λ3k0[γ](λx λx

prime) (628)

G0[γλ](x ν xprime νprime) = λ3G0[γ](λx ν λxprime νprime) (629)

A general n-th order contribution to the interacting Greenrsquos function contains

bull n interaction lines

bull 2n+1 G0 lines

bull 2n interior points

A variable substitution in the integrals x rarr xλ then yields a factor of λminus3 for each inner nodeIn addition this variable substitutions will give a factor of λ for each Coulomb interaction lineCombining these observations we see that an arbitrary n-th order contribution scales as follows

Gξ(n)[γλ](x ν xprime νprime) = λn+3Gξ(n)[γ](λx ν λxprime νprime) (630)

A final variable substitution in Eq(67) then yields

Ω(n)c [γλ] = λnΩ(n)

c [γ] (631)

Using Eqs (544) and (545) to define Wc[γ] and Sc[γ] we get

W (n)c [γλ] = λnW (n)

c [γ] = nΩ(n)c [γλ] (632)

τS(n)c [γλ] = λnτS(n)

c [γ] = (nminus 1)Ω(n)c [γλ] (633)

We immediately see that if a diagram fulfills Ω(n)c [γ] lt 0 then the exact relations (58) (59)

(510)(547) (548) and (549) are fulfilledTo show that Eqs (534) (535) and (536) are also fulfilled we prove the validity of Eqs (531)

and (532) as restated in the following




Taking the difference of the left (L) and right (R) sides of Eq (634) and using Eqs (632) and(633) we get

LminusR = (1minus nλnminus1 + (nminus 1)λn)Ωc[γ] (636)

part

partλ(LminusR) = n(nminus 1)(λnminus1 minus λnminus2) (637)

We see that for λ = 1 L minus R = 0 Furthermore the derivative with respect to λ is bigger 0 forλ lt 1 and smaller 0 for λ gt 1 It follows that LminusR le 0 For Eq (635) we get

LminusR = (λn minus nλ+ (nminus 1))Ωc[γ] (638)

part

partλ(LminusR) = n(λnminus1 minus 1) (639)

59135


The same argument as before again proves Lminus R le 0 These two results prove the validity of Eqs(531) and (532) and therefore of Eqs (534) (535) and (536)

It is now also possible to show the equivalence of Eqs (557) and (558) for these approximationsWe therefore investigate an arbitrary nth-order contribution to Gλ under scaling of temperature andinteraction Again using Eqs (61) and (64) we acquire

k0(ητ)[γ](x xprime) = ηk0(τ)[γ](x x

prime) (640)

G0(ητ)[γ](x ν xprime νprime) = G0(τ)[γ](x ην xprime ηνprime) (641)

Because the Coulomb interaction is instantaneous Gξ(n) contains only n temperature integrationsTherefore

Gξ(n)(ητ κw)[γ](x ν xprime νprime) = ηminusnκnGξ(n)(τ w)[γ](x ην xprime ηνprime) (642)

From Eq (67) we finally arrive at

W (n)c (ητ κw)[γ] = η1minusnκnW (n)

c (τ w)[γ] (643)

which fulfills Eq (554) and therefore proves the equivalence of Eqs (557) and (558) The previousresults can be combined to get a complete description of the scaling behaviour of an arbitrary nth-order contribution to the grand potential functional

Ω(n)c (ητ κw)[γλ] = η1minusnκnλnΩ(n)

c (τ w)[γ] (644)


Employing the fact that in FT-RDMFT there is a Kohn-Sham system we derived a methodology toconstruct approximate functionals by using diagramatic techniques from FT-MBPT Because in thestandard FT-MBPT treatment the chemical potential is held fixed the interacting and noninteractingsystems will not have the same eq-1RDM In FT-RDMFT however we need to determine the variouscontributions to the grand potential for a given 1RDM ie the goal is to construct a functional ofthe 1RDM using FT-RDMFT We solved this problem by introducing an additional one-particlepotential which renders the 1RDM invariant under the perturbation We investigated the resultingperturbative expansion and showed that for every choice of contributing interaction diagrams the one-particle contribution can be uniquely determined We also showed that the first-order functional hasthe same form as the zero-temperature one with an inclusion of the noninteracting entropy functionalwhich justifies the assumption of Section 424 Additionally we pointed out the importance of ourinvestigation of eq-V-representability for the considerations in this section Finally we investigatedhow the resulting approximate functionals behave under scaling of coordinates temperature andinteraction strength and showed that under a few conditions they fulfill all exact properties derivedin Section 5

After these theoretical considerations one has to investigate how different choices of diagramsor classes of diagrams perform when applied to model and real systems We do so in Section 86considering both second-order Born as well as RPA diagrams

Continuing our considerations regarding systems in canonical equilibrium from Section 45 werestate that the exponent of the eq-SDO in a canonical ensemble contains only N -particle contribu-tions This is rather unfortunate because the finite temperature version of Wickrsquos theorem explicitlyrelies on the interplay of states of different particle numbers and is therefore invalid for canonicalensembles Accordingly our perturbative methods will not be valid for the construction of functionalsfor the description of systems in canonical equilibrium

60135

7 Numerical treatment

We have shown in Section 421 that it is possible to describe a quantum mechanical system ingrand canonical equilibrium with the help of the 1RDM One can find the eq-1RDM by a mini-mization of a grand potential functional Ω[γ] over a certain set of 1RDMs This set was shown inSection 422 to be easily determined by simple constraints on the ONs and NOs of the 1RDMs (seeEqs(219)-(221)) To approximate the exact functional one could now either use the method weintroduced in Section 6 or simply guess a functional guided by physical insight and tested againstexact properties eg the ones derived in Section 5 Regardless of the origin of a functional one willeventually have to minimize it Because at zero temperature there exists no Kohn-Sham system inRDMFT (see Thm 31 Section 31) one usually resorts to direct minimization schemes [105 106]However because at finite temperature there exists a Kohn-Sham system (see Section 423) thishandicap disappears and we will be able to construct a self-consistent (sc) minimization scheme forFT-RDMFT Furthermore we will show that this procedure can also be used to minimize functionalsfrom zero-temperature RDMFT to arbitrary accuracy Therein one will construct a noninteractingsystem at finite temperature which reproduces the eq-1RDM of an interacting system at zero tem-perature The remainder of this section is structured as follows

Section Description

71 We are going to review the main concept of minimization via a self-consistent schemeemploying a noninteracting effective system This will help in understanding the capa-bilities and limitations of our procedure It also paves the way for improvements in ourscheme like the introduction of a temperature tensor in Section 73

72 We will then define the Hamiltonian describing the effective noninteracting system ingrand canonical equilibrium It will be given completely in terms of the derivatives ofthe functional Ω[γ] wrt the ONs and NOs

73 Inspired by considerations in Section 71 we are able to greatly improve the adaptabilityof the effective system This will be done by introducing a state-dependent temperaturetensor for the effective noninteracting system We then present a computational schemefor a self-consistent minimization procedure employing this noninteracting system

74 To analytically investigate our minimization scheme we can do a perturbative small-step expansion We derive requirements on the effective temperature tensor to ensurea decrease of the functional Ω[γ] at each iteration step

75 Paying tribute to the possible inaccuracy of numerical derivatives we then derive twoderivative-based convergence measures which allow to judge the convergence of theminimization with respect to ONs and NOs

76 Finally we are going to investigate the performance of the Kohn-Sham minimizationscheme We therefore implemented it in the FP-LAPW-Code Elk [107] As an examplewe minimize a common RDMFT functional applied to LiH and discuss achievementsand shortcomings


71 Key idea of self-consistent minimization

We are now going to have a closer look at the idea of a minimization by employing an arbitrarynoninteracting system The functional which we want to minimize shall be denoted by Ω[γ] and itsdomain shall be given by ΓN If one considers the functional as a functional of ONs and NOs thenit describes a grand potential surface on the set of allowed ni and φi For a given 1RDM γk thequestion arising is where one would assume the next 1RDM γk+1 so that one will move downwards

61135

Numerical treatment

γ

F [γ]

γ0γ1γ2γ3

Ω(0)effΩ

(1)effΩ

(2)eff

Ω

Figure 71 Iterative minimization of Ω[γ] (red dashed line) by employing effective noninteracting

functionals Ω(n)eff [γ] (blue solid lines)

on the grand potential surface A standard approach to this problem is the use of steepest-descentmethods Therein one will use the first derivatives with respect to the ONs and NOs and takea step along the smallest slope The main problem of this approach is the incorporation of theauxiliary conditions on the ONs and NOs These are the boundedness of the ONs 0 le ni le 1the particle number conservation

sumi ni = N and most importantly because most difficulty the

orthonormality of the NOs Usually the orthonormality of the NOs will be enforced by applyingan orthonomalization algorithm to the NOs after they have been modified using the informationprovided by the functional derivatives δΩ[γ]δφi These orthonormalization procedures can changeseveral orbitals quite significantly which can lead to a slow convergence of the minimization routines

The main idea of a self-consistent minimization scheme is now to approximate the grand potentialsurface by a simpler one whose minimum incorporating all auxiliary constraints can be foundeasily In our situation we take the information about the derivatives of Ω[γ] at γk and constructan effective noninteracting system in grand canonical equilibrium whose grand potential functionalΩeff has the same functional derivative in γk The minimum of this grand potential surface is foundby a diagonalisation of the effective Hamiltonian and an occupation of the new ONs according tothe Fermi-Dirac distribution The resulting eq-1RDM will then serve as the starting point γk+1 forthe subsequent iteration This idea is schematically sketched in Figure 71 We should like to pointout that this method intrinsically incorporates the constraints on the ONs and NOs and we will nothave to apply subsequent orthonormalizations or the like The success of this scheme of courserelies on the similarity of the grand potential surfaces of Ω[γ] and Ωeff [γ] However we know fromTheorem 410 that both Ω[γ] and Ωeff [γ] are upwards convex This ensures that one will eventuallyarrive at the minimum of Ω given appropriately small step lengths These considerations also showthe necessity of an upwards convexity of approximate functionals If this feature was not satisfieda minimization might end up in a local minimum This problem however is prevalent in mostminimization schemes which use only the information of the first derivatives

We will now proceed to derive the variational equations guiding the determination of γk+1

62135

Numerical treatment

72 Effective Hamiltonian

We will restate the interacting as well as the noninteracting grand potential functionals in thefollowing (see Eqs (456)-(461))

Ω[γ] = Ωk[γ] + Vext[γ] + ΩHxc[γ]minus microN [γ]minus 1βS0[γ] (71)

Ωeff [γ] = Ωk[γ] + Veff [γ]minus micro0N [γ]minus 1β0S0[γ] (72)

We combined the Hartree exchange and correlation contributions to the grand potential intoΩHxc[γ] We explicitly allow the Kohn-Sham system to have a different temperature β0 from theinteracting one The effective noninteracting system is now constructed so that the derivatives ofthe interacting as well as of the noninteracting functional (Eqs (71) and (72)) coincide

δΩeff [γ]

δγ(x xprime)=

δΩ[γ]

δγ(x xprime) (73)

The effective Hamiltonian in spatial representation then becomes

heff [γ](x xprime) = t[γ](x xprime) + vext[γ](x x

prime) + vHxc[γ](x xprime)+

(1

β0minus 1

β

)σ[γ](x xprime) + (micro0 minus micro)δ(x minus xprime) (74)

where

vHxc[γ](x xprime) =

δΩHxc[γ]

δγ(x xprime)(75)

σ[γ](x xprime) =δS0[γ]

δγ(x xprime) (76)

We want to use the chain rule for the functional derivative We therefore need the derivatives of theONs and NOs with respect to γ They can be obtained using first-order perturbation theory leadingto

δnkδγ(xprime x)

= φlowastk(xprime)φk(x) (77)

δφk(y)

δγ(xprime x)=sum

l 6=k

φlowastl (xprime)φk(x)

nk minus nlφl(y) (78)

δφlowastk(y)

δγ(xprime x)=sum

l 6=k

φlowastk(xprime)φl(x)

nk minus nlφlowastl (y) (79)

In the following it will be useful to work in the basis of NOs An arbitrary function g(x xprime) is thenrepresented by gij where

gij =

intdxdxprimeφlowasti (x)g(x x

prime)φj(xprime) (710)

The effective Hamiltonian is then represented by its matrix elements heff ij in the basis of the NOs

heff ij = δij

(partΩ[γ]

partni+ micro0 +

σiβ0

)+

1minus δijni minus nj

intdy

(δΩ[γ]

δφi(y)φj(y)minus

δΩ[γ]

δφlowastj (y)φlowasti (y)

) (711)

where the entropic contribution σi is given by

σi =partS0[γ]

partni= ln

(1minus nini

) (712)

63135

Numerical treatment

The offdiagonal elements are exactly those Pernal [109] derived in her approach for the derivation ofan effective potential for RDMFT They are also simply related to the ones Piris and Ugalde [106]introduced in their method for an orbital minimization It has to be noted however that in ourapproach the diagonal elements are not free to choose but are determined by the thermodynamicensemble We also see that one can control the change in the 1RDM by tuning β0 If β0 was smallie if the corresponding effective temperature was high the diagonal part of Heff will be bigger

compared to the offdiagonal parts Therefore after a diagonalisation of Heff the orbitals will change

less Accordingly in the limit of β rarr 0 the offdiagonal elements of Heff can be neglected and thediagonal elements will be given by the entropic contribution σiβ ie h

effij = δijεi = δijσiβ If

one would now starting from a set of ONs ni construct a new set of ONs nprimei from this effectiveHamiltonian via Eqs (452) and (453) then one finds that the ONs are left invariant ie nprimei = niWe will further investigate the behaviour of our self-consistent minimization scheme for small β0 inSection 74

Before we construct a self-consistent scheme for the minimization of Ω[γ] in the following we willshow how the concept of a temperature tensor greatly enhances the adaptability of the Kohn-Shamsystem which will improve the performance of the minimization procedure

73 Temperature tensor

So far we treated temperature as a single parameter in the definition of our effective HamiltonianWe will show now how the concept of a temperature tensor crucially increases our variational freedomThe following considerations will be instructive In a self-consistent minimization scheme for a given1RDM we construct a known (noninteracting) functional whose first derivative coincides with theone from the interacting functional For a fixed β0 the parameter responsible for this fitting ismicro0 β0 can now be varied to modify how narrow the noninteracting grand potential surface willbe However second derivatives with respect to the ONs may differ quite substantially and a valueof β0 which describes the grand potential surface wrt one ON well might describe others quitebadly A simple example would be a quadratic two-state model functional Ω[n1 n2] without orbitaldependence

Ω[n1 n2] =α1

2(n1 minus 05)2 +

α2

2(n2 minus 05)2 (713)

= Ω1[n1] + Ω2[n2] (714)

A choice of α1 = 50 and α2 = 1 leads to heff 11 = ε1 = minus0225 + micro and heff 22 = ε2 = 000450 + microin Eq (711) The corresponding projected grand potential surfaces are plotted in Figure 73 forβ0 = 011 As one can see the choice of β0 = 011 models the first grand potential surface quite wellbut the second one fails to be reproduced One would like to have some sort of adaptive β0 whichcan be related to the second derivatives However before one can use such a construct one has toconfirm that it corresponds to a grand potential surface whose minimum can easily be found

We are now going to show that this is possible by a slight variation of the definition of grandcanonical ensembles We consider the following generalized SDO-grand potential functional

G[D] = TrD(B(H minus microN) + ln D) (715)

where B is an arbitrary hermitean operator on the Fock-space The same proof as in [11] now leadsto the following variational principle

G[D] ge G[Deq] (716)

where the equality is only fulfilled if D = Deq

Deq = eminusB(HminusmicroN)Zeq (717)

Zeq = TreminusB(HminusmicroN) (718)

64135

Numerical treatment

0

2

4

6

8

10

0 02 04 06 08 1

n1

Ω[n

1]

Ω1[n1]Ωeff1[n1]

72a First ON α1 = 50

0

005

010

015

020

0 02 04 06 08 1

n2

Ω[n

2]

Ω2[n2]Ωeff2[n2]

72b Second ON α2 = 1

Figure 72 Projected grand potential surfaces for the model of Eq (713) with α1 = 50 α2 = 1 β0 =011

For a noninteracting Hamiltonian and a B for which [B H] = 0 the Fermi Dirac relation reads

ni =1

eβi(εiminusmicro) + 1(719)

εi minus micro =1

βiln

(1minus nini

) (720)

where βi denotes the i-th eigenvalue of B This leads to the following expression for the grandpotential

Ω[γ] =sum

i

(ni(εi minus micro) +

1

βi(ni lnni + (1minus ni) ln(1minus ni))

)(721)

=sum

i

Ωi[ni βi] (722)

Where in the case of a scalar temperature we just had one parameter to construct our effectivenoninteracting system we now have one for each ON A straightformard utilization of this freedomwould be to let the second derivatives of the energy functional with respect to the ONs of theinteracting functional and the noninteracting one be proportional to each other ie

βi = ηpart2S0[γ]

partn2i

part2Ω[γ]partn2i

(723)

= η1

ni(1minus ni)

(part2Ω[γ]

partn2i

)minus1

(724)

where η the proportionality factor is the only global parameter In our model (Eq (713)) thisyields

βi =η

αi

1

ni(1minus ni) (725)

η = 1 lets the second derivatives of interacting and noninteracting functional be equal whereas anincrease (decrease) of η leads to a spreading (compression) of the noninteracting grand potentialsurface As can be seen from Figure 73 with a good choice of η (in our model η = 05) one canreproduce the different grand potential surfaces simultaneously

65135

Numerical treatment

0

2

4

6

8

10

0 02 04 06 08 1

n1

Ω[n

1]

Ω1[n1]

Ωeff1[n1]

73a First ON β0 = 5

0

005

010

015

020

0 02 04 06 08 1

n2

Ω[n

2]

Ω2[n2]

Ωeff2[n2]

73b Second ON β2 = 555

Figure 73 Projected grand potential surfaces for the model of Eq (713) with α1 = 50 α2 = 1The choice of η = 05 leads to β1 = 011 and β2 = 555

We can now construct a self-consistent scheme for the minimization of Ω[γ] which we sketch inFigure 74 Every iteration of this scheme requires a change in NOs and therefore an update ofthe derivatives wrt the NOs This might prove to be expensive and one might be interested in aprocedure which only minimizes the ONs and leaves the NOs invariant Fortunately the minimizationscheme can easily be modified to accomplish this task The effective Hamiltonian Heff is simplyassumed to be diagonal ie one only populates the diagonal elements following Eq (711) Theeigenvalues ie the diagonal elements of Heff will then yield a new set of ONs via Eq (719)Assigning the new ONs to the frozen orbitals is straightforward because a given ON ni leads toa specific εprimei = heffii which in turn leads to an unambiguously defined nprimei The last step in theminimization scheme the mixing of 1RDMs is straightforward because ΓN is a convex set

74 Small step investigation

We showed in the previous considerations that one can employ the Kohn-Sham system in FT-RDMFTto construct a self-consistent minimization scheme However this does not ensure that an applicationof this scheme will actually lead to a minimum of the functional This is a common problem ofminimization schemes but in the following we are going to show that for small step lengths ourmethod will definitely lead to a decrease of the value of the functional under consideration As weargued at the end of Section 72 choosing a smaller β0 will lead to smaller changes in ONs and NOsStarting from a given 1RDM γ we therefore apply first-order perturbation theory to get the modified1RDM γprime By the virtue of Eq (711) γ leads to the effective Hamiltonian Heff A diagonalizationunder the assumptions of first-order perturbation theory then yields the following new eigenvaluesεprimei and eigenstates φprimei

εprimei = heff ii (726)

φprimei(x) = φi(x) +sum

j 6=i

heff ji

εi minus εjφj(x) (727)

The new ONs resulting from our modified eigenenergies become

nprimei =1

1 + eβi(εprimeiminusmicrominus∆micro) (728)

66135

Numerical treatment

1) diagonalize γ(k)(x xprime) =sum

i n(k)i φ

(k)lowasti (xprime)φ(k)i (x)

2) fromn(k)i

and

φ(k)i

calculate H

(k)eff by Eq (711)

3) diagonalize H(k)eff to get a new set of

ε(k+1)i

and

φ(k+1)i (x)

4) constructn(k+1)i

using Eq (720) and find micro

(k+1)0 so that

sumi n

(k+1)i = N

5) fromφ(k+1)i (x)

and

n(k+1)i

build new 1RDM γ(x xprime)

6) mixing of γ(x xprime) with γ(k)(x xprime) yields γ(k+1)(x xprime)

Figure 74 Self-consistent minimization scheme in FT-RDMFT

where one had to introduce the chemical potential correction ∆micro to ensure particle number conser-vation With Eq (712) one gets

nprimei =ni

ni + (1minus ni)eβi

(partΩpartni

minus∆micro

) (729)

Expanding ∆micro in orders of βi and subsequently Eq (729) for small βi we get

δni = nprimei minus ni (730)

= βini(ni minus 1)

(partΩ[γ]

partniminus∆(0)

micro

) (731)

This result is very similar to the steepest-descent method with an additional factor of ni(ni minus 1)

This additional term tries to keep the ONs in the allowed set 0 lt ni lt 1 ∆(0)micro can now be found by

the requirement of particle number conservation

∆(0)micro =

sumi βini(ni minus 1)partΩ[γ]

partnisumi βini(ni minus 1)

(732)

67135

Numerical treatment

The overall change in the 1RDM up to first order in βi is then given by

∆γij = γprimeij minus δijni (733)

= δijδni + (1minus δij)ni minus njεi minus εj

heff ij (734)

The grand potential changes accordingly as

∆Ω =

intdxdxprime

δΩ[γ]

δγ(x xprime)∆γ(xprime x) (735)

=sum

ij

δΩ[γ]

δγij∆γji (736)

=sum

i

δnipartΩ[γ]

partni︸︷︷︸

∆Ω1

+sum

i6=j

ni minus njεi minus εj

|heff ij |2

︸︷︷︸∆Ω2

(737)

We see that the grand potential change ∆Ω separates into two parts ∆Ω1 determined by the changein ONs and ∆Ω2 coming from the change in NOs In the following we are going to investigate thesetwo different contributions separately

741 Occupation number (ON) contribution

We will now show that the first term in Eq (737) which is due to the change in ONs is alwaysnegative for appropriately small step lengths

∆Ω1 =sum

i

δnipartΩ[γ]

partni(738)

=sum

i

βini(ni minus 1)

(partΩ[γ]

partniminus∆(0)

micro

)partΩ[γ]

partni(739)

=sum

i

βini(ni minus 1)

(partΩ[γ]

partni

)2

minus

(sumi βini(ni minus 1)partΩ[γ]

partni

)2

sumi βini(ni minus 1)

(740)

For brevity we introduce ci =βini(niminus1)sumi βini(niminus1) which leads to

∆Ω1 =

sum

j

βjnj(nj minus 1)

sum

i

ci

(partΩ[γ]

partni

)2

minus(sum

k

ckpartΩ[γ]

partnk

)2 (741)

=

sum

j

βjnj(nj minus 1)

sum

i

ci

(partΩ[γ]

partniminussum

k

ckpartΩ[γ]

partnk

)2

(742)

Because every ON ni fulfills 0 lt ni lt 1 and every βi is greater 0 this leads to the conclusion

∆Ω1 le 0 (743)

742 Natural orbital (NO) contribution

We can now turn to the second term in Eq (737) which represents the functional change due tothe change in NOs

∆Ω2 =sum

i6=j


|heff ij |2 (744)

68135

Numerical treatment

By using Eq (720) this transforms to

∆Ω2 =sum

i6=j

ni minus nj1βi

ln(

1minusni

ni

)minus 1

βjln(

1minusnj

nj

) |heff ij |2 (745)

We see that for an arbitrary choice of βi we cannot ensure the negativity of ∆Ω2 But if we use aconstant β0 we get

∆Ω2 = β0sum

i6=j

ni minus njln(

nj(1minusni)ni(1minusnj)

) |heff ij |2 (746)

which is nonpositive

∆Ω2 le 0 (747)

We have shown that for small enough βi the ON change will always decrease the grand potentialregardless of whether one chooses a constant temperature or a temperature tensor When consideringchanges in the NOs one has to fall back to constant temperature to ensure a decrease of the functionalvalue

75 Convergence measures

We have now all the necessary tools at hand to iteratively minimize a functional Ω[γ] We needhowever some measures to judge if a calculation is converged There are two main reasons whyusing the grand potential itself as convergence measure is disadvantageous

Firstly often the calculation of derivatives is not accurate and consequently a derivative-basedminimization may lead to a fixpoint where Ω[γ] is not minimal This leads to a sign change of theconvergence measure and implementing the strict decrease of grand potential as a requirement ofthe minimization procedure will then lead to a result depending on the starting point To illustratethis problem we consider a simple parabola Ω(x) = ax2 whose minimum is at x = 0 Let us nowassume that the calculation of the derivative is only approximate and leads to a constant error δ Thederivatives then do not describe the surface given by Ω(x) but rather one defined by Ω(x) = ax2+δxwhose minimum will be at x = minusd(2a) If one would now approach this minimum coming fromthe left with a series of positions xi then Ω(xi) will be monotonously decreasing At the pointof convergence its value will be Ω(x) = d2(4a) If on the other hand we would have approachedthe minimum coming from the right ie starting at x = infin then we would have passed throughthe minimum of Ω(x) and the series Ω(xi) would exhibit a valley Incorporating the grand potentialΩ(x) or rather the change in grand potential as convergence measure would therefore prevent theminimization to approach the true minimum of Ω(x) and lead to two different points of convergencedepending on the starting position

Secondly because the true minimal grand potential is not known one would have to judgeconvergence from the change in Ω[γ] after iterating the minimization routine ie a small change inΩ[γ] indicates a relative closeness to the real minimum This might pose a problem if the minimumof the grand potential surface as defined by the derivatives of Ω[γ] is very shallow or worse if theminimization procedure leads to a slow approach to the minimum An example for such a situationis discussed in Section 763

Because of these problems we would rather use a strictly positive convergence measure whichgoes to 0 if the 1RDM approaches the minimum of the grand potential surface as defined by thederivatives of Ω[γ] We will establish our choice of convergence measures on the following twoobservations

bull In the minimum the derivatives with respect to the ONs will be equal for unpinned states

69135

Numerical treatment

bull In the minimum the effective Hamiltonian Heff will be diagonal

The first observation allows us to define a convergence measure χ2n for a minimization with respect

to the ONs

χ2n =

1

Nunpinned

Nunpinnedsum

i

(partΩ

partniminus micro

)2

(748)

micro =1

Nunpinned

Nunpinnedsum

i

partΩ

partni(749)

The second statement leads to the following definition of χ2φ as a convergence measure for a mini-

mization with respect to the NOs

χ2φ =

1

N minus 1

sumNi6=j |heff ij |2sum

i ε2i

(750)

If a minimization is converging both measures should approach 0

76 Sample calculations

We test the self-consistent procedure for the case of solid LiH at zero temperature by using theFP-LAPW code Elk [107] The exchange-correlation energy will be modelled by the α functionalEα

xc[γ] as defined in Section 311

Eαxc[γ] = minus

1

2

sum

ij

nαi nαj



prime) (751)

We choose this functional because it exhibits several properties making it difficult to be minimizedAs in the case of the HEG in Section 32 it will lead to several fully occupied ie pinned statesThese lie on the boundary of the domain of the effective grand potential functional Ωeff [γ] Theminimization of the α functional is therefore a good test for the minimization scheme leading toboundary minima Furthermore the α functional exhibits divergencies in the derivatives wrt theONs for ni rarr 0 If in the minimum there will be ONs close to 0 (and there will be if one considersenough NOs) this might lead to convergence problems of the minimization We will now investigatethe performance of the self-consistent minimization scheme wrt ON- and NO-convergence

761 Occupation number (ON) minimization

We have minimized the α functional for α = 0565 with three methods First we used the steepest-descent method as implemented in Elk then we used the self-consistent FT-RDMFT minimizationwith constant β0 and finally we used a temperature tensor βi of the form of Eq (724) withparameter η In all three methods we chose all parameters to achieve fastest convergence Theresults which are shown in Figures 75a-75f show that both self-consistent minimizations lead toa faster convergence than steepest-descent A dramatic increase in the speed of convergence is thenachieved by employing a temperature tensor The slow decrease of χ2

n in Figures 75d and 75f forthe steepest-descent- and constant β0-methods can be attributed to the following fact For these twomethods the ONs which will be pinned at the equilibrium approach their final values quite slowlyTherefore their derivatives contribute to χ2

n via Eq (748) even after several iterations

70135

Numerical treatment

minus794

minus793

minus792

minus791

minus790

minus789

minus788

0 5 10 15 20

it

E(Ha)

τ = 20β0 = 30η = 10

75a Energy 1 k-point

1eminus25

1eminus20

1eminus15

1eminus10

1eminus05

1

0 20 40 60 80 100

it

χ2 n

τ = 20β0 = 30η = 10

75b ON-Variance 1 k-point

minus808

minus806

minus804

minus802

minus800

0 5 10 15 20

it

E(Ha)

τ = 20β0 = 30η = 10

75c Energy 2x2x2 k-points

1eminus30

1eminus25

1eminus20

1eminus15

1eminus10

1eminus05

1

0 20 40 60 80 100

it

χ2 n τ = 20

β0 = 30η = 10

75d ON-Variance 2x2x2 k-points

minus812

minus810

minus808

minus806

minus804

minus802

0 5 10 15 20

it

E(Ha)

τ = 10β0 = 30η = 10

75e Energy 3x3x3 k-points

1eminus30

1eminus25

1eminus20

1eminus15

1eminus10

1eminus05

1

0 20 40 60 80 100

it

χ2 n

τ = 10β0 = 30η = 10

75f ON-Variance 3x3x3 k-points

Figure 75 Energy E and ON-convergence measure χn for ON minimizations of the α functionalwith α = 0565 applied to LiH The red short dashed lines stand for a steepest-descent minimizationthe blue solid ones for a sc-Kohn-Sham minimization with constant β0 and the black long dashedones for a sc-Kohn-Sham minimization with adaptive βi τ denotes the value for he parameterldquotaurdmnrdquo in Elk whereas β0 and η are defined via Eqs (72) and (723)

71135

Numerical treatment

762 Full minimization

We can now turn to the problem of minimizing E[γ] with respect to both ONs and NOs Wefind that the overall performance of this full minimization is greatly improved by introducing aON-minimization after every NO-minimization step (see Figure 74) Because we have seen in theprevious section that this can be done very efficiently and effectively this increases the runtime ofa full minimization run only negligibly The deeper reason for the improvement of the convergenceby inclusion of an ON-minimization is the following It might happen that two states φi and φjhave similar eigenvalues in Heff but considerably different ONs A diagonalisation of Heff mightthen yield a strong mixing between these states If the ONs were not updated one might be ledaway from the minimum of the energy functional A subsequent ON-minimization remedies thisproblem and assigns the optimal ON for each NO We show a sketch of the full minimization schemein Figure 76 An application of this scheme to LiH then leads to the results depicted in Figure77 Again we see a tremendous increase in speed and accuracy for the self-consistent minimizationscheme compared to the steepest-descent method The steepest-descent method shows a very slowconvergence which can be attributed to the orthonormalization of NOs Furthermore the increaseof the energy curves in Figures 77c and 77e is due to the approximative nature of the derivativesAs we have pointed out before the minimization procedure is only guided by the values of thederivatives (see Eq (711)) and it will not minimize the energy surface defined by E[γ] but ratherone defined by the approximate derivatives Coincidentally the starting point for the minimizationof E[γ] as shown in Figure 76 leads to a path to the minimum of the approximate energy surfacewhich leads through a part of the exact surface which has a lower value Coming from anotherstarting point this would not necessarily have been the case This starting point dependence alwaysexists if the derivatives are approximate which solidifies the argument that the energy should not beused as convergence measure

One full effective Hamiltonian diagonal-ization with constant β0

ON minimization with βi-tensor

cycle

Figure 76 Full minimization scheme

72135

Numerical treatment

minus79445

minus79440

minus79435

minus79430

minus79425

0 100 200 300 400 500

it

E(Ha)

τ = 10

β0 = 20


1eminus24

1eminus20

1eminus16

1eminus12

1eminus08

1eminus04

0 100 200 300 400 500

it

χ2 φ

τ = 10β0 = 20

77b NO-Variance 1 k-point

minus8085

minus8084

minus8083

minus8082

minus8081

minus8080

0 100 200 300 400 500

it

E(Ha)

τ = 10

β0 = 10


1eminus13

1eminus11

1eminus09

1eminus07

1eminus05

0 100 200 300 400 500

it

χ2 φ

τ = 10

β0 = 10

77d NO-Variance 2x2x2 k-points

minus81285

minus81280

minus81275

minus81270

minus81265

minus81260

minus81255

0 100 200 300 400 500

it

E(Ha) τ = 10

β0 = 10


1eminus13

1eminus11

1eminus09

1eminus07

1eminus05

0 100 200 300 400 500

it

χ2 φ

τ = 10

β0 = 10

77f NO-Variance 3x3x3 k-points

Figure 77 Energy E and NO-convergence measure χφ for NO minimizations of the α functionalwith α = 0565 applied to LiH Both variables are plotted against the number of NO changes Aftereach change in NO there follows a complete ON minimization The red dashed lines stand fora steepest-descent minimization whereas the blue solid ones depict a sc-Kohn-Sham minimizationwith constant β0 τ denotes the parameter value for taurdmc in Elk whereas β0 is defined via Eq(72)

73135

Numerical treatment

minus81284

minus81282

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minus81278

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0 100 200 300 400 500

it

E(Ha)

β0 = 10β0 = 20β0 = 50

78a Energy 3x3x3 k-points

1eminus13

1eminus11

1eminus09

1eminus07

1eminus05

0 100 200 300 400 500

it

χ2 φ

β0 = 10

β0 = 20

β0 = 50

78b NO-Variance 3x3x3 k-points

Figure 78 NO-convergence for different effective temperatures


In this section we introduced a self-consistent minimization scheme in the theoretical framework ofFT-RDMFT We then defined measures which allow us to judge the convergence of a calculationwithout having to resort to the energy This was necessary because the numerical derivatives areusually not accurate enough We could show that this self-consistent procedure is superior in manyaspects compared to the steepest-descent method especially considering a minimization wrt theNOs The important parameter in the minimization scheme is the effective temperature β0 andthe speed of convergence crucially depends on it In Figure 78 we show the behaviour of theminimization scheme for three different choices of β0 β0 = 1 represents the optimal value ie thevalue for which the convergence measure χ2

φ decreases the fastest We see that the energy reachesits fixpoint after approximately 300 iterations An increase of β0 to β0 = 2 seemingly speeds up theenergy convergence but from χ2

φ one can see that after about 100 iterations the minimization failsto diagonalize heff any further The changes in the 1RDM whose amplitudes are determined by β0become too big and the 1RDM jumps around the fixpoint of the energy Without considering χ2

φthis would have been difficult to detect which illustrates the importance of a convergence measurewhich is independent of the energy value One might argue that this choice of β0 still leads to afixpoint very close to the optimal one but this cannot be ensured for all problems and all choices ofβ0 and therefore has to be seen in the actual example as rather accidental ie fortunate A furtherincrease of β0 to β0 = 5 then exposes this problem more clearly The energy apparently reaches afixpoint But this fixpoint is considerably above the optimal one Just having the energy at handthis might have been difficult to detect But χ2

φ directly shows that the minimization is far frombeing converged

One important feature which can be extracted from Figure 78 is that all three parameters leadto a similar energy vs iteration curve Apparently a minimization-run with β0 being too big is ableto lead to the vicinity of the fixpoint A utilization of this fact would now be to use an adaptive β0rather than a constant one One could start with a big β0 till the energy does not change anymoreand than decrease β0 until χ2

φ surpasses the convergence threshold

74135

8 Applications

The initial success of DFT can be attributed mainly to the remarkable performance of the LDAAs pointed out in Section 32 a similar formulation in the framework of RDMFT is complicated bythe fact that one would have to consider a HEG subject to nonlocal external potentials This is inprinciple possible via Monte-Carlo techniques but it has not been carried out so far The testingof RDMFT functionals could therefore only be done for HEGs with local external potentials Ingrand canonical ensembles the situation becomes even worse The calculation of the exact equi-librium grand potential of a HEG at finite temperature is in principle possible via the method ofPath integral Monte-Carlo However the fermion sign problem renders these calculations extremelydifficult and expensive at low temperatures [80] Therefore to the best of our knowledge there existsno parametrization of the equilibrium grand potential or free energy of a HEG at finite tempera-ture Accordingly one has to resort to the description of grand canonical systems by approximatemeans There exists wide variety of different approaches to accomplish this task These include theintroduction of approximate model interactions [85 110] the utilization of the dielectric formula-tion (employing various approximations including the hypernetted chain approximation [19 111]the modified convolution approximation [112 113] and the equation-of-motion approach of SingwiTosi Land and Sjolander [114 115 116]) the mapping of quantum systems to classical systemsat finite temperature [20 117] and the utilization of FT-MBPT including non-diagrammatic localfield corrections to the RPA [118 119 120 121 122 123 124] Because there are no exact results aquantitative testing of approximate FT-RDMFT functionals is not possible However one can stillinvestigate their qualitative behaviour Because we will be interested in magnetic phase transitionswe will focus on the qualitatively correct description of the phase diagram of the HEG We willstructure the remainder of this section as follows

Section Description

81 We will start with a short clarification of the nomenclature used concerning phasetransitions and phase diagrams

82 We will then investigate the HEG replacing the Coulomb interaction by a contactinteraction We will calculate the phase diagram and discuss shortcomings of this ap-proximation

83 As a second approach we will review how standard FT-MBPT approaches the problemcorrecting preliminary results by [119] and calculating the magnetic phase diagram ofthe HEG in collinear spin configuration

84 Subsequently we will incorporate our novel approach of FT-RDMFT Using the first-orderexchange-only functional we will calculate the magnetic phase diagram for bothcollinear spin configuration as well as planar spin spiral states

85 Using the equivalence of the first-order FT-RDMFT functional with FT-HF theory wewill investigate the temperature dependence of the quasi-particle spectrum for bothcollinear and spin spiral configurations

86 Closing we will focus on the problem of the description of correlation effects Startingwith the inclusion of correlation functionals from DFT we will proceed to employ phe-nomenological correlation functionals from RDMFT at zero temperature Subsequentlywe will incoporate the perturbative methodology as derived in Section 62 and show thata straightforward utilization might lead to the problem of variational collapse Finallywe will present a procedure to avoid this collapse utilizing exact properties of the finitetemperature polarization propagator


75135

Applications

81 Magnetic phase transitions in the HEG

In this section eventually we want to test FT-RDMFT applied to the HEG In particular we wantto calculate the free energy magnetic phase diagram This however needs some clarification Inthe development of FT-RDMFT in Section 4 we explicitly considered grand canonical ensemblesThere were two main reasons for this restriction The first one concerns the Kohn-Sham system forcanonical ensembles From Section 45 we know that the set of eq-V-representable 1RDMs also in thecase of canonical ensembles lies dense in the set of ensemble-N-representable 1RDMs However it isnot clear how one could recover the Kohn-Sham potential from a given 1RDM The inversion in Eq(453) is valid only for grand canonical ensembles The second reason was that the FT-MBPT is alsodefined only for grand canonical ensembles A formulation of a FT-MBPT for canonical ensemblesbreaks down in the proof of the corresponding Wickrsquos theorem This proof explicitly requires thatthe traces involved are defined on the full Fock space and not just on an N -particle Hilbert spaceThis does not disprove the possibility of a perturbation theory for canonical ensembles It just showsthat the methods derived for grand canonical ensembles cannot be translated easily Accordinglythe variational principle refers to the grand potential rather than the free energy of the systemWe therefore have to assume that the system is in the thermodynamic limit which will allow us tocalculate the canonical free energy F c from the grand canonical Ωgc grand potential as

F c = Ωgc + microN (81)

However in the case of small finite systems embedded in a temperature bath this assumption neednot be valid and might be a source of possible deviations between calculations and measurementsrequiring further investigation [126]

The phase diagram is now constructed by calculation and comparison of free energies of differentphases When considering phase transitions we will use a terminology in the spirit of Ehrenfestscharacterization ie if the derivative of the free energy changes discontinuously wrt a certainorder parameter then we call it a first-order phase transition If on the other hand it changescontinuously then we call it a second-order phase transition In Figure 81 we show an examplecoming from the first-order functional of FT-RDMFT as discussed later in Section 84 Thereinwe assumed a collinear spin configuration where the polarization ξeq of the eq-state is the orderparameter We then calculated the free energy for various values of the Wigner-Seitz radius rs andthe polarization ξ We denote the minimal free energy for fixed density by the thick black lineWe see that with an increase of rs we first encounter a first-order phase transition between theunpolarized and the polarized state Increasing rs even further will ulimately lead to a second-orderphase transition back to the unpolarized state

Although the exact magnetic phase diagram of the HEG in collinear spin configuration is notknown we can still derive general properties of it from the following considerations Let us firstconsider the zero-temperature case In collinear configuration we expect that the kinetic energyfavours an unpolarized gs This is due to the fact that the kinetic energy operator in Eq (22)is a spin-independent one-particle operator An eigenstate to the kinetic energy operator of spincharacter uarr will therefore yield the same energy expectation value as the corresponding darr-state Wenow assume that the states ψiσ are ordered wrt their kinetic energy expectation values If nowψ0uarr is occupied we see from the previous considerations that it is favourable to occupy ψ0darr nextrather than ψ1uarr The same argument extends to all spin-independent one-particle HamiltoniansThe interaction on the other hand corresponds to a two-particle operator (see Eq(24)) We areconcerned with the Coulomb interaction whose spatial representation reads w(x xprime) = 1|r minus rprime|Because of the divergence of the interaction for r rarr rprime one can now argue that the interaction yieldsa big energy contribution if the different wavefunctions have a big spatial overlap The two statesψiuarr and ψidarr have maximal spatial overlap and it is therefore resonable to state that the Coulombinteraction generally favours a polarized configuration The explicit form of the kinetic energy leadsto an approximate rminus2

s dependence To estimate the density dependence of the Coulomb interactionwe can as a first step consider a classical system With the inclusion of an appropriate background

76135

Applications

minus40

minus50

minus45

5

10

15

20

0

02

08

06

04

10

rs(au)

ξF(10minus

2Ha)

Figure 81 Free energy wrt the polarization ξ and rs at T = 5000K from the first-orderFT-RDMFT functional The black line denotes the equilibrium free energy for a fixed rs Atrs asymp 58au a first-order phase transition between paramagnetic and ferromagnetic phases takesplace An increase in rs then leads to a second-order phase transition back to the paramagneticstate

charge we can then deduce from the form of w(x xprime) that it shows an rminus1s behaviour which is also

the leading term in the quantum case For an increase of rs we therefore expect a magnetic phasetransition at a critical density rc between an unpolarized configuration and a polarized one Thenature of this phase transition cannot be determined by the previous simple arguments and has tobe investigated in detail Monte-Carlo calculations confirm the previous considerations suggesting acontinuous phase transition at a critical Wigner-Seitz radius of around rc asymp 60au ([48]rc asymp 75au[85]rc asymp 50au)

We can now investigate how the situation will change when the entropy as defined in Eq (43)is included The entropy value is determined solely by the statistical weights wi from Eq (26)

S[D] =sum

i

wi lnwi (82)

We have shown in Eq (418) in Section 411 that S[D] is a convex functional of the SDO D Thisimplies that generally the entropy favours a configuration where more states are occupied In theunpolarized configuration there are twice as many states to be occupied compared to the polarizedconfiguration which leads to the conclusion that the entropy generally favours a paramagnetic con-figuration As mentioned before the entropy depends only on the statistical weights wi and showsno explicit density dependence Because of the rminus2

s and rminus1s dependencies of the kinetic energy and

the Coulomb interaction we expect the relative effect of entropy to be increased with increasing rsFurthermore we expect the phase transitions to take place when the entropy contributions becomescomparable to the energy The characteristic energy of a system is given by the Fermi energy εFand the corresponding Fermi temperature is TF = εF kB (see Eqs (228) and (229)) Becausethe kinetic energy has less relative influence for bigger rs we also expect that the phase transitionfrom polarized to unpolarized configurations happens at larger relative temperatures t = TTF forincreasing rs We summarize our expectations in Table 81 and show a sketch of a generic phase

77135

Applications

1 At zero temperature with increasing rs there will be a phase transition between unpolarizedand polarized configurations

2 The phase transition at zero temperature is continuous ie of second order

3 An increase in temperature will eventually favour an unpolarized state

4 The effect of temperature is increased for increasing rs leading to a faster decrease of a polarizedphase

5 The phase transitions occur for temperatures comparable to the Fermi temperature TF

6 For increasing rs the phase transition from polarized to unpolarized configurations occurs forhigher values of the temperatures compared to the Fermi temperature TF

Table 81 General assumptions about behaviour of the phase diagram of the HEG in collinearconfiguration

diagram fulfilling our conclusions in Figure 82 We should like to emphasize again that the previousconsiderations are rather general and the resulting assumptions are of qualitative nature We do notclaim that eg for small temperatures a slight increase in temperature might not lead to a favouri-sation of a phase which is more structured as we will actually find out in Section 842 consideringa spin spiral configuration

In the remainder of this section we will call an unpolarized state paramagnetic (PM) and afully polarized collinear state ferromagnetic (FM) A partially polarized state will be denoted bythe abbreviation PP Furthermore from now on we will consider energy densities rather than total

FMPP

PM

r s(au)

T (K)

rc (rTc Tc)

TF

Figure 82 Expected qualitative behaviour of the phase diagram of the HEG in collinear configura-tion FM ferromagnetic phase PM paramagnetic phase PP partially polarized phase The dottedblue line denotes the Fermi temperature TF

78135

Applications

energies Not to overload the notation we will denote these densities just as the total energies IeΩ now refers to the grand potential density

After these clarifications we will now review previous approximations for the spin-dependent freeenergy of a HEG The resulting phase diagrams will then be interpreted and qualitatively comparedto the general expected properties as listed in Table 81 and shown in Figure 82

82 Homogeneous electron gas with contact interaction

We assume a HEG in collinear spin configuration The momentum distributions of the spin channelsshall be denoted by nσ(k) The density n is then given by

n =

intd3k

(2π)3(nuarr(k) + ndarr(k)) (83)

The different contributions to the free energy density of a noninteracting system are given entirelyin terms of the momentum distribution

Ωkσ =1

n

intd3k

(2π)3k2

2nσ(k) (84)

Ωextσ =1

n

intd3k

(2π)3nσ(k)vextσ(k) (85)

nσ =1

n

intd3k

(2π)3nσ(k) (86)

S0σ = minus 1

n

intd3k

(2π)3(nσ(k) lnnσ(k) + (1minus nσ(k) ln(1minus nσ(k))) (87)

F =sum

σ

(Ωkσ +Ωextσ minus 1βS0σ) (88)

As mentioned in Section 81 without interaction or external potential the HEG will always beparamagnetic because kinetic energy as well as entropy favour this configuration

One way to include interaction was proposed by Zong Lin and Ceperley [85] They argue thatthe interaction is sufficiently screened to be effectively modelled by a Stoner-type contact interactionTaking account of the Pauli principle this interaction explicitly couples only particles of oppositespin Its spatial representation therefore behaves like

wS(x xprime) prop (1minus δσ1σ2)δ(r minus rprime) (89)

At this point we should like to point out some inconsistencies in the literature In Eq (8) ofRef [85] the rs-dependence of the interaction is denoted to be 2 orders higher than the one ofthe kinetic energy This however would lead to the result that the interaction energy per particlewould be density independent just like the entropy Furthermore ΩS

W is intrinsically favouring aspin-polarized state Therefore for big rs where the kinetic energy will be negligible there willultimately form a balance between the interaction and the entropy which will eventually lead to theunphysical result of a nonvanishing ferromagnetic phase This is not shown in Figure 8 of Ref [85]and we believe that Eq (8) of that reference contains a typo To get the correct density dependenceof the contact interaction we use that the Coulomb interaction is prop rminus1

s whereas the δ-function isprop rminus3

s We therefore set

wS(x xprime) prop r2s(1minus δσ1σ2)δ(r minus rprime) (810)

This interaction is now included as a first-order perturbation on top of the noninteracting HEGIntroducing the interaction strength parameter g the interaction grand potential is then simplywritten with the help of the polarization ξ as

ΩSW = grminus1

s (1minus ξ2) (811)

79135

Applications

FM

PP

PM

0 20 40 6010 30 50

60

80

100

120

140

40

160

200

180

r s(au)

T (K)

rc(rc Tc)

TF

Figure 83 Collinear phase diagram of the HEG for a contact interaction FM ferromagnetic phasePM paramagnetic phase PP partially polarized phase The dotted blue line denotes the Fermitemperature TF

If g is chosen big enough there will be a ferromagnetic region in the phase diagram The strengthof the interaction is now determined by the requirement of the reproduction of the zero-temperaturecritical Wigner-Seitz radius rc asymp 60au which yields g asymp 00102 The resulting phase diagram isshown in Figure 83 We see that the simple Stoner model is capable of reproducing most of the exactproperties from Table 81 However the critical temperature Tc at rc is too low compared to theFermi temperature In the following sections we are going to investigate if we can get a qualitativelybetter phase diagram by more eleborate theoretical means

83 FT-MBPT

A different approach to the calculation of the phase diagram of the HEG employs FT-MBPT acompact review of which can be found eg in [104] or [127] We will restate the expression for thenoninteracting free energy F0

F0 = Ωk minus 1βS0 (812)

The kinetic energy and entropy contributions are given in Eqs (84) and (87) The momentumdistribution in FT-MBPT is given by the Fermi distribution of the noninteracting system

n(k) =1

1 + eβ(ε(k)minusmicro)(813)

ε(k) =k2

2(814)

It is justified to assume that most of the interesting physical properties of a metallic system aremainly determined by the behaviour at the Fermi surface It will therefore prove to be helpful towork with reduced variables for temperature and momentum

t =T

TF k =

k

kF(815)

80135

Applications

The polarization-dependent quantities are given by

tσ = t(1plusmn ξ)minus 23 kσ = k(1plusmn ξ)minus 1

3 (816)

Under these transformations the momentum distribution becomes

nσ(k) =1

1 + ek2tminusα(tσ)

(817)

where the fugacity α = βmicro only depends on the reduced temperature t rather than on both T andkF separately [128] and is determined by the following equation

2

3tminus

32 =

int infin

0

dx

radicx

1 + exminusα(818)

A further simplification can be achieved by the introduction of the Fermi-Dirac integrals Fj(x) [129]

Fj(x) =1

Γ(j + 1)

int infin

0

dzzj

1 + ezminusx(819)

Together with the reduced variables they allow the noninteracting contributions to the free energyto be written in a more compact form

nσ =3

8π

12 t

32F 1

2(ασ) (820)

Ekσ =9

32π

12 k2F t

52F 3

2(ασ) (821)

microσ =1

2k2F tασ (822)

S0σ =

(15

16π

12 t

32F 3

2(ασ)minus ασnσ

)(823)

Relation (818) then becomes

4

3πminus 1

2 tminus 3

2σ = F 1

2(ασ) (824)

The Fermi-Dirac integrals or the closely related Polylogarithms are included in most modern nu-merical libraries (eg GSL [130]) This makes the calculation of the noninteracting contributions tothe free energy very efficient We can now investigate the effect of the inclusion of different diagramsfrom FT-MBPT

831 Exchange-only

Including the first-order diagram leads to the following expression for the free energy

F = F0 +

int 1

0

dλ

λ

︸︷︷︸rarrΩx

(825)

= Ωk minus 1βS0 +Ωx (826)

The exchange diagram is linear in the coupling constant λ and we can therefore drop the integrationThe exchange contribution of spin channel σ to the free energy is then given as

Ωxσ = minus 1

n

intd3k

(2π)3d3q

(2π)3W (k minus q)nσ(k)nσ(q) (827)

81135

Applications

0

02

04

06

08

1

0 1 2 3 4 5

t

I x(t)

84a Exchange

0

05

1

15

2

25

3

0 1 2 3 4 5

t

I 2b(t)

84b Second-order Born

Figure 84 Universal functions Ix(t) and I2b(t) as defined in Eqs (830) and (842)

with the Fourier transform of the Coulomb interaction

W (k) =4π

k2 (828)

When considering reduced variables we see that the kF -dependence can be separated from theintegral in Eq (827) We can therefore introduce a universal function Ix(t) leading to the followingexpressions [128]

Ωxσ(t) = minus(1plusmn ξ)43 kF

3

8πminus1Ix(tσ) (829)

Ix(t) =

intdx

intdy

x

1 + ex2

tminusα(t)

y

1 + ey2

tminusα(t)

ln(x+ y)2

(xminus y)2 (830)

We show Ix(t) in Figure 84a I(x) is a smooth monotonically decreasing function with known hightemperature limit 49tminus1 It can therefore efficiently be parametrized or tabulated A density changecan then be incorporated into Ωxσ by an appropriate change of t and a change of the prefactor inEq (829) This makes the calculation of the exchange contribution very fast

We investigated the magnetic phase transitions for collinear spin configuration and show a sketchof the phase diagram in Figure 85

We see that we recover the zero-temperature Hartree-Fock critical density of rc asymp 554au Thiswas to be expected because at zero temperature the first-order zero-temperature many-body pertur-bation theory (MBPT) approximation is known to be equivalent to the Hartree-Fock approximationAn increase in temperature now leads to an increase of the effect of entropy which itself favoursa paramagnetic configuration Therefore above the critical temperature Tc asymp 9800K we will onlyencounter the paramagnetic equilibrium configuration Due to the 1rs-dependence of Ωx the phasediagram in Figure 85 reproduces several features expected from our general considerations listed inTable 81 and sketched in Figure 82 However the zero temperature phase transition is first-orderand the resulting critical Wigner-Seitz radius is one order of magnitude below the ones expectedfrom quantum Monte-Carlo calculations (rc asymp 60au) We conclude that higher-order contributionsin the perturbative expansion of the free energy are required for a more accurate description of thephase diagram As a first choice we will consider the famous RPA diagrams in the following

82135

Applications

0 2000 4000 6000 8000 100000

2

4

6

8

10

12

14

16

18

20

PP

PM

FMr s(au)

T (K)

rc(rTc Tc)

TF

Figure 85 Collinear phase diagram of the HEG from FT-MBPT including only the exchange con-tribution FM ferromagnetic phase PM paramagnetic phase PP partially polarized phase Thethick dashed red line denotes first-order phase transitions The dotted blue line denotes the Fermitemperature TF

832 Exchange + RPA

The RPA ring-diagrams were shown to reproduce the high-temperature low-density classical limitas well as the zero-temperature high-density quantum limit of the correlation energy ie the ln rsdependence for small rs [131 132] An inclusion of these diagrams then leads to the following freeenergy expression

F = F0 +

int 1

0

dλ

λ

︸︷︷︸

rarrΩx

+

︸︷︷︸rarrΩr

(831)

= Ωk minus 1βS0 +Ωx +Ωr (832)

Carrying out the coupling constant integration the RPA contribution becomes

Ωr =1

n

1

2β

intd3q

(2π)3

infinsum

a=minusinfin

(ln(1minusW (q)χ(q νa)) +W (q)χ(q νa)

) (833)

where the discrete frequencies νa are given by

νa =2πa

β (834)

83135

Applications

0 2 4 6 8 10

minus05

minus04

minus03

minus02

minus01

00 2 4 6 8 10

minus025

minus020

minus015

minus010

0 2 4 6 8 10

minus009

minus008

minus007

minus006

minus005

minus004

0 2 4 6 8 10

minus0011

minus0010

minus0009

minus0008

tt

Ωr(t)

Ωr(t) Ω

r (t)Ω

r (t)

rs = 025 rs = 114

rs = 540 rs = 117

Figure 86 RPA contributions to the free energy density of a HEG at finite temperature The linesdenote our fully numerical results The dots show the results from Ref [119]

The polarization insertion χ(q νa) is given as a sum over its different spin contributions χ(q νa) =sumσ χσ(q νa)

χσ(q νa) = minusint

d3k

(2π)3nσ(k + q)minus nσ(k)

iνa minus (εσ(k + q)minus εσ(k))(835)

We will now again change to reduced variables Using the spatial invariance of the system we areable to integrate over angles in Eqs (833) and (835) [123]

Ωr =k2F t

8π2

intdqq2

infinsum

a=minusinfin


)(836)

χσ(q νa) = minus1

(2π)2kFσ

q

intdk

k

1 + ek2tσminusα(tσ)

ln

((2πatσ)

2 + (q2 + 2kq)2

(2πatσ)2 + (q2 minus 2kq)2

) (837)

Gupta and Rajagopal calculated the RPA contributions for the unpolarized HEG in 1982 ([119 121])Due to the tremendous increase in computational power since these preliminary calculations we wereable to fully calculate the integrals in Eqs (836) and (837) for arbitrary choices of parameters Ourresults are limited only by the accuracy of the numerical integration routines We show our findingsas well as the results from Refs [119] and [121] in Figure 86 For t rarr 0 we reproduce the valuesreported from Gupta and Rajagopal For t gt 0 however our results start to differ and increasinglyso for bigger rs Nonetheless because we did not use any approximations in our calculation of theintegrals in Eqs (836) and (837) we expect our results to be accurate Whereas the exchangecontribution to the free energy decreases with temperature from Figure 86 we see that the RPA-correlation contribution for low temperatures gains influence and reaches its maximum absolute valueat around the Fermi temperature This illustrates the need for an inclusion of correlation effects inthe description of warm matter

As a test of our finite-temperature calculations we investigated the low-temperature behaviour ofΩr We therefore calculated Ωr for three small temperatures t = 0001 0005 001 and extrapolatedto t = 0 using a quadratical fit We compare these extrapolated results with explicit zero-temperature

84135

Applications

minus1eminus3

minus5eminus4

1eminus3

5eminus4

0

1 10 100

δ(r sξ)

rs(au)

ξ = 00

ξ = 02

ξ = 04

ξ = 06

ξ = 08

ξ = 10

Figure 87 Relative deviation of RPA energies from a zero-temperature calculation [133] and ourextrapolated finite-temperature results

0 50 100 200150 250

10

15

20

25

30

35

40

45

50

55

60

PP

PM

FM

300 350

r s(au)

T (K)

rc(rTc Tc)

TF

Figure 88 Collinear phase diagram of the HEG from FT-MBPT including the exchange and RPAcontributions FM ferromagnetic phase PM paramagnetic phase PP partially polarized phaseThe thick dashed red line denotes first-order phase transitions The dotted blue line denotes theFermi temperature TF

calculations from Vosko and Perdew [133] We show the relative deviation in Figure 87 and pointout that over the whole range of densities and polarizations the relative deviation lies below 0001This backs up our confidence in the accuracy of our numerical results

In Figure 88 we show the phase diagram of the HEG resulting from the exchange plus RPA freeenergy in Eq (832) The zero temperature phase transition stays first-order and its critical densityrc increases to asymp 18au However the critical temperature Tc is asymp 270K which is far below thecorresponding Fermi temperature TF asymp 1600K Apparently the inclusion of RPA diagrams overes-timates the ambition of the HEG to assume a paramagnetic configuration To remedy this problemwe will in the following include the second-order Born diagrams in our free energy expression

85135

Applications

833 Exchange + RPA + second-order Born

As was pointed out in Ref [134] the second-order Born diagram is the only diagram of second-ordercontributing to the free energy if one requires the noninteracting and interacting systems to have thesame particle number The other second-order diagrams cancel This is very similar to the derivationof the methodology in Section 6 There we found that the other second-order diagrams disappearif one requires the two systems to exhibit the same 1RDM which of course also leads to the sameparticle number The resulting expression for the free energy is now

F = F0 +

int 1

0

dλ

λ

︸︷︷︸

rarrΩx

+

︸︷︷︸rarrΩ2b

+

︸︷︷︸rarrΩr

(838)

= Ωk minus 1βS0 +Ωx +Ω2b +Ωr (839)

Carrying out the coupling constant integration in Eq (838) (or rather replacing it by a factor of12 for Ω2b) we arrive at the following expression for the second-order Born contribution from spinchannel σ

Ω2bσ =1

n2minus10πminus9

intd3k

intd3p

intd3q

W (q)W (k + p+ q)nσ(k)nσ(p)(1minus nσ(k + q))(1minus nσ(p+ q))

ε(k + q) + ε(p+ q)minus ε(k)minus ε(p) (840)

As for the exchange contribution the calculation of the second-order Born (Ω2b) contribution ishugely simplified by the fact that by changing to reduced variables the explicit kF -dependence canbe separated from the momentum integrals

Ω2bσ(t) =(1plusmn ξ)

2Ω2bσ(0)I2b(tσ) (841)

I2b(t) = minus3

Ω2bσ(0)2minus6πminus5

intd3k

intd3p

intd3q

n(q)n(k + p+ q)(1minus n(k + q))(1minus n(p+ q))

k2p2k middot p(842)

Ω2bσ(0) =

(ln 2

6minus 3

4π2ζ(3)

) (843)

where ζ is Riemannrsquos Zeta-function The zero-temperature limit of Ω2b was calculated analytically inRef [135] and the high-temperature limit of I2b(t) can be shown to be sim tminus32 For a translationallyinvariant system the dimensionality of the integral in Eq (842) can be reduced to six In AppendixA6 we present a form of Ω2bσ which is appropriate for numerical integration We calculated theuniversal functional for various values of t and show it in Figure 84b We note that as in thecase of Ix(t) we can parametrize or tabulate I2b(t) which leads to a tremendous decrease of thecomputational cost for the calculation of Ω2b One important observation we should like to pointout here is that Ω2bσ is independent of the density

The phase diagram corresponding to the free energy approximation from Eq (839) is shownin Figure 89 For T = 0 we recover the RPA critical Wigner-Seitz radius of rc asymp 18au Anincrease of the temperature then leads to a phase transition from a paramagnetic to a ferromagneticconfiguration in the density range of rs isin 4 19au which is in contrast to the third propertyfrom Table 81 Furthermore the critical radius rTc is reduced to rTc asymp 4au and the criticaltemperature is increased to Tc asymp 16000K To understand these unexpected effects we show the

86135

Applications

0 4000 8000 120000

4

8

12

16

20

24

28

32

36

40

16000FM

PP

PMr s(au)

T (K)

TF

(rTc Tc)rc

Figure 89 Collinear phase diagram of the HEG from FT-MBPT including exchange RPA andsecond-order Born contributions FM ferromagnetic phase PM paramagnetic phase PP partiallypolarized phase The thick dashed red line denotes first-order phase transitions The dotted blueline denotes the Fermi temperature TF

contributions from the noninteracting free energy the exchange energy the RPA energy and thesecond-order Born contribution in Figure 810 for rs = 7au and different polarizations We see thatthe noninteracting as well as the RPA contributions always favour a paramagnetic configurationwhereas the exchange contribution favours a ferromagnetic one Furthermore the energy differencesbetween different polarizations stay almost constant for small temperatures The second-order Bornapproximation on the other hand shows a qualitatively different behaviour For T = 0 the reducedtemperatures for the spin channels are the same (tuarr = tdarr = 0) From Eq (841) we then concludethat Ω2b = Ω2buarr+Ω2bdarr is independent of the polarization This explains why for T = 0 we recover theRPA critical density Increasing the temperature then leads to a favourisation of the ferromagneticconfiguration As mentioned before the energy differences of different polarizations coming from theother contributions stay almost invariant for small temperatures This then explains the formationof a ferromagnetic configuration with increasing temperature for rs slightly below rc We can alsosee from Figure 810 that the second-order Born contribution at about t = 05 starts to favour theparamagnetic configuration Therefore a further increase of the temperature will eventually yielda paramagnetic eq-state We conclude this investigation by summarizing that the inclusion of thesecond-order Born approximation leads to a worse phase diagram compared to our assumptionsfrom Figure 82 It both leads to a distinctive temperature-induced paramagnetic to ferromagneticphase transition for rs isin 4 19au and reduces the critical density rTc to a value even below theHartree-Fock result

After this investigation of several FT-MBPT approximations on the magnetic phase diagramof the HEG in collinear spin configuration we will turn to the description via our new theory ofFT-RDMFT

87135

Applications

minus15

minus10

minus5

0

5

minus8

minus6

minus4

minus2

0 05 10 15

minus7

minus6

minus5

minus4

minus3

0 05 10 15 20

3

4

5

6

7

F0(10minus

2Ha) Ω

x(10 minus

2Ha)

Ωr(10minus

2Ha) Ω

2b (10 minus

2Ha)

tt

ξ = 0ξ = 05ξ = 08ξ = 10

Figure 810 Clockwise from top left noninteracting free energy exchange RPA and second-orderBorn contributions at rs = 7au (TF asymp 11900K) versus the temperature T of the HEG for differentpolarizations ξ

84 FT-RDMFT

We investigate the performance of our new theory of FT-RDMFT by choosing the first-order func-tional ie the exchange-only functional from Sec 6 as our free energy functional One reasonfor this choice is that the results produce most of the features one might be interested in in theinvestigation of the full system eg first and second-order phase transitions between both collinearand spin spiral phases Its simplicity also allows a more direct understanding and interpretationserving as a guideline for more involved functionals Another reason for the choice of this functionalis that its minimization was shown in Section 425 to be equivalent to a solution of the FT-HFequations The FT-HF solution can then serve as a starting point for various approximations inother theoretical frameworks Because of its conceptual importance the temperature dependence ofthe free energy of the HEG in FT-HF has been investigated [136 137] but a thorough calculationof the corresponding phase diagram including collinear and spin spiral phases to the best of ourknowledge has not been carried out so far substantiating the importance of our calculations

The expression for the free energy in the first-order FT-RDMFT approximation reads

F [γ] = F0[γ] +

int 1

0

dλ

λ

︸︷︷︸rarrΩx

(844)

= Ωk[γ]minus 1βS0[γ] + Ωx[γ] (845)

The definitions for the different contributions are given in Eqs (84)-(87)(827) But in contrastto the FT-MBPT treatment in FT-RDMFT the momentum distribution n(k) refers to the effectiveKohn-Sham system and is subject to optimization The choice of NOs then determines the symmetryof the system In this work we will focus on NOs representing a collinear spin configuration in Section841 and a planar spin wave configuration in Section 842

88135

Applications

841 Collinear spin configuration

The NOs describing a collinear spin state are plane waves By discretizing we assume the ONs to beconstant in the small volumes Vi around the sample k-points ki and hence the 1RDM is representedby the following expression

γ(r minus rprime) =sum

σi

nσi

int

Vi

d3k

(2π)3eikmiddot(rminusrprime) (846)

Therefore the integrals in Eqs (84)-(87)(827) become sums and we arrive at the final expressionfor the free energy functional

F [nσi] = Ωk[nσi]minus 1βS0[nσi] + Ωx[nσi] (847)

The kinetic energy Ωk the entropy S0 and the exchange energy Ωx are given by

Ωk[nσi] =sum

σi

nσiti (848)

S0[nσi] = minussum

σi

(nσi ln(nσi) + (1minus nσi) ln(1minus nσi))ωi (849)

Ωx[nσi] = minus1

2

sum

σij

nσinσjKij (850)

where the integral weights are

ti =1

n

int

Vi

d3k

(2π)3k2

2(851)

Kij =1

n

int

Vi

dk31(2π)3

int

Vj

dk32(2π)3

4π

(k minus kprime)2(852)

ωi =1

n

int

Vi

d3k

(2π)3 (853)

The minimization of F [nσi] wrt the ONs has then to be performed under the constraints ofparticle number and polarization conservation

sumi nσi = nσ and the fermionic requirement 0 le

nσi le 1 We employ the minimization methods derived in Section 7 which have proved to be veryefficient and effective

We now determine the eq-polarization as explained in Section 81 and depicted in Figure 81 Thegeneral behaviour of the phase transitions is the same compared to the exchange-only treatment fromFT-MBPT in Section 831 As we illustrate in Figure 811 an increase of rs leads to a first-orderphase transition between a paramagnetic and a ferromagnetic state and a subsequent second-orderphase transition back to the paramagnetic state An increase of the temperature then leads to anincrease of the Wigner-Seitz radius where the first-order phase transition occurs and to a decreaseof the Wigner-Seitz radius where the paramagnetic configuration is assumed again Interestinglythe nature of the phase transitions does not change even at high temperatures

In order to study the various contributions to the free energy separately we plotted kineticenergy exchange contribution and entropy in Figure 812 We see that the entropy and exchangecontributions always show a monotonically decreasing behaviour wrt an increase in the polarizationξ The kinetic energy however which is known at zero temperature to be monotonically increasingwith ξ actually becomes decreasing for high values of the reduced temperature t = TTF

As we will see in the following this somewhat counterintuitive effect is due to the fact that theexchange contribution hinders the temperature-induced smoothing of the momentum distributionsand stronger so for the ferromagnetic configuration To elucidate this argument we choose a density

89135

Applications

4 6 8 10 12 14 160

02

04

06

08

10

T=0T=5000T=8000T=10000

ξ min

rs(au)

Figure 811 Equilibrium polarization ξmin of the HEG for different rs and T Above T = 6000Kthere is no fully polarized equilibrium state

05

10

15

20

0

0 02 04 06 08

minus4

0 02 04 06 08 1

minus3

0

3

6

1

2

3

minus3

minus2

minus1

ξξ

Ωk(10minus

2Ha) Ω

x(10 minus

2Ha)S

∆F(10 minus

3Ha)

t = 0t = 10t = 16t = 22

Figure 812 Clockwise from bottom left entropy kinetic energy exchange energy and free energy atrs = 15au (TF = 2590K) versus the polarization ξ of the HEG for different reduced temperaturest The free energy is origin-shifted for demonstration purposes

which will yield a ferromagnetic solution at zero temperature (rs = 15au) Thermodynamic vari-ables for both paramagnetic and ferromagnetic configurations as functions of the temperature areshown in Figure 813 The curves denoted by ldquoFDrdquo correspond to the FT-MBPT expressions iethe FT-RDMFT functionals applied to Fermi-Dirac momentum distributions with the appropriatetemperature The FT-RDMFT functional is then minimized to give the curves denoted by ldquoHFrdquoThe differences of the energies for ferromagnetic and paramagnetic configurations are then includedas the ldquo∆1minus0rdquo curves

∆1minus0 = Ω(ξ = 1)minus Ω(ξ = 0) (854)

We see that in the case of a noninteracting system ie for the ldquoFDrdquo curves the kinetic energiesof paramagnetic and ferromagnetic configurations approach each other but do not cross The factthat they converge for high temperatures can qualitatively be understood by the concept of different

90135

Applications

0

1

2

3

4

0 1 2 0 1 2 3

minus8

minus6

minus4

minus2

minus1

0

1

2

3

4

minus3

minus2

minus1

tt

Ωk(10minus

2Ha) Ω

x(10 minus

2Ha)

SF(10 minus

2Ha)

FD ξ = 0

FD ξ = 1

HF ξ = 0

HF ξ = 1

FD ∆1minus0

HF ∆1minus0

Figure 813 Clockwise from bottom left entropy kinetic energy exchange energy and free energyfor the HF and the noninteracting functionals at rs = 15au (TF = 2590K) and ξ = (0 1) versus thereduced temperature t of the HEG The black dotted and the grey shaded lines denote the differencesbetween the ferro- and paramagnetic configurations The free energy differences are scaled by 10The behaviour of the HF-curves can be explained by different ldquoeffectiverdquo Fermi temperatures

ldquoeffectiverdquo Fermi temperatures T lowastF of the configurations For the paramagnetic configuration at zero

temperature the ONs occupy two Fermi spheres of radii kuFuarr = kuFdarr = kF one for each spin channelBecause in the ferromagnetic situation the ONs are restriced to only one spin channel there will alsobe only one Fermi sphere with increased radius kpFuarr = 2

13 kF (kpFdarr = 0) Because the kinetic energies

are proportional tosum

σ(kFσ)2 this explains the favourisation of the paramagnetic configuration at

zero temperature An increase in temperature will now lead to a smoothing of the Fermi sphereie the momentum distributions and therefore to an overall increase of the kinetic energy Thequickness of the smoothing is determined mainly by the characteristic energy of the system ie theFermi energy or correspondingly the Fermi temperature Following from the arguments abovethe paramagnetic configuration exhibits a smaller Fermi temperature when compared to the ferro-magnetic configuration This implies that the corresponding momentum distributions are smoothedmore quickly which in turn leads to a relative increase of the kinetic energy of the paramagneticconfiguration This effect hower as we can see from Figure 813 is not big enough to let the kineticenergy curves cross and they converge for T rarrinfin

The situation changes if one includes the exchange contribution and minimizes wrt the momen-tum distribution The exchange contribution by itself is known to be minimal for a ferromagneticconfiguration with a momentum distribution describing a sharp Fermi sphere Because it does notcouple different spin channels this also implies that for general polarizations the favourable config-urations are the ones describing sharp Fermi spheres of appropriate radii Ωx therefore counteractsthe effect of temperature which can be interpreted as an increase of the effective Fermi temperatureT lowastF As argued this increase of T lowast

F is stronger for the ferromagnetic configuration which in addi-tion to our considerations of noninteracting systems above then leads to a cross-over of the kinetic

91135

Applications

Figure 814 Phase diagram of the HEG for collinear spin configuration for the first-orderFT-RDMFT functional For T lt Tc when increasing rs the HEG shows both first and second-order phase transitions The red dashed line denotes TF

energy curves in the FT-RDMFT treatment These arguments can be applied to both entropic aswell as exchange contributions but there the ferromagnetic configuration exhibits a lower value forzero temperature preventing a crossing of energy curves The behaviour of the free energy is morecomplicated because the entropy enters negatively It has to be pointed out however that in thecase of ldquoFDrdquo-momentum distributions ie in first order FT-MBPT an increase of temperaturefirst leads to an increase in the free energy before an eventually monotonic decrease This is ratherunphysical and can be appointed to the fact that the ldquoFDrdquo-momentum distribution is not acquiredby any sort of variational principle The ldquoHFrdquo-momentum distribution one the other hand is ex-plicitly determined by a minimization procedure which leads to the qualitatively correct monotonicaldecrease of the free energy with temperature

After these considerations we now calculate the equilibrium polarization of the HEG for a widerange of densities and temperatures The resulting magnetic phase diagram is shown in Figure 814The critical Wigner-Seitz radius rc marking the zero-temperature magnetic transition between theparamagnetic and ferromagnetic phases is the zero-temperature Hartree-Fock result of rc asymp 554auOn increasing the temperature the ferromagnetic phase gets reduced until after a critical point itvanishes The corresponding temperature Tc is calculated to be at about 10500K while the criticalWigner-Seitz radius rTc becomes 71au A comparison of the phase diagrams for the exchange-onlyFT-MBPT treatment and the actual first-order FT-RDMFT functional shows that the ferromagneticphase is slightly increased for the FT-RDMFT treament This can be explained on the basis ofthe previous considerations For temperatures similar or higher than the Fermi temperature theFT-RDMFT treatment favours a ferromagnetic configuration when compared to the FT-MBPTThis concludes our investigation of the phase diagram of the HEG in collinear spin configuration forthe first-order functional The effect of correlation is then discussed in Section 86

We will in the following consider the explicitly noncollinear basis of planar spin spirals (PSS)

92135

Applications

842 Planar spin spirals (PSS)

Although collinear plane waves as NOs respect the symmetry of the system they do not necessarilyyield the lowest free energy Another possible choice are spin density waves (SDW) which areexplicitly noncollinear but are still describable with a small set of parameters The theoreticalinterest in SDW dates back to the beginning of modern many-body physics It was in 1962 whenOverhauser [138] proved that for the electron gas at zero temperature in Hartree-Fock approximationa state describing a SDW will yield energies below the energies provided by the collinear phases

Using the framework of FT-RDMFT we will now be able to investigate how the situation changesfor a HEG in grand canonical equilibrium restricting ourselves to the description of pure spin waves(SW) An investigation of SWs within zero-temperature RDMFT was done by Eich et al[139] andwe will repeat the most important concepts in the following The NOs describing a SW are given by

φ1k(r) =

(cos(Θk

2

)eminusiqmiddotr2

sin(Θk

2

)eiqmiddotr2

)eikmiddotrradicV

(855)

φ2k(r) =

(minus sin

(Θk

2

)eminusiqmiddotr2

cos(Θk

2

)eiqmiddotr2

)eikmiddotrradicV (856)

where the wavevector q and the azimuthal angle Θk of the spin spirals are subject to variation Vdenotes the Volume of the space under consideration To distinguish the SW index from the spinindex we will denote it by b in the following In the collinear basis the 1RDM then becomes

γuarruarr(r rprime qΘk) =

intd3k

(2π)3

(n1k cos

2

(Θk

2

)+ n2k sin

2

(Θk

2

))ei(kminusq2)middot(rminusrprime) (857)

γuarrdarr(r rprime qΘk) =

intd3k

(2π)31

2(n1k minus n2k) sin (Θk) e

ikmiddot(rminusrprime)eiq2middot(r+rprime) (858)

γdarruarr(r rprime qΘk) =

intd3k

(2π)31


ikmiddot(rminusrprime)eminusiq2middot(r+rprime) (859)

γdarrdarr(r rprime qΘk) =

intd3k

(2π)3

(n1k sin

2

(Θk

2

)+ n2k cos

2

(Θk

2

))ei(k+q2)middot(rminusrprime) (860)

The exchange contribution to the free energy is given by Eq (466) as

Ωx[γ] = minus1

2

sum

σ1σ2

intd3rd3rprimew(r minus rprime)γσ1σ2

(r rprime)γσ2σ1(rprime r) (861)

For our choice of SWs as NOs this becomes for the Coulomb interaction

Ωx[γ] = minus1

2n

intd3k1(2π)3

d3k2(2π)3

4π

|k1 minus k2|2[sum

b

(nbk1nbk2

) cos2(Θk1minusΘk2

2

)+ (n1k1

n2k2+ n2k1

n1k2) sin2

(Θk1minusΘk2

2

)] (862)

Interestingly the exchange energy contribution shows no explicit dependence on the SW wavevectorq We will from now on assume that q is parallel to the z-axis which suggests the use of cylindricalcoordinates A vector k is then given by k = (kz kρ φ) We finally arrive at an expression for the

93135

Applications

free energy functional appropriate for numerical minimization

F [q nbi Θi] =sum

bi

nbiti +q2

8minus q

sum

i

(n1i minus n2i) cos(Θi)Qi+

1βsum

bi

(nbi ln(nbi) + (1minus nbi) ln(1minus nbi)

)ωiminus

1

2

sum

bij

(nbinbj) cos2

(Θi minusΘj

2

)Kij minus

sum

ij

(n1in2j) sin2

(Θi minusΘj

2

)Kij (863)

The integral weights are defined in Eqs (851)-(853) with the additional

Qi =1

2n

int

Vi

d3k

(2π)3kz (864)

The magnetization of the HEG for SW-NOs is given by

m(r) = minus

A cos(q middot r)A sin(q middot r)

B

(865)

The two amplitudes A and B are determined via

A =1

2

intd3k

(2π)3(n1k minus n2k) sin(Θk) (866)

B =1

2

intd3k

(2π)3(n1k minus n2k) cos(Θk) (867)

From now on we will restrict ourselves to planar SWs ie SWs for which the magnetization of theHEG in z-direction vanishes As argued in [110] and [139] this planar configuration has a lowerenergy than a conical SW with nonvanishing z-component These SWs will be called planar spinspirals (PSS) and the amplitude A of the PSS configuration becomes the order parameter

The requirement of a vanishing z-component can be met by simple constraints on the ONs andNOs

nbkρminuskz= nbkρkz

(868)

Θkρplusmn|kz| =π

2(1∓ akρ|kz|) (869)

As for the collinear case we can define a PSS polarization ξPSS using N1 =sum

i n1i and N2 =sum

i n2i

ξPSS =N1 minusN2

N1 +N2(870)

If we now choose q = 0 and akρ|kz| = 1 then the PSS unpolarized state ξPSS = 0 corresponds to

the paramagnetic state and the PSS polarized one ξPSS = 1 describes the ferromagnetic solutionTo understand the effect of an increase of q it is instructive to consider a fully polarized nonin-

teracting system We show a sketch of the q-dependence of the Fermi surface in Figure 815 Forq = 0 the ONs describe a Fermi sphere of radius 2

13 kF around k = 0 If one increases q one can

derive from the first three terms in Eq (863) and the symmetry relations (868) (869) that theFermi sphere will divide symmetrically along the z-direction If q supersedes 2kF then there will be

94135

Applications

kρkρkρkρ

kzkzkzkz

qkF = 0 qkF = 10 qkF = 18 qkF = 2

kF kF kF kF

-kF -kF -kF -kF

q2

q2

q2

- q2- q2 - q2

Figure 815 Sketch of q-dependence of Fermi surface of a fully polarized PSS state

5

6

7

minus112

minus105

minus98

minus91

0 05 1 15 2 25

0

0 05 1 15 2 25 3

minus49

minus48

minus47

minus46

5

10

15

qkFqkF

Ωk(10minus

2Ha)

Ωx(10 minus

2Ha)

S(10minus

2)

F(10 minus

2Ha)

T = 0KT = 2500KT = 4000KT = 5000K

Figure 816 Clockwise from bottom left entropy kinetic energy exchange energy and free energyat rs = 500 (TF = 23300K) versus the PSS q-vector The minimal free energies are denoted by thearrows

two distinct Fermi spheres with radius kF centered at k = minusq2 and k = q2 respectively and thisconfiguration represents a paramagnetic collinear configuration If one now includes temperatureeffects then the momentum distribution around the Fermi surface will be smoothed To reproducethe paramagnetic collinear configuration one therefore has to consider the limit q rarrinfin

We can now minimize the free energy functional wrt the ONs and orbital angles For all ofour calculations we found that the fully polarized PSS state yields the lowest free energy In thefollowing we will therefore always consider fully polarized PSS configurations The dependence ofthe entropy and the other thermodynamic variables on q at several temperatures is shown in Figure816 The entropy displays a monotonically increasing almost linear dependence which in turn leadsto an increase of the optimal q-vector (the q for which the free energy is minimal) This situation isdepicted for rs = 550au in Figure 817 where we show the free energy and the amplitude of a PSSstate wrt q for several temperatures close to the critical temperature TPSS

c For temperaturesbelow TPSS

c the free energy exhibits a minimum for finite q If the temperature is increased aboveTPSSc qmin instantaneously jumps to a value bigger than 2kF letting the amplitude of the PSS

vanish The jump in q at TPSSc therefore marks a first-order phase transition of the PSS phase

where the amplitude of the optimal PSS is approaching zero instantaneously In his work on SDWs

95135

Applications

0

1

2

3

4

5

1 15 2 25minus494

minus493

minus492

minus491

minus490

minus489

qkF

F(10minus

2Ha)

A(10minus

4au)

T = 4700K t = 0244T = 4800K t = 0249T = 4900K t = 0254T = 5000K t = 0260

Figure 817 Amplitude A and free energy F for different q and T for rs = 55au(TF = 19200K)The amplitudes change only slightly when increasing the temperature but the optimal q increases(arrows) discontinuously letting the optimal amplitude Aopt vanish This denotes a first-order phasetransition in the PSS phase

in an electron gas Overhauser [138] made the conjecture that on increasing the temperature theoptimal amplitude of the SDW state will approach zero continuously giving rise to a second-orderphase transition Our results disprove the validity of this conjecture for PSS states for densities withWigner-Seitz radii above rs asymp 4au

We show the dependence of the amplitude of the optimal PSS wrt T for several rs in Figure818 For small q the amplitude increases slightly when increasing the temperature which seems tobe surprising at first Temperature is usually expected to favour states of higher disorder and onecould therefore expect the amplitude to be reduced However in our calculations we encounter thefollowing behaviour of the Fermi surface For qkF lt 2 the zero-temperature Fermi surface assumesan hourglass-like shape as depicted in Figure 815 We find that an increase of the temperature forsmall temperatures smoothes the Fermi surface but also leads to an increase of occupation at theldquowaistrdquo of the hourglass The angles are usually closer to π2 for smaller values of kz which thenleads to an increase of the PSS amplitude A via Eq 866 Furthermore the free energy gain fromthis effect is bigger for smaller q which also leads to a decrease of the optimal q-vector increasing theamplitude even further For higher temperatures these two effects become less important leadingto the eventual decrease of the optimal PSS-phase

We have to point out however the numerical uncertainty of these findings The energy differencesfor slightly changed momentum distributions are very small but the amplitude shows a much strongerdependence This also explains the considerable variance of the calculated optimal amplitudes asshown in Figure 818 However we did a careful minimization of the functional with several k-point-mesh refinements and believe that our numerical findings are quantitatively correct We haveargued that for small temperatures the amplitude of a PSS state increases The optimal q-vectoron the other hand changes only slightly Therefore for small temperatures the optimal amplitudeincreases It was already mentioned in [139] that at zero temperature the amplitude of the PSSdecreases with decreasing rs Below a critical radius rPSS

c = 35 we cannot resolve the amplitudeof the PSS anymore This reduction of the amplitude is mainly due to the fact that the energydifference between the paramagnetic and ferromagnetic phases increases and therefore the optimalq approaches 2kF (see Figure 817) When considering finite temperatures this leads to a decrease

96135

Applications

0 1000 2000 3000 4000 50000

1

2

3

4

T (K)

Aopt(10minus

4au)

rs = 55rs = 50rs = 45rs = 40

Figure 818 Amplitudes Aopt of the quilibrium PSS phases for different rs(40 45 50 55au)and T The corresponding Fermi temperatures are 36400 28800 23300 19200K

of the critical temperature TPSSc as shown in Table 82

For those rs for which a polarized configuration is favourable over an unpolarized one a formationof a PSS was found to increase the free energy

In Figure 819 we combine the results from the collinear NOs and the PSS NOs calculations toget a schematic diagram of the magnetic phase diagram of the HEG for the first-order FT-RDMFTfunctional

The magnetic phase transition between paramagnetic and polarized phases for increasing rs wasshown to be first-order while the transition from polarized to paramagnetic phases for higher rs issecond-order As can be seen from Figure 819 the second-order phase transitions occurs close tothe Fermi temperature TF In summary the collinear phase diagram resulting from the first-orderfunctional in FT-RDMFT reproduces most features listed in Table 81 but it exhbits a first-orderphase transition at zero temperature Furthermore it fails to reproduce a qualitatively correct rcThe PSS phase shows a much stronger density dependence then the collinear phase and it is strongestfor 4 le rs le 6 In this range we also see a temperature-driven first-order phase transition betweena PSS state and the paramagnetic collinear state

This concludes our investigation of the phase diagram of the HEG for the first-order FT-RDMFTfunctional We have shown in Section 425 that the minimization of this first-order functional isequivalent to a solution of the FT-HF equations In the following section we will therefore investigatethe temperature dependence of the FT-HF single-particle dispersion relation for collinear as well asfor PSS configurations

rc(au) 40 45 50 55

Tc(K) 2700 3400 4100 5000

Table 82 Critical temperatures TPSSc above which no equilibrium PSS phase was found

97135

Applications

0 2000 4000 6000 8000 100000

2

4

6

8

10

12

14

16

18

20

PSS

FMPP

PM

r s(au)

T (K)

rc(rTc Tc)

TPSSc

TF

Figure 819 Phase diagram of the HEG in first-order FT-RDMFT (FT-HF) approximation FMferromagnetic collinear phase PM paramagnetic collinear phase PP partially polarized collinearphase PSS planar spin spiral state The thick dashed red line denotes first-order phase transitionsThe dotted blue line denotes the Fermi temperature TF

85 FT-HF dispersion relations

At zero temperature the HF dispersion relations for a collinear spin configuration can be calculatedanalytically to yield

εσ(k) =k2

2minus 2

πfd(kkFσ) (871)

fd(x) denotes the dimension-dependent corrections In three dimensions it reads

f3(x) =1

2+

1minus x24x

log

∣∣∣∣1 + x

1minus x

∣∣∣∣ (872)

The two-dimensional correction is given by

f2(x) =

E(x) x lt 1

x(E(1x

)minus(1minus 1

x2

)K(1x

)) x ge 1

where E(x) and K(x) are the complete elliptic integrals of the first and second kindAt finite temperature because of the entropy the minimum of the free energy functional will

always be assumed on the interior of ΓN (see Eq (222)) ie there will be no pinned statesTherefore at the minimum of F [ni] from Eq (847) the following relation holds

partF

partni= 0 =

partΩk

partniminus 1

β

partS0

partni+partΩx

partni (873)

98135

Applications

minus4

minus3

minus2

minus1

0

1

2

3

4

0 05 1 15 2 25

kkF

ε σ(kk

F)ε F

k2200105

12510100

820a Three dimensions

minus8

minus6

minus4

minus2

0

2

4

0 05 1 15 2 25

kkF

ε σ(kk

F)ε F

k220

0105

12510100

820b Two dimensions

Figure 820 Hartree-Fock dispersion relation for the HEG in collinear spin configuration in threeand two dimensions at rs = 5au(TF = 23300K) for different values of t = TTF The black dottedline denotes the noninteracting relation ε(k) = k22

With the explict form of S0[ni] of Eq (849) this yields

εi minus micro =partΩk

partni+partΩx

partni (874)

which can be easily accessed numerically We calculated the FT-HF dispersion relation for the fullypolarized HEG in three and two dimensions for rs = 5au and various values of t The results areshown in Figure 820 Because of the full polarization the Fermi surface is at kkF = 2

1d where d is

the dimension of the HEG under consideration Because the exchange contribution does not couplethe different spin channels in collinear configuration the unoccupied band shows the noninteractingdispersion relation k22 For the occupied band at zero temperature we recover the exact relations togood agreement An increase of the temperature then first lets the ldquovalleyrdquo in the dispersion relationvanish and then secondly lets the eigenenergies approach the noninteracting relation therefore closingthe HF-gap This can be understood from the thermal smoothing of the momentum distributioninduced by the entropy Because NOs of very different quantum numbers generally have only smalloverlap this leads to a decrease of the influence of the exchange contribution compared to the kineticenergy

As a second choice for the NOs of the 1RDM we will now consider PSS states as defined in Section842 The single-particle eigenenergies can be determined in the same way as for the collinear case(see Eq (874)) We choose the same Wigner-Seitz radius rs = 5au as in Figure 820 to allowfor a comparison between the dispersion relations of collinear and PSS configurations Consideringa fully polarized PSS configuration with q = 16 we yield the dispersion relation shown in Figure821 In Figure 821a we show the zero-temperature PSS dispersion (T=0) the collinear ferromag-netic (FM) dispersion and the PSS dispersion corresponding to a noninteracting system (ni) Thenoninteracting PSS energies e0b(k) are given simply by

e0b(k) =

(k plusmn q

2

)2

2minus q2

8=

1

2(k2 plusmn kq) (875)

As a first difference to the collinear case we see that the unoccupied band at zero temperature doesnot follow the noninteracting relation This is due to the fact that the exchange contribution inEq (862) explicitly couples both PSS channels In Figure 821b we then demonstrate the effect oftemperature on the PSS dispersion relation As for the collinear configuration the temperature firstlets the ldquovalleyrdquo in the dispersion relation of the occupied band disappear With a further increase

99135

Applications

minus4

minus2

0

2

4

6

8

10

0 05 1 15 2

kkF

ε σ(kk

F)ε F

ni

T = 0

FM

821a Limiting behaviour

minus4

minus2

0

2

4

6

8

10

0 05 1 15 2

kkF

ε(kk

F)ε F

091787

821b Temperature dependence

Figure 821 Hartree-Fock dispersion relation for the PSS phase with q = 16 in the HEG at rs =5au(TF = 23300K) for different values of t = TTF Figure 821a The red short-dashed lines (ni)denote the noninteracting dispersions ε(kkF )εF = (k2 plusmn kq)2 The blue long-dashed lines (FM)represent the ferromagnetic collinear dispersions The black solid line shows the PSS dispersion atT = 0 Figure 821b Dispersion relations for fully polarized PSS configurations at different reducedtemperatures

of the temperature both unoccupied as well as occupied bands then approach the noninteractingdispersion relation

The method of FT-RDMFT proved to be an efficient tool for the description of the eq-propertiesof the HEG at finite temperature in FT-HF approximation An inclusion of correlation can now beachieved by an addition of an appropriate functional Ωc to the first-order functional as defined inEq (850) In the following we are going to investigate several choices of correlation functionalsin the theoretical framework of FT-RDMFT and study their effect on the collinear magnetic phasediagram

86 Correlation in FT-RDMFT

As a first class of candidates for correlation functionals in FT-RDMFT we will now investigatecorrelation functionals from DFT both from zero as well as from finite temperature frameworks

861 DFT-LSDA correlation

The employment of correlation functionals from DFT in FT-RDMFT basically approximates the full1RDM dependence of the correlation functional as a density-dependent one

Ωc[γ] = ΩDFTc [n] (876)

We should like to point out that this approach exhibits an important intrinsic flaw This becomesclear if one reviews the minimization procedure for the case of local external potentials and separatesthe variation over 1RDMs in a combined variation over densities and a variation over the class of1RDMs which yield a certain density

Feq = minγF [γ] = min

nminγrarrn

F [γ] (877)

= minn

(Ωext[n] + ΩH [n] + Ωc[n] + min

γrarrn(Ωk[γ]minus kBTS0[γ] + Ωx[γ])

) (878)

As we assumed the correlation contribution to be independent of the particular form of the 1RDMas long as the diagonal ie the density stays the same we can take it out of the minimization

100135

Applications

over 1RDMs We find that the minimization over the 1RDMs now only contains kinetic exchangeand entropy contributions If the density n refers to a solution of the finite temperature Hartree-Fock equations to some external potential then we have shown in Section 425 that the constrainedminimization of this combination of functionals gives the corresponding Hartree-Fock momentumdistribution The resulting 1RDM will in this sense be ldquouncorrelatedrdquo The severe problem withthis result can be understood by considering the zero-temperature limit Any choice of correlationfunctional in FT-RDMFT which just depends on the density will lead to a step function as themomentum distribution for low temperatures As mentioned this argument hinges on the assumptionthat the density n is a Hartree-Fock density of some external potential In the case of the HEG thisis obviously justified but for general systems with nonlocal external potential it is not clear if alldensities share this property Nonetheless the previous considerations show that for a wide class ofexternal potentials ie all which lead to a gs-density which is the Hartree-Fock density to someother external potential the optimal 1RDM will be uncorrelated This knowledge of the nature ofthe failure of DFT-functionals allows us to at least partially take account of these shortcomings inthe following

We are going to employ the correlation-energy of the electron gas as parametrized in the PWCAapproximation Remembering the conceptual difference between RDMFT and DFT ie the knowl-edge of the exact kinetic energy functional one might be tempted to remove the kinetic contributionfrom the PWCA-functional before employing it as a correlation-energy functional in RDMFT Thisprocedure however would be incorrect as will be explained below

Based on our previous discussion of the intrinsic problems of the utilization of DFT functionalsin RDMFT we know that the minimizing momentum distribution will be ldquouncorrelatedrdquo ie willreproduce the Hartree-Fock solution which becomes a step function n0(k) in the T rarr 0 limit If weremoved the kinetic contribution the correlation-energy functional would be given by the interactioncontribution WPWCA

c alone and the free energy at density n would read

FTrarr0= Ωk[n

0(k)] + Ωx[n0(k)] +WPWCA

c (n) (879)

Because the momentum distribution is a step function this expression reproduces exactly the Monte-Carlo results without the kinetic contribution This poses a problem because removing the kineticcontribution from a DFT-correlation functional yields an overall bad approximation A straightfor-ward remedy of this problem is the re-inclusion of the kinetic contribution to accomodate for theldquouncorrelatedrdquo nature of the momentum distribution Cashing in on our previous discussion we aretherefore able to construct a FT-RDMFT functional which reproduces the exact Monte-Carlo resultsin the T rarr 0 limit The resulting phase diagram is shown in Figure 822

By construction we recover a critical Wigner-Seitz radius of rc asymp 70au However we encounterseveral properties not expected from our considerations in Section 81 In the radius range of 20auto 70au with an increase of temperature we encounter a first-order phase transition to a partiallypolarized state A further increase of temperature then leads to a second-order phase transition backto the paramagnetic configuration Considering the density range defined by 70au lt rs lt 90authe situation becomes worse Starting from the ferromagnetic configuration at zero temperaturewe encounter first a first-order phase transition to a paramagnetic state then another first-orderphase transition to a partially polarized state and then another second-order transition back to theparamagnetic configuration We attribute the appearance of the big partially polarized phase fortemperatures above TF to the fact that the 1RDM is ldquouncorrelatedrdquo and therefore the entropy isunderestimated It would be interesting to investigate how a FT-LSDA from DFT would performin the FT-RDMFT framework but unfortunately this theory does not exist so far We will thereforeresort to an approximate inclusion of temperature effects by employing the temperature-dependentRPA approximation from FT-MBPT (see Section 83)

101135

Applications

0 1000 1500 2000 2500 30000

20

40

60

80

100

120

140

160

180

200

FM

PP

PM

500

T (K)

r s(au)

TF

(rTc Tc)rc

Figure 822 Collinear phase diagram of the HEG for the DFT-LSDA functional FM ferromagneticphase PM paramagnetic phase PP partially polarized phase The dotted blue line denotes theFermi temperature TF

862 FT-DFT-RPA correlation

From our investigation of the FT-MBPT contributions on the phase diagram of the HEG in Sections832 and 833 we deduce that an inclusion of the second-order Born approximation does moreharm than good We therefore include only the finite-temperature RPA contribution as defined inEqs (836) and (837) The resulting phase diagram is shown in Figure 823 As can be seen from

0 20 40 60 80 100 120 140

15

20

25

30

35

40

45

FM

PP

PM

T (K)

r s(au)

(rTc Tc)

rc

Figure 823 Collinear phase diagram of the HEG for the FT-DFT-RPA functional The thick dashedred line denotes first-order phase transitions TF is too big to fit in the graph

102135

Applications

these results the inclusion of the DFT-RPA contribution leads to a decrease of the ferromagneticphase The shape of the phase diagram stays the same as for the Hartree-Fock functional Itexhibits both characteristic phase transitions a first-order and a second-order one Because the1RDM is rdquouncorrelatedrdquo at zero temperature the momentum distribution will be the same as inthe FT-MBPT treatment namely a step function Therefore we recover the critical RPA Wigner-Seitz radius of rc asymp 18au The critical temperature Tc becomes asymp 120K which is an order ofmagnitude smaller than the Fermi temperature TF asymp 1300K of the corresponding critical radiusrTc asymp 21au When we compare the phase diagram from our FT-RDMFT treatment in Figure 823with the corresponding one from FT-MBPT in Figure 88 we see that the ferromagnetic phase isreduced This is in accordance with our investigation of the behaviour of the energy contributionsin Section 841 (see Figure 813) Because the ferromagnetic phase in FT-MBPT RPA exists at atemperature much smaller than TF the FT-RDMFT variation of the momentum distribution willlead to a favourisation of the paramagnetic configuration compared to the FT-MBPT treatment

This concludes our investigation of the effect of DFT correlation functionals in FT-RDMFT Aswe have seen the uncorrelated nature of the resulting eq-1RDM leads to several properties of theresulting phase diagrams which do not match our expectation from Section 81 These were eg theappearance of partially polarized phases above the Fermi temperature or critical temperatures farbelow the Fermi temperature We will now try to remedy these problems by considering correlationfunctionals from the zero-temperature version of RDMFT

863 Zero-temperature RDMFT correlation

Both DFT methods for the inclusion of correlation effects described so far exhibit severe shortcomingsand qualitatively wrong results with respect to the equilibrium momentum distributions and phasediagrams As a third option we therefore employ correlation functionals from zero-temperatureRDMFT These functionals by construction are independent of the temperature and thereforehave to be seen as first steps towards the development of truly temperature-dependent correlationfunctionals in the framework of FT-RDMFT

At first we will consider the α functional as defined in Eq (316) with f(ni nj) = (ninj)α

We already mentioned in Section 3 that because of its spin-channel seperability the α functional isintrinsically incapable of accurately reproducing the equilibrium free energies of both unpolarizedas well as polarized HEGs together However because it is a successful representative of zero-temperature xc functionals we investigate its finite-temperature performance hoping to gain someinsight into its behaviour at arbitrary temperatures including the zero-temperature case For ourdiscretized 1RDM the xc functional contribution becomes

Ωαxc[nσi] =

sum

σij

nασinασjKij (880)

with Kij given by Eq (852) Repeating some of our statements from Section 3 α = 1 reproducesthe first-order approximation neglecting correlation completely Choosing α = 05 corresponds tothe Muller functional [70 71] which at zero temperature is known to overestimate the correlationcontribution in spin-unpolarized situations (see Figure 31) We calculated the phase diagram forseveral values of α and show the values for rc Tc r

Tc in Table 83 α = 055 was shown in [31] to

reproduce the correlation energy of the HEG for 02 lt rs lt 90 quite accuratelyTo understand the effect of the α functional on the phase diagram it is instructive to review

the important aspects of the first-order functional mentioned at the end of the previous section Adecreasing α now has two main effects

First of all because 0 le ni le 1 and α lt 1

Ωαxc[γ] le Ωx[γ] (881)

Therefore the α functional yields a bigger contribution to the free energy than the first-order func-tional

103135

Applications

α 1 9 8 7 6 55

rc(au) 556 545 539 528 470 asymp5Tc(K) 10500 9400 8500 7200 6500 4600

rTc (au) 710 725 775 825 900 950

Table 83 Critical temperatures and densities for various values of α The energy differences decreasewith decreasing α For α = 55 we could not get an accurate estimate for rc within our energyresolution

0 05 1 15 2 250

02

04

06

08

1

kkf

nk

HF T = 0HFα = 08α = 0565ni

Figure 824 Equilibrium momentum distributions for a paramagnetic configuration at rs = 10auThe solid black line denotes the Hartree-Fock (or α = 1) solution at zero temperature The othergraphs show the equilibrium momentum distributions for T = 5000K for different values of α Thenoninteracting solution is given by the red 2-dash-1-dot line

The second effect is due to the nonanalyticity of Ωαxc for vanishing ONs As explained in Section 3

this leads to an occupation of all ONs and therefore to a smoothing of the momentum distribution ofthe eq-1RDM We show two representative distributions in Figure 824 Such a smoother momentumdistribution leads to a bigger entropy contribution which in turn favours the paramagnetic phase

These two effects compete and from our results we see that for low temperatures the first onedominates leading to a decrease of the critical density rc as shown in Table 83 With increasingtemperature the second effect finally overcomes the first effect and therefore with decreasing α thecritical temperature Tc and the critial Wigner-Seitz radius rTc increase

When considering PSS states it was shown in [139] that an increase of α leads to a decrease ofthe amplitudes of the PSS states at zero temperature In accordance with these results we couldnot observe any PSS phase for α lt 08 Furthermore as mentioned before the smoothing of themomentum distribution due to the decrease of α increases the effect of temperature when comparedto the first-order FT-RDMFT functional A decrease of α therefore leads to a reduction of the criticaltemperature of the PSS phase The values of TPSS

c for some values of α are given in Table 84From these results we see that the α functional is capable of capturing some effects of correlation

α 1 095 09 085

TPSSc (K) 4500 3000 2000 500

Table 84 TPSSc for various values of α

104135

Applications

0 1000 2000 3000 4000 5000

10

20

30

40

50

60

70

80

FM

PP

PM

T (K)

r s(au)

TF

(rTc Tc)

rc

Figure 825 Collinear phase diagram of the HEG for the BOW-TIE functional FM ferromagneticphase PM paramagnetic phase PP partially polarized phase The thick dashed red line denotes afirst-order phase transition The dotted blue line denotes the Fermi temperature TF

also at finite temperature like the broadening of the momentum distribution and the reductionof the ferromagnetic phase It fails however to increase rc which is known to be one order ofmagnitude higher than the Hartree-Fock result Making the parameter α polarization-dependentdoes not manage to increase rc As we have seen in Section 3 the choice of α(ξ) which reproducesthe correlation energies best fails to reproduce a magnetic phase transition Therefore as a secondexample of a zero-temperature RDMFT correlation functional we are now going to investigate theBOW functional as defined in Section 33 To recover the magnetic phase transition we use theTIE-modification from Eq (330) which yields rc asymp 30au and reproduces the correlation energiesover the whole range of densities and polarizations quite accurately The resulting phase diagramfor collinear spin configuration is shown in Figure 825 It reproduces most the features expected(see Table 81 and Figure 82)

The investigation of PSS states in the BOW functional cannot be easily achieved by a changeof the prefactors in the exchange contribution This can be understood from the following Let usrecapitulate the exchange contribution to the free energy of Eq (861) We now want to include anadditional coupling between the uarruarr and darrdarr contributions of the 1RDM We model this additionalterm as

Ω+c [γ] = minus

1

2

intd3rd3rprimew(r minus rprime)(γuarruarr(r r

prime)γdarrdarr(rprime r) + γdarrdarr(r r

prime)γuarruarr(rprime r)) (882)

Using the SDW-1RDM as defined in Eqs (860)-(860) one gets

Ω+c [γ] = minus

1

2n

intd3k1(2π)3

d3k2(2π)3

4π

|k1 minus k2 minus q|2(n1k1

cos2(Θk1

2

)+ n2k1

sin2(Θk1

2

))(n2k2

cos2(Θk2

2

)+ n1k2

sin2(Θk2

2

))(883)

We see that in contrast to the exchange contribution this correlation functional explicitly dependson the PSS wavevector q Carrying out the product in the integrand one will then acquire fourproducts of two ONs These could then be replaced by f(ninj) as presented in Section 311 to

105135

Applications

model correlation effects The implementation and testing of these functionals is left for furtherstudies and will not be dealt with in this work

As a final approach to the problem of describing correlation effects in FT-RDMFT we will nowconsider the perturbative methodology as derived in Section 6 We will show that a straightforwardinclusion of a given subset diagrams is fairly complicated and could lead to a variational collapse

864 FT-RDMFT perturbation expansion

Whereas the contributions ΩkΩext S0 and Ωx depend only on the momentum distribution directlyΩ2b and Ωr (and all higher-order diagrams which exhibit more independent momenta than noninter-acting Greenrsquos functions) also depend explicitly on the Kohn-Sham energies This becomes a problemin FT-RDMFT as we will show in the following Using the fundamental one-to-one correspondencenharr vKS we see that n determines vKS which determines ε(k) Hence the ε(k) are functionals of nand Ωr can be viewed as an implicit functional of ε(k) To get a rough idea of the behaviour of thisfunctional we parametrize the Kohn-Sham energies in the following way

ε(mk) =k2

2m(884)

m is the variational parameter and can be interpreted as an effective mass We note that the fugacityshows a simple dependence on m namely α(tm) = α(mt) and the momentum distribution becomes

n(tmk) =1

1 + ek2

mtminusα(mt)

= n(mt 1k) (885)

The general effect of m on the momentum distribution is then that a decrease of m leads to asmoother momentum distribution We are therefore able to investigate qualitatively the behaviourof FT-RDMFT functionals under a change in n(k) The contributions up to first-order show thefollowing dependence on m

Ωk(tm) = Ωk(mt 1) (886)

Ωext(tm) = Ωext(mt 1) (887)

N(tm) = N(mt 1) (888)

S0(tm) = S0(mt 1) (889)

Ωx(tm) = Ωx(mt 1) (890)

F (tm) = Ek(mt 1) + Ωext(mt 1)minus 1βS0(mt 1) + Ωx(mt 1) (891)

It has to be noted that because of the temperature prefactor of S0 F (tm) 6= F (mt 1) The second-order Born diagram as well as the RPA contribution shows different behaviour because of theexplicit dependence on ε(mk)

Ω2bσ(tm) = mΩ2bσ(mt 1) (892)

χσ(tm q νa) = mχσ(mt 1 q νa) (893)

This choice of diagrams leads to severe problems as we will show in the following We assumet ltlt 1m gtgt 1 andmt ltlt 1 A small change inm will therefore change the momentum distributiononly negligibly and all contributions up to first-order in the interaction will not change strongly Wecan see from Figure 86 that Ωr then decreases for increasingm Ωr therefore favours a more ldquowashed-outrdquo momentum distribution Considering now the limit t rarr 0 one can choose m arbitrarily bigleading to a divergence of Ωr An inclusion of Ωr only in addition to the first-order functional willtherefore lead to a variational collapse for t rarr 0 We will now investigate the behaviour of Ω2b inour parametrization With the same restrictions on m and t as before we see from Figure 84 that

106135

Applications

Ω2b increases with increasing m and therefore favours a less ldquowashed-outrdquo momentum distributionΩ2b and Ωr now compete in a variational minimization wrt m and the problem of variationalcollapse disappears for this choice of diagrams However our calculations show that the results showa qualitatively incorrect density dependence for trarr 0 This is due to the fact that Ω2b is independentof the density With increasing rs it therefore gains influence when compared to the Ωr-contributionThis then leads to a less ldquowashed-outrdquo momentum distribution which is in contrast to the expectedexact behaviour When considering high densities ie rs rarr 0 the situation is reversed Ωr isdominant over Ω2b which leads to a more ldquowashed-outrdquo momentum distribution Furthermore thedominance of Ωr will lead to a big value for m which in turn via Eq (893) leads to free energieswhich are orders of magnitude below the FT-MBPT results

A solution to the wrong behaviour for small densities could be a screening of the second-orderBorn contribution Ω2b We implemented an RPA screening of Ω2b and found no improvement forour results - a fact not too surprising considering that the screening vanishes for mrarr 0

WRPA(tm) =W

1minusmWχ(mt 1)(894)

The wrong behaviour for small rs could be corrected by a restriction on the variational parameterm since one knows that m = 1 reproduces the correct result for rs rarr 0 But this is against theideal of having a functional which is capable of reproducing the equilibrium properties of a systemby a free variation It would also not be clear how one would restrict the momentum distributionfor different parametrizations

Taking the previous considerations into account we come to the conclusion that an inclusion ofhigher-order diagrams in the methodology derived in Section 6 will most likely lead to ill-behavedfunctionals yielding wrong energies and momentum distributions This does not affect the validity ofthe results from Section 6 It shows only that the utilization of a subset of diagrams in a variationalscheme carries the danger of leading to a variational collapse This fact is to be attributed to thetotal freedom of choice for ε(k) by the inclusion of nonlocal potentials a fact also recently pointedout in the context of GW [140] As a general recipe for avoiding the problem of variational collapseone might model the perturbative expressions by approximations using the momentum distributionsalone We are going to investigate several first steps in this direction in the following

As we have seen before the divergence in Ωr stems from the fact that the eigenenergies εσ(k) ap-pear exlicitely in the definition of χσ A first guess to remedy this problem is to fix these eigenenergiesto the noninteracting values

εσ(k) = ε0σ(k) =k2

2(895)

We have implemented this correlation functional and found that it yields qualitatively incorrectresults The correlation functional in this approximation favours less washed out momentum distri-butions

This drawback lets us formulate a functional resting on exact properties of the polarization prop-agator As a first step in our approximation process we will model χ(q νa) as being frequency inde-pendent much in the spirit of the Coulomb hole plus screened exchange approximation (COHSEX)[141] Using the relation

partn(k)

partε(k)= minusβn(k)(1minus n(k)) (896)

the q rarr 0 limit of the kernel of χ(q 0) can be calculated up to second order to give

n(k + q)minus n(k)ε(k + q)minus ε(k) =

partn(k)

partε(k)

1 +

β(n(k)minus 12 )sum

ij qiqjpartε(k)partki

partε(k)partkjsum

i qipartε(k)partki

(897)

107135

Applications

We assume the ONs and eigenenergies to be point symmetrical around the origin ie n(k) = n(minusk)and ε(k) = ε(minusk) Accordingly the second term in Eq (897) vanishes in the integration over kThe polarization propagator in the limit q rarr 0 therefore becomes

χσ(q)qrarr0minusrarr minusβ

intd3k

(2π)3nσ(k)(1minus nσ(k)) +O(q2) (898)

We now need to describe the momentum dependence for big q We restate the main expressionfor the RPA correlation grand potential Eq (836)

Ωr =1

β(2π)minus2

intdqq2 ln (1minusW (q)χ(q)) +W (q)χ(q) (899)

It is now possible to estimate the behaviour of Ωr If W (q)χ(q) is big the logarithm in the kernel ofΩr becomes negligible The main object defining whether or not W (q)χ(q) is big or small is β(q2)We therefore split the integral in Eq (899) in two parts one where q lt q =

radicβ and one where

q gt q

Ωr asymp Ω0r +Ωinfin

r (8100)

Ω0r =

1

β(2π)minus2

int q

0

dqq2(W (q)χ(q)) (8101)

=4π

β(2π)minus2

int radicβ

0

dqχ(q)) (8102)

Ωinfinr = minus 1

β(2π)minus2

int infin

radicβ

dqq2(W (q)χ(q))2

2 (8103)

It can be deduced from Eq (8102) that in the limit T rarr 0 ie β rarr infin Ω0r diverges unless χ(q)

behaves like

χ(q)qrarrinfinminusrarrprop qminusκ (8104)

κ ge 2 (8105)

We incorporate the findings from the investigation of the exact limits of χ in the following KAPPA-approximation to the polarization propagator in Eq (899)

χκ[n(k)](q) =sum

σ

χκσ[n(k)](q) (8106)

χκσ[n(k)](q) = minus

β

1 + (qkF )κ

intd3k

(2π)3nσ(k)(1minus nσ(k)) (8107)

= minus β

1 + (qkF )κ(nσ minus 〈n2σ〉

) (8108)

where

〈n2σ〉 =int

d3k

(2π)3n2σ(k) (8109)

The prefactor of β ensures that Ωr stays nonvanishing in the limit β rarr infin The occurrence of(nσ minus 〈n2σ〉

)in this approximation shows that χκ

σ[n(k)](q) will favour a more washed out momentumdistribution

We calculated the correlation-energy in the zero-temperature limit for several values of κ andshow the results in Figure 826 The momentum distributions resulting from Ωκ

r [γ] for κ = 29 areshown in Figure 827

108135

Applications

01 1 10 100

0

minus002

minus004

minus006

minus008

minus010

minus012

rs(au)

e c(Ha)

PWCAκ = 29κ = 27κ = 25κ = 23

Figure 826 Correlation-energy of the three-dimensional electron gas for the KAPPA-functional forvarious values of κ The black line denotes Monte-Carlo results in the PWCA [52] parametrization

Unfortunately the KAPPA-functional becomes spin-channel seperable in the T rarr 0 limit As inour discussion of zero temperature RDMFT we can make the parameter polarization dependent todescribe the partially polarized HEG correctly (see Table 85) However for the same reasons as inthe previous discussions the descreasing parameter for decreasing polarization leads to a favourisationof the unpolarized phase in the low density limit and therefore to the absence of a phase transition

0

02

04

06

08

10

0 05 10 15 20 25 30

kkF

n(kk

F)

rs = 1 GZ

rs = 5 GZ

rs = 10 GZ

rs = 1 κ

rs = 5 κ

rs = 10 κ

Figure 827 Momentum distributions of the HEG for rs isin 1 5 10au from the parametrizationby Gori-Giorgi and Ziesche (GZ) [55] and from the KAPPA-functional for κ = 29

KAPPA-functional

ξ 00 01 02 03 04 05 06 07 08 09 10

κ 29 29 29 30 30 30 31 31 31 32 32

Table 85 Best parameters to describe the correlation-energy of a HEG in three dimensions in therange 1 lt rs lt 100 for the KAPPA-functional

109135

Applications

κ c

KAPPA-TIE 32 024

Table 86 Best parameters for the KAPPA-TIE-functional as defined in Eq (8107)

0 200 400 600 800 1000

10

20

30

40

50

60

70

80

01200

FM

PP

PM

T (K)

r s(au)

TF

(rTc Tc)

Figure 828 Collinear phase diagram of the HEG for the KAPPA-TIE functional There are nofirst-order phase transitions The dotted blue line denotes the Fermi temperature TF

We therefore propose a modification similar to the BOW-TIE functional named KAPPA-TIE as inEq (330) to recover a favourisation of the ferromagnetic phase for low densities

χκminusTIEσ [n(k)](q) = minusχκ

σ[n(k)](q)minus cβ

1 + q2k2F

intd3k

(2π)3nuarr(k)ndarr(k)(1minus nuarr(k)ndarr(k)) (8110)

The parameters which lead to an accurate description of correlation energies for arbitrary polarizationover the whole range of densities and a decrease in the critical density are shown in Table 86

We conclude our work by showing the phase diagram as resulting from the KAPPA-TIE functionalwith this choice of parameters in Figure 828 We see that the unphysical instantaneous quantumphase transition disappears and an increase in temperature slightly favours a partially polarizedconfiguration over a paramagnetic one In summary the KAPPA-TIE functional constitutes anintrinsically temperature dependent true FT-RDMFT functional which describes the correlation-energy of a HEG at zero temperature with arbitrary polarization accurately yields a qualtitativelyimproved momentum distribution as compared to previously used RDMFT functionals and leadsto a reasonable magnetic collinear phase diagram


In this final section of our work we investigated the phase diagram of the three-dimensional HEGBecause the exact phase diagram is not known we collected several properties we believe are prevalentin the exact phase diagram We then reviewed existing methods for the calculation of the free energyof a HEG in grand canonical equilibrium The first constituted a replacement of the Coulombinteraction by a model contact interaction which leads to a qualitatively correct phase diagram

We then employed FT-MBPT approximations We chose exchange-only exchange+RPA andexchange+RPA+second-order Born diagrams and investigated the resulting phase diagrams Both

110135

Applications

exchange-only and exchange+RPA phase diagrams exhibit the correct shape when compared to ourassumptions However both exhibit a critical Wigner-Seitz radius which is too low and additionallythe exchange+RPA phase diagram leads to a critical temperature far below the Fermi temperatureThe inclusion of the second-order Born diagram leads to several undesired properties of the phasediagram the reasons of which we discussed

After these applications of existing methods we turned to the description of the free energyvia our novel framework of FT-RDMFT We focussed on the first-order functional whose solutionwe had shown before to be equivalent to the FT-HF approximation The resulting phase diagramis similar to the exchange-only diagram from FT-MBPT exhibiting both the good overall shapeas well as the bad critical density However the FT-RDMFT treatment was able to remedy someintrinsic problems of the FT-MBPT method These were eg the equal smoothing of the momentumdistribution of the different spin channels for partially polarized configurations and the increase ofthe free energy for low temperatures

Besides the investigation of the collinear spin phase we also considered a phase exhibiting achiral spin symmetry which we termed PSS states We showed that in contrast to a conjecture byOverhauser the PSS states exhibit a first-order phase transition when the temperature is increasedThis phase transition could be attributed to an instantaneous jump in the optimal wavevector ofthe PSS state rather than to the reduction of the amplitude by temperature Interestingly wefound that at small temperatures the amplitude of the PSS state actually increases compared to thezero-temperature amplitude

The fact that the first-order FT-RDMFT functional is capable of reproducing the FT-HF solutionwas utilized to calculate the temperature dependence of the HF single particle dispersion relationWe found that for the three- and two-dimensional HEG in collinear spin configuration as well asfor the three-dimensional HEG in PSS configuration an increase in temperature first leads to avanishing of the characteristic ldquovalleysrdquo in the dispersion relations This might serve as an argumentwhy for medium temperatures a description via an effective mass might be justified A furtherincrease in temperature then lets the eigenenergies approach the noninteracting dispersion relationas expected

Crawling to the finish line we studied the effect of correlation effects in the framework ofFT-RDMFT We could show that an inclusion of correlation functionals from DFT will inevitablylead to uncorrelated eq-1RDMs The inclusion of the LSDA functional leads to several features ofthe phase diagram which are expected to be qualitatively wrong like the subsequent phase transi-tions from ferromagnetic to paramagnetic to partially polarized to paramagnetic The inclusion ofFT-DFT-RPA on the other hand reproduced the shape of the phase diagram correctly but under-estimated both the critical Wigner-Seitz radius as well as the critical temperature in comparison tothe Fermi temperature Subsequently we employed correlation functionals from zero-temperatureRDMFT The phase diagram resulting from the spin-separable α functional shows no improvementover the HF phase diagram It leads to a slight decrease of the critical Wigner-Seitz radius as wellas to a vanishing of the PSS phase To overcome these problems and to get a qualitatively correctphase diagram we then investigated the phenomenologically derived BOW-TIE functional from Sec-tion 33 By construction it leads to a critical Wigner-Seitz radius comparable to the exact oneAdditionally we find that the resulting phase diagram also exhibits most properties we expect fromthe exact one Finally we considered true temperature dependent functionals depending on the full1RDM As a backdraw we found that the inclusion of higher orders in a perturbative treatmentas derived in Section 6 is very complicated and will most likely lead to ill-behaved functionals Asa first remedy to this problem we proposed a method similar to the existing COHSEX approxi-mations relying on several exact properties of the polarization propagator The resulting KAPPAfunctional yields a reasonable collinear magnetic phase diagram and qualitatively correct momentumdistributions We summarize the agreement of the several approximations we investigated with theexact properties from Table 81 in Table 87

We had already seen in Section 33 that the BOW functional is also capable of a qualitatively

111135

Applications

(1) (2) (3) (4) (5) (6) rc

Stoner (Figure 83) X

FT-MBPT

x-only (Figure 85) X X

x+RPA (Figure 88) X X X

x+RPA+second Born (Figure 89) X X X

FT-RDMFT

first-order (FT-HF) (Figure 819) X X

x+DFT-LSDA (Figure 822) X

x+FT-DFT-RPA (Figure 823) X X X

BOW-TIE (Figure 825) X X X

KAPPA-TIE (Figure 828) X

Table 87 Agreement of phase diagrams for different approximations with the properties from Table81 An X in one of the first six columns denotes properties which are not fulfilled An X in column7 means that the critical density at zero temperature is not reproduced

correct description of the momentum distribution at zero temperature As a task for the futurewe therefore propose the implementation and testing of the BOW and BOW-TIE functionals aswell as of the KAPPA and KAPPA-TIE functionals for real physical systems Questions of greattheoretical interest concern then the dependence of the eq-symmetry on the temperature as well asthe temperature dependence of the quasi-particle spectrum in solids

112135

A APPENDIX

A1 Probabilistic interpretation of the ONs of the 1RDM

We are now going to investigate the ONs of a eq-1RDMwhich stems from a eq-SDO D =sum

i wi|ψi〉〈ψi|The 1RDM is given as

γ(x xprime) =sum

i

wi〈ψi|Ψ+(xprime)Ψ(x)|ψi〉 (A1)

=sum

i

niφlowasti (x

prime)φi(x) (A2)

The NOs φ can be used to build Slater determinants |χ〉 which form a basis of the Hilbert spaceWe can now expand |ψi〉 in terms of |χi〉

|ψi〉 =sum

l

cij |χj〉 (A3)

cij = 〈χi|ψj〉 (A4)

The SDO then becomes

D =sum

ijk

wiclowastijcik|χj〉〈χk| (A5)

Because we used the eigenfunctions of γ as basis for the Slater determinants the expansion of theproduct of the field operators ψ+(xprime)ψ(x) in terms of creation and annihilation operators (a+ a) willyield no offdiagonal contributions in the expectation value of Eq (A1) Using that |χi〉 are Slaterdeterminants Eq(A1) then becomes

γ(x xprime) =sum

i

sum

jk

wj |cjk|2〈χk|ni|χk〉

φlowasti (x

prime)φi(x) (A6)

This expression has a probabilistic interpretation if one identifies |cjk|2 with P (χk|ψj) ie theconditional probability of finding |χk〉 in a measurement if we know that the system is in the state|ψj〉 Additionally 〈χk|ni|χk〉 can be interpreted as P (φi|χk) the conditional probability that |φi〉is part of the Slater determinant |χk〉 And finally wj can be seen as the probability P (ψi) ie theprobability that |ψj〉 contributes to the eq-SDO This leads to the following representation of theeq-1RDM

γ(x xprime) =sum

i

sum

jk

P (ψj)P (χk|ψj)P (φi|χk)

φlowasti (x

prime)φi(x) (A7)

One can now use the result from probability theory that the probability of event A is given as a sumover the probability of event B times the conditional probability of A given B

P (A) =sum

B

P (A|B)P (B) (A8)

We then arrive at the following equation

γ(x xprime) =sum

i

P (φi)φlowasti (x

prime)φi(x) (A9)

Therefore ni = P (φα) can be interpreted as the probability that the actual state of the systemdescribed by D contains |φi〉 in an expansion of the form (A5) with the weights wi The situationbecomes simpler at zero temperature where the SDO is a single projection operator on the gs of thesystem ie wi = δi0

113135

Appendix

A2 Equilibrium ONs in general systems

As we have pointed out in Section 423 ONs of an eq-1RDM of a noninteracting system are givenby

ni =1

1 + eβ(εiminusmicro)(A10)

and will therefore be strictly between 0 and 1 We will now show that the latter is also true for theoccupation numbers of eq-1RDMs of arbitrary systems including interacting ones

We start from the spectral representation of the eq-1RDM

γ(x xprime) =sum

i

niφlowasti (x

prime)φi(x) (A11)

The occupation number operator ni is now defined as

ni = c+i ci (A12)

where c+i (ci) creates (annihilates) the natural orbital φiAn arbitrary occupation number of the eq-1RDM in grand canonical equilibrium can then be

written as

ni = TrDni =sum

e

we〈Ψe|ni|Ψe〉 (A13)

The Ψe are eigenfunctions of the Hamiltonian of the system and form a basis of the underlyingHilbert space Another basis is formed by the Slater determinants Φα which are constructed outof the natural orbitals φi of the eq-1RDM of the system The transformation between these basesis governed by the expansion coefficients ceα via

Ψe =sum

α

ceαΦα (A14)

Due to completeness and normalization of the Ψe and Φα the coefficients fulfill

sum

e

|ceα|2 =sum

α

|ceα|2 = 1 (A15)

Expanding the Ψe in Eq (A13) in terms of the Φα then leads to

ni =sum

e

we

sum

αβ

clowasteαceβ〈Φα|ni|Φβ〉 (A16)

Since the Slater determinants Φα are constructed to be eigenfunctions of the occupation numberoperator ni this reduces to

ni =sum

α

(sum

e

we|ceα|2)

︸︷︷︸fα

〈Φα|ni|Φα〉︸︷︷︸giα

(A17)

Using Eq (A15) and the properties of the thermal weights we gt 0 andsum

e we = 1 we see that

fα gt 0 (A18)sum

α

fα = 1 (A19)

114135

Appendix

The factors giα are equal to 1 if the natural orbital φi appears in the Slater determinant Φα and0 otherwise The summation over α corresponds to a summation over a basis of the Hilbert spacewhich is in the situation of a grand canonical ensemble the Fock space Therefore for a fixed i therewill be at least one α so that giα = 1 and at least one α for which giα = 0 Applying this resulttogether with Eqs (A18) and (A19) to Eq (A17) yields the desired inequality relations

0 lt ni lt 1 (A20)

A3 Zero-temperature mapping between potentials and wavefunctions

To investigate the nature of the mapping between the nonlocal external potential and the corre-sponding gs-wavefunction we will consider the relationship between the potential and the gs-1RDMand then use Gilbertrsquos theorem [26] to translate the results

Assume an arbitrary Hamiltonian H with gs-1RDM γgs

H = T + V + W (A21)

γgs(x xprime) =

sum

i

niφlowasti (x

prime)φi(x) (A22)

Due to Gilbertrsquos theorem the wavefunction can be formally written as a functional of the 1RDMand therefore an energy functional E[γ] can be defined as

E[γ] = 〈Ψ[γ]|H|Ψ[γ]〉 (A23)

From the variational principle this functional is minimal for the gs-1RDM γgs(x xprime) and therefore

the following relations hold

partE[γ]

partni

∣∣∣∣γgs

=

ai gt micro ni = 0

micro 0 lt ni lt 1

bi lt micro ni = 1

(A24)

where micro can be identified as the chemical potential of the system We will now be able to showthat the mapping between potential and gs-1RDM is one-to-one if and only if there are no pinnedoccupation numbers ie occupation numbers equal to 0 or 1 This will be done in two steps Atfirst we show that in the case of unpinned occupation numbers the external potential is uniquelydetermined up to a constant Secondly we will consider gs-1RDMs with pinned occupation numbersand show that one can explicitly construct infinitely many potentials which leave the gs-1RDMinvariantbull unpinned states

The absence of pinned states allows us to use the following Euler-Lagrange equation

δE[γ]

δγ(x xprime)

∣∣∣∣γgs

= microδN [γ]

δγ(x xprime)

∣∣∣∣γgs

= microδ(x xprime) (A25)

The addition of an arbitrary external potential contribution U [γ] =intdxdxprimeu(xprime x)γ(x xprime) to the

energy functional Eu[γ] = E[γ] + U [γ] then yields

δEu[γ]

δγ(x xprime)

∣∣∣∣γgs

= microδ(x xprime) + u(x xprime) (A26)

If we now claim that the minimizing 1RDM from E[γ] is also the optimal one for Eu[γ] we candeduce that the only possible choice of u(x xprime) which leaves the gs-1RDM invariant is

u(x xprime) = cδ(x xprime) (A27)

115135

Appendix

bull pinned statesAs in the case of pinned occupation numbers the minimum of E[γ] will be at the boundary of thedomain we cannot use Eq (A25) It would be possible to adjust the Euler-Lagrange equationby an incorporation of Kuhn-Tucker multipliers [94] but there is a simpler way as described in thefollowing

The fact that for pinned states the derivatives are allowed to be different from micro allows usto construct a one-particle potential which leaves the gs-1RDM invariant This potential shall begoverned by the generally nonlocal kernel u(x xprime) By choosing it to be diagonal in the basis ofnatural orbitals of the gs-1RDM we ensure that the orbitals will not change upon addition of thepotential For simplicity we choose only one component to be non-vanishing

u(x xprime) = uφlowastα(xprime)φα(x) (A28)

The new energy functional is then given by

Eα[γ] = E[γ] +

intdxdxprimeu(x xprime)γ(xprime x) (A29)

and the derivative with respect to the occupation numbers becomes

partEα[γ]

partni

∣∣∣∣γgs

=

ai + uδiα ni = 0

micro+ uδiα 0 lt ni lt 1

bi + uδiα ni = 1

(A30)

These considerations can now be employed to show the ambiguity of the external potential inRDMFT for groundstates with pinned occupation numbers For simplicity we will now assume thatthere is exactly one pinned occupation number with nβ = 0 We then construct a potential of theform of Eq (A28) with α = β From Eq (A30) we see that every choice of u gt micro minus aβ leads toa situation where the β-orbital exhibits a derivative bigger than micro leaving the gs-1RDM invariantFor one pinned ocuppation number nβ = 1 we can choose u lt microminus bβ which lets the derivative of theβ-orbital always surpass micro which again leads to the same gs-1RDM When considering gs-1RDMwith several pinned states these arguments are readily translated which proves the ambiguity of theone-particle potential for gs-1RDM with pinned occupation numbers

A4 Noninteracting grand canonical equilibrium

We are now going to investigate the grand canonical equilibrium of a noninteracting system Wewill show that one can express the ONs and NOs of the eq-1RDM easily via the eigenvalues andeigenfunctions of the noninteracting one-particle Hamiltonian Furthermore we will show that boththe grand potential Ω0 as well as the entropy S0 are simple functionals of the ONs of the eq-1RDMγ The eq-SDO D is given by

D =eminusβ(HminusmicroN)

Z=sum

i

wi|χi〉〈χi| (A31)

where the noninteracting one-particle Hamiltonian H(1) is given in second quantization as

H =

intdxdxprimeh(x xprime)Ψ+(xprime)Ψ(x) (A32)

H commutes with the 1RDM operator γ given by

γ =

intdxdxprimeγ(x xprime)Ψ+(xprime)Ψ(x) (A33)

116135

Appendix

from which we can deduce that h(x xprime) and γ(x xprime) have the same eigenfunctions The eq-1RDM γwill be denoted by

γ(x xprime) =sum

i

wi〈χi|Ψ+(xprime)Ψ(x)|χi〉 (A34)

=sum

i

niφlowasti (x

prime)φi(x) (A35)

We will now express the field operators Ψ(x) in the basis of NOs of γ

Ψ(x) =sum

i

ciφi(x) (A36)

Knowing that with this choice Ψ+(xprime)Ψ(x) will yield no offdiagonal contributions in γ we canreformulate Eq (A34)

γ(x xprime) = Tr

eminusβ(HminusmicroN)

ZΨ+(xprime)Ψ(x)

(A37)

=1

Z

sum

i

φlowasti (xprime)φi(x)

sum

n1n2

〈n1 n2 |eminusβsum

j nj(εjminusmicro)ni|n1 n2 〉 (A38)

=1

Z

sum

i


sum

n1n2

〈n1|eminusβn1(ε1minusmicro)|n1〉〈ni|eminusβnα(εiminusmicro)ni|ni〉 (A39)

=1

Z

sum

i


prod

j

sum

nj

〈nj |eminusβnj(εjminusmicro)|nj〉

〈ni|e

minusβni(εiminusmicro)ni|ni〉〈ni|eminusβni(εiminusmicro)|ni〉

(A40)

Carrying out the summation over nj one gets

γ(x xprime) =1

Z

sum

i


prod

j

(1 + eminusβ(εjminusmicro))

1

(1 + eβ(εiminusmicro)) (A41)

The partition function Z can now be expressed as

Z = Treminusβ(HminusmicroN)

(A42)

=sum

n1n2

〈n1 n2 |prod

i

eminusβni(εiminusmicro)|n1 n2 〉 (A43)

=prod

i

sum

ni

〈ni|eminusβni(εiminusmicro)|ni〉 (A44)

=prod

i

(1 + eminusβ(εiminusmicro)) (A45)

We see that the partition function exactly cancels one factor in Eq A41 yielding our final resultfor the eq-1RDM of a noninteracting system in grand canonical equilibrium

γ(x xprime) =sum

i

niφlowasti (x

prime)φi(x) (A46)

ni =1

1 + eβ(εiminusmicro) (A47)

117135

Appendix

This relation can simply be inverted to yield

εi minus micro =1

βln

(1minus nini

) (A48)

The grand potential is generally related to the partition function via

Ω = minus 1

βlnZ (A49)

Using Eqs (A45) and (A48) we arrive at

Ω0 = minus 1

β

sum

i

ln(1 + eminusβ(εiminusmicro)

)(A50)

=1

β

sum

i

ln(1minus ni) (A51)

The entropy can then be found by using the following thermodynamic relation

S =part

partTΩ

∣∣∣∣micro

(A52)

The derivative of the ONs with respect to the temperature is given as

part

partTni

∣∣∣∣micro

=1

Tni(1minus ni) ln

(1minus nini

)(A53)

which leads to our final result for the noninteracting entropy in terms of the ONs of the eq-1RDM

S0 = minussum

i

(ni lnni + (1minus ni) ln(1minus ni)) (A54)

A5 Feynman rules

The rules for the evaluation of the graphical contributions in Section 6 are the common Feynmanrules from FT-MBPT with an inclusion of the additional effective one-particle potential veff Theyare listed in the following

1 For fixed numbers n and m draw all topologically different connected diagrams with n inter-action m effective potential and 2n+ 1 +m particle lines

2 For every particle line going from point 2 to point 1 assign a noninteracting Greenrsquos functionG0(x1 τ1 x2 τ2)

3 For every two-particle interaction line going from point 2 to point 1 assign a factor w(x1 τ1 x2 τ2)

4 For every one-particle effective-potential line going from point 2 to point 1 assign a factorveff (x1 x2)

5 Integrate all internal variablesintdxint β

0dτ

6 Take the trace over spins

7 Multiply the diagram by (minus1)n(minus1)F where F is the number of closed fermion loops

8 Greenrsquos functions at equal times are evaluated as G0(x τ xprime τ) = G0(x τ xprime τ+)

118135

Appendix

A6 Second-order Born diagram

Starting from Eq (840) the second-order Born contribution to the free energy of a HEG becomes

Ω2b =1

n2minus10πminus9

intd3k

intd3p

intd3q

W (q)W (k + p+ q)nknp(1minus nk+q)(1minus np+q)

εk+q + εp+q minus εk minus εp(A55)

=1

n2minus6πminus7

intd3k

intd3p

intd3q

nknp(1minus nk+q)(1minus np+q)

q2(k + p+ q)2(q2 + q middot (k + p))(A56)

This can be reformulated with the help of the following variable transformations

krarr q (A57)

prarr minus(k + p+ q) (A58)

q rarr k (A59)

Ω2b = minus1

n2minus6πminus7

intd3k

intd3p

intd3q


k2p2k middot p(A60)

Changing to spherical coordinates we can integrate over θk φk and φp We will be left with thefollowing six-dimensional integral

Ω2b = minus1

n2minus2πminus5

int infin

0

dk

int infin

0

dp

int infin

0

dq

int π

0

dθp

int π

0

dθq

int π2

minusπ2

dφq

q2 tan(θp) sin(θq)

kpn(q)n(k + p + q)(1 minus n(k + q))(1 minus n(p + q)) (A61)

For numerical purposes it is now helpful to define the following function F (kp q θp) which issemipositive

F (kp q θp) =

int π

0

dθq sin(θq)(1minus n(k + q))

int π2

minusπ2

dφqn(k + p+ q)(1minus n(p+ q)) (A62)

To further simplify notation we introduce the function G(k p q) by the following equation

G(k p q) = q2fq

int π2

0

dθp tan(π2+ θp

)(F(k p q

π

2+ θq

)minus F

(k p q

π

2minus θq

)) (A63)

The previous definitions in combination with variable transformations of the kind k rarr ln k then leadto the following expression for Ω2b

Ω2b = minus1

n2minus2πminus5

int infin

minusinfindk

int infin

minusinfindp

int infin

0

dqG(ek ep q) (A64)

The momenta k p and q can now be interpreted as coordinates The integration therefore runs overhalf the space spanned by k p and q One can therefore introduce spherical coordinates for thisthree-dimensional space

k(x) = x cos(θx) (A65)

p(x) = x sin(θx) sin(φx) (A66)

q(x) = x sin(θx) cos(φx) (A67)

This then leads to our final representation of Ω2b which is suitable for numerical integration

Ω2b = minus1

n2minus2πminus5

int infin

0

dxx2int π

0

dθx sin(θx)

int π2

minusπ2

dφxG(ek(x) ep(x) q(x)) (A68)

119135

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C Deutsche Kurzfassung

Reduzierte-Dichtematrix-Funktionaltheorie (RDMFT) ist ein vielversprechender Kandidat fur dieBehandlung von Fragestellungen in der Quantenmechanik welche durch konventionelle Dichtefunk-tionaltheorie (DFT) nur schwer zu behandeln sind Dazu gehoren die Berechnung von Dissoziations-kurven von Molekulen sowie die Bestimmung der elektronischen Energielucke in Ubergangsmetall-Oxiden Bei der Beschreibung des homogenen Elektronengases welches eines der bedeutensten Mo-dellsysteme fur die Familie der Funktionaltheorien darstellt liefern bestehende Funktionale allerdingskeine akkuraten Ergebnisse In Teil 3 dieser Arbeit haben wir daher ein RDMFT-Funktional herge-leitet welches sowohl die Korrelationsenergie mit hoher Genauigkeit als auch die Grundzustands-Impulsverteilung qualitativ korrekt wiedergibt

Ein weiteres Gebiet auf dem DFT eher geringe Erfolge vorzuweisen hat ist die Beschreibung quan-tenmechanischer Eigenschaften von Vielteilchensystemen im thermodynamischen Gleichgewicht InTeil 4 der vorliegenden Arbeit untersuchen wir daher die Moglichkeit eine Erweiterung von RDMFTauf groszligkanonische Systeme vorzunehmen Ausgehend vom Variationsprinzip konnten wir zeigendass ein bijektiver Zusammenhang zwischen dem statistischen Dichteoperator (SDO) im thermischenGleichgewicht und der dazugehorigen 1-reduzierten Dichtematrix (1RDM) besteht Dies ermoglichtdie Bestimmung der Gleichgewichts-1RDM durch eine variationelle Funktionaltheorie (FT-RDMFT)Sowohl die SDOs als auch die 1RDMs konnen als Elemente eines Banachraumes verstanden werdenwas uns einerseits ermoglichte wichtige Eigenschaften der FT-RDMFT-Funktionale herzuleiten undandererseits den Defininitionsbereich dieser Funktionale auf eine einfach zu charakterisierende Mengevon 1RDMs zu erweitern

Um die Entwicklung von genaherten Funktionalen in FT-RDMFT zu erleichtern haben wir inTeil 5 dieser Arbeit Eigenschaften der exakten Funktionale untersucht Als besonders fruchtbarhat sich hierbei die Methode der Skalierung von Parametern erwiesen Dadurch war es uns moglichmehrere Beziehungen der verschiedenen Korrelationsbeitrage untereinander herzuleiten sowie exakteAussagen uber das Skalierungsverhalten der einzelnen Funktionale zu treffen Diese Resultate konnennun benutzt werden um bestehende genaherte Funktionale in FT-RDMFT zu testen oder um alsLeitlinien fur die Entwicklung neuer Funktionale zu dienen

Die Entwicklung einer lokalen Dichtenaherung (LDA) wie sie mit uberwaltigendem Erfolg inDFT angewandt wird ist in FT-RDMFT vorerst nicht moglich da keine Monte-Carlo Ergebnissefur thermodynamische Potentiale von Systemen mit nochtlokalem externen Potential in thermody-namischem Gleichgewicht existieren In Teil 6 der vorliegenden Arbeit entwickelten wir daher einestorungstheoretische Methodik zur Herleitung genaherter Funtionale Wir konnten zeigen dass dieMinimierung des resultierenden Funktionals erster Ordnung aquivalent zur Losung der temperatu-rabhangigen Hartree-Fock Gleichungen ist

Es gibt im Rahmen von FT-RDMFT im Gegensatz zur Grundzustands-RDMFT ein effektivesnicht wechselwirkendes System welches die Gleichgewichts-1RDM eines wechselwirkenden Systemsreproduziert Dies konnten wir in Teil 7 unserer Arbeit nutzen um eine selbstkonsistente Minimie-rungsmethode fur Funktionale aus sowohl FT-RDMFT als auch RDMFT zu entwickeln Durch dieAnwendung auf ein Beispielsystem konnten wir die Effektivitat und Effizienz dieser Methode belegen

Nachdem wir in den vorhergegangenen Teilen eine ausfuhrliche Behandlung der theoretischenGrundlagen von FT-RDMFT vorgenommen haben untersuchten wir in Teil 8 unserer Arbeit dieAuswirkung von FT-RDMFT auf das homogene Elektronengas So konnten wir das Phasendia-gramm fur das Funktional erster Ordnung bestimmen wobei wir sowohl kollineare als auch spi-ralformige Spin-Konfigurationen untersucht haben FT-RDMFT ermoglichte es uns weiterhin dieTemperaturabhangigkeit der zugehorigen Quasiteilchen-Dispersionsrelationen zu untersuchen Diesist von groszligem Interesse fur die Anwendung auf reale Vielteilchensysteme zB Festkorper Abschlie-szligend haben wir verschiedene Moglichkeiten der Behandlung von Korrelationseffekten in FT-RDMFTuntersucht Dabei war es uns moglich ausgehend von exakten Eigenschaften des Polarisationspro-pagators ein FT-RDMFT-Korrelationsfunktional abzuleiten welches zu einem qualitativ korrektenPhasendiagramm fuhrt

131135

132135

D Publications

1 Discontinuities of the Chemical Potential in Reduced Density Matrix Functional Theory

N N Lathiotakis S Sharma N Helbig J K Dewhurst M A L Marques F G Eich TBaldsiefen A Zacarias and E K U GrossZeitschrift fur Physikalische Chemie 224(03-04) p 467-480 (2010)httpdxdoiorg101524zpch20106118

2 Minimization procedure in reduced density matrix functional theory by means of an effective

noninteracting system

T Baldsiefen and E K U GrossComputational and Theoretical Chemistry in press (2012)httpdxdoiorg101016jcomptc201209001

3 Reduced density matrix functional theory at finite temperature I Theoretical foundations

T Baldsiefen and E K U GrossarXiv12084703 (2012) submitted to Phys Rev A

httparxivorgabs12084703

4 Reduced density matrix functional theory at finite temperature II Application to the electron

gas Exchange only

T Baldsiefen F G Eich and E K U GrossarXiv12084705 (2012) submitted to Phys Rev A


5 Reduced density matrix functional theory at finite temperature III Application to the electron

gas Correlation effects



6 Properties of exact functionals in finite-temperature reduced density matrix functional theory

T Baldsiefen and E K U Grosssubmitted to Phys Rev A

133135

134135

E Acknowledgements

ldquohoc loco qui propter animi voluptates coli dicunt ea studia non intellegunt idcirco

esse ea propter se expetenda quod nulla utilitate obiecta delectentur animi atque ipsa

scientia etiamsi incommodatura sit gaudeantrdquo

Cicero De Finibus Bonorum et Malorum Liber Quintus XIX 50

First of all I would like to thank Prof Gross for giving me the opportunity to follow my scientificinterest in a quite unconstrained way Although this might not always have been the most productivepath it surely was a very challenging and satisfying one

I thank all members of the committee for the time they invested in reviewing my work especi-ally Prof Schotte who accepted the burden of being my second referee

Although their fields of research have been quite disjunct from the ones dealt with in this workseveral members of the Gross workgroup took some real interest and spent considerable amounts oftime in fruitful discussions In this context I especially thank F G Eich for his interest and input

As this work was done in both Berlin and Halle(Saale) I would like to thank all members of thestaff at the physics institute of the Freie Universitat Berlin and the Max-Planck Institute fur Mi-krostrukturphysik in Halle(Saale) Without their efforts there would hardly be any time left forresearch

I would also like to thank Ms Muriel Hopwood for proofreading my thesis

Finally I would like to point out my sincere gratitude to my family and friends whose supportwas a great help in this sometimes daunting adventure

Tim

135135

Title

Abstract

Table of Contents

List of Figures

List of Tables


Introduction

Mathematical prerequisites

Hamiltonian and spin notation

Statistical density operator (SDO)

1-reduced density matrix (1RDM)

Homogeneous electron gas (HEG)

Zero-temperature RDMFT

Theoretical foundations

Exchange-correlation functionals

The problem of the construction of an RDMFT-LDA

BOW functional

The problem of the description of magnetic systems

Summary and outlook

Finite-temperature RDMFT (FT-RDMFT)

Physical situation and standard quantum mechanical treatment

Properties of thermodynamic variables

Mathematical foundation of FT-RDMFT

Existence of grand potential functional Omega[gamma]

Lieb formulation of the universal functional

Kohn-Sham system for FT-RDMFT

Functionals in FT-RDMFT

Finite-temperature Hartree-Fock (FT-HF)

Properties of the universal functional

Existence of minimum

Lower semicontinuity

Convexity

Eq-V-representability

Relation to microcanonical and canonical formulations

Summary and outlook

Properties of the exact FT-RDMFT functionals

Negativity of the correlation functionals

Adiabatic connection formula

Uniform coordiante scaling

Temperature and interaction scaling

Summary and outlook

Constructing approximate functionals

Why not use standard FT-MBPT

Methodology of modified perturbation theory

Elimination of V_eff

Importance of eq-V-representability

Scaling behaviour

Summary and outlook

Numerical treatment

Key idea of self-consistent minimization

Effective Hamiltonian

Temperature tensor

Small step investigation

Occupation number (ON) contribution

Natural orbital (NO) contribution

Convergence measures

Sample calculations

Occupation number (ON) minimization

Full minimization

Summary and outlook

Applications

Magnetic phase transitions in the HEG

Homogeneous electron gas with contact interaction

FT-MBPT

Exchange-only

Exchange + RPA

Exchange + RPA + second-order Born

FT-RDMFT

Collinear spin configuration

Planar spin spirals (PSS)

FT-HF dispersion relations

Correlation in FT-RDMFT

DFT-LSDA correlation

FT-DFT-RPA correlation

Zero-temperature RDMFT correlation

FT-RDMFT perturbation expansion

Summary and outlook

APPENDIX

Probabilistic interpretation of the ONs of the 1RDM

Equilibrium ONs in general systems

Zero-temperature mapping between potentials and wavefunctions

Noninteracting grand canonical equilibrium

Feynman rules

Second-order Born diagram

Bibliography

Deutsche Kurzfassung

Publications

Acknowledgements

Erstgutacher Prof Dr E K U Gross

Zweitgutachter Prof Dr K Schotte

Datum der Disputation 31102012




Berlin den 10102012

Abstract






Table of Contents

List of Figures v

List of Tables vii


1 Introduction 1










i

TABLE OF CONTENTS














B Bibliography 121


D Publications 133

ii

TABLE OF CONTENTS


iii

List of Figures









v

LIST OF FIGURES


vi

List of Tables







vii













EQ Equilibrium

FM Ferromagnetic







GS Ground state



HF Hartree-Fock

HS Hilbert-Schmidt







ix


MC Monte-Carlo


NO Natural orbitals



PM Paramagnetic







SC Self-consistent




SW Spin waves



x

1 Introduction







1135

1 INTRODUCTION











2135

1 INTRODUCTION




3135

4135



Section Description


2223






H = T + W + Vext (21)


intdx =

sumσ



T =

intdx lim

xprimerarrx

(minusnabla

2

2


Vext =



W =



w(x xprime) =1




D =sum


i

wi = 1 (26)



(27)

5135

Prerequisites

T

T1

DN


T

TradicN

ΓN




N =




(29)


AHS =

radicTrA+A (210)


T1 =A | AHS le 1

(211)






γ(x xprime) =sum

i

niφlowasti (x

prime)φi(x) (213)

6135

Prerequisites


Φi(r) =

(φi1(r)

φi2(r)

) (214)


γ(r rprime) =sum

i

niΦdaggeri (r


=

(γ11(r r



prime)

)=sum

i

ni

(φlowasti1(r



prime)φi2(r)

)(216)


Φi1(r) =

(φi1(r)

0

)Φi2(r) =

(0

φi2(r)

)(217)



sum

i

niσφlowastiσ(r





sumi niφ

lowasti (x



i

ni = N (220)





(222)

7135

Prerequisites




radicN

TradicN =

A | AHS le

radicN

(223)





n(x) = γ(x x) (224)


nσ(r) = γσσ(r r) (225)





rs =

(3

4πn

)13

(226)

8135

Prerequisites


kF =(3π2n

) 13 =

(9

4π

) 13

rminus1s (227)

εF =k2F2

(228)

TF =εFkB

(229)



(230)









Enuc [n(r)] =



Vext σσprime =



9135

10135




Section Description








D = |Ψgs〉〈Ψgs| (31)


E = Ek[γ] + Vext[γ] +W [Γ(2)] (32)

Ek[γ] =

intdx lim

xprimerarrx

(minusnabla

2

2

)γ(xprime x) (33)

Vext[γ] =



W [Γ(2)] =




2TrDΨ+(x4)Ψ

+(x3)Ψ+(x1)Ψ

+(x2) (36)

11135

ZT-RDMFT


γ(x xprime) =2

N minus 1

intdx2Γ

(2)(x x2xprime x2) (37)



W [Γ(2)] =W [γ] (38)





prime) (310)






12135

ZT-RDMFT




W [γ] = EH [γ] + Ex[γ] + Ec[γ] (311)

EH [γ] =1

2


Ex[γ] = minus1

2








13135

ZT-RDMFT


Exc[γ] = Ex[γ] + Ec[γ] (315)

Exc[γ] = minus1

2

sum

ij

f(ni nj)



prime) (316)











14135

ZT-RDMFT



intd3RεHEG

c [γ(kR)] (317)



γ(kR) =















15135

ZT-RDMFT

10010101001

minus030

minus020

minus010

0

minus005

minus015

minus025

rs(au)

e c(Ha)


MLα (α = 055)




33 BOW functional




16135

ZT-RDMFT

0

02

04

06

08

10

0 02 04 06 08 10

ninj

f

fHF






2

sum

ij

fBOW (ni nj α)



prime) (319)








17135

ZT-RDMFT

00 05 10 15 2000

02

04

06

08

10

kkF

n(kk

F)

rs = 05

rs = 1

GZBOW

GZBOW

00 05 10 15 2000

02

04

06

08

10

kkF

n(kk

F)

rs = 5

rs = 10

GZBOW

GZBOW


10010101001

minus08

minus07

minus06

minus05

minus04

minus03

minus02

minus01

0


MLα (α = 06)

rs(au)

e c(Ha)




18135

ZT-RDMFT










E(nuarr ndarr) =1







19135

ZT-RDMFT

001 01 1 10 100minus010

minus008

minus006

minus004

minus002

0

rs(au)

∆F(Ha)

ξ00204060810


001 01 1 10 100

0

002

004

006

008

010

rs(au)

∆P(Ha)

ξ00204060810


0

01

02

03

04

001 01 1 10 100rs(au)

δF

00204

060810


minus08

minus06

minus04

minus02

0

001 01 1 10 100rs(au)

δP

00204

060810





δF =∆F

Wc(328)

δP =∆P

Wc (329)



20135

ZT-RDMFT

3D

ξ 00 01 02 03 04 05 06 07 08 09 10

α 056 056 056 057 057 057 058 059 061 063 066

BOW 061 061 061 062 062 063 063 064 065 067 069

2D

ξ 00 01 02 03 04 05 06 07 08 09 10

α 063 063 063 063 064 064 065 066 067 07 074

BOW 066 066 066 067 067 068 069 071 073 076 080





2

sum

ijσ


csum

ij

((niuarrnjdarr)

αU



21135

ZT-RDMFT

αP αU c

BOW-TIE 070 20 019









22135




Section Description







23135

FT-RDMFT


H


∆Nmicro

∆E β



H =

infinoplus

n=0

Shotimesn (41)










24135

FT-RDMFT

F [D]

D

downwards convex

upwards convex

42a Convexity

F [D]

D








γ(x xprime) = Tr

eminusβ(T+V+W )

ZΨ+(xprime)Ψ(x)

(48)



(49)



T (see Section 23)





25135

FT-RDMFT





F [Dk] (412)













S[A B] = supλ

((1λ)


))(419)




S[Ak Bk] (420)

26135

FT-RDMFT


TrA = supP

TrP A (421)

TrP A le PTrA (422)






S[A B] = supP λ

((1λ)tr


))(425)


S[Ak Bk] (426)





Ω[Dk] (427)


S[Dk] (428)






27135

FT-RDMFT

Deq

Deq

Deq

γeq

γeq neq

neq

neq

+ =rArr










U =


βlnZ

Z prime (430)





minus 1

βlnZ

Z prime = u11 = u22 = u11 + u22 (431)



28135

FT-RDMFT







eq) (433)


leads to



eq) (435)
















Ω[γeq] =







Ωeq = minDisinDN

Ω[D] (443)

29135

FT-RDMFT


Ωeq = minγisinΓN

infDisinDNrarrγ

Ω[D[γ]] (444)


Ωeq = minγisinΓN

(F[γ] +





TrD(T + W + 1β ln D) (446)


Ω[γ] = F[γ] +




Ωeq = minγisinγN

Ω[γ] (448)



δF [γ]

δγ(xprime x)

∣∣∣∣γeq




30135

FT-RDMFT





H(1)s =

sum

i

εi|φi〉〈φi| (450)


γeq(x xprime) =

sum

i

niφlowasti (x

prime)φi(x) (451)


ni =1



εi minus micro =1

βln

(1minus nini

) (453)



veff (x xprime) =

sum

ij






31135

FT-RDMFT


Ωk[γ] =

intdxprime lim

xrarrxprime

(minusnabla

2

2

)γ(x xprime) (456)

N [γ] =

intdxγ(x x) (457)

Ωext[γ] =



ΩH [γ] =1

2


Ωx[γ] = minus1

2


S0[γ] = minussum

i







Ω[γ] =

intdxprime lim

xrarrxprime

(minusnabla

2

2

)γ(x xprime) +

intdxvext(x x


1

2


2


1βsum

i



δΩ[γ]



0 =

intdxprimeφi(x

prime)δΩ[γ]

δγ(xprime x)(465)

=

(minusnabla

2

2

)φi(x) +

intdxprimevext(x x



prime)+



32135

FT-RDMFT





Ω[γ] = F [γ] +
















L sub DN )]


F [γ] = TrD(H0 + 1β ln D) (470)


33135

FT-RDMFT


limkrarrinfin

TrDk(H0 + 1β ln Dk) = F [γ] (471)












Mf =









34135

FT-RDMFT



infinsum

i=M+1









TrD(H0 + 1β ln D) = F [γ] (479)






F [γk] (480)


F [γ] le TrD(H0 + 1β ln D) (481)


F [γk] (482)

35135

FT-RDMFT

433 Convexity









infγisinΓN

(F [γ] +

intv0γ

)= F [γ0] +

intv0γ0 (484)






36135

FT-RDMFT








F [γ] +

intvγ ge F [γ0] +

intvγ0 (487)

infγisinΓN

(F [γ] +

intvγ

)ge F [γ0] +

intvγ0 (488)



F [γ0] +

intv0γ0 = inf

γisinΓN

(F [γ] +

intv0γ

)(489)






37135

FT-RDMFT





intv0γ



1 γk rarr γ




∆0 = F [γ0] +

intv0γ0 minus inf

γisinΓN

F [γ] +

intv0γ

(492)








38135

FT-RDMFT




Dc =sum

α


α

wαN = 1 (495)


F [D] = TrD(H + 1β ln D) (496)


Dceq =

eminusβH

TreminusβH (497)



TrD(T + W + 1β ln D) (498)


F [γ] = Fc[γ] + Vext[γ] (499)



F [γ] = Ω[γ] + microN [γ] (4100)

39135

FT-RDMFT

T

T1

DN

DV

DNL


T

TradicN

ΓN

ΓV








DNL




40135

FT-RDMFT







1minus1larrrarr ΓN




solved by

leads to

solved by

leads to

solved by

enables

used for

allows


41135

42135



Section Description









Ω0[D0] lt Ω0[Dw] (52)








43135

Exact properties


Sc[γ] lt 0


Ωc[γ] lt 0

(58)

(59)

(510)




Hλ = T + V0 + λW (511)




)(512)

= minDrarrγ


)minus Tr

DV λ[γ]

) (513)





)minus Tr

D[γ]V λ[γ]

(514)


dΩλ[γ]

dλ= Tr

Dλ[γ]

(W +

d

dλV λ[γ]

)minus d

dλTrD[γ]V λ[γ]

(515)


Ω[γ] = Ω0[γ] +

int 1

0

dλTrDλ[γ]W

(516)


Ωc[γ] =

int 1

0

dλ

λWλ

c [γ] (517)



44135

Exact properties

minus4 minus2 2 4

minus02

minus01

01

02

z

Ψ210(0 0 z)

51a Wavefunction

minus4 minus2 2 4

001

002

003

004

0z

ρ210(0 0 z)

51b Density







ψλ(r1 rN ) = 〈r1 rN |U(λ)|ψ〉 (518)

= λ3N2ψ(λr1 λrN ) (519)


45135

Exact properties


〈ψλ|Oλ|ψλ〉 = 〈ψ|O|ψ〉 (520)


Tλ =1

λ2T (521)

Wλ =1

λW (522)

Nλ = N (523)


T [γλ] = λ2Ωk[γ] (524)

ΩH [γλ] = λΩH [γ] (525)

Ωx[γλ] = λΩx[γ] (526)

S0[γλ] = S0[γ] (527)





get


λ[γλ]) (529)

TrD 1λ[γλ](W 1

λ+ τ ln D 1


λ+ τ ln D[γ]) (530)











(535)

(536)

46135

Exact properties





(537)

(538)

(539)


0 =d

dλ

(Ω[D 1

λ[γλ]]

)∣∣∣λ=1

(540)


0 =d

dλTr

D 1

λ[γλ]

(1

λW 1

λ+ τ ln(D 1

λ[γλ]

)∣∣∣∣λ=1

(541)


W [γ] =d



Wc[γλ] = λd

dλΩc[γλ] (543)


Ωc[γλ] =

int λ

0

dmicro



Sc[γλ] = λ2(part

partλ

1

λΩc[γλ]

)(545)


d

dλτSc[γλ] = λ

d

dλ

(1

λWc[γλ]

) (546)


d

dλΩc[γλ] =

1

λWc[γλ] lt 0

d

dλWc[γλ] lt

d

dλΩc[γλ] lt 0

d

dλSc[γλ] lt 0

(547)

(548)

(549)

47135

Exact properties

Ωc[γλ]

λ1

minusτSc[γ]

0


Ωc[γλ]




d2


1

λ2Wc[γλ] +

1

λ

d


1

λ2Wc[γλ] +

1

λ

d

dλΩc[γλ] (550)

d2








Dλ(τ w)[γ] = D(λ2τ λw)[γλ] (553)

48135

Exact properties




λ2Sc(τλ

2 wλ)[γ]


(554)

(555)

(556)


Ωc(τ w)[γ] =

int 1

0

dλ

λWc(τ λw)[γ] (557)

Ωc(τ w)[γ] =

int 1

0








49135

50135



Section Description










prime) =sumi εiφ

lowasti (x



i

φi(x)φlowasti (x


ni ν gt νprime




i

niφlowasti (x

prime)φi(x) (62)

51135


W + Veff




ni =1


εi =1

βln

(1minus nini

)+ micro (64)




52135




Ω[γ] = Ω0[γ] +

int 1

0

dλTrDλ[γ]W

(65)




〈λW 〉λ =1

2

intdxdxprime lim

νprimerarrν+





〈λW 〉λ =1

2

intdxdxprime lim

νprimerarrν+




1

2

intd3xd3xprime


)γ(x xprime) (68)






〈λW 〉unifλ =1

2

Σlowast minus

(610)


53135


λw(r rprime)

vλ(x xprime)






veff = v(1)eff + v

(2)eff + (611)



0

0+

=

0

0+

+

0

0+

Σlowast (612)


0 =

0

0+

Σlowast (613)

54135


+ + + +

+ + + + + +

+ + + +

+ + + + + +



(1)

0

0+

= minus

0

0+

minus

0+

0

(614)

55135


intdydyprime

int β

0


= minusintdydyprime

int β

0


int β

0



v(1)eff (x x



= minus minus (617)



potential v(2)eff

(2)

0

0+

= minus

0+

0

minus

0+

0

(618)


(2)

0

0+

= minus

0

0+

M (619)


intdydyprime

int β

0


= minusintdydyprime

int β

0


56135



v(2)eff =

sumij φi(x)φ

lowastj (x

prime)int β



int β

0eν


where

Mij(ν νprime) =








M =

sumij φi(x)φ

lowastj (x

prime)int β



int β

0eν








Ω(1)xc [γ] = + (625)


Ωxc[γ] =1

2


2


57135



Ω(2)xc [γ] = + + + (627)







Ω[γ]

γΓV ΓNΓV ΓV

(γV EV

) (γN EN

)


Ω[γ]

γΓN ΓV

(γV EV

) (γN EN

)



58135






prime) (628)




bull 2n+1 G0 lines






c [γ] (631)



c [γ] = nΩ(n)c [γλ] (632)


c [γ] = (nminus 1)Ω(n)c [γλ] (633)









part




part


59135





prime) (640)






c (τ w)[γ] (643)



c (τ w)[γ] (644)





60135



Section Description










61135

Numerical treatment

γ

F [γ]

γ0γ1γ2γ3

Ω(0)effΩ

(1)effΩ

(2)eff

Ω








62135

Numerical treatment






δΩeff [γ]

δγ(x xprime)=

δΩ[γ]

δγ(x xprime) (73)




(1

β0minus 1

β


where


δΩHxc[γ]

δγ(x xprime)(75)


δγ(x xprime) (76)


δnkδγ(xprime x)


δφk(y)

δγ(xprime x)=sum

l 6=k



δφlowastk(y)

δγ(xprime x)=sum

l 6=k




gij =




heff ij = δij

(partΩ[γ]

partni+ micro0 +

σiβ0

)+


intdy

(δΩ[γ]

δφi(y)φj(y)minus

δΩ[γ]


) (711)


σi =partS0[γ]

partni= ln

(1minus nini

) (712)

63135

Numerical treatment









Ω[n1 n2] =α1

2(n1 minus 05)2 +

α2

2(n2 minus 05)2 (713)

= Ω1[n1] + Ω2[n2] (714)









64135

Numerical treatment

0

2

4

6

8

10

0 02 04 06 08 1

n1

Ω[n

1]

Ω1[n1]Ωeff1[n1]


0

005

010

015

020

0 02 04 06 08 1

n2

Ω[n

2]

Ω2[n2]Ωeff2[n2]




ni =1


εi minus micro =1

βiln

(1minus nini

) (720)


Ω[γ] =sum

i


1


)(721)

=sum

i

Ωi[ni βi] (722)


βi = ηpart2S0[γ]

partn2i

part2Ω[γ]partn2i

(723)

= η1

ni(1minus ni)

(part2Ω[γ]

partn2i

)minus1

(724)


βi =η

αi

1

ni(1minus ni) (725)


65135

Numerical treatment

0

2

4

6

8

10

0 02 04 06 08 1

n1

Ω[n

1]

Ω1[n1]

Ωeff1[n1]


0

005

010

015

020

0 02 04 06 08 1

n2

Ω[n

2]

Ω2[n2]

Ωeff2[n2]








j 6=i

heff ji



nprimei =1


66135

Numerical treatment


i n(k)i φ


2) fromn(k)i

and

φ(k)i

calculate H

(k)eff by Eq (711)


ε(k+1)i

and

φ(k+1)i (x)

4) constructn(k+1)i


(k+1)0 so that

sumi n

(k+1)i = N

5) fromφ(k+1)i (x)

and

n(k+1)i





nprimei =ni


(partΩpartni

minus∆micro

) (729)



= βini(ni minus 1)

(partΩ[γ]

partniminus∆(0)

micro

) (731)




∆(0)micro =



(732)

67135

Numerical treatment




heff ij (734)


∆Ω =

intdxdxprime

δΩ[γ]


=sum

ij

δΩ[γ]

δγij∆γji (736)

=sum

i

δnipartΩ[γ]

partni︸︷︷︸

∆Ω1

+sum

i6=j


|heff ij |2

︸︷︷︸∆Ω2

(737)




∆Ω1 =sum

i

δnipartΩ[γ]

partni(738)

=sum

i

βini(ni minus 1)

(partΩ[γ]

partniminus∆(0)

micro

)partΩ[γ]

partni(739)

=sum

i

βini(ni minus 1)

(partΩ[γ]

partni

)2

minus


partni

)2


(740)


∆Ω1 =

sum

j

βjnj(nj minus 1)

sum

i

ci

(partΩ[γ]

partni

)2

minus(sum

k

ckpartΩ[γ]

partnk

)2 (741)

=

sum

j

βjnj(nj minus 1)

sum

i

ci

(partΩ[γ]

partniminussum

k

ckpartΩ[γ]

partnk

)2

(742)


∆Ω1 le 0 (743)



∆Ω2 =sum

i6=j


|heff ij |2 (744)

68135

Numerical treatment


∆Ω2 =sum

i6=j

ni minus nj1βi

ln(

1minusni

ni

)minus 1

βjln(

1minusnj

nj

) |heff ij |2 (745)


∆Ω2 = β0sum

i6=j

ni minus njln(


) |heff ij |2 (746)


∆Ω2 le 0 (747)








69135

Numerical treatment



to the ONs

χ2n =

1

Nunpinned

Nunpinnedsum

i

(partΩ

partniminus micro

)2

(748)

micro =1

Nunpinned

Nunpinnedsum

i

partΩ

partni(749)



χ2φ =

1

N minus 1


i ε2i

(750)





Eαxc[γ] = minus

1

2

sum

ij

nαi nαj



prime) (751)






70135

Numerical treatment

minus794

minus793

minus792

minus791

minus790

minus789

minus788

0 5 10 15 20

it

E(Ha)

τ = 20β0 = 30η = 10


1eminus25

1eminus20

1eminus15

1eminus10

1eminus05

1

0 20 40 60 80 100

it

χ2 n

τ = 20β0 = 30η = 10


minus808

minus806

minus804

minus802

minus800

0 5 10 15 20

it

E(Ha)

τ = 20β0 = 30η = 10


1eminus30

1eminus25

1eminus20

1eminus15

1eminus10

1eminus05

1

0 20 40 60 80 100

it

χ2 n τ = 20

β0 = 30η = 10


minus812

minus810

minus808

minus806

minus804

minus802

0 5 10 15 20

it

E(Ha)

τ = 10β0 = 30η = 10


1eminus30

1eminus25

1eminus20

1eminus15

1eminus10

1eminus05

1

0 20 40 60 80 100

it

χ2 n

τ = 10β0 = 30η = 10



71135

Numerical treatment





cycle


72135

Numerical treatment

minus79445

minus79440

minus79435

minus79430

minus79425

0 100 200 300 400 500

it

E(Ha)

τ = 10

β0 = 20


1eminus24

1eminus20

1eminus16

1eminus12

1eminus08

1eminus04

0 100 200 300 400 500

it

χ2 φ

τ = 10β0 = 20


minus8085

minus8084

minus8083

minus8082

minus8081

minus8080

0 100 200 300 400 500

it

E(Ha)

τ = 10

β0 = 10


1eminus13

1eminus11

1eminus09

1eminus07

1eminus05

0 100 200 300 400 500

it

χ2 φ

τ = 10

β0 = 10


minus81285

minus81280

minus81275

minus81270

minus81265

minus81260

minus81255

0 100 200 300 400 500

it

E(Ha) τ = 10

β0 = 10


1eminus13

1eminus11

1eminus09

1eminus07

1eminus05

0 100 200 300 400 500

it

χ2 φ

τ = 10

β0 = 10



73135

Numerical treatment

minus81284

minus81282

minus81280

minus81278

minus81276

minus81274

minus81272

0 100 200 300 400 500

it

E(Ha)

β0 = 10β0 = 20β0 = 50


1eminus13

1eminus11

1eminus09

1eminus07

1eminus05

0 100 200 300 400 500

it

χ2 φ

β0 = 10

β0 = 20

β0 = 50











74135

8 Applications


Section Description








75135

Applications








76135

Applications

minus40

minus50

minus45

5

10

15

20

0

02

08

06

04

10

rs(au)

ξF(10minus

2Ha)





S[D] =sum

i

wi lnwi (82)




77135

Applications










FMPP

PM

r s(au)

T (K)

rc (rTc Tc)

TF


78135

Applications





n =

intd3k



Ωkσ =1

n

intd3k

(2π)3k2

2nσ(k) (84)

Ωextσ =1

n

intd3k


nσ =1

n

intd3k

(2π)3nσ(k) (86)

S0σ = minus 1

n

intd3k


F =sum

σ








s We therefore set



ΩSW = grminus1

s (1minus ξ2) (811)

79135

Applications

FM

PP

PM

0 20 40 6010 30 50

60

80

100

120

140

40

160

200

180

r s(au)

T (K)

rc(rc Tc)

TF



83 FT-MBPT




n(k) =1


ε(k) =k2

2(814)


t =T

TF k =

k

kF(815)

80135

Applications



3 (816)


nσ(k) =1


(817)


2

3tminus

32 =

int infin

0

dx

radicx

1 + exminusα(818)


Fj(x) =1

Γ(j + 1)

int infin

0

dzzj

1 + ezminusx(819)


nσ =3

8π

12 t

32F 1

2(ασ) (820)

Ekσ =9

32π

12 k2F t

52F 3

2(ασ) (821)

microσ =1

2k2F tασ (822)

S0σ =

(15

16π

12 t

32F 3


)(823)


4

3πminus 1

2 tminus 3

2σ = F 1

2(ασ) (824)


831 Exchange-only


F = F0 +

int 1

0

dλ

λ

︸︷︷︸rarrΩx

(825)



Ωxσ = minus 1

n

intd3k

(2π)3d3q


81135

Applications

0

02

04

06

08

1

0 1 2 3 4 5

t

I x(t)

84a Exchange

0

05

1

15

2

25

3

0 1 2 3 4 5

t

I 2b(t)




W (k) =4π

k2 (828)



3


Ix(t) =

intdx

intdy

x

1 + ex2

tminusα(t)

y

1 + ey2

tminusα(t)

ln(x+ y)2

(xminus y)2 (830)




82135

Applications

0 2000 4000 6000 8000 100000

2

4

6

8

10

12

14

16

18

20

PP

PM

FMr s(au)

T (K)

rc(rTc Tc)

TF


832 Exchange + RPA


F = F0 +

int 1

0

dλ

λ

︸︷︷︸

rarrΩx

+

︸︷︷︸rarrΩr

(831)



Ωr =1

n

1

2β

intd3q

(2π)3

infinsum

a=minusinfin


) (833)


νa =2πa

β (834)

83135

Applications

0 2 4 6 8 10

minus05

minus04

minus03

minus02

minus01

00 2 4 6 8 10

minus025

minus020

minus015

minus010

0 2 4 6 8 10

minus009

minus008

minus007

minus006

minus005

minus004

0 2 4 6 8 10

minus0011

minus0010

minus0009

minus0008

tt

Ωr(t)

Ωr(t) Ω

r (t)Ω

r (t)

rs = 025 rs = 114

rs = 540 rs = 117




d3k




Ωr =k2F t

8π2

intdqq2

infinsum

a=minusinfin


)(836)


(2π)2kFσ

q

intdk

k


ln

((2πatσ)

2 + (q2 + 2kq)2


) (837)



84135

Applications

minus1eminus3

minus5eminus4

1eminus3

5eminus4

0

1 10 100

δ(r sξ)

rs(au)

ξ = 00

ξ = 02

ξ = 04

ξ = 06

ξ = 08

ξ = 10


0 50 100 200150 250

10

15

20

25

30

35

40

45

50

55

60

PP

PM

FM

300 350

r s(au)

T (K)

rc(rTc Tc)

TF




85135

Applications



F = F0 +

int 1

0

dλ

λ

︸︷︷︸

rarrΩx

+


+

︸︷︷︸rarrΩr

(838)



Ω2bσ =1

n2minus10πminus9

intd3k

intd3p

intd3q





2Ω2bσ(0)I2b(tσ) (841)

I2b(t) = minus3


intd3k

intd3p

intd3q


k2p2k middot p(842)

Ω2bσ(0) =

(ln 2

6minus 3

4π2ζ(3)

) (843)



86135

Applications

0 4000 8000 120000

4

8

12

16

20

24

28

32

36

40

16000FM

PP

PMr s(au)

T (K)

TF

(rTc Tc)rc




87135

Applications

minus15

minus10

minus5

0

5

minus8

minus6

minus4

minus2

0 05 10 15

minus7

minus6

minus5

minus4

minus3

0 05 10 15 20

3

4

5

6

7

F0(10minus

2Ha) Ω

x(10 minus

2Ha)

Ωr(10minus

2Ha) Ω

2b (10 minus

2Ha)

tt

ξ = 0ξ = 05ξ = 08ξ = 10


84 FT-RDMFT



F [γ] = F0[γ] +

int 1

0

dλ

λ

︸︷︷︸rarrΩx

(844)



88135

Applications




σi

nσi

int

Vi

d3k





Ωk[nσi] =sum

σi

nσiti (848)

S0[nσi] = minussum

σi


Ωx[nσi] = minus1

2

sum

σij

nσinσjKij (850)


ti =1

n

int

Vi

d3k

(2π)3k2

2(851)

Kij =1

n

int

Vi

dk31(2π)3

int

Vj

dk32(2π)3

4π


ωi =1

n

int

Vi

d3k

(2π)3 (853)







89135

Applications

4 6 8 10 12 14 160

02

04

06

08

10

T=0T=5000T=8000T=10000

ξ min

rs(au)


05

10

15

20

0

0 02 04 06 08

minus4

0 02 04 06 08 1

minus3

0

3

6

1

2

3

minus3

minus2

minus1

ξξ

Ωk(10minus

2Ha) Ω

x(10 minus

2Ha)S

∆F(10 minus

3Ha)

t = 0t = 10t = 16t = 22



∆1minus0 = Ω(ξ = 1)minus Ω(ξ = 0) (854)


90135

Applications

0

1

2

3

4

0 1 2 0 1 2 3

minus8

minus6

minus4

minus2

minus1

0

1

2

3

4

minus3

minus2

minus1

tt

Ωk(10minus

2Ha) Ω

x(10 minus

2Ha)

SF(10 minus

2Ha)

FD ξ = 0

FD ξ = 1

HF ξ = 0

HF ξ = 1

FD ∆1minus0

HF ∆1minus0










91135

Applications





92135

Applications




φ1k(r) =

(cos(Θk

2

)eminusiqmiddotr2

sin(Θk

2

)eiqmiddotr2

)eikmiddotrradicV

(855)

φ2k(r) =

(minus sin

(Θk

2

)eminusiqmiddotr2

cos(Θk

2

)eiqmiddotr2




intd3k

(2π)3

(n1k cos

2

(Θk

2

)+ n2k sin

2

(Θk

2



intd3k

(2π)31




intd3k

(2π)31




intd3k

(2π)3

(n1k sin

2

(Θk

2

)+ n2k cos

2

(Θk

2



Ωx[γ] = minus1

2

sum

σ1σ2




Ωx[γ] = minus1

2n

intd3k1(2π)3

d3k2(2π)3

4π

|k1 minus k2|2[sum

b

(nbk1nbk2


2

)+ (n1k1

n2k2+ n2k1

n1k2) sin2

(Θk1minusΘk2

2

)] (862)


93135

Applications


F [q nbi Θi] =sum

bi

nbiti +q2

8minus q

sum

i


1βsum

bi


)ωiminus

1

2

sum

bij

(nbinbj) cos2

(Θi minusΘj

2

)Kij minus

sum

ij

(n1in2j) sin2

(Θi minusΘj

2

)Kij (863)


Qi =1

2n

int

Vi

d3k

(2π)3kz (864)


m(r) = minus


B

(865)


A =1

2

intd3k


B =1

2

intd3k





(868)

Θkρplusmn|kz| =π

2(1∓ akρ|kz|) (869)


i n1i and N2 =sum

i n2i

ξPSS =N1 minusN2

N1 +N2(870)






94135

Applications

kρkρkρkρ

kzkzkzkz

qkF = 0 qkF = 10 qkF = 18 qkF = 2

kF kF kF kF

-kF -kF -kF -kF

q2

q2

q2

- q2- q2 - q2


5

6

7

minus112

minus105

minus98

minus91

0 05 1 15 2 25

0

0 05 1 15 2 25 3

minus49

minus48

minus47

minus46

5

10

15

qkFqkF

Ωk(10minus

2Ha)

Ωx(10 minus

2Ha)

S(10minus

2)

F(10 minus

2Ha)

T = 0KT = 2500KT = 4000KT = 5000K








95135

Applications

0

1

2

3

4

5

1 15 2 25minus494

minus493

minus492

minus491

minus490

minus489

qkF

F(10minus

2Ha)

A(10minus

4au)

T = 4700K t = 0244T = 4800K t = 0249T = 4900K t = 0254T = 5000K t = 0260






96135

Applications

0 1000 2000 3000 4000 50000

1

2

3

4

T (K)

Aopt(10minus

4au)

rs = 55rs = 50rs = 45rs = 40







rc(au) 40 45 50 55

Tc(K) 2700 3400 4100 5000


97135

Applications

0 2000 4000 6000 8000 100000

2

4

6

8

10

12

14

16

18

20

PSS

FMPP

PM

r s(au)

T (K)

rc(rTc Tc)

TPSSc

TF




εσ(k) =k2

2minus 2

πfd(kkFσ) (871)


f3(x) =1

2+

1minus x24x

log

∣∣∣∣1 + x

1minus x

∣∣∣∣ (872)


f2(x) =

E(x) x lt 1

x(E(1x

)minus(1minus 1

x2

)K(1x

)) x ge 1



partF

partni= 0 =

partΩk

partniminus 1

β

partS0

partni+partΩx

partni (873)

98135

Applications

minus4

minus3

minus2

minus1

0

1

2

3

4

0 05 1 15 2 25

kkF

ε σ(kk

F)ε F

k2200105

12510100


minus8

minus6

minus4

minus2

0

2

4

0 05 1 15 2 25

kkF

ε σ(kk

F)ε F

k220

0105

12510100

820b Two dimensions




partni+partΩx

partni (874)


1d where d is



e0b(k) =

(k plusmn q

2

)2

2minus q2

8=

1



99135

Applications

minus4

minus2

0

2

4

6

8

10

0 05 1 15 2

kkF

ε σ(kk

F)ε F

ni

T = 0

FM


minus4

minus2

0

2

4

6

8

10

0 05 1 15 2

kkF

ε(kk

F)ε F

091787









Ωc[γ] = ΩDFTc [n] (876)



nminγrarrn

F [γ] (877)

= minn



) (878)


100135

Applications





FTrarr0= Ωk[n


c (n) (879)



101135

Applications

0 1000 1500 2000 2500 30000

20

40

60

80

100

120

140

160

180

200

FM

PP

PM

500

T (K)

r s(au)

TF

(rTc Tc)rc




0 20 40 60 80 100 120 140

15

20

25

30

35

40

45

FM

PP

PM

T (K)

r s(au)

(rTc Tc)

rc


102135

Applications







Ωαxc[nσi] =

sum

σij









103135

Applications

α 1 9 8 7 6 55

rc(au) 556 545 539 528 470 asymp5Tc(K) 10500 9400 8500 7200 6500 4600

rTc (au) 710 725 775 825 900 950


0 05 1 15 2 250

02

04

06

08

1

kkf

nk

HF T = 0HFα = 08α = 0565ni







α 1 095 09 085

TPSSc (K) 4500 3000 2000 500


104135

Applications

0 1000 2000 3000 4000 5000

10

20

30

40

50

60

70

80

FM

PP

PM

T (K)

r s(au)

TF

(rTc Tc)

rc




Ω+c [γ] = minus

1

2





Ω+c [γ] = minus

1

2n

intd3k1(2π)3

d3k2(2π)3

4π


cos2(Θk1

2

)+ n2k1

sin2(Θk1

2

))(n2k2

cos2(Θk2

2

)+ n1k2

sin2(Θk2

2

))(883)


105135

Applications





ε(mk) =k2

2m(884)


n(tmk) =1

1 + ek2

mtminusα(mt)

= n(mt 1k) (885)


Ωk(tm) = Ωk(mt 1) (886)


N(tm) = N(mt 1) (888)

S0(tm) = S0(mt 1) (889)

Ωx(tm) = Ωx(mt 1) (890)



Ω2bσ(tm) = mΩ2bσ(mt 1) (892)



106135

Applications



WRPA(tm) =W





εσ(k) = ε0σ(k) =k2

2(895)



partn(k)




partn(k)

partε(k)

1 +



partε(k)partkjsum

i qipartε(k)partki

(897)

107135

Applications



intd3k

(2π)3nσ(k)(1minus nσ(k)) +O(q2) (898)


Ωr =1

β(2π)minus2




q gt q


r (8100)

Ω0r =

1

β(2π)minus2

int q

0

dqq2(W (q)χ(q)) (8101)

=4π

β(2π)minus2

int radicβ

0

dqχ(q)) (8102)

Ωinfinr = minus 1

β(2π)minus2

int infin

radicβ

dqq2(W (q)χ(q))2

2 (8103)


behaves like


κ ge 2 (8105)


χκ[n(k)](q) =sum

σ

χκσ[n(k)](q) (8106)


β

1 + (qkF )κ

intd3k

(2π)3nσ(k)(1minus nσ(k)) (8107)

= minus β


) (8108)

where

〈n2σ〉 =int

d3k

(2π)3n2σ(k) (8109)






108135

Applications

01 1 10 100

0

minus002

minus004

minus006

minus008

minus010

minus012

rs(au)

e c(Ha)

PWCAκ = 29κ = 27κ = 25κ = 23



0

02

04

06

08

10

0 05 10 15 20 25 30

kkF

n(kk

F)

rs = 1 GZ

rs = 5 GZ

rs = 10 GZ

rs = 1 κ

rs = 5 κ

rs = 10 κ


KAPPA-functional

ξ 00 01 02 03 04 05 06 07 08 09 10

κ 29 29 29 30 30 30 31 31 31 32 32


109135

Applications

κ c

KAPPA-TIE 32 024


0 200 400 600 800 1000

10

20

30

40

50

60

70

80

01200

FM

PP

PM

T (K)

r s(au)

TF

(rTc Tc)





1 + q2k2F

intd3k







110135

Applications







111135

Applications

(1) (2) (3) (4) (5) (6) rc


FT-MBPT




FT-RDMFT








112135

A APPENDIX




γ(x xprime) =sum

i


=sum

i

niφlowasti (x

prime)φi(x) (A2)


|ψi〉 =sum

l

cij |χj〉 (A3)



D =sum

ijk



γ(x xprime) =sum

i

sum

jk


φlowasti (x

prime)φi(x) (A6)


γ(x xprime) =sum

i

sum

jk


φlowasti (x

prime)φi(x) (A7)


P (A) =sum

B

P (A|B)P (B) (A8)


γ(x xprime) =sum

i

P (φi)φlowasti (x

prime)φi(x) (A9)


113135

Appendix



ni =1




γ(x xprime) =sum

i

niφlowasti (x

prime)φi(x) (A11)


ni = c+i ci (A12)


written as

ni = TrDni =sum

e



Ψe =sum

α

ceαΦα (A14)


sum

e

|ceα|2 =sum

α

|ceα|2 = 1 (A15)


ni =sum

e

we

sum

αβ



ni =sum

α

(sum

e

we|ceα|2)

︸︷︷︸fα


(A17)



fα gt 0 (A18)sum

α

fα = 1 (A19)

114135

Appendix


0 lt ni lt 1 (A20)




H = T + V + W (A21)

γgs(x xprime) =

sum

i

niφlowasti (x

prime)φi(x) (A22)


E[γ] = 〈Ψ[γ]|H|Ψ[γ]〉 (A23)



partE[γ]

partni

∣∣∣∣γgs

=

ai gt micro ni = 0

micro 0 lt ni lt 1

bi lt micro ni = 1

(A24)



δE[γ]

δγ(x xprime)

∣∣∣∣γgs

= microδN [γ]

δγ(x xprime)

∣∣∣∣γgs




δEu[γ]

δγ(x xprime)

∣∣∣∣γgs




115135

Appendix





Eα[γ] = E[γ] +



partEα[γ]

partni

∣∣∣∣γgs

=

ai + uδiα ni = 0


bi + uδiα ni = 1

(A30)





Z=sum

i



H =



γ =


116135

Appendix


γ(x xprime) =sum

i


=sum

i

niφlowasti (x

prime)φi(x) (A35)


Ψ(x) =sum

i

ciφi(x) (A36)


γ(x xprime) = Tr


ZΨ+(xprime)Ψ(x)

(A37)

=1

Z

sum

i


sum

n1n2



=1

Z

sum

i


sum

n1n2


=1

Z

sum

i


prod

j

sum

nj


〈ni|e


(A40)


γ(x xprime) =1

Z

sum

i


prod

j


1




(A42)

=sum

n1n2

〈n1 n2 |prod

i


=prod

i

sum

ni


=prod

i



γ(x xprime) =sum

i

niφlowasti (x

prime)φi(x) (A46)

ni =1


117135

Appendix


εi minus micro =1

βln

(1minus nini

) (A48)


Ω = minus 1

βlnZ (A49)


Ω0 = minus 1

β

sum

i


)(A50)

=1

β

sum

i

ln(1minus ni) (A51)


S =part

partTΩ

∣∣∣∣micro

(A52)


part

partTni

∣∣∣∣micro

=1

Tni(1minus ni) ln

(1minus nini

)(A53)


S0 = minussum

i


A5 Feynman rules







0dτ




118135

Appendix



Ω2b =1

n2minus10πminus9

intd3k

intd3p

intd3q



=1

n2minus6πminus7

intd3k

intd3p

intd3q




krarr q (A57)


q rarr k (A59)

Ω2b = minus1

n2minus6πminus7

intd3k

intd3p

intd3q


k2p2k middot p(A60)


Ω2b = minus1

n2minus2πminus5

int infin

0

dk

int infin

0

dp

int infin

0

dq

int π

0

dθp

int π

0

dθq

int π2

minusπ2

dφq




F (kp q θp) =

int π

0


int π2

minusπ2



G(k p q) = q2fq

int π2

0

dθp tan(π2+ θp

)(F(k p q

π

2+ θq

)minus F

(k p q

π

2minus θq

)) (A63)


Ω2b = minus1

n2minus2πminus5

int infin

minusinfindk

int infin

minusinfindp

int infin

0

dqG(ek ep q) (A64)






Ω2b = minus1

n2minus2πminus5

int infin

0

dxx2int π

0

dθx sin(θx)

int π2

minusπ2


119135

120135

B Bibliography

References






3865





PhysRevLett832628




PhysRevLett87133004


10106312987202


0304-8853(79)90333-0


PhysRevB5514975

121135

Bibliography



100257001





6216536










0921-4526(91)90075-P


dxdoiorg10106311906203

122135

Bibliography



PhysRevA79040501












322010


dxdoiorg1010631449326


101002qua560360864



123135

Bibliography



1103PhysRev971474




566






PhysRevB4513244



447879






124135

Bibliography




171257


101103PhysRevA64042512





org101098rspa20021027



235121




0375-9601(84)91034-X


00268970110070243


866


057

125135

Bibliography



PhysRevA77032509


106312899328









dxdoiorg1010631479623


0305-447089021



PhysRevE66036703




126135

Bibliography










47R1591




0378-4371(83)90254-6


pssb2221240140








org10106312888550

127135

Bibliography


1002jcc21225







PhysRevA34433





589











128135

Bibliography


0370-1573(82)90077-1








PhysRevE77021120



2064



softwaregsl




org101139p75-176



19664730108


129135

Bibliography







PhysRev139A796

130135








131135

132135

D Publications










gas Exchange only









133135

134135

E Acknowledgements











Tim

135135

Title

Abstract

Table of Contents

List of Figures

List of Tables


Introduction










BOW functional


Summary and outlook













Convexity



Summary and outlook






Summary and outlook






Scaling behaviour

Summary and outlook

Numerical treatment



Temperature tensor





Sample calculations


Full minimization

Summary and outlook

Applications



FT-MBPT

Exchange-only

Exchange + RPA


FT-RDMFT









Summary and outlook

APPENDIX





Feynman rules


Bibliography


Publications

Acknowledgements




Berlin den 10102012

Abstract






Table of Contents

List of Figures v

List of Tables vii


1 Introduction 1










i

TABLE OF CONTENTS














B Bibliography 121


D Publications 133

ii

TABLE OF CONTENTS


iii

List of Figures









v

LIST OF FIGURES


vi

List of Tables







vii













EQ Equilibrium

FM Ferromagnetic







GS Ground state



HF Hartree-Fock

HS Hilbert-Schmidt







ix


MC Monte-Carlo


NO Natural orbitals



PM Paramagnetic







SC Self-consistent




SW Spin waves



x

1 Introduction







1135

1 INTRODUCTION











2135

1 INTRODUCTION




3135

4135



Section Description


2223






H = T + W + Vext (21)


intdx =

sumσ



T =

intdx lim

xprimerarrx

(minusnabla

2

2


Vext =



W =



w(x xprime) =1




D =sum


i

wi = 1 (26)



(27)

5135

Prerequisites

T

T1

DN


T

TradicN

ΓN




N =




(29)


AHS =

radicTrA+A (210)


T1 =A | AHS le 1

(211)






γ(x xprime) =sum

i

niφlowasti (x

prime)φi(x) (213)

6135

Prerequisites


Φi(r) =

(φi1(r)

φi2(r)

) (214)


γ(r rprime) =sum

i

niΦdaggeri (r


=

(γ11(r r



prime)

)=sum

i

ni

(φlowasti1(r



prime)φi2(r)

)(216)


Φi1(r) =

(φi1(r)

0

)Φi2(r) =

(0

φi2(r)

)(217)



sum

i

niσφlowastiσ(r





sumi niφ

lowasti (x



i

ni = N (220)





(222)

7135

Prerequisites




radicN

TradicN =

A | AHS le

radicN

(223)





n(x) = γ(x x) (224)


nσ(r) = γσσ(r r) (225)





rs =

(3

4πn

)13

(226)

8135

Prerequisites


kF =(3π2n

) 13 =

(9

4π

) 13

rminus1s (227)

εF =k2F2

(228)

TF =εFkB

(229)



(230)









Enuc [n(r)] =



Vext σσprime =



9135

10135




Section Description








D = |Ψgs〉〈Ψgs| (31)


E = Ek[γ] + Vext[γ] +W [Γ(2)] (32)

Ek[γ] =

intdx lim

xprimerarrx

(minusnabla

2

2

)γ(xprime x) (33)

Vext[γ] =



W [Γ(2)] =




2TrDΨ+(x4)Ψ

+(x3)Ψ+(x1)Ψ

+(x2) (36)

11135

ZT-RDMFT


γ(x xprime) =2

N minus 1

intdx2Γ

(2)(x x2xprime x2) (37)



W [Γ(2)] =W [γ] (38)





prime) (310)






12135

ZT-RDMFT




W [γ] = EH [γ] + Ex[γ] + Ec[γ] (311)

EH [γ] =1

2


Ex[γ] = minus1

2








13135

ZT-RDMFT


Exc[γ] = Ex[γ] + Ec[γ] (315)

Exc[γ] = minus1

2

sum

ij

f(ni nj)



prime) (316)











14135

ZT-RDMFT



intd3RεHEG

c [γ(kR)] (317)



γ(kR) =















15135

ZT-RDMFT

10010101001

minus030

minus020

minus010

0

minus005

minus015

minus025

rs(au)

e c(Ha)


MLα (α = 055)




33 BOW functional




16135

ZT-RDMFT

0

02

04

06

08

10

0 02 04 06 08 10

ninj

f

fHF






2

sum

ij

fBOW (ni nj α)



prime) (319)








17135

ZT-RDMFT

00 05 10 15 2000

02

04

06

08

10

kkF

n(kk

F)

rs = 05

rs = 1

GZBOW

GZBOW

00 05 10 15 2000

02

04

06

08

10

kkF

n(kk

F)

rs = 5

rs = 10

GZBOW

GZBOW


10010101001

minus08

minus07

minus06

minus05

minus04

minus03

minus02

minus01

0


MLα (α = 06)

rs(au)

e c(Ha)




18135

ZT-RDMFT










E(nuarr ndarr) =1







19135

ZT-RDMFT

001 01 1 10 100minus010

minus008

minus006

minus004

minus002

0

rs(au)

∆F(Ha)

ξ00204060810


001 01 1 10 100

0

002

004

006

008

010

rs(au)

∆P(Ha)

ξ00204060810


0

01

02

03

04

001 01 1 10 100rs(au)

δF

00204

060810


minus08

minus06

minus04

minus02

0

001 01 1 10 100rs(au)

δP

00204

060810





δF =∆F

Wc(328)

δP =∆P

Wc (329)



20135

ZT-RDMFT

3D

ξ 00 01 02 03 04 05 06 07 08 09 10

α 056 056 056 057 057 057 058 059 061 063 066

BOW 061 061 061 062 062 063 063 064 065 067 069

2D

ξ 00 01 02 03 04 05 06 07 08 09 10

α 063 063 063 063 064 064 065 066 067 07 074

BOW 066 066 066 067 067 068 069 071 073 076 080





2

sum

ijσ


csum

ij

((niuarrnjdarr)

αU



21135

ZT-RDMFT

αP αU c

BOW-TIE 070 20 019









22135




Section Description







23135

FT-RDMFT


H


∆Nmicro

∆E β



H =

infinoplus

n=0

Shotimesn (41)










24135

FT-RDMFT

F [D]

D

downwards convex

upwards convex

42a Convexity

F [D]

D








γ(x xprime) = Tr

eminusβ(T+V+W )

ZΨ+(xprime)Ψ(x)

(48)



(49)



T (see Section 23)





25135

FT-RDMFT





F [Dk] (412)













S[A B] = supλ

((1λ)


))(419)




S[Ak Bk] (420)

26135

FT-RDMFT


TrA = supP

TrP A (421)

TrP A le PTrA (422)






S[A B] = supP λ

((1λ)tr


))(425)


S[Ak Bk] (426)





Ω[Dk] (427)


S[Dk] (428)






27135

FT-RDMFT

Deq

Deq

Deq

γeq

γeq neq

neq

neq

+ =rArr










U =


βlnZ

Z prime (430)





minus 1

βlnZ

Z prime = u11 = u22 = u11 + u22 (431)



28135

FT-RDMFT







eq) (433)


leads to



eq) (435)
















Ω[γeq] =







Ωeq = minDisinDN

Ω[D] (443)

29135

FT-RDMFT


Ωeq = minγisinΓN

infDisinDNrarrγ

Ω[D[γ]] (444)


Ωeq = minγisinΓN

(F[γ] +





TrD(T + W + 1β ln D) (446)


Ω[γ] = F[γ] +




Ωeq = minγisinγN

Ω[γ] (448)



δF [γ]

δγ(xprime x)

∣∣∣∣γeq




30135

FT-RDMFT





H(1)s =

sum

i

εi|φi〉〈φi| (450)


γeq(x xprime) =

sum

i

niφlowasti (x

prime)φi(x) (451)


ni =1



εi minus micro =1

βln

(1minus nini

) (453)



veff (x xprime) =

sum

ij






31135

FT-RDMFT


Ωk[γ] =

intdxprime lim

xrarrxprime

(minusnabla

2

2

)γ(x xprime) (456)

N [γ] =

intdxγ(x x) (457)

Ωext[γ] =



ΩH [γ] =1

2


Ωx[γ] = minus1

2


S0[γ] = minussum

i







Ω[γ] =

intdxprime lim

xrarrxprime

(minusnabla

2

2

)γ(x xprime) +

intdxvext(x x


1

2


2


1βsum

i



δΩ[γ]



0 =

intdxprimeφi(x

prime)δΩ[γ]

δγ(xprime x)(465)

=

(minusnabla

2

2

)φi(x) +

intdxprimevext(x x



prime)+



32135

FT-RDMFT





Ω[γ] = F [γ] +
















L sub DN )]


F [γ] = TrD(H0 + 1β ln D) (470)


33135

FT-RDMFT


limkrarrinfin

TrDk(H0 + 1β ln Dk) = F [γ] (471)












Mf =









34135

FT-RDMFT



infinsum

i=M+1









TrD(H0 + 1β ln D) = F [γ] (479)






F [γk] (480)


F [γ] le TrD(H0 + 1β ln D) (481)


F [γk] (482)

35135

FT-RDMFT

433 Convexity









infγisinΓN

(F [γ] +

intv0γ

)= F [γ0] +

intv0γ0 (484)






36135

FT-RDMFT








F [γ] +

intvγ ge F [γ0] +

intvγ0 (487)

infγisinΓN

(F [γ] +

intvγ

)ge F [γ0] +

intvγ0 (488)



F [γ0] +

intv0γ0 = inf

γisinΓN

(F [γ] +

intv0γ

)(489)






37135

FT-RDMFT





intv0γ



1 γk rarr γ




∆0 = F [γ0] +

intv0γ0 minus inf

γisinΓN

F [γ] +

intv0γ

(492)








38135

FT-RDMFT




Dc =sum

α


α

wαN = 1 (495)


F [D] = TrD(H + 1β ln D) (496)


Dceq =

eminusβH

TreminusβH (497)



TrD(T + W + 1β ln D) (498)


F [γ] = Fc[γ] + Vext[γ] (499)



F [γ] = Ω[γ] + microN [γ] (4100)

39135

FT-RDMFT

T

T1

DN

DV

DNL


T

TradicN

ΓN

ΓV








DNL




40135

FT-RDMFT







1minus1larrrarr ΓN




solved by

leads to

solved by

leads to

solved by

enables

used for

allows


41135

42135



Section Description









Ω0[D0] lt Ω0[Dw] (52)








43135

Exact properties


Sc[γ] lt 0


Ωc[γ] lt 0

(58)

(59)

(510)




Hλ = T + V0 + λW (511)




)(512)

= minDrarrγ


)minus Tr

DV λ[γ]

) (513)





)minus Tr

D[γ]V λ[γ]

(514)


dΩλ[γ]

dλ= Tr

Dλ[γ]

(W +

d

dλV λ[γ]

)minus d

dλTrD[γ]V λ[γ]

(515)


Ω[γ] = Ω0[γ] +

int 1

0

dλTrDλ[γ]W

(516)


Ωc[γ] =

int 1

0

dλ

λWλ

c [γ] (517)



44135

Exact properties

minus4 minus2 2 4

minus02

minus01

01

02

z

Ψ210(0 0 z)

51a Wavefunction

minus4 minus2 2 4

001

002

003

004

0z

ρ210(0 0 z)

51b Density







ψλ(r1 rN ) = 〈r1 rN |U(λ)|ψ〉 (518)

= λ3N2ψ(λr1 λrN ) (519)


45135

Exact properties


〈ψλ|Oλ|ψλ〉 = 〈ψ|O|ψ〉 (520)


Tλ =1

λ2T (521)

Wλ =1

λW (522)

Nλ = N (523)


T [γλ] = λ2Ωk[γ] (524)

ΩH [γλ] = λΩH [γ] (525)

Ωx[γλ] = λΩx[γ] (526)

S0[γλ] = S0[γ] (527)





get


λ[γλ]) (529)

TrD 1λ[γλ](W 1

λ+ τ ln D 1


λ+ τ ln D[γ]) (530)











(535)

(536)

46135

Exact properties





(537)

(538)

(539)


0 =d

dλ

(Ω[D 1

λ[γλ]]

)∣∣∣λ=1

(540)


0 =d

dλTr

D 1

λ[γλ]

(1

λW 1

λ+ τ ln(D 1

λ[γλ]

)∣∣∣∣λ=1

(541)


W [γ] =d



Wc[γλ] = λd

dλΩc[γλ] (543)


Ωc[γλ] =

int λ

0

dmicro



Sc[γλ] = λ2(part

partλ

1

λΩc[γλ]

)(545)


d

dλτSc[γλ] = λ

d

dλ

(1

λWc[γλ]

) (546)


d

dλΩc[γλ] =

1

λWc[γλ] lt 0

d

dλWc[γλ] lt

d

dλΩc[γλ] lt 0

d

dλSc[γλ] lt 0

(547)

(548)

(549)

47135

Exact properties

Ωc[γλ]

λ1

minusτSc[γ]

0


Ωc[γλ]




d2


1

λ2Wc[γλ] +

1

λ

d


1

λ2Wc[γλ] +

1

λ

d

dλΩc[γλ] (550)

d2








Dλ(τ w)[γ] = D(λ2τ λw)[γλ] (553)

48135

Exact properties




λ2Sc(τλ

2 wλ)[γ]


(554)

(555)

(556)


Ωc(τ w)[γ] =

int 1

0

dλ

λWc(τ λw)[γ] (557)

Ωc(τ w)[γ] =

int 1

0








49135

50135



Section Description










prime) =sumi εiφ

lowasti (x



i

φi(x)φlowasti (x


ni ν gt νprime




i

niφlowasti (x

prime)φi(x) (62)

51135


W + Veff




ni =1


εi =1

βln

(1minus nini

)+ micro (64)




52135




Ω[γ] = Ω0[γ] +

int 1

0

dλTrDλ[γ]W

(65)




〈λW 〉λ =1

2

intdxdxprime lim

νprimerarrν+





〈λW 〉λ =1

2

intdxdxprime lim

νprimerarrν+




1

2

intd3xd3xprime


)γ(x xprime) (68)






〈λW 〉unifλ =1

2

Σlowast minus

(610)


53135


λw(r rprime)

vλ(x xprime)






veff = v(1)eff + v

(2)eff + (611)



0

0+

=

0

0+

+

0

0+

Σlowast (612)


0 =

0

0+

Σlowast (613)

54135


+ + + +

+ + + + + +

+ + + +

+ + + + + +



(1)

0

0+

= minus

0

0+

minus

0+

0

(614)

55135


intdydyprime

int β

0


= minusintdydyprime

int β

0


int β

0



v(1)eff (x x



= minus minus (617)



potential v(2)eff

(2)

0

0+

= minus

0+

0

minus

0+

0

(618)


(2)

0

0+

= minus

0

0+

M (619)


intdydyprime

int β

0


= minusintdydyprime

int β

0


56135



v(2)eff =

sumij φi(x)φ

lowastj (x

prime)int β



int β

0eν


where

Mij(ν νprime) =








M =

sumij φi(x)φ

lowastj (x

prime)int β



int β

0eν








Ω(1)xc [γ] = + (625)


Ωxc[γ] =1

2


2


57135



Ω(2)xc [γ] = + + + (627)







Ω[γ]

γΓV ΓNΓV ΓV

(γV EV

) (γN EN

)


Ω[γ]

γΓN ΓV

(γV EV

) (γN EN

)



58135






prime) (628)




bull 2n+1 G0 lines






c [γ] (631)



c [γ] = nΩ(n)c [γλ] (632)


c [γ] = (nminus 1)Ω(n)c [γλ] (633)









part




part


59135





prime) (640)






c (τ w)[γ] (643)



c (τ w)[γ] (644)





60135



Section Description










61135

Numerical treatment

γ

F [γ]

γ0γ1γ2γ3

Ω(0)effΩ

(1)effΩ

(2)eff

Ω








62135

Numerical treatment






δΩeff [γ]

δγ(x xprime)=

δΩ[γ]

δγ(x xprime) (73)




(1

β0minus 1

β


where


δΩHxc[γ]

δγ(x xprime)(75)


δγ(x xprime) (76)


δnkδγ(xprime x)


δφk(y)

δγ(xprime x)=sum

l 6=k



δφlowastk(y)

δγ(xprime x)=sum

l 6=k




gij =




heff ij = δij

(partΩ[γ]

partni+ micro0 +

σiβ0

)+


intdy

(δΩ[γ]

δφi(y)φj(y)minus

δΩ[γ]


) (711)


σi =partS0[γ]

partni= ln

(1minus nini

) (712)

63135

Numerical treatment









Ω[n1 n2] =α1

2(n1 minus 05)2 +

α2

2(n2 minus 05)2 (713)

= Ω1[n1] + Ω2[n2] (714)









64135

Numerical treatment

0

2

4

6

8

10

0 02 04 06 08 1

n1

Ω[n

1]

Ω1[n1]Ωeff1[n1]


0

005

010

015

020

0 02 04 06 08 1

n2

Ω[n

2]

Ω2[n2]Ωeff2[n2]




ni =1


εi minus micro =1

βiln

(1minus nini

) (720)


Ω[γ] =sum

i


1


)(721)

=sum

i

Ωi[ni βi] (722)


βi = ηpart2S0[γ]

partn2i

part2Ω[γ]partn2i

(723)

= η1

ni(1minus ni)

(part2Ω[γ]

partn2i

)minus1

(724)


βi =η

αi

1

ni(1minus ni) (725)


65135

Numerical treatment

0

2

4

6

8

10

0 02 04 06 08 1

n1

Ω[n

1]

Ω1[n1]

Ωeff1[n1]


0

005

010

015

020

0 02 04 06 08 1

n2

Ω[n

2]

Ω2[n2]

Ωeff2[n2]








j 6=i

heff ji



nprimei =1


66135

Numerical treatment


i n(k)i φ


2) fromn(k)i

and

φ(k)i

calculate H

(k)eff by Eq (711)


ε(k+1)i

and

φ(k+1)i (x)

4) constructn(k+1)i


(k+1)0 so that

sumi n

(k+1)i = N

5) fromφ(k+1)i (x)

and

n(k+1)i





nprimei =ni


(partΩpartni

minus∆micro

) (729)



= βini(ni minus 1)

(partΩ[γ]

partniminus∆(0)

micro

) (731)




∆(0)micro =



(732)

67135

Numerical treatment




heff ij (734)


∆Ω =

intdxdxprime

δΩ[γ]


=sum

ij

δΩ[γ]

δγij∆γji (736)

=sum

i

δnipartΩ[γ]

partni︸︷︷︸

∆Ω1

+sum

i6=j


|heff ij |2

︸︷︷︸∆Ω2

(737)




∆Ω1 =sum

i

δnipartΩ[γ]

partni(738)

=sum

i

βini(ni minus 1)

(partΩ[γ]

partniminus∆(0)

micro

)partΩ[γ]

partni(739)

=sum

i

βini(ni minus 1)

(partΩ[γ]

partni

)2

minus


partni

)2


(740)


∆Ω1 =

sum

j

βjnj(nj minus 1)

sum

i

ci

(partΩ[γ]

partni

)2

minus(sum

k

ckpartΩ[γ]

partnk

)2 (741)

=

sum

j

βjnj(nj minus 1)

sum

i

ci

(partΩ[γ]

partniminussum

k

ckpartΩ[γ]

partnk

)2

(742)


∆Ω1 le 0 (743)



∆Ω2 =sum

i6=j


|heff ij |2 (744)

68135

Numerical treatment


∆Ω2 =sum

i6=j

ni minus nj1βi

ln(

1minusni

ni

)minus 1

βjln(

1minusnj

nj

) |heff ij |2 (745)


∆Ω2 = β0sum

i6=j

ni minus njln(


) |heff ij |2 (746)


∆Ω2 le 0 (747)








69135

Numerical treatment



to the ONs

χ2n =

1

Nunpinned

Nunpinnedsum

i

(partΩ

partniminus micro

)2

(748)

micro =1

Nunpinned

Nunpinnedsum

i

partΩ

partni(749)



χ2φ =

1

N minus 1


i ε2i

(750)





Eαxc[γ] = minus

1

2

sum

ij

nαi nαj



prime) (751)






70135

Numerical treatment

minus794

minus793

minus792

minus791

minus790

minus789

minus788

0 5 10 15 20

it

E(Ha)

τ = 20β0 = 30η = 10


1eminus25

1eminus20

1eminus15

1eminus10

1eminus05

1

0 20 40 60 80 100

it

χ2 n

τ = 20β0 = 30η = 10


minus808

minus806

minus804

minus802

minus800

0 5 10 15 20

it

E(Ha)

τ = 20β0 = 30η = 10


1eminus30

1eminus25

1eminus20

1eminus15

1eminus10

1eminus05

1

0 20 40 60 80 100

it

χ2 n τ = 20

β0 = 30η = 10


minus812

minus810

minus808

minus806

minus804

minus802

0 5 10 15 20

it

E(Ha)

τ = 10β0 = 30η = 10


1eminus30

1eminus25

1eminus20

1eminus15

1eminus10

1eminus05

1

0 20 40 60 80 100

it

χ2 n

τ = 10β0 = 30η = 10



71135

Numerical treatment





cycle


72135

Numerical treatment

minus79445

minus79440

minus79435

minus79430

minus79425

0 100 200 300 400 500

it

E(Ha)

τ = 10

β0 = 20


1eminus24

1eminus20

1eminus16

1eminus12

1eminus08

1eminus04

0 100 200 300 400 500

it

χ2 φ

τ = 10β0 = 20


minus8085

minus8084

minus8083

minus8082

minus8081

minus8080

0 100 200 300 400 500

it

E(Ha)

τ = 10

β0 = 10


1eminus13

1eminus11

1eminus09

1eminus07

1eminus05

0 100 200 300 400 500

it

χ2 φ

τ = 10

β0 = 10


minus81285

minus81280

minus81275

minus81270

minus81265

minus81260

minus81255

0 100 200 300 400 500

it

E(Ha) τ = 10

β0 = 10


1eminus13

1eminus11

1eminus09

1eminus07

1eminus05

0 100 200 300 400 500

it

χ2 φ

τ = 10

β0 = 10



73135

Numerical treatment

minus81284

minus81282

minus81280

minus81278

minus81276

minus81274

minus81272

0 100 200 300 400 500

it

E(Ha)

β0 = 10β0 = 20β0 = 50


1eminus13

1eminus11

1eminus09

1eminus07

1eminus05

0 100 200 300 400 500

it

χ2 φ

β0 = 10

β0 = 20

β0 = 50











74135

8 Applications


Section Description








75135

Applications








76135

Applications

minus40

minus50

minus45

5

10

15

20

0

02

08

06

04

10

rs(au)

ξF(10minus

2Ha)





S[D] =sum

i

wi lnwi (82)




77135

Applications










FMPP

PM

r s(au)

T (K)

rc (rTc Tc)

TF


78135

Applications





n =

intd3k



Ωkσ =1

n

intd3k

(2π)3k2

2nσ(k) (84)

Ωextσ =1

n

intd3k


nσ =1

n

intd3k

(2π)3nσ(k) (86)

S0σ = minus 1

n

intd3k


F =sum

σ








s We therefore set



ΩSW = grminus1

s (1minus ξ2) (811)

79135

Applications

FM

PP

PM

0 20 40 6010 30 50

60

80

100

120

140

40

160

200

180

r s(au)

T (K)

rc(rc Tc)

TF



83 FT-MBPT




n(k) =1


ε(k) =k2

2(814)


t =T

TF k =

k

kF(815)

80135

Applications



3 (816)


nσ(k) =1


(817)


2

3tminus

32 =

int infin

0

dx

radicx

1 + exminusα(818)


Fj(x) =1

Γ(j + 1)

int infin

0

dzzj

1 + ezminusx(819)


nσ =3

8π

12 t

32F 1

2(ασ) (820)

Ekσ =9

32π

12 k2F t

52F 3

2(ασ) (821)

microσ =1

2k2F tασ (822)

S0σ =

(15

16π

12 t

32F 3


)(823)


4

3πminus 1

2 tminus 3

2σ = F 1

2(ασ) (824)


831 Exchange-only


F = F0 +

int 1

0

dλ

λ

︸︷︷︸rarrΩx

(825)



Ωxσ = minus 1

n

intd3k

(2π)3d3q


81135

Applications

0

02

04

06

08

1

0 1 2 3 4 5

t

I x(t)

84a Exchange

0

05

1

15

2

25

3

0 1 2 3 4 5

t

I 2b(t)




W (k) =4π

k2 (828)



3


Ix(t) =

intdx

intdy

x

1 + ex2

tminusα(t)

y

1 + ey2

tminusα(t)

ln(x+ y)2

(xminus y)2 (830)




82135

Applications

0 2000 4000 6000 8000 100000

2

4

6

8

10

12

14

16

18

20

PP

PM

FMr s(au)

T (K)

rc(rTc Tc)

TF


832 Exchange + RPA


F = F0 +

int 1

0

dλ

λ

︸︷︷︸

rarrΩx

+

︸︷︷︸rarrΩr

(831)



Ωr =1

n

1

2β

intd3q

(2π)3

infinsum

a=minusinfin


) (833)


νa =2πa

β (834)

83135

Applications

0 2 4 6 8 10

minus05

minus04

minus03

minus02

minus01

00 2 4 6 8 10

minus025

minus020

minus015

minus010

0 2 4 6 8 10

minus009

minus008

minus007

minus006

minus005

minus004

0 2 4 6 8 10

minus0011

minus0010

minus0009

minus0008

tt

Ωr(t)

Ωr(t) Ω

r (t)Ω

r (t)

rs = 025 rs = 114

rs = 540 rs = 117




d3k




Ωr =k2F t

8π2

intdqq2

infinsum

a=minusinfin


)(836)


(2π)2kFσ

q

intdk

k


ln

((2πatσ)

2 + (q2 + 2kq)2


) (837)



84135

Applications

minus1eminus3

minus5eminus4

1eminus3

5eminus4

0

1 10 100

δ(r sξ)

rs(au)

ξ = 00

ξ = 02

ξ = 04

ξ = 06

ξ = 08

ξ = 10


0 50 100 200150 250

10

15

20

25

30

35

40

45

50

55

60

PP

PM

FM

300 350

r s(au)

T (K)

rc(rTc Tc)

TF




85135

Applications



F = F0 +

int 1

0

dλ

λ

︸︷︷︸

rarrΩx

+


+

︸︷︷︸rarrΩr

(838)



Ω2bσ =1

n2minus10πminus9

intd3k

intd3p

intd3q





2Ω2bσ(0)I2b(tσ) (841)

I2b(t) = minus3


intd3k

intd3p

intd3q


k2p2k middot p(842)

Ω2bσ(0) =

(ln 2

6minus 3

4π2ζ(3)

) (843)



86135

Applications

0 4000 8000 120000

4

8

12

16

20

24

28

32

36

40

16000FM

PP

PMr s(au)

T (K)

TF

(rTc Tc)rc




87135

Applications

minus15

minus10

minus5

0

5

minus8

minus6

minus4

minus2

0 05 10 15

minus7

minus6

minus5

minus4

minus3

0 05 10 15 20

3

4

5

6

7

F0(10minus

2Ha) Ω

x(10 minus

2Ha)

Ωr(10minus

2Ha) Ω

2b (10 minus

2Ha)

tt

ξ = 0ξ = 05ξ = 08ξ = 10


84 FT-RDMFT



F [γ] = F0[γ] +

int 1

0

dλ

λ

︸︷︷︸rarrΩx

(844)



88135

Applications




σi

nσi

int

Vi

d3k





Ωk[nσi] =sum

σi

nσiti (848)

S0[nσi] = minussum

σi


Ωx[nσi] = minus1

2

sum

σij

nσinσjKij (850)


ti =1

n

int

Vi

d3k

(2π)3k2

2(851)

Kij =1

n

int

Vi

dk31(2π)3

int

Vj

dk32(2π)3

4π


ωi =1

n

int

Vi

d3k

(2π)3 (853)







89135

Applications

4 6 8 10 12 14 160

02

04

06

08

10

T=0T=5000T=8000T=10000

ξ min

rs(au)


05

10

15

20

0

0 02 04 06 08

minus4

0 02 04 06 08 1

minus3

0

3

6

1

2

3

minus3

minus2

minus1

ξξ

Ωk(10minus

2Ha) Ω

x(10 minus

2Ha)S

∆F(10 minus

3Ha)

t = 0t = 10t = 16t = 22



∆1minus0 = Ω(ξ = 1)minus Ω(ξ = 0) (854)


90135

Applications

0

1

2

3

4

0 1 2 0 1 2 3

minus8

minus6

minus4

minus2

minus1

0

1

2

3

4

minus3

minus2

minus1

tt

Ωk(10minus

2Ha) Ω

x(10 minus

2Ha)

SF(10 minus

2Ha)

FD ξ = 0

FD ξ = 1

HF ξ = 0

HF ξ = 1

FD ∆1minus0

HF ∆1minus0










91135

Applications





92135

Applications




φ1k(r) =

(cos(Θk

2

)eminusiqmiddotr2

sin(Θk

2

)eiqmiddotr2

)eikmiddotrradicV

(855)

φ2k(r) =

(minus sin

(Θk

2

)eminusiqmiddotr2

cos(Θk

2

)eiqmiddotr2




intd3k

(2π)3

(n1k cos

2

(Θk

2

)+ n2k sin

2

(Θk

2



intd3k

(2π)31




intd3k

(2π)31




intd3k

(2π)3

(n1k sin

2

(Θk

2

)+ n2k cos

2

(Θk

2



Ωx[γ] = minus1

2

sum

σ1σ2




Ωx[γ] = minus1

2n

intd3k1(2π)3

d3k2(2π)3

4π

|k1 minus k2|2[sum

b

(nbk1nbk2


2

)+ (n1k1

n2k2+ n2k1

n1k2) sin2

(Θk1minusΘk2

2

)] (862)


93135

Applications


F [q nbi Θi] =sum

bi

nbiti +q2

8minus q

sum

i


1βsum

bi


)ωiminus

1

2

sum

bij

(nbinbj) cos2

(Θi minusΘj

2

)Kij minus

sum

ij

(n1in2j) sin2

(Θi minusΘj

2

)Kij (863)


Qi =1

2n

int

Vi

d3k

(2π)3kz (864)


m(r) = minus


B

(865)


A =1

2

intd3k


B =1

2

intd3k





(868)

Θkρplusmn|kz| =π

2(1∓ akρ|kz|) (869)


i n1i and N2 =sum

i n2i

ξPSS =N1 minusN2

N1 +N2(870)






94135

Applications

kρkρkρkρ

kzkzkzkz

qkF = 0 qkF = 10 qkF = 18 qkF = 2

kF kF kF kF

-kF -kF -kF -kF

q2

q2

q2

- q2- q2 - q2


5

6

7

minus112

minus105

minus98

minus91

0 05 1 15 2 25

0

0 05 1 15 2 25 3

minus49

minus48

minus47

minus46

5

10

15

qkFqkF

Ωk(10minus

2Ha)

Ωx(10 minus

2Ha)

S(10minus

2)

F(10 minus

2Ha)

T = 0KT = 2500KT = 4000KT = 5000K








95135

Applications

0

1

2

3

4

5

1 15 2 25minus494

minus493

minus492

minus491

minus490

minus489

qkF

F(10minus

2Ha)

A(10minus

4au)

T = 4700K t = 0244T = 4800K t = 0249T = 4900K t = 0254T = 5000K t = 0260






96135

Applications

0 1000 2000 3000 4000 50000

1

2

3

4

T (K)

Aopt(10minus

4au)

rs = 55rs = 50rs = 45rs = 40







rc(au) 40 45 50 55

Tc(K) 2700 3400 4100 5000


97135

Applications

0 2000 4000 6000 8000 100000

2

4

6

8

10

12

14

16

18

20

PSS

FMPP

PM

r s(au)

T (K)

rc(rTc Tc)

TPSSc

TF




εσ(k) =k2

2minus 2

πfd(kkFσ) (871)


f3(x) =1

2+

1minus x24x

log

∣∣∣∣1 + x

1minus x

∣∣∣∣ (872)


f2(x) =

E(x) x lt 1

x(E(1x

)minus(1minus 1

x2

)K(1x

)) x ge 1



partF

partni= 0 =

partΩk

partniminus 1

β

partS0

partni+partΩx

partni (873)

98135

Applications

minus4

minus3

minus2

minus1

0

1

2

3

4

0 05 1 15 2 25

kkF

ε σ(kk

F)ε F

k2200105

12510100


minus8

minus6

minus4

minus2

0

2

4

0 05 1 15 2 25

kkF

ε σ(kk

F)ε F

k220

0105

12510100

820b Two dimensions




partni+partΩx

partni (874)


1d where d is



e0b(k) =

(k plusmn q

2

)2

2minus q2

8=

1



99135

Applications

minus4

minus2

0

2

4

6

8

10

0 05 1 15 2

kkF

ε σ(kk

F)ε F

ni

T = 0

FM


minus4

minus2

0

2

4

6

8

10

0 05 1 15 2

kkF

ε(kk

F)ε F

091787









Ωc[γ] = ΩDFTc [n] (876)



nminγrarrn

F [γ] (877)

= minn



) (878)


100135

Applications





FTrarr0= Ωk[n


c (n) (879)



101135

Applications

0 1000 1500 2000 2500 30000

20

40

60

80

100

120

140

160

180

200

FM

PP

PM

500

T (K)

r s(au)

TF

(rTc Tc)rc




0 20 40 60 80 100 120 140

15

20

25

30

35

40

45

FM

PP

PM

T (K)

r s(au)

(rTc Tc)

rc


102135

Applications







Ωαxc[nσi] =

sum

σij









103135

Applications

α 1 9 8 7 6 55

rc(au) 556 545 539 528 470 asymp5Tc(K) 10500 9400 8500 7200 6500 4600

rTc (au) 710 725 775 825 900 950


0 05 1 15 2 250

02

04

06

08

1

kkf

nk

HF T = 0HFα = 08α = 0565ni







α 1 095 09 085

TPSSc (K) 4500 3000 2000 500


104135

Applications

0 1000 2000 3000 4000 5000

10

20

30

40

50

60

70

80

FM

PP

PM

T (K)

r s(au)

TF

(rTc Tc)

rc




Ω+c [γ] = minus

1

2





Ω+c [γ] = minus

1

2n

intd3k1(2π)3

d3k2(2π)3

4π


cos2(Θk1

2

)+ n2k1

sin2(Θk1

2

))(n2k2

cos2(Θk2

2

)+ n1k2

sin2(Θk2

2

))(883)


105135

Applications





ε(mk) =k2

2m(884)


n(tmk) =1

1 + ek2

mtminusα(mt)

= n(mt 1k) (885)


Ωk(tm) = Ωk(mt 1) (886)


N(tm) = N(mt 1) (888)

S0(tm) = S0(mt 1) (889)

Ωx(tm) = Ωx(mt 1) (890)



Ω2bσ(tm) = mΩ2bσ(mt 1) (892)



106135

Applications



WRPA(tm) =W





εσ(k) = ε0σ(k) =k2

2(895)



partn(k)




partn(k)

partε(k)

1 +



partε(k)partkjsum

i qipartε(k)partki

(897)

107135

Applications



intd3k

(2π)3nσ(k)(1minus nσ(k)) +O(q2) (898)


Ωr =1

β(2π)minus2




q gt q


r (8100)

Ω0r =

1

β(2π)minus2

int q

0

dqq2(W (q)χ(q)) (8101)

=4π

β(2π)minus2

int radicβ

0

dqχ(q)) (8102)

Ωinfinr = minus 1

β(2π)minus2

int infin

radicβ

dqq2(W (q)χ(q))2

2 (8103)


behaves like


κ ge 2 (8105)


χκ[n(k)](q) =sum

σ

χκσ[n(k)](q) (8106)


β

1 + (qkF )κ

intd3k

(2π)3nσ(k)(1minus nσ(k)) (8107)

= minus β


) (8108)

where

〈n2σ〉 =int

d3k

(2π)3n2σ(k) (8109)






108135

Applications

01 1 10 100

0

minus002

minus004

minus006

minus008

minus010

minus012

rs(au)

e c(Ha)

PWCAκ = 29κ = 27κ = 25κ = 23



0

02

04

06

08

10

0 05 10 15 20 25 30

kkF

n(kk

F)

rs = 1 GZ

rs = 5 GZ

rs = 10 GZ

rs = 1 κ

rs = 5 κ

rs = 10 κ


KAPPA-functional

ξ 00 01 02 03 04 05 06 07 08 09 10

κ 29 29 29 30 30 30 31 31 31 32 32


109135

Applications

κ c

KAPPA-TIE 32 024


0 200 400 600 800 1000

10

20

30

40

50

60

70

80

01200

FM

PP

PM

T (K)

r s(au)

TF

(rTc Tc)





1 + q2k2F

intd3k







110135

Applications







111135

Applications

(1) (2) (3) (4) (5) (6) rc


FT-MBPT




FT-RDMFT








112135

A APPENDIX




γ(x xprime) =sum

i


=sum

i

niφlowasti (x

prime)φi(x) (A2)


|ψi〉 =sum

l

cij |χj〉 (A3)



D =sum

ijk



γ(x xprime) =sum

i

sum

jk


φlowasti (x

prime)φi(x) (A6)


γ(x xprime) =sum

i

sum

jk


φlowasti (x

prime)φi(x) (A7)


P (A) =sum

B

P (A|B)P (B) (A8)


γ(x xprime) =sum

i

P (φi)φlowasti (x

prime)φi(x) (A9)


113135

Appendix



ni =1




γ(x xprime) =sum

i

niφlowasti (x

prime)φi(x) (A11)


ni = c+i ci (A12)


written as

ni = TrDni =sum

e



Ψe =sum

α

ceαΦα (A14)


sum

e

|ceα|2 =sum

α

|ceα|2 = 1 (A15)


ni =sum

e

we

sum

αβ



ni =sum

α

(sum

e

we|ceα|2)

︸︷︷︸fα


(A17)



fα gt 0 (A18)sum

α

fα = 1 (A19)

114135

Appendix


0 lt ni lt 1 (A20)




H = T + V + W (A21)

γgs(x xprime) =

sum

i

niφlowasti (x

prime)φi(x) (A22)


E[γ] = 〈Ψ[γ]|H|Ψ[γ]〉 (A23)



partE[γ]

partni

∣∣∣∣γgs

=

ai gt micro ni = 0

micro 0 lt ni lt 1

bi lt micro ni = 1

(A24)



δE[γ]

δγ(x xprime)

∣∣∣∣γgs

= microδN [γ]

δγ(x xprime)

∣∣∣∣γgs




δEu[γ]

δγ(x xprime)

∣∣∣∣γgs




115135

Appendix





Eα[γ] = E[γ] +



partEα[γ]

partni

∣∣∣∣γgs

=

ai + uδiα ni = 0


bi + uδiα ni = 1

(A30)





Z=sum

i



H =



γ =


116135

Appendix


γ(x xprime) =sum

i


=sum

i

niφlowasti (x

prime)φi(x) (A35)


Ψ(x) =sum

i

ciφi(x) (A36)


γ(x xprime) = Tr


ZΨ+(xprime)Ψ(x)

(A37)

=1

Z

sum

i


sum

n1n2



=1

Z

sum

i


sum

n1n2


=1

Z

sum

i


prod

j

sum

nj


〈ni|e


(A40)


γ(x xprime) =1

Z

sum

i


prod

j


1




(A42)

=sum

n1n2

〈n1 n2 |prod

i


=prod

i

sum

ni


=prod

i



γ(x xprime) =sum

i

niφlowasti (x

prime)φi(x) (A46)

ni =1


117135

Appendix


εi minus micro =1

βln

(1minus nini

) (A48)


Ω = minus 1

βlnZ (A49)


Ω0 = minus 1

β

sum

i


)(A50)

=1

β

sum

i

ln(1minus ni) (A51)


S =part

partTΩ

∣∣∣∣micro

(A52)


part

partTni

∣∣∣∣micro

=1

Tni(1minus ni) ln

(1minus nini

)(A53)


S0 = minussum

i


A5 Feynman rules







0dτ




118135

Appendix



Ω2b =1

n2minus10πminus9

intd3k

intd3p

intd3q



=1

n2minus6πminus7

intd3k

intd3p

intd3q




krarr q (A57)


q rarr k (A59)

Ω2b = minus1

n2minus6πminus7

intd3k

intd3p

intd3q


k2p2k middot p(A60)


Ω2b = minus1

n2minus2πminus5

int infin

0

dk

int infin

0

dp

int infin

0

dq

int π

0

dθp

int π

0

dθq

int π2

minusπ2

dφq




F (kp q θp) =

int π

0


int π2

minusπ2



G(k p q) = q2fq

int π2

0

dθp tan(π2+ θp

)(F(k p q

π

2+ θq

)minus F

(k p q

π

2minus θq

)) (A63)


Ω2b = minus1

n2minus2πminus5

int infin

minusinfindk

int infin

minusinfindp

int infin

0

dqG(ek ep q) (A64)






Ω2b = minus1

n2minus2πminus5

int infin

0

dxx2int π

0

dθx sin(θx)

int π2

minusπ2


119135

120135

B Bibliography

References






3865





PhysRevLett832628




PhysRevLett87133004


10106312987202


0304-8853(79)90333-0


PhysRevB5514975

121135

Bibliography



100257001





6216536










0921-4526(91)90075-P


dxdoiorg10106311906203

122135

Bibliography



PhysRevA79040501












322010


dxdoiorg1010631449326


101002qua560360864



123135

Bibliography



1103PhysRev971474




566






PhysRevB4513244



447879






124135

Bibliography




171257


101103PhysRevA64042512





org101098rspa20021027



235121




0375-9601(84)91034-X


00268970110070243


866


057

125135

Bibliography



PhysRevA77032509


106312899328









dxdoiorg1010631479623


0305-447089021



PhysRevE66036703




126135

Bibliography










47R1591




0378-4371(83)90254-6


pssb2221240140








org10106312888550

127135

Bibliography


1002jcc21225







PhysRevA34433





589











128135

Bibliography


0370-1573(82)90077-1








PhysRevE77021120



2064



softwaregsl




org101139p75-176



19664730108


129135

Bibliography







PhysRev139A796

130135








131135

132135

D Publications










gas Exchange only









133135

134135

E Acknowledgements











Tim

135135

Title

Abstract

Table of Contents

List of Figures

List of Tables


Introduction










BOW functional


Summary and outlook













Convexity



Summary and outlook






Summary and outlook






Scaling behaviour

Summary and outlook

Numerical treatment



Temperature tensor





Sample calculations


Full minimization

Summary and outlook

Applications



FT-MBPT

Exchange-only

Exchange + RPA


FT-RDMFT









Summary and outlook

APPENDIX





Feynman rules


Bibliography


Publications

Acknowledgements

Abstract






Table of Contents

List of Figures v

List of Tables vii


1 Introduction 1










i

TABLE OF CONTENTS














B Bibliography 121


D Publications 133

ii

TABLE OF CONTENTS


iii

List of Figures









v

LIST OF FIGURES


vi

List of Tables







vii













EQ Equilibrium

FM Ferromagnetic







GS Ground state



HF Hartree-Fock

HS Hilbert-Schmidt







ix


MC Monte-Carlo


NO Natural orbitals



PM Paramagnetic







SC Self-consistent




SW Spin waves



x

1 Introduction







1135

1 INTRODUCTION











2135

1 INTRODUCTION




3135

4135



Section Description


2223






H = T + W + Vext (21)


intdx =

sumσ



T =

intdx lim

xprimerarrx

(minusnabla

2

2


Vext =



W =



w(x xprime) =1




D =sum


i

wi = 1 (26)



(27)

5135

Prerequisites

T

T1

DN


T

TradicN

ΓN




N =




(29)


AHS =

radicTrA+A (210)


T1 =A | AHS le 1

(211)






γ(x xprime) =sum

i

niφlowasti (x

prime)φi(x) (213)

6135

Prerequisites


Φi(r) =

(φi1(r)

φi2(r)

) (214)


γ(r rprime) =sum

i

niΦdaggeri (r


=

(γ11(r r



prime)

)=sum

i

ni

(φlowasti1(r



prime)φi2(r)

)(216)


Φi1(r) =

(φi1(r)

0

)Φi2(r) =

(0

φi2(r)

)(217)



sum

i

niσφlowastiσ(r





sumi niφ

lowasti (x



i

ni = N (220)





(222)

7135

Prerequisites




radicN

TradicN =

A | AHS le

radicN

(223)





n(x) = γ(x x) (224)


nσ(r) = γσσ(r r) (225)





rs =

(3

4πn

)13

(226)

8135

Prerequisites


kF =(3π2n

) 13 =

(9

4π

) 13

rminus1s (227)

εF =k2F2

(228)

TF =εFkB

(229)



(230)









Enuc [n(r)] =



Vext σσprime =



9135

10135




Section Description








D = |Ψgs〉〈Ψgs| (31)


E = Ek[γ] + Vext[γ] +W [Γ(2)] (32)

Ek[γ] =

intdx lim

xprimerarrx

(minusnabla

2

2

)γ(xprime x) (33)

Vext[γ] =



W [Γ(2)] =




2TrDΨ+(x4)Ψ

+(x3)Ψ+(x1)Ψ

+(x2) (36)

11135

ZT-RDMFT


γ(x xprime) =2

N minus 1

intdx2Γ

(2)(x x2xprime x2) (37)



W [Γ(2)] =W [γ] (38)





prime) (310)






12135

ZT-RDMFT




W [γ] = EH [γ] + Ex[γ] + Ec[γ] (311)

EH [γ] =1

2


Ex[γ] = minus1

2








13135

ZT-RDMFT


Exc[γ] = Ex[γ] + Ec[γ] (315)

Exc[γ] = minus1

2

sum

ij

f(ni nj)



prime) (316)











14135

ZT-RDMFT



intd3RεHEG

c [γ(kR)] (317)



γ(kR) =















15135

ZT-RDMFT

10010101001

minus030

minus020

minus010

0

minus005

minus015

minus025

rs(au)

e c(Ha)


MLα (α = 055)




33 BOW functional




16135

ZT-RDMFT

0

02

04

06

08

10

0 02 04 06 08 10

ninj

f

fHF






2

sum

ij

fBOW (ni nj α)



prime) (319)








17135

ZT-RDMFT

00 05 10 15 2000

02

04

06

08

10

kkF

n(kk

F)

rs = 05

rs = 1

GZBOW

GZBOW

00 05 10 15 2000

02

04

06

08

10

kkF

n(kk

F)

rs = 5

rs = 10

GZBOW

GZBOW


10010101001

minus08

minus07

minus06

minus05

minus04

minus03

minus02

minus01

0


MLα (α = 06)

rs(au)

e c(Ha)




18135

ZT-RDMFT










E(nuarr ndarr) =1







19135

ZT-RDMFT

001 01 1 10 100minus010

minus008

minus006

minus004

minus002

0

rs(au)

∆F(Ha)

ξ00204060810


001 01 1 10 100

0

002

004

006

008

010

rs(au)

∆P(Ha)

ξ00204060810


0

01

02

03

04

001 01 1 10 100rs(au)

δF

00204

060810


minus08

minus06

minus04

minus02

0

001 01 1 10 100rs(au)

δP

00204

060810





δF =∆F

Wc(328)

δP =∆P

Wc (329)



20135

ZT-RDMFT

3D

ξ 00 01 02 03 04 05 06 07 08 09 10

α 056 056 056 057 057 057 058 059 061 063 066

BOW 061 061 061 062 062 063 063 064 065 067 069

2D

ξ 00 01 02 03 04 05 06 07 08 09 10

α 063 063 063 063 064 064 065 066 067 07 074

BOW 066 066 066 067 067 068 069 071 073 076 080





2

sum

ijσ


csum

ij

((niuarrnjdarr)

αU



21135

ZT-RDMFT

αP αU c

BOW-TIE 070 20 019









22135




Section Description







23135

FT-RDMFT


H


∆Nmicro

∆E β



H =

infinoplus

n=0

Shotimesn (41)










24135

FT-RDMFT

F [D]

D

downwards convex

upwards convex

42a Convexity

F [D]

D








γ(x xprime) = Tr

eminusβ(T+V+W )

ZΨ+(xprime)Ψ(x)

(48)



(49)



T (see Section 23)





25135

FT-RDMFT





F [Dk] (412)













S[A B] = supλ

((1λ)


))(419)




S[Ak Bk] (420)

26135

FT-RDMFT


TrA = supP

TrP A (421)

TrP A le PTrA (422)






S[A B] = supP λ

((1λ)tr


))(425)


S[Ak Bk] (426)





Ω[Dk] (427)


S[Dk] (428)






27135

FT-RDMFT

Deq

Deq

Deq

γeq

γeq neq

neq

neq

+ =rArr










U =


βlnZ

Z prime (430)





minus 1

βlnZ

Z prime = u11 = u22 = u11 + u22 (431)



28135

FT-RDMFT







eq) (433)


leads to



eq) (435)
















Ω[γeq] =







Ωeq = minDisinDN

Ω[D] (443)

29135

FT-RDMFT


Ωeq = minγisinΓN

infDisinDNrarrγ

Ω[D[γ]] (444)


Ωeq = minγisinΓN

(F[γ] +





TrD(T + W + 1β ln D) (446)


Ω[γ] = F[γ] +




Ωeq = minγisinγN

Ω[γ] (448)



δF [γ]

δγ(xprime x)

∣∣∣∣γeq




30135

FT-RDMFT





H(1)s =

sum

i

εi|φi〉〈φi| (450)


γeq(x xprime) =

sum

i

niφlowasti (x

prime)φi(x) (451)


ni =1



εi minus micro =1

βln

(1minus nini

) (453)



veff (x xprime) =

sum

ij






31135

FT-RDMFT


Ωk[γ] =

intdxprime lim

xrarrxprime

(minusnabla

2

2

)γ(x xprime) (456)

N [γ] =

intdxγ(x x) (457)

Ωext[γ] =



ΩH [γ] =1

2


Ωx[γ] = minus1

2


S0[γ] = minussum

i







Ω[γ] =

intdxprime lim

xrarrxprime

(minusnabla

2

2

)γ(x xprime) +

intdxvext(x x


1

2


2


1βsum

i



δΩ[γ]



0 =

intdxprimeφi(x

prime)δΩ[γ]

δγ(xprime x)(465)

=

(minusnabla

2

2

)φi(x) +

intdxprimevext(x x



prime)+



32135

FT-RDMFT





Ω[γ] = F [γ] +
















L sub DN )]


F [γ] = TrD(H0 + 1β ln D) (470)


33135

FT-RDMFT


limkrarrinfin

TrDk(H0 + 1β ln Dk) = F [γ] (471)












Mf =









34135

FT-RDMFT



infinsum

i=M+1









TrD(H0 + 1β ln D) = F [γ] (479)






F [γk] (480)


F [γ] le TrD(H0 + 1β ln D) (481)


F [γk] (482)

35135

FT-RDMFT

433 Convexity









infγisinΓN

(F [γ] +

intv0γ

)= F [γ0] +

intv0γ0 (484)






36135

FT-RDMFT








F [γ] +

intvγ ge F [γ0] +

intvγ0 (487)

infγisinΓN

(F [γ] +

intvγ

)ge F [γ0] +

intvγ0 (488)



F [γ0] +

intv0γ0 = inf

γisinΓN

(F [γ] +

intv0γ

)(489)






37135

FT-RDMFT





intv0γ



1 γk rarr γ




∆0 = F [γ0] +

intv0γ0 minus inf

γisinΓN

F [γ] +

intv0γ

(492)








38135

FT-RDMFT




Dc =sum

α


α

wαN = 1 (495)


F [D] = TrD(H + 1β ln D) (496)


Dceq =

eminusβH

TreminusβH (497)



TrD(T + W + 1β ln D) (498)


F [γ] = Fc[γ] + Vext[γ] (499)



F [γ] = Ω[γ] + microN [γ] (4100)

39135

FT-RDMFT

T

T1

DN

DV

DNL


T

TradicN

ΓN

ΓV








DNL




40135

FT-RDMFT







1minus1larrrarr ΓN




solved by

leads to

solved by

leads to

solved by

enables

used for

allows


41135

42135



Section Description









Ω0[D0] lt Ω0[Dw] (52)








43135

Exact properties


Sc[γ] lt 0


Ωc[γ] lt 0

(58)

(59)

(510)




Hλ = T + V0 + λW (511)




)(512)

= minDrarrγ


)minus Tr

DV λ[γ]

) (513)





)minus Tr

D[γ]V λ[γ]

(514)


dΩλ[γ]

dλ= Tr

Dλ[γ]

(W +

d

dλV λ[γ]

)minus d

dλTrD[γ]V λ[γ]

(515)


Ω[γ] = Ω0[γ] +

int 1

0

dλTrDλ[γ]W

(516)


Ωc[γ] =

int 1

0

dλ

λWλ

c [γ] (517)



44135

Exact properties

minus4 minus2 2 4

minus02

minus01

01

02

z

Ψ210(0 0 z)

51a Wavefunction

minus4 minus2 2 4

001

002

003

004

0z

ρ210(0 0 z)

51b Density







ψλ(r1 rN ) = 〈r1 rN |U(λ)|ψ〉 (518)

= λ3N2ψ(λr1 λrN ) (519)


45135

Exact properties


〈ψλ|Oλ|ψλ〉 = 〈ψ|O|ψ〉 (520)


Tλ =1

λ2T (521)

Wλ =1

λW (522)

Nλ = N (523)


T [γλ] = λ2Ωk[γ] (524)

ΩH [γλ] = λΩH [γ] (525)

Ωx[γλ] = λΩx[γ] (526)

S0[γλ] = S0[γ] (527)





get


λ[γλ]) (529)

TrD 1λ[γλ](W 1

λ+ τ ln D 1


λ+ τ ln D[γ]) (530)











(535)

(536)

46135

Exact properties





(537)

(538)

(539)


0 =d

dλ

(Ω[D 1

λ[γλ]]

)∣∣∣λ=1

(540)


0 =d

dλTr

D 1

λ[γλ]

(1

λW 1

λ+ τ ln(D 1

λ[γλ]

)∣∣∣∣λ=1

(541)


W [γ] =d



Wc[γλ] = λd

dλΩc[γλ] (543)


Ωc[γλ] =

int λ

0

dmicro



Sc[γλ] = λ2(part

partλ

1

λΩc[γλ]

)(545)


d

dλτSc[γλ] = λ

d

dλ

(1

λWc[γλ]

) (546)


d

dλΩc[γλ] =

1

λWc[γλ] lt 0

d

dλWc[γλ] lt

d

dλΩc[γλ] lt 0

d

dλSc[γλ] lt 0

(547)

(548)

(549)

47135

Exact properties

Ωc[γλ]

λ1

minusτSc[γ]

0


Ωc[γλ]




d2


1

λ2Wc[γλ] +

1

λ

d


1

λ2Wc[γλ] +

1

λ

d

dλΩc[γλ] (550)

d2








Dλ(τ w)[γ] = D(λ2τ λw)[γλ] (553)

48135

Exact properties




λ2Sc(τλ

2 wλ)[γ]


(554)

(555)

(556)


Ωc(τ w)[γ] =

int 1

0

dλ

λWc(τ λw)[γ] (557)

Ωc(τ w)[γ] =

int 1

0








49135

50135



Section Description










prime) =sumi εiφ

lowasti (x



i

φi(x)φlowasti (x


ni ν gt νprime




i

niφlowasti (x

prime)φi(x) (62)

51135


W + Veff




ni =1


εi =1

βln

(1minus nini

)+ micro (64)




52135




Ω[γ] = Ω0[γ] +

int 1

0

dλTrDλ[γ]W

(65)




〈λW 〉λ =1

2

intdxdxprime lim

νprimerarrν+





〈λW 〉λ =1

2

intdxdxprime lim

νprimerarrν+




1

2

intd3xd3xprime


)γ(x xprime) (68)






〈λW 〉unifλ =1

2

Σlowast minus

(610)


53135


λw(r rprime)

vλ(x xprime)






veff = v(1)eff + v

(2)eff + (611)



0

0+

=

0

0+

+

0

0+

Σlowast (612)


0 =

0

0+

Σlowast (613)

54135


+ + + +

+ + + + + +

+ + + +

+ + + + + +



(1)

0

0+

= minus

0

0+

minus

0+

0

(614)

55135


intdydyprime

int β

0


= minusintdydyprime

int β

0


int β

0



v(1)eff (x x



= minus minus (617)



potential v(2)eff

(2)

0

0+

= minus

0+

0

minus

0+

0

(618)


(2)

0

0+

= minus

0

0+

M (619)


intdydyprime

int β

0


= minusintdydyprime

int β

0


56135



v(2)eff =

sumij φi(x)φ

lowastj (x

prime)int β



int β

0eν


where

Mij(ν νprime) =








M =

sumij φi(x)φ

lowastj (x

prime)int β



int β

0eν








Ω(1)xc [γ] = + (625)


Ωxc[γ] =1

2


2


57135



Ω(2)xc [γ] = + + + (627)







Ω[γ]

γΓV ΓNΓV ΓV

(γV EV

) (γN EN

)


Ω[γ]

γΓN ΓV

(γV EV

) (γN EN

)



58135






prime) (628)




bull 2n+1 G0 lines






c [γ] (631)



c [γ] = nΩ(n)c [γλ] (632)


c [γ] = (nminus 1)Ω(n)c [γλ] (633)









part




part


59135





prime) (640)






c (τ w)[γ] (643)



c (τ w)[γ] (644)





60135



Section Description










61135

Numerical treatment

γ

F [γ]

γ0γ1γ2γ3

Ω(0)effΩ

(1)effΩ

(2)eff

Ω








62135

Numerical treatment






δΩeff [γ]

δγ(x xprime)=

δΩ[γ]

δγ(x xprime) (73)




(1

β0minus 1

β


where


δΩHxc[γ]

δγ(x xprime)(75)


δγ(x xprime) (76)


δnkδγ(xprime x)


δφk(y)

δγ(xprime x)=sum

l 6=k



δφlowastk(y)

δγ(xprime x)=sum

l 6=k




gij =




heff ij = δij

(partΩ[γ]

partni+ micro0 +

σiβ0

)+


intdy

(δΩ[γ]

δφi(y)φj(y)minus

δΩ[γ]


) (711)


σi =partS0[γ]

partni= ln

(1minus nini

) (712)

63135

Numerical treatment









Ω[n1 n2] =α1

2(n1 minus 05)2 +

α2

2(n2 minus 05)2 (713)

= Ω1[n1] + Ω2[n2] (714)









64135

Numerical treatment

0

2

4

6

8

10

0 02 04 06 08 1

n1

Ω[n

1]

Ω1[n1]Ωeff1[n1]


0

005

010

015

020

0 02 04 06 08 1

n2

Ω[n

2]

Ω2[n2]Ωeff2[n2]




ni =1


εi minus micro =1

βiln

(1minus nini

) (720)


Ω[γ] =sum

i


1


)(721)

=sum

i

Ωi[ni βi] (722)


βi = ηpart2S0[γ]

partn2i

part2Ω[γ]partn2i

(723)

= η1

ni(1minus ni)

(part2Ω[γ]

partn2i

)minus1

(724)


βi =η

αi

1

ni(1minus ni) (725)


65135

Numerical treatment

0

2

4

6

8

10

0 02 04 06 08 1

n1

Ω[n

1]

Ω1[n1]

Ωeff1[n1]


0

005

010

015

020

0 02 04 06 08 1

n2

Ω[n

2]

Ω2[n2]

Ωeff2[n2]








j 6=i

heff ji



nprimei =1


66135

Numerical treatment


i n(k)i φ


2) fromn(k)i

and

φ(k)i

calculate H

(k)eff by Eq (711)


ε(k+1)i

and

φ(k+1)i (x)

4) constructn(k+1)i


(k+1)0 so that

sumi n

(k+1)i = N

5) fromφ(k+1)i (x)

and

n(k+1)i





nprimei =ni


(partΩpartni

minus∆micro

) (729)



= βini(ni minus 1)

(partΩ[γ]

partniminus∆(0)

micro

) (731)




∆(0)micro =



(732)

67135

Numerical treatment




heff ij (734)


∆Ω =

intdxdxprime

δΩ[γ]


=sum

ij

δΩ[γ]

δγij∆γji (736)

=sum

i

δnipartΩ[γ]

partni︸︷︷︸

∆Ω1

+sum

i6=j


|heff ij |2

︸︷︷︸∆Ω2

(737)




∆Ω1 =sum

i

δnipartΩ[γ]

partni(738)

=sum

i

βini(ni minus 1)

(partΩ[γ]

partniminus∆(0)

micro

)partΩ[γ]

partni(739)

=sum

i

βini(ni minus 1)

(partΩ[γ]

partni

)2

minus


partni

)2


(740)


∆Ω1 =

sum

j

βjnj(nj minus 1)

sum

i

ci

(partΩ[γ]

partni

)2

minus(sum

k

ckpartΩ[γ]

partnk

)2 (741)

=

sum

j

βjnj(nj minus 1)

sum

i

ci

(partΩ[γ]

partniminussum

k

ckpartΩ[γ]

partnk

)2

(742)


∆Ω1 le 0 (743)



∆Ω2 =sum

i6=j


|heff ij |2 (744)

68135

Numerical treatment


∆Ω2 =sum

i6=j

ni minus nj1βi

ln(

1minusni

ni

)minus 1

βjln(

1minusnj

nj

) |heff ij |2 (745)


∆Ω2 = β0sum

i6=j

ni minus njln(


) |heff ij |2 (746)


∆Ω2 le 0 (747)








69135

Numerical treatment



to the ONs

χ2n =

1

Nunpinned

Nunpinnedsum

i

(partΩ

partniminus micro

)2

(748)

micro =1

Nunpinned

Nunpinnedsum

i

partΩ

partni(749)



χ2φ =

1

N minus 1


i ε2i

(750)





Eαxc[γ] = minus

1

2

sum

ij

nαi nαj



prime) (751)






70135

Numerical treatment

minus794

minus793

minus792

minus791

minus790

minus789

minus788

0 5 10 15 20

it

E(Ha)

τ = 20β0 = 30η = 10


1eminus25

1eminus20

1eminus15

1eminus10

1eminus05

1

0 20 40 60 80 100

it

χ2 n

τ = 20β0 = 30η = 10


minus808

minus806

minus804

minus802

minus800

0 5 10 15 20

it

E(Ha)

τ = 20β0 = 30η = 10


1eminus30

1eminus25

1eminus20

1eminus15

1eminus10

1eminus05

1

0 20 40 60 80 100

it

χ2 n τ = 20

β0 = 30η = 10


minus812

minus810

minus808

minus806

minus804

minus802

0 5 10 15 20

it

E(Ha)

τ = 10β0 = 30η = 10


1eminus30

1eminus25

1eminus20

1eminus15

1eminus10

1eminus05

1

0 20 40 60 80 100

it

χ2 n

τ = 10β0 = 30η = 10



71135

Numerical treatment





cycle


72135

Numerical treatment

minus79445

minus79440

minus79435

minus79430

minus79425

0 100 200 300 400 500

it

E(Ha)

τ = 10

β0 = 20


1eminus24

1eminus20

1eminus16

1eminus12

1eminus08

1eminus04

0 100 200 300 400 500

it

χ2 φ

τ = 10β0 = 20


minus8085

minus8084

minus8083

minus8082

minus8081

minus8080

0 100 200 300 400 500

it

E(Ha)

τ = 10

β0 = 10


1eminus13

1eminus11

1eminus09

1eminus07

1eminus05

0 100 200 300 400 500

it

χ2 φ

τ = 10

β0 = 10


minus81285

minus81280

minus81275

minus81270

minus81265

minus81260

minus81255

0 100 200 300 400 500

it

E(Ha) τ = 10

β0 = 10


1eminus13

1eminus11

1eminus09

1eminus07

1eminus05

0 100 200 300 400 500

it

χ2 φ

τ = 10

β0 = 10



73135

Numerical treatment

minus81284

minus81282

minus81280

minus81278

minus81276

minus81274

minus81272

0 100 200 300 400 500

it

E(Ha)

β0 = 10β0 = 20β0 = 50


1eminus13

1eminus11

1eminus09

1eminus07

1eminus05

0 100 200 300 400 500

it

χ2 φ

β0 = 10

β0 = 20

β0 = 50











74135

8 Applications


Section Description








75135

Applications








76135

Applications

minus40

minus50

minus45

5

10

15

20

0

02

08

06

04

10

rs(au)

ξF(10minus

2Ha)





S[D] =sum

i

wi lnwi (82)




77135

Applications










FMPP

PM

r s(au)

T (K)

rc (rTc Tc)

TF


78135

Applications





n =

intd3k



Ωkσ =1

n

intd3k

(2π)3k2

2nσ(k) (84)

Ωextσ =1

n

intd3k


nσ =1

n

intd3k

(2π)3nσ(k) (86)

S0σ = minus 1

n

intd3k


F =sum

σ








s We therefore set



ΩSW = grminus1

s (1minus ξ2) (811)

79135

Applications

FM

PP

PM

0 20 40 6010 30 50

60

80

100

120

140

40

160

200

180

r s(au)

T (K)

rc(rc Tc)

TF



83 FT-MBPT




n(k) =1


ε(k) =k2

2(814)


t =T

TF k =

k

kF(815)

80135

Applications



3 (816)


nσ(k) =1


(817)


2

3tminus

32 =

int infin

0

dx

radicx

1 + exminusα(818)


Fj(x) =1

Γ(j + 1)

int infin

0

dzzj

1 + ezminusx(819)


nσ =3

8π

12 t

32F 1

2(ασ) (820)

Ekσ =9

32π

12 k2F t

52F 3

2(ασ) (821)

microσ =1

2k2F tασ (822)

S0σ =

(15

16π

12 t

32F 3


)(823)


4

3πminus 1

2 tminus 3

2σ = F 1

2(ασ) (824)


831 Exchange-only


F = F0 +

int 1

0

dλ

λ

︸︷︷︸rarrΩx

(825)



Ωxσ = minus 1

n

intd3k

(2π)3d3q


81135

Applications

0

02

04

06

08

1

0 1 2 3 4 5

t

I x(t)

84a Exchange

0

05

1

15

2

25

3

0 1 2 3 4 5

t

I 2b(t)




W (k) =4π

k2 (828)



3


Ix(t) =

intdx

intdy

x

1 + ex2

tminusα(t)

y

1 + ey2

tminusα(t)

ln(x+ y)2

(xminus y)2 (830)




82135

Applications

0 2000 4000 6000 8000 100000

2

4

6

8

10

12

14

16

18

20

PP

PM

FMr s(au)

T (K)

rc(rTc Tc)

TF


832 Exchange + RPA


F = F0 +

int 1

0

dλ

λ

︸︷︷︸

rarrΩx

+

︸︷︷︸rarrΩr

(831)



Ωr =1

n

1

2β

intd3q

(2π)3

infinsum

a=minusinfin


) (833)


νa =2πa

β (834)

83135

Applications

0 2 4 6 8 10

minus05

minus04

minus03

minus02

minus01

00 2 4 6 8 10

minus025

minus020

minus015

minus010

0 2 4 6 8 10

minus009

minus008

minus007

minus006

minus005

minus004

0 2 4 6 8 10

minus0011

minus0010

minus0009

minus0008

tt

Ωr(t)

Ωr(t) Ω

r (t)Ω

r (t)

rs = 025 rs = 114

rs = 540 rs = 117




d3k




Ωr =k2F t

8π2

intdqq2

infinsum

a=minusinfin


)(836)


(2π)2kFσ

q

intdk

k


ln

((2πatσ)

2 + (q2 + 2kq)2


) (837)



84135

Applications

minus1eminus3

minus5eminus4

1eminus3

5eminus4

0

1 10 100

δ(r sξ)

rs(au)

ξ = 00

ξ = 02

ξ = 04

ξ = 06

ξ = 08

ξ = 10


0 50 100 200150 250

10

15

20

25

30

35

40

45

50

55

60

PP

PM

FM

300 350

r s(au)

T (K)

rc(rTc Tc)

TF




85135

Applications



F = F0 +

int 1

0

dλ

λ

︸︷︷︸

rarrΩx

+


+

︸︷︷︸rarrΩr

(838)



Ω2bσ =1

n2minus10πminus9

intd3k

intd3p

intd3q





2Ω2bσ(0)I2b(tσ) (841)

I2b(t) = minus3


intd3k

intd3p

intd3q


k2p2k middot p(842)

Ω2bσ(0) =

(ln 2

6minus 3

4π2ζ(3)

) (843)



86135

Applications

0 4000 8000 120000

4

8

12

16

20

24

28

32

36

40

16000FM

PP

PMr s(au)

T (K)

TF

(rTc Tc)rc




87135

Applications

minus15

minus10

minus5

0

5

minus8

minus6

minus4

minus2

0 05 10 15

minus7

minus6

minus5

minus4

minus3

0 05 10 15 20

3

4

5

6

7

F0(10minus

2Ha) Ω

x(10 minus

2Ha)

Ωr(10minus

2Ha) Ω

2b (10 minus

2Ha)

tt

ξ = 0ξ = 05ξ = 08ξ = 10


84 FT-RDMFT



F [γ] = F0[γ] +

int 1

0

dλ

λ

︸︷︷︸rarrΩx

(844)



88135

Applications




σi

nσi

int

Vi

d3k





Ωk[nσi] =sum

σi

nσiti (848)

S0[nσi] = minussum

σi


Ωx[nσi] = minus1

2

sum

σij

nσinσjKij (850)


ti =1

n

int

Vi

d3k

(2π)3k2

2(851)

Kij =1

n

int

Vi

dk31(2π)3

int

Vj

dk32(2π)3

4π


ωi =1

n

int

Vi

d3k

(2π)3 (853)







89135

Applications

4 6 8 10 12 14 160

02

04

06

08

10

T=0T=5000T=8000T=10000

ξ min

rs(au)


05

10

15

20

0

0 02 04 06 08

minus4

0 02 04 06 08 1

minus3

0

3

6

1

2

3

minus3

minus2

minus1

ξξ

Ωk(10minus

2Ha) Ω

x(10 minus

2Ha)S

∆F(10 minus

3Ha)

t = 0t = 10t = 16t = 22



∆1minus0 = Ω(ξ = 1)minus Ω(ξ = 0) (854)


90135

Applications

0

1

2

3

4

0 1 2 0 1 2 3

minus8

minus6

minus4

minus2

minus1

0

1

2

3

4

minus3

minus2

minus1

tt

Ωk(10minus

2Ha) Ω

x(10 minus

2Ha)

SF(10 minus

2Ha)

FD ξ = 0

FD ξ = 1

HF ξ = 0

HF ξ = 1

FD ∆1minus0

HF ∆1minus0










91135

Applications





92135

Applications




φ1k(r) =

(cos(Θk

2

)eminusiqmiddotr2

sin(Θk

2

)eiqmiddotr2

)eikmiddotrradicV

(855)

φ2k(r) =

(minus sin

(Θk

2

)eminusiqmiddotr2

cos(Θk

2

)eiqmiddotr2




intd3k

(2π)3

(n1k cos

2

(Θk

2

)+ n2k sin

2

(Θk

2



intd3k

(2π)31




intd3k

(2π)31




intd3k

(2π)3

(n1k sin

2

(Θk

2

)+ n2k cos

2

(Θk

2



Ωx[γ] = minus1

2

sum

σ1σ2




Ωx[γ] = minus1

2n

intd3k1(2π)3

d3k2(2π)3

4π

|k1 minus k2|2[sum

b

(nbk1nbk2


2

)+ (n1k1

n2k2+ n2k1

n1k2) sin2

(Θk1minusΘk2

2

)] (862)


93135

Applications


F [q nbi Θi] =sum

bi

nbiti +q2

8minus q

sum

i


1βsum

bi


)ωiminus

1

2

sum

bij

(nbinbj) cos2

(Θi minusΘj

2

)Kij minus

sum

ij

(n1in2j) sin2

(Θi minusΘj

2

)Kij (863)


Qi =1

2n

int

Vi

d3k

(2π)3kz (864)


m(r) = minus


B

(865)


A =1

2

intd3k


B =1

2

intd3k





(868)

Θkρplusmn|kz| =π

2(1∓ akρ|kz|) (869)


i n1i and N2 =sum

i n2i

ξPSS =N1 minusN2

N1 +N2(870)






94135

Applications

kρkρkρkρ

kzkzkzkz

qkF = 0 qkF = 10 qkF = 18 qkF = 2

kF kF kF kF

-kF -kF -kF -kF

q2

q2

q2

- q2- q2 - q2


5

6

7

minus112

minus105

minus98

minus91

0 05 1 15 2 25

0

0 05 1 15 2 25 3

minus49

minus48

minus47

minus46

5

10

15

qkFqkF

Ωk(10minus

2Ha)

Ωx(10 minus

2Ha)

S(10minus

2)

F(10 minus

2Ha)

T = 0KT = 2500KT = 4000KT = 5000K








95135

Applications

0

1

2

3

4

5

1 15 2 25minus494

minus493

minus492

minus491

minus490

minus489

qkF

F(10minus

2Ha)

A(10minus

4au)

T = 4700K t = 0244T = 4800K t = 0249T = 4900K t = 0254T = 5000K t = 0260






96135

Applications

0 1000 2000 3000 4000 50000

1

2

3

4

T (K)

Aopt(10minus

4au)

rs = 55rs = 50rs = 45rs = 40







rc(au) 40 45 50 55

Tc(K) 2700 3400 4100 5000


97135

Applications

0 2000 4000 6000 8000 100000

2

4

6

8

10

12

14

16

18

20

PSS

FMPP

PM

r s(au)

T (K)

rc(rTc Tc)

TPSSc

TF




εσ(k) =k2

2minus 2

πfd(kkFσ) (871)


f3(x) =1

2+

1minus x24x

log

∣∣∣∣1 + x

1minus x

∣∣∣∣ (872)


f2(x) =

E(x) x lt 1

x(E(1x

)minus(1minus 1

x2

)K(1x

)) x ge 1



partF

partni= 0 =

partΩk

partniminus 1

β

partS0

partni+partΩx

partni (873)

98135

Applications

minus4

minus3

minus2

minus1

0

1

2

3

4

0 05 1 15 2 25

kkF

ε σ(kk

F)ε F

k2200105

12510100


minus8

minus6

minus4

minus2

0

2

4

0 05 1 15 2 25

kkF

ε σ(kk

F)ε F

k220

0105

12510100

820b Two dimensions




partni+partΩx

partni (874)


1d where d is



e0b(k) =

(k plusmn q

2

)2

2minus q2

8=

1



99135

Applications

minus4

minus2

0

2

4

6

8

10

0 05 1 15 2

kkF

ε σ(kk

F)ε F

ni

T = 0

FM


minus4

minus2

0

2

4

6

8

10

0 05 1 15 2

kkF

ε(kk

F)ε F

091787









Ωc[γ] = ΩDFTc [n] (876)



nminγrarrn

F [γ] (877)

= minn



) (878)


100135

Applications





FTrarr0= Ωk[n


c (n) (879)



101135

Applications

0 1000 1500 2000 2500 30000

20

40

60

80

100

120

140

160

180

200

FM

PP

PM

500

T (K)

r s(au)

TF

(rTc Tc)rc




0 20 40 60 80 100 120 140

15

20

25

30

35

40

45

FM

PP

PM

T (K)

r s(au)

(rTc Tc)

rc


102135

Applications







Ωαxc[nσi] =

sum

σij









103135

Applications

α 1 9 8 7 6 55

rc(au) 556 545 539 528 470 asymp5Tc(K) 10500 9400 8500 7200 6500 4600

rTc (au) 710 725 775 825 900 950


0 05 1 15 2 250

02

04

06

08

1

kkf

nk

HF T = 0HFα = 08α = 0565ni







α 1 095 09 085

TPSSc (K) 4500 3000 2000 500


104135

Applications

0 1000 2000 3000 4000 5000

10

20

30

40

50

60

70

80

FM

PP

PM

T (K)

r s(au)

TF

(rTc Tc)

rc




Ω+c [γ] = minus

1

2





Ω+c [γ] = minus

1

2n

intd3k1(2π)3

d3k2(2π)3

4π


cos2(Θk1

2

)+ n2k1

sin2(Θk1

2

))(n2k2

cos2(Θk2

2

)+ n1k2

sin2(Θk2

2

))(883)


105135

Applications





ε(mk) =k2

2m(884)


n(tmk) =1

1 + ek2

mtminusα(mt)

= n(mt 1k) (885)


Ωk(tm) = Ωk(mt 1) (886)


N(tm) = N(mt 1) (888)

S0(tm) = S0(mt 1) (889)

Ωx(tm) = Ωx(mt 1) (890)



Ω2bσ(tm) = mΩ2bσ(mt 1) (892)



106135

Applications



WRPA(tm) =W





εσ(k) = ε0σ(k) =k2

2(895)



partn(k)




partn(k)

partε(k)

1 +



partε(k)partkjsum

i qipartε(k)partki

(897)

107135

Applications



intd3k

(2π)3nσ(k)(1minus nσ(k)) +O(q2) (898)


Ωr =1

β(2π)minus2




q gt q


r (8100)

Ω0r =

1

β(2π)minus2

int q

0

dqq2(W (q)χ(q)) (8101)

=4π

β(2π)minus2

int radicβ

0

dqχ(q)) (8102)

Ωinfinr = minus 1

β(2π)minus2

int infin

radicβ

dqq2(W (q)χ(q))2

2 (8103)


behaves like


κ ge 2 (8105)


χκ[n(k)](q) =sum

σ

χκσ[n(k)](q) (8106)


β

1 + (qkF )κ

intd3k

(2π)3nσ(k)(1minus nσ(k)) (8107)

= minus β


) (8108)

where

〈n2σ〉 =int

d3k

(2π)3n2σ(k) (8109)






108135

Applications

01 1 10 100

0

minus002

minus004

minus006

minus008

minus010

minus012

rs(au)

e c(Ha)

PWCAκ = 29κ = 27κ = 25κ = 23



0

02

04

06

08

10

0 05 10 15 20 25 30

kkF

n(kk

F)

rs = 1 GZ

rs = 5 GZ

rs = 10 GZ

rs = 1 κ

rs = 5 κ

rs = 10 κ


KAPPA-functional

ξ 00 01 02 03 04 05 06 07 08 09 10

κ 29 29 29 30 30 30 31 31 31 32 32


109135

Applications

κ c

KAPPA-TIE 32 024


0 200 400 600 800 1000

10

20

30

40

50

60

70

80

01200

FM

PP

PM

T (K)

r s(au)

TF

(rTc Tc)





1 + q2k2F

intd3k







110135

Applications







111135

Applications

(1) (2) (3) (4) (5) (6) rc


FT-MBPT




FT-RDMFT








112135

A APPENDIX




γ(x xprime) =sum

i


=sum

i

niφlowasti (x

prime)φi(x) (A2)


|ψi〉 =sum

l

cij |χj〉 (A3)



D =sum

ijk



γ(x xprime) =sum

i

sum

jk


φlowasti (x

prime)φi(x) (A6)


γ(x xprime) =sum

i

sum

jk


φlowasti (x

prime)φi(x) (A7)


P (A) =sum

B

P (A|B)P (B) (A8)


γ(x xprime) =sum

i

P (φi)φlowasti (x

prime)φi(x) (A9)


113135

Appendix



ni =1




γ(x xprime) =sum

i

niφlowasti (x

prime)φi(x) (A11)


ni = c+i ci (A12)


written as

ni = TrDni =sum

e



Ψe =sum

α

ceαΦα (A14)


sum

e

|ceα|2 =sum

α

|ceα|2 = 1 (A15)


ni =sum

e

we

sum

αβ



ni =sum

α

(sum

e

we|ceα|2)

︸︷︷︸fα


(A17)



fα gt 0 (A18)sum

α

fα = 1 (A19)

114135

Appendix


0 lt ni lt 1 (A20)




H = T + V + W (A21)

γgs(x xprime) =

sum

i

niφlowasti (x

prime)φi(x) (A22)


E[γ] = 〈Ψ[γ]|H|Ψ[γ]〉 (A23)



partE[γ]

partni

∣∣∣∣γgs

=

ai gt micro ni = 0

micro 0 lt ni lt 1

bi lt micro ni = 1

(A24)



δE[γ]

δγ(x xprime)

∣∣∣∣γgs

= microδN [γ]

δγ(x xprime)

∣∣∣∣γgs




δEu[γ]

δγ(x xprime)

∣∣∣∣γgs




115135

Appendix





Eα[γ] = E[γ] +



partEα[γ]

partni

∣∣∣∣γgs

=

ai + uδiα ni = 0


bi + uδiα ni = 1

(A30)





Z=sum

i



H =



γ =


116135

Appendix


γ(x xprime) =sum

i


=sum

i

niφlowasti (x

prime)φi(x) (A35)


Ψ(x) =sum

i

ciφi(x) (A36)


γ(x xprime) = Tr


ZΨ+(xprime)Ψ(x)

(A37)

=1

Z

sum

i


sum

n1n2



=1

Z

sum

i


sum

n1n2


=1

Z

sum

i


prod

j

sum

nj


〈ni|e


(A40)


γ(x xprime) =1

Z

sum

i


prod

j


1




(A42)

=sum

n1n2

〈n1 n2 |prod

i


=prod

i

sum

ni


=prod

i



γ(x xprime) =sum

i

niφlowasti (x

prime)φi(x) (A46)

ni =1


117135

Appendix


εi minus micro =1

βln

(1minus nini

) (A48)


Ω = minus 1

βlnZ (A49)


Ω0 = minus 1

β

sum

i


)(A50)

=1

β

sum

i

ln(1minus ni) (A51)


S =part

partTΩ

∣∣∣∣micro

(A52)


part

partTni

∣∣∣∣micro

=1

Tni(1minus ni) ln

(1minus nini

)(A53)


S0 = minussum

i


A5 Feynman rules







0dτ




118135

Appendix



Ω2b =1

n2minus10πminus9

intd3k

intd3p

intd3q



=1

n2minus6πminus7

intd3k

intd3p

intd3q




krarr q (A57)


q rarr k (A59)

Ω2b = minus1

n2minus6πminus7

intd3k

intd3p

intd3q


k2p2k middot p(A60)


Ω2b = minus1

n2minus2πminus5

int infin

0

dk

int infin

0

dp

int infin

0

dq

int π

0

dθp

int π

0

dθq

int π2

minusπ2

dφq




F (kp q θp) =

int π

0


int π2

minusπ2



G(k p q) = q2fq

int π2

0

dθp tan(π2+ θp

)(F(k p q

π

2+ θq

)minus F

(k p q

π

2minus θq

)) (A63)


Ω2b = minus1

n2minus2πminus5

int infin

minusinfindk

int infin

minusinfindp

int infin

0

dqG(ek ep q) (A64)






Ω2b = minus1

n2minus2πminus5

int infin

0

dxx2int π

0

dθx sin(θx)

int π2

minusπ2


119135

120135

B Bibliography

References






3865





PhysRevLett832628




PhysRevLett87133004


10106312987202


0304-8853(79)90333-0


PhysRevB5514975

121135

Bibliography



100257001





6216536










0921-4526(91)90075-P


dxdoiorg10106311906203

122135

Bibliography



PhysRevA79040501












322010


dxdoiorg1010631449326


101002qua560360864



123135

Bibliography



1103PhysRev971474




566






PhysRevB4513244



447879






124135

Bibliography




171257


101103PhysRevA64042512





org101098rspa20021027



235121




0375-9601(84)91034-X


00268970110070243


866


057

125135

Bibliography



PhysRevA77032509


106312899328









dxdoiorg1010631479623


0305-447089021



PhysRevE66036703




126135

Bibliography










47R1591




0378-4371(83)90254-6


pssb2221240140








org10106312888550

127135

Bibliography


1002jcc21225







PhysRevA34433





589











128135

Bibliography


0370-1573(82)90077-1








PhysRevE77021120



2064



softwaregsl




org101139p75-176



19664730108


129135

Bibliography







PhysRev139A796

130135








131135

132135

D Publications










gas Exchange only









133135

134135

E Acknowledgements











Tim

135135

Title

Abstract

Table of Contents

List of Figures

List of Tables


Introduction










BOW functional


Summary and outlook













Convexity



Summary and outlook






Summary and outlook






Scaling behaviour

Summary and outlook

Numerical treatment



Temperature tensor





Sample calculations


Full minimization

Summary and outlook

Applications



FT-MBPT

Exchange-only

Exchange + RPA


FT-RDMFT









Summary and outlook

APPENDIX





Feynman rules


Bibliography


Publications

Acknowledgements

Table of Contents

List of Figures v

List of Tables vii


1 Introduction 1










i

TABLE OF CONTENTS














B Bibliography 121


D Publications 133

ii

TABLE OF CONTENTS


iii

List of Figures









v

LIST OF FIGURES


vi

List of Tables







vii













EQ Equilibrium

FM Ferromagnetic







GS Ground state



HF Hartree-Fock

HS Hilbert-Schmidt







ix


MC Monte-Carlo


NO Natural orbitals



PM Paramagnetic







SC Self-consistent




SW Spin waves



x

1 Introduction







1135

1 INTRODUCTION











2135

1 INTRODUCTION




3135

4135



Section Description


2223






H = T + W + Vext (21)


intdx =

sumσ



T =

intdx lim

xprimerarrx

(minusnabla

2

2


Vext =



W =



w(x xprime) =1




D =sum


i

wi = 1 (26)



(27)

5135

Prerequisites

T

T1

DN


T

TradicN

ΓN




N =




(29)


AHS =

radicTrA+A (210)


T1 =A | AHS le 1

(211)






γ(x xprime) =sum

i

niφlowasti (x

prime)φi(x) (213)

6135

Prerequisites


Φi(r) =

(φi1(r)

φi2(r)

) (214)


γ(r rprime) =sum

i

niΦdaggeri (r


=

(γ11(r r



prime)

)=sum

i

ni

(φlowasti1(r



prime)φi2(r)

)(216)


Φi1(r) =

(φi1(r)

0

)Φi2(r) =

(0

φi2(r)

)(217)



sum

i

niσφlowastiσ(r





sumi niφ

lowasti (x



i

ni = N (220)





(222)

7135

Prerequisites




radicN

TradicN =

A | AHS le

radicN

(223)





n(x) = γ(x x) (224)


nσ(r) = γσσ(r r) (225)





rs =

(3

4πn

)13

(226)

8135

Prerequisites


kF =(3π2n

) 13 =

(9

4π

) 13

rminus1s (227)

εF =k2F2

(228)

TF =εFkB

(229)



(230)









Enuc [n(r)] =



Vext σσprime =



9135

10135




Section Description








D = |Ψgs〉〈Ψgs| (31)


E = Ek[γ] + Vext[γ] +W [Γ(2)] (32)

Ek[γ] =

intdx lim

xprimerarrx

(minusnabla

2

2

)γ(xprime x) (33)

Vext[γ] =



W [Γ(2)] =




2TrDΨ+(x4)Ψ

+(x3)Ψ+(x1)Ψ

+(x2) (36)

11135

ZT-RDMFT


γ(x xprime) =2

N minus 1

intdx2Γ

(2)(x x2xprime x2) (37)



W [Γ(2)] =W [γ] (38)





prime) (310)






12135

ZT-RDMFT




W [γ] = EH [γ] + Ex[γ] + Ec[γ] (311)

EH [γ] =1

2


Ex[γ] = minus1

2








13135

ZT-RDMFT


Exc[γ] = Ex[γ] + Ec[γ] (315)

Exc[γ] = minus1

2

sum

ij

f(ni nj)



prime) (316)











14135

ZT-RDMFT



intd3RεHEG

c [γ(kR)] (317)



γ(kR) =















15135

ZT-RDMFT

10010101001

minus030

minus020

minus010

0

minus005

minus015

minus025

rs(au)

e c(Ha)


MLα (α = 055)




33 BOW functional




16135

ZT-RDMFT

0

02

04

06

08

10

0 02 04 06 08 10

ninj

f

fHF






2

sum

ij

fBOW (ni nj α)



prime) (319)








17135

ZT-RDMFT

00 05 10 15 2000

02

04

06

08

10

kkF

n(kk

F)

rs = 05

rs = 1

GZBOW

GZBOW

00 05 10 15 2000

02

04

06

08

10

kkF

n(kk

F)

rs = 5

rs = 10

GZBOW

GZBOW


10010101001

minus08

minus07

minus06

minus05

minus04

minus03

minus02

minus01

0


MLα (α = 06)

rs(au)

e c(Ha)




18135

ZT-RDMFT










E(nuarr ndarr) =1







19135

ZT-RDMFT

001 01 1 10 100minus010

minus008

minus006

minus004

minus002

0

rs(au)

∆F(Ha)

ξ00204060810


001 01 1 10 100

0

002

004

006

008

010

rs(au)

∆P(Ha)

ξ00204060810


0

01

02

03

04

001 01 1 10 100rs(au)

δF

00204

060810


minus08

minus06

minus04

minus02

0

001 01 1 10 100rs(au)

δP

00204

060810





δF =∆F

Wc(328)

δP =∆P

Wc (329)



20135

ZT-RDMFT

3D

ξ 00 01 02 03 04 05 06 07 08 09 10

α 056 056 056 057 057 057 058 059 061 063 066

BOW 061 061 061 062 062 063 063 064 065 067 069

2D

ξ 00 01 02 03 04 05 06 07 08 09 10

α 063 063 063 063 064 064 065 066 067 07 074

BOW 066 066 066 067 067 068 069 071 073 076 080





2

sum

ijσ


csum

ij

((niuarrnjdarr)

αU



21135

ZT-RDMFT

αP αU c

BOW-TIE 070 20 019









22135




Section Description







23135

FT-RDMFT


H


∆Nmicro

∆E β



H =

infinoplus

n=0

Shotimesn (41)










24135

FT-RDMFT

F [D]

D

downwards convex

upwards convex

42a Convexity

F [D]

D








γ(x xprime) = Tr

eminusβ(T+V+W )

ZΨ+(xprime)Ψ(x)

(48)



(49)



T (see Section 23)





25135

FT-RDMFT





F [Dk] (412)













S[A B] = supλ

((1λ)


))(419)




S[Ak Bk] (420)

26135

FT-RDMFT


TrA = supP

TrP A (421)

TrP A le PTrA (422)






S[A B] = supP λ

((1λ)tr


))(425)


S[Ak Bk] (426)





Ω[Dk] (427)


S[Dk] (428)






27135

FT-RDMFT

Deq

Deq

Deq

γeq

γeq neq

neq

neq

+ =rArr










U =


βlnZ

Z prime (430)





minus 1

βlnZ

Z prime = u11 = u22 = u11 + u22 (431)



28135

FT-RDMFT







eq) (433)


leads to



eq) (435)
















Ω[γeq] =







Ωeq = minDisinDN

Ω[D] (443)

29135

FT-RDMFT


Ωeq = minγisinΓN

infDisinDNrarrγ

Ω[D[γ]] (444)


Ωeq = minγisinΓN

(F[γ] +





TrD(T + W + 1β ln D) (446)


Ω[γ] = F[γ] +




Ωeq = minγisinγN

Ω[γ] (448)



δF [γ]

δγ(xprime x)

∣∣∣∣γeq




30135

FT-RDMFT





H(1)s =

sum

i

εi|φi〉〈φi| (450)


γeq(x xprime) =

sum

i

niφlowasti (x

prime)φi(x) (451)


ni =1



εi minus micro =1

βln

(1minus nini

) (453)



veff (x xprime) =

sum

ij






31135

FT-RDMFT


Ωk[γ] =

intdxprime lim

xrarrxprime

(minusnabla

2

2

)γ(x xprime) (456)

N [γ] =

intdxγ(x x) (457)

Ωext[γ] =



ΩH [γ] =1

2


Ωx[γ] = minus1

2


S0[γ] = minussum

i







Ω[γ] =

intdxprime lim

xrarrxprime

(minusnabla

2

2

)γ(x xprime) +

intdxvext(x x


1

2


2


1βsum

i



δΩ[γ]



0 =

intdxprimeφi(x

prime)δΩ[γ]

δγ(xprime x)(465)

=

(minusnabla

2

2

)φi(x) +

intdxprimevext(x x



prime)+



32135

FT-RDMFT





Ω[γ] = F [γ] +
















L sub DN )]


F [γ] = TrD(H0 + 1β ln D) (470)


33135

FT-RDMFT


limkrarrinfin

TrDk(H0 + 1β ln Dk) = F [γ] (471)












Mf =









34135

FT-RDMFT



infinsum

i=M+1









TrD(H0 + 1β ln D) = F [γ] (479)






F [γk] (480)


F [γ] le TrD(H0 + 1β ln D) (481)


F [γk] (482)

35135

FT-RDMFT

433 Convexity









infγisinΓN

(F [γ] +

intv0γ

)= F [γ0] +

intv0γ0 (484)






36135

FT-RDMFT








F [γ] +

intvγ ge F [γ0] +

intvγ0 (487)

infγisinΓN

(F [γ] +

intvγ

)ge F [γ0] +

intvγ0 (488)



F [γ0] +

intv0γ0 = inf

γisinΓN

(F [γ] +

intv0γ

)(489)






37135

FT-RDMFT





intv0γ



1 γk rarr γ




∆0 = F [γ0] +

intv0γ0 minus inf

γisinΓN

F [γ] +

intv0γ

(492)








38135

FT-RDMFT




Dc =sum

α


α

wαN = 1 (495)


F [D] = TrD(H + 1β ln D) (496)


Dceq =

eminusβH

TreminusβH (497)



TrD(T + W + 1β ln D) (498)


F [γ] = Fc[γ] + Vext[γ] (499)



F [γ] = Ω[γ] + microN [γ] (4100)

39135

FT-RDMFT

T

T1

DN

DV

DNL


T

TradicN

ΓN

ΓV








DNL




40135

FT-RDMFT







1minus1larrrarr ΓN




solved by

leads to

solved by

leads to

solved by

enables

used for

allows


41135

42135



Section Description









Ω0[D0] lt Ω0[Dw] (52)








43135

Exact properties


Sc[γ] lt 0


Ωc[γ] lt 0

(58)

(59)

(510)




Hλ = T + V0 + λW (511)




)(512)

= minDrarrγ


)minus Tr

DV λ[γ]

) (513)





)minus Tr

D[γ]V λ[γ]

(514)


dΩλ[γ]

dλ= Tr

Dλ[γ]

(W +

d

dλV λ[γ]

)minus d

dλTrD[γ]V λ[γ]

(515)


Ω[γ] = Ω0[γ] +

int 1

0

dλTrDλ[γ]W

(516)


Ωc[γ] =

int 1

0

dλ

λWλ

c [γ] (517)



44135

Exact properties

minus4 minus2 2 4

minus02

minus01

01

02

z

Ψ210(0 0 z)

51a Wavefunction

minus4 minus2 2 4

001

002

003

004

0z

ρ210(0 0 z)

51b Density







ψλ(r1 rN ) = 〈r1 rN |U(λ)|ψ〉 (518)

= λ3N2ψ(λr1 λrN ) (519)


45135

Exact properties


〈ψλ|Oλ|ψλ〉 = 〈ψ|O|ψ〉 (520)


Tλ =1

λ2T (521)

Wλ =1

λW (522)

Nλ = N (523)


T [γλ] = λ2Ωk[γ] (524)

ΩH [γλ] = λΩH [γ] (525)

Ωx[γλ] = λΩx[γ] (526)

S0[γλ] = S0[γ] (527)





get


λ[γλ]) (529)

TrD 1λ[γλ](W 1

λ+ τ ln D 1


λ+ τ ln D[γ]) (530)











(535)

(536)

46135

Exact properties





(537)

(538)

(539)


0 =d

dλ

(Ω[D 1

λ[γλ]]

)∣∣∣λ=1

(540)


0 =d

dλTr

D 1

λ[γλ]

(1

λW 1

λ+ τ ln(D 1

λ[γλ]

)∣∣∣∣λ=1

(541)


W [γ] =d



Wc[γλ] = λd

dλΩc[γλ] (543)


Ωc[γλ] =

int λ

0

dmicro



Sc[γλ] = λ2(part

partλ

1

λΩc[γλ]

)(545)


d

dλτSc[γλ] = λ

d

dλ

(1

λWc[γλ]

) (546)


d

dλΩc[γλ] =

1

λWc[γλ] lt 0

d

dλWc[γλ] lt

d

dλΩc[γλ] lt 0

d

dλSc[γλ] lt 0

(547)

(548)

(549)

47135

Exact properties

Ωc[γλ]

λ1

minusτSc[γ]

0


Ωc[γλ]




d2


1

λ2Wc[γλ] +

1

λ

d


1

λ2Wc[γλ] +

1

λ

d

dλΩc[γλ] (550)

d2








Dλ(τ w)[γ] = D(λ2τ λw)[γλ] (553)

48135

Exact properties




λ2Sc(τλ

2 wλ)[γ]


(554)

(555)

(556)


Ωc(τ w)[γ] =

int 1

0

dλ

λWc(τ λw)[γ] (557)

Ωc(τ w)[γ] =

int 1

0








49135

50135



Section Description










prime) =sumi εiφ

lowasti (x



i

φi(x)φlowasti (x


ni ν gt νprime




i

niφlowasti (x

prime)φi(x) (62)

51135


W + Veff




ni =1


εi =1

βln

(1minus nini

)+ micro (64)




52135




Ω[γ] = Ω0[γ] +

int 1

0

dλTrDλ[γ]W

(65)




〈λW 〉λ =1

2

intdxdxprime lim

νprimerarrν+





〈λW 〉λ =1

2

intdxdxprime lim

νprimerarrν+




1

2

intd3xd3xprime


)γ(x xprime) (68)






〈λW 〉unifλ =1

2

Σlowast minus

(610)


53135


λw(r rprime)

vλ(x xprime)






veff = v(1)eff + v

(2)eff + (611)



0

0+

=

0

0+

+

0

0+

Σlowast (612)


0 =

0

0+

Σlowast (613)

54135


+ + + +

+ + + + + +

+ + + +

+ + + + + +



(1)

0

0+

= minus

0

0+

minus

0+

0

(614)

55135


intdydyprime

int β

0


= minusintdydyprime

int β

0


int β

0



v(1)eff (x x



= minus minus (617)



potential v(2)eff

(2)

0

0+

= minus

0+

0

minus

0+

0

(618)


(2)

0

0+

= minus

0

0+

M (619)


intdydyprime

int β

0


= minusintdydyprime

int β

0


56135



v(2)eff =

sumij φi(x)φ

lowastj (x

prime)int β



int β

0eν


where

Mij(ν νprime) =








M =

sumij φi(x)φ

lowastj (x

prime)int β



int β

0eν








Ω(1)xc [γ] = + (625)


Ωxc[γ] =1

2


2


57135



Ω(2)xc [γ] = + + + (627)







Ω[γ]

γΓV ΓNΓV ΓV

(γV EV

) (γN EN

)


Ω[γ]

γΓN ΓV

(γV EV

) (γN EN

)



58135






prime) (628)




bull 2n+1 G0 lines






c [γ] (631)



c [γ] = nΩ(n)c [γλ] (632)


c [γ] = (nminus 1)Ω(n)c [γλ] (633)









part




part


59135





prime) (640)






c (τ w)[γ] (643)



c (τ w)[γ] (644)





60135



Section Description










61135

Numerical treatment

γ

F [γ]

γ0γ1γ2γ3

Ω(0)effΩ

(1)effΩ

(2)eff

Ω








62135

Numerical treatment






δΩeff [γ]

δγ(x xprime)=

δΩ[γ]

δγ(x xprime) (73)




(1

β0minus 1

β


where


δΩHxc[γ]

δγ(x xprime)(75)


δγ(x xprime) (76)


δnkδγ(xprime x)


δφk(y)

δγ(xprime x)=sum

l 6=k



δφlowastk(y)

δγ(xprime x)=sum

l 6=k




gij =




heff ij = δij

(partΩ[γ]

partni+ micro0 +

σiβ0

)+


intdy

(δΩ[γ]

δφi(y)φj(y)minus

δΩ[γ]


) (711)


σi =partS0[γ]

partni= ln

(1minus nini

) (712)

63135

Numerical treatment









Ω[n1 n2] =α1

2(n1 minus 05)2 +

α2

2(n2 minus 05)2 (713)

= Ω1[n1] + Ω2[n2] (714)









64135

Numerical treatment

0

2

4

6

8

10

0 02 04 06 08 1

n1

Ω[n

1]

Ω1[n1]Ωeff1[n1]


0

005

010

015

020

0 02 04 06 08 1

n2

Ω[n

2]

Ω2[n2]Ωeff2[n2]




ni =1


εi minus micro =1

βiln

(1minus nini

) (720)


Ω[γ] =sum

i


1


)(721)

=sum

i

Ωi[ni βi] (722)


βi = ηpart2S0[γ]

partn2i

part2Ω[γ]partn2i

(723)

= η1

ni(1minus ni)

(part2Ω[γ]

partn2i

)minus1

(724)


βi =η

αi

1

ni(1minus ni) (725)


65135

Numerical treatment

0

2

4

6

8

10

0 02 04 06 08 1

n1

Ω[n

1]

Ω1[n1]

Ωeff1[n1]


0

005

010

015

020

0 02 04 06 08 1

n2

Ω[n

2]

Ω2[n2]

Ωeff2[n2]








j 6=i

heff ji



nprimei =1


66135

Numerical treatment


i n(k)i φ


2) fromn(k)i

and

φ(k)i

calculate H

(k)eff by Eq (711)


ε(k+1)i

and

φ(k+1)i (x)

4) constructn(k+1)i


(k+1)0 so that

sumi n

(k+1)i = N

5) fromφ(k+1)i (x)

and

n(k+1)i





nprimei =ni


(partΩpartni

minus∆micro

) (729)



= βini(ni minus 1)

(partΩ[γ]

partniminus∆(0)

micro

) (731)




∆(0)micro =



(732)

67135

Numerical treatment




heff ij (734)


∆Ω =

intdxdxprime

δΩ[γ]


=sum

ij

δΩ[γ]

δγij∆γji (736)

=sum

i

δnipartΩ[γ]

partni︸︷︷︸

∆Ω1

+sum

i6=j


|heff ij |2

︸︷︷︸∆Ω2

(737)




∆Ω1 =sum

i

δnipartΩ[γ]

partni(738)

=sum

i

βini(ni minus 1)

(partΩ[γ]

partniminus∆(0)

micro

)partΩ[γ]

partni(739)

=sum

i

βini(ni minus 1)

(partΩ[γ]

partni

)2

minus


partni

)2


(740)


∆Ω1 =

sum

j

βjnj(nj minus 1)

sum

i

ci

(partΩ[γ]

partni

)2

minus(sum

k

ckpartΩ[γ]

partnk

)2 (741)

=

sum

j

βjnj(nj minus 1)

sum

i

ci

(partΩ[γ]

partniminussum

k

ckpartΩ[γ]

partnk

)2

(742)


∆Ω1 le 0 (743)



∆Ω2 =sum

i6=j


|heff ij |2 (744)

68135

Numerical treatment


∆Ω2 =sum

i6=j

ni minus nj1βi

ln(

1minusni

ni

)minus 1

βjln(

1minusnj

nj

) |heff ij |2 (745)


∆Ω2 = β0sum

i6=j

ni minus njln(


) |heff ij |2 (746)


∆Ω2 le 0 (747)








69135

Numerical treatment



to the ONs

χ2n =

1

Nunpinned

Nunpinnedsum

i

(partΩ

partniminus micro

)2

(748)

micro =1

Nunpinned

Nunpinnedsum

i

partΩ

partni(749)



χ2φ =

1

N minus 1


i ε2i

(750)





Eαxc[γ] = minus

1

2

sum

ij

nαi nαj



prime) (751)






70135

Numerical treatment

minus794

minus793

minus792

minus791

minus790

minus789

minus788

0 5 10 15 20

it

E(Ha)

τ = 20β0 = 30η = 10


1eminus25

1eminus20

1eminus15

1eminus10

1eminus05

1

0 20 40 60 80 100

it

χ2 n

τ = 20β0 = 30η = 10


minus808

minus806

minus804

minus802

minus800

0 5 10 15 20

it

E(Ha)

τ = 20β0 = 30η = 10


1eminus30

1eminus25

1eminus20

1eminus15

1eminus10

1eminus05

1

0 20 40 60 80 100

it

χ2 n τ = 20

β0 = 30η = 10


minus812

minus810

minus808

minus806

minus804

minus802

0 5 10 15 20

it

E(Ha)

τ = 10β0 = 30η = 10


1eminus30

1eminus25

1eminus20

1eminus15

1eminus10

1eminus05

1

0 20 40 60 80 100

it

χ2 n

τ = 10β0 = 30η = 10



71135

Numerical treatment





cycle


72135

Numerical treatment

minus79445

minus79440

minus79435

minus79430

minus79425

0 100 200 300 400 500

it

E(Ha)

τ = 10

β0 = 20


1eminus24

1eminus20

1eminus16

1eminus12

1eminus08

1eminus04

0 100 200 300 400 500

it

χ2 φ

τ = 10β0 = 20


minus8085

minus8084

minus8083

minus8082

minus8081

minus8080

0 100 200 300 400 500

it

E(Ha)

τ = 10

β0 = 10


1eminus13

1eminus11

1eminus09

1eminus07

1eminus05

0 100 200 300 400 500

it

χ2 φ

τ = 10

β0 = 10


minus81285

minus81280

minus81275

minus81270

minus81265

minus81260

minus81255

0 100 200 300 400 500

it

E(Ha) τ = 10

β0 = 10


1eminus13

1eminus11

1eminus09

1eminus07

1eminus05

0 100 200 300 400 500

it

χ2 φ

τ = 10

β0 = 10



73135

Numerical treatment

minus81284

minus81282

minus81280

minus81278

minus81276

minus81274

minus81272

0 100 200 300 400 500

it

E(Ha)

β0 = 10β0 = 20β0 = 50


1eminus13

1eminus11

1eminus09

1eminus07

1eminus05

0 100 200 300 400 500

it

χ2 φ

β0 = 10

β0 = 20

β0 = 50











74135

8 Applications


Section Description








75135

Applications








76135

Applications

minus40

minus50

minus45

5

10

15

20

0

02

08

06

04

10

rs(au)

ξF(10minus

2Ha)





S[D] =sum

i

wi lnwi (82)




77135

Applications










FMPP

PM

r s(au)

T (K)

rc (rTc Tc)

TF


78135

Applications





n =

intd3k



Ωkσ =1

n

intd3k

(2π)3k2

2nσ(k) (84)

Ωextσ =1

n

intd3k


nσ =1

n

intd3k

(2π)3nσ(k) (86)

S0σ = minus 1

n

intd3k


F =sum

σ








s We therefore set



ΩSW = grminus1

s (1minus ξ2) (811)

79135

Applications

FM

PP

PM

0 20 40 6010 30 50

60

80

100

120

140

40

160

200

180

r s(au)

T (K)

rc(rc Tc)

TF



83 FT-MBPT




n(k) =1


ε(k) =k2

2(814)


t =T

TF k =

k

kF(815)

80135

Applications



3 (816)


nσ(k) =1


(817)


2

3tminus

32 =

int infin

0

dx

radicx

1 + exminusα(818)


Fj(x) =1

Γ(j + 1)

int infin

0

dzzj

1 + ezminusx(819)


nσ =3

8π

12 t

32F 1

2(ασ) (820)

Ekσ =9

32π

12 k2F t

52F 3

2(ασ) (821)

microσ =1

2k2F tασ (822)

S0σ =

(15

16π

12 t

32F 3


)(823)


4

3πminus 1

2 tminus 3

2σ = F 1

2(ασ) (824)


831 Exchange-only


F = F0 +

int 1

0

dλ

λ

︸︷︷︸rarrΩx

(825)



Ωxσ = minus 1

n

intd3k

(2π)3d3q


81135

Applications

0

02

04

06

08

1

0 1 2 3 4 5

t

I x(t)

84a Exchange

0

05

1

15

2

25

3

0 1 2 3 4 5

t

I 2b(t)




W (k) =4π

k2 (828)



3


Ix(t) =

intdx

intdy

x

1 + ex2

tminusα(t)

y

1 + ey2

tminusα(t)

ln(x+ y)2

(xminus y)2 (830)




82135

Applications

0 2000 4000 6000 8000 100000

2

4

6

8

10

12

14

16

18

20

PP

PM

FMr s(au)

T (K)

rc(rTc Tc)

TF


832 Exchange + RPA


F = F0 +

int 1

0

dλ

λ

︸︷︷︸

rarrΩx

+

︸︷︷︸rarrΩr

(831)



Ωr =1

n

1

2β

intd3q

(2π)3

infinsum

a=minusinfin


) (833)


νa =2πa

β (834)

83135

Applications

0 2 4 6 8 10

minus05

minus04

minus03

minus02

minus01

00 2 4 6 8 10

minus025

minus020

minus015

minus010

0 2 4 6 8 10

minus009

minus008

minus007

minus006

minus005

minus004

0 2 4 6 8 10

minus0011

minus0010

minus0009

minus0008

tt

Ωr(t)

Ωr(t) Ω

r (t)Ω

r (t)

rs = 025 rs = 114

rs = 540 rs = 117




d3k




Ωr =k2F t

8π2

intdqq2

infinsum

a=minusinfin


)(836)


(2π)2kFσ

q

intdk

k


ln

((2πatσ)

2 + (q2 + 2kq)2


) (837)



84135

Applications

minus1eminus3

minus5eminus4

1eminus3

5eminus4

0

1 10 100

δ(r sξ)

rs(au)

ξ = 00

ξ = 02

ξ = 04

ξ = 06

ξ = 08

ξ = 10


0 50 100 200150 250

10

15

20

25

30

35

40

45

50

55

60

PP

PM

FM

300 350

r s(au)

T (K)

rc(rTc Tc)

TF




85135

Applications



F = F0 +

int 1

0

dλ

λ

︸︷︷︸

rarrΩx

+


+

︸︷︷︸rarrΩr

(838)



Ω2bσ =1

n2minus10πminus9

intd3k

intd3p

intd3q





2Ω2bσ(0)I2b(tσ) (841)

I2b(t) = minus3


intd3k

intd3p

intd3q


k2p2k middot p(842)

Ω2bσ(0) =

(ln 2

6minus 3

4π2ζ(3)

) (843)



86135

Applications

0 4000 8000 120000

4

8

12

16

20

24

28

32

36

40

16000FM

PP

PMr s(au)

T (K)

TF

(rTc Tc)rc




87135

Applications

minus15

minus10

minus5

0

5

minus8

minus6

minus4

minus2

0 05 10 15

minus7

minus6

minus5

minus4

minus3

0 05 10 15 20

3

4

5

6

7

F0(10minus

2Ha) Ω

x(10 minus

2Ha)

Ωr(10minus

2Ha) Ω

2b (10 minus

2Ha)

tt

ξ = 0ξ = 05ξ = 08ξ = 10


84 FT-RDMFT



F [γ] = F0[γ] +

int 1

0

dλ

λ

︸︷︷︸rarrΩx

(844)



88135

Applications




σi

nσi

int

Vi

d3k





Ωk[nσi] =sum

σi

nσiti (848)

S0[nσi] = minussum

σi


Ωx[nσi] = minus1

2

sum

σij

nσinσjKij (850)


ti =1

n

int

Vi

d3k

(2π)3k2

2(851)

Kij =1

n

int

Vi

dk31(2π)3

int

Vj

dk32(2π)3

4π


ωi =1

n

int

Vi

d3k

(2π)3 (853)







89135

Applications

4 6 8 10 12 14 160

02

04

06

08

10

T=0T=5000T=8000T=10000

ξ min

rs(au)


05

10

15

20

0

0 02 04 06 08

minus4

0 02 04 06 08 1

minus3

0

3

6

1

2

3

minus3

minus2

minus1

ξξ

Ωk(10minus

2Ha) Ω

x(10 minus

2Ha)S

∆F(10 minus

3Ha)

t = 0t = 10t = 16t = 22



∆1minus0 = Ω(ξ = 1)minus Ω(ξ = 0) (854)


90135

Applications

0

1

2

3

4

0 1 2 0 1 2 3

minus8

minus6

minus4

minus2

minus1

0

1

2

3

4

minus3

minus2

minus1

tt

Ωk(10minus

2Ha) Ω

x(10 minus

2Ha)

SF(10 minus

2Ha)

FD ξ = 0

FD ξ = 1

HF ξ = 0

HF ξ = 1

FD ∆1minus0

HF ∆1minus0










91135

Applications





92135

Applications




φ1k(r) =

(cos(Θk

2

)eminusiqmiddotr2

sin(Θk

2

)eiqmiddotr2

)eikmiddotrradicV

(855)

φ2k(r) =

(minus sin

(Θk

2

)eminusiqmiddotr2

cos(Θk

2

)eiqmiddotr2




intd3k

(2π)3

(n1k cos

2

(Θk

2

)+ n2k sin

2

(Θk

2



intd3k

(2π)31




intd3k

(2π)31




intd3k

(2π)3

(n1k sin

2

(Θk

2

)+ n2k cos

2

(Θk

2



Ωx[γ] = minus1

2

sum

σ1σ2




Ωx[γ] = minus1

2n

intd3k1(2π)3

d3k2(2π)3

4π

|k1 minus k2|2[sum

b

(nbk1nbk2


2

)+ (n1k1

n2k2+ n2k1

n1k2) sin2

(Θk1minusΘk2

2

)] (862)


93135

Applications


F [q nbi Θi] =sum

bi

nbiti +q2

8minus q

sum

i


1βsum

bi


)ωiminus

1

2

sum

bij

(nbinbj) cos2

(Θi minusΘj

2

)Kij minus

sum

ij

(n1in2j) sin2

(Θi minusΘj

2

)Kij (863)


Qi =1

2n

int

Vi

d3k

(2π)3kz (864)


m(r) = minus


B

(865)


A =1

2

intd3k


B =1

2

intd3k





(868)

Θkρplusmn|kz| =π

2(1∓ akρ|kz|) (869)


i n1i and N2 =sum

i n2i

ξPSS =N1 minusN2

N1 +N2(870)






94135

Applications

kρkρkρkρ

kzkzkzkz

qkF = 0 qkF = 10 qkF = 18 qkF = 2

kF kF kF kF

-kF -kF -kF -kF

q2

q2

q2

- q2- q2 - q2


5

6

7

minus112

minus105

minus98

minus91

0 05 1 15 2 25

0

0 05 1 15 2 25 3

minus49

minus48

minus47

minus46

5

10

15

qkFqkF

Ωk(10minus

2Ha)

Ωx(10 minus

2Ha)

S(10minus

2)

F(10 minus

2Ha)

T = 0KT = 2500KT = 4000KT = 5000K








95135

Applications

0

1

2

3

4

5

1 15 2 25minus494

minus493

minus492

minus491

minus490

minus489

qkF

F(10minus

2Ha)

A(10minus

4au)

T = 4700K t = 0244T = 4800K t = 0249T = 4900K t = 0254T = 5000K t = 0260






96135

Applications

0 1000 2000 3000 4000 50000

1

2

3

4

T (K)

Aopt(10minus

4au)

rs = 55rs = 50rs = 45rs = 40







rc(au) 40 45 50 55

Tc(K) 2700 3400 4100 5000


97135

Applications

0 2000 4000 6000 8000 100000

2

4

6

8

10

12

14

16

18

20

PSS

FMPP

PM

r s(au)

T (K)

rc(rTc Tc)

TPSSc

TF




εσ(k) =k2

2minus 2

πfd(kkFσ) (871)


f3(x) =1

2+

1minus x24x

log

∣∣∣∣1 + x

1minus x

∣∣∣∣ (872)


f2(x) =

E(x) x lt 1

x(E(1x

)minus(1minus 1

x2

)K(1x

)) x ge 1



partF

partni= 0 =

partΩk

partniminus 1

β

partS0

partni+partΩx

partni (873)

98135

Applications

minus4

minus3

minus2

minus1

0

1

2

3

4

0 05 1 15 2 25

kkF

ε σ(kk

F)ε F

k2200105

12510100


minus8

minus6

minus4

minus2

0

2

4

0 05 1 15 2 25

kkF

ε σ(kk

F)ε F

k220

0105

12510100

820b Two dimensions




partni+partΩx

partni (874)


1d where d is



e0b(k) =

(k plusmn q

2

)2

2minus q2

8=

1



99135

Applications

minus4

minus2

0

2

4

6

8

10

0 05 1 15 2

kkF

ε σ(kk

F)ε F

ni

T = 0

FM


minus4

minus2

0

2

4

6

8

10

0 05 1 15 2

kkF

ε(kk

F)ε F

091787









Ωc[γ] = ΩDFTc [n] (876)



nminγrarrn

F [γ] (877)

= minn



) (878)


100135

Applications





FTrarr0= Ωk[n


c (n) (879)



101135

Applications

0 1000 1500 2000 2500 30000

20

40

60

80

100

120

140

160

180

200

FM

PP

PM

500

T (K)

r s(au)

TF

(rTc Tc)rc




0 20 40 60 80 100 120 140

15

20

25

30

35

40

45

FM

PP

PM

T (K)

r s(au)

(rTc Tc)

rc


102135

Applications







Ωαxc[nσi] =

sum

σij









103135

Applications

α 1 9 8 7 6 55

rc(au) 556 545 539 528 470 asymp5Tc(K) 10500 9400 8500 7200 6500 4600

rTc (au) 710 725 775 825 900 950


0 05 1 15 2 250

02

04

06

08

1

kkf

nk

HF T = 0HFα = 08α = 0565ni







α 1 095 09 085

TPSSc (K) 4500 3000 2000 500


104135

Applications

0 1000 2000 3000 4000 5000

10

20

30

40

50

60

70

80

FM

PP

PM

T (K)

r s(au)

TF

(rTc Tc)

rc




Ω+c [γ] = minus

1

2





Ω+c [γ] = minus

1

2n

intd3k1(2π)3

d3k2(2π)3

4π


cos2(Θk1

2

)+ n2k1

sin2(Θk1

2

))(n2k2

cos2(Θk2

2

)+ n1k2

sin2(Θk2

2

))(883)


105135

Applications





ε(mk) =k2

2m(884)


n(tmk) =1

1 + ek2

mtminusα(mt)

= n(mt 1k) (885)


Ωk(tm) = Ωk(mt 1) (886)


N(tm) = N(mt 1) (888)

S0(tm) = S0(mt 1) (889)

Ωx(tm) = Ωx(mt 1) (890)



Ω2bσ(tm) = mΩ2bσ(mt 1) (892)



106135

Applications



WRPA(tm) =W





εσ(k) = ε0σ(k) =k2

2(895)



partn(k)




partn(k)

partε(k)

1 +



partε(k)partkjsum

i qipartε(k)partki

(897)

107135

Applications



intd3k

(2π)3nσ(k)(1minus nσ(k)) +O(q2) (898)


Ωr =1

β(2π)minus2




q gt q


r (8100)

Ω0r =

1

β(2π)minus2

int q

0

dqq2(W (q)χ(q)) (8101)

=4π

β(2π)minus2

int radicβ

0

dqχ(q)) (8102)

Ωinfinr = minus 1

β(2π)minus2

int infin

radicβ

dqq2(W (q)χ(q))2

2 (8103)


behaves like


κ ge 2 (8105)


χκ[n(k)](q) =sum

σ

χκσ[n(k)](q) (8106)


β

1 + (qkF )κ

intd3k

(2π)3nσ(k)(1minus nσ(k)) (8107)

= minus β


) (8108)

where

〈n2σ〉 =int

d3k

(2π)3n2σ(k) (8109)






108135

Applications

01 1 10 100

0

minus002

minus004

minus006

minus008

minus010

minus012

rs(au)

e c(Ha)

PWCAκ = 29κ = 27κ = 25κ = 23



0

02

04

06

08

10

0 05 10 15 20 25 30

kkF

n(kk

F)

rs = 1 GZ

rs = 5 GZ

rs = 10 GZ

rs = 1 κ

rs = 5 κ

rs = 10 κ


KAPPA-functional

ξ 00 01 02 03 04 05 06 07 08 09 10

κ 29 29 29 30 30 30 31 31 31 32 32


109135

Applications

κ c

KAPPA-TIE 32 024


0 200 400 600 800 1000

10

20

30

40

50

60

70

80

01200

FM

PP

PM

T (K)

r s(au)

TF

(rTc Tc)





1 + q2k2F

intd3k







110135

Applications







111135

Applications

(1) (2) (3) (4) (5) (6) rc


FT-MBPT




FT-RDMFT








112135

A APPENDIX




γ(x xprime) =sum

i


=sum

i

niφlowasti (x

prime)φi(x) (A2)


|ψi〉 =sum

l

cij |χj〉 (A3)



D =sum

ijk



γ(x xprime) =sum

i

sum

jk


φlowasti (x

prime)φi(x) (A6)


γ(x xprime) =sum

i

sum

jk


φlowasti (x

prime)φi(x) (A7)


P (A) =sum

B

P (A|B)P (B) (A8)


γ(x xprime) =sum

i

P (φi)φlowasti (x

prime)φi(x) (A9)


113135

Appendix



ni =1




γ(x xprime) =sum

i

niφlowasti (x

prime)φi(x) (A11)


ni = c+i ci (A12)


written as

ni = TrDni =sum

e



Ψe =sum

α

ceαΦα (A14)


sum

e

|ceα|2 =sum

α

|ceα|2 = 1 (A15)


ni =sum

e

we

sum

αβ



ni =sum

α

(sum

e

we|ceα|2)

︸︷︷︸fα


(A17)



fα gt 0 (A18)sum

α

fα = 1 (A19)

114135

Appendix


0 lt ni lt 1 (A20)




H = T + V + W (A21)

γgs(x xprime) =

sum

i

niφlowasti (x

prime)φi(x) (A22)


E[γ] = 〈Ψ[γ]|H|Ψ[γ]〉 (A23)



partE[γ]

partni

∣∣∣∣γgs

=

ai gt micro ni = 0

micro 0 lt ni lt 1

bi lt micro ni = 1

(A24)



δE[γ]

δγ(x xprime)

∣∣∣∣γgs

= microδN [γ]

δγ(x xprime)

∣∣∣∣γgs




δEu[γ]

δγ(x xprime)

∣∣∣∣γgs




115135

Appendix





Eα[γ] = E[γ] +



partEα[γ]

partni

∣∣∣∣γgs

=

ai + uδiα ni = 0


bi + uδiα ni = 1

(A30)





Z=sum

i



H =



γ =


116135

Appendix


γ(x xprime) =sum

i


=sum

i

niφlowasti (x

prime)φi(x) (A35)


Ψ(x) =sum

i

ciφi(x) (A36)


γ(x xprime) = Tr


ZΨ+(xprime)Ψ(x)

(A37)

=1

Z

sum

i


sum

n1n2



=1

Z

sum

i


sum

n1n2


=1

Z

sum

i


prod

j

sum

nj


〈ni|e


(A40)


γ(x xprime) =1

Z

sum

i


prod

j


1




(A42)

=sum

n1n2

〈n1 n2 |prod

i


=prod

i

sum

ni


=prod

i



γ(x xprime) =sum

i

niφlowasti (x

prime)φi(x) (A46)

ni =1


117135

Appendix


εi minus micro =1

βln

(1minus nini

) (A48)


Ω = minus 1

βlnZ (A49)


Ω0 = minus 1

β

sum

i


)(A50)

=1

β

sum

i

ln(1minus ni) (A51)


S =part

partTΩ

∣∣∣∣micro

(A52)


part

partTni

∣∣∣∣micro

=1

Tni(1minus ni) ln

(1minus nini

)(A53)


S0 = minussum

i


A5 Feynman rules







0dτ




118135

Appendix



Ω2b =1

n2minus10πminus9

intd3k

intd3p

intd3q



=1

n2minus6πminus7

intd3k

intd3p

intd3q




krarr q (A57)


q rarr k (A59)

Ω2b = minus1

n2minus6πminus7

intd3k

intd3p

intd3q


k2p2k middot p(A60)


Ω2b = minus1

n2minus2πminus5

int infin

0

dk

int infin

0

dp

int infin

0

dq

int π

0

dθp

int π

0

dθq

int π2

minusπ2

dφq




F (kp q θp) =

int π

0


int π2

minusπ2



G(k p q) = q2fq

int π2

0

dθp tan(π2+ θp

)(F(k p q

π

2+ θq

)minus F

(k p q

π

2minus θq

)) (A63)


Ω2b = minus1

n2minus2πminus5

int infin

minusinfindk

int infin

minusinfindp

int infin

0

dqG(ek ep q) (A64)






Ω2b = minus1

n2minus2πminus5

int infin

0

dxx2int π

0

dθx sin(θx)

int π2

minusπ2


119135

120135

B Bibliography

References






3865





PhysRevLett832628




PhysRevLett87133004


10106312987202


0304-8853(79)90333-0


PhysRevB5514975

121135

Bibliography



100257001





6216536










0921-4526(91)90075-P


dxdoiorg10106311906203

122135

Bibliography



PhysRevA79040501












322010


dxdoiorg1010631449326


101002qua560360864



123135

Bibliography



1103PhysRev971474




566






PhysRevB4513244



447879






124135

Bibliography




171257


101103PhysRevA64042512





org101098rspa20021027



235121




0375-9601(84)91034-X


00268970110070243


866


057

125135

Bibliography



PhysRevA77032509


106312899328









dxdoiorg1010631479623


0305-447089021



PhysRevE66036703




126135

Bibliography










47R1591




0378-4371(83)90254-6


pssb2221240140








org10106312888550

127135

Bibliography


1002jcc21225







PhysRevA34433





589











128135

Bibliography


0370-1573(82)90077-1








PhysRevE77021120



2064



softwaregsl




org101139p75-176



19664730108


129135

Bibliography







PhysRev139A796

130135








131135

132135

D Publications










gas Exchange only









133135

134135

E Acknowledgements











Tim

135135

Title

Abstract

Table of Contents

List of Figures

List of Tables


Introduction










BOW functional


Summary and outlook













Convexity



Summary and outlook






Summary and outlook






Scaling behaviour

Summary and outlook

Numerical treatment



Temperature tensor





Sample calculations


Full minimization

Summary and outlook

Applications



FT-MBPT

Exchange-only

Exchange + RPA


FT-RDMFT









Summary and outlook

APPENDIX





Feynman rules


Bibliography


Publications

Acknowledgements

TABLE OF CONTENTS














B Bibliography 121


D Publications 133

ii

TABLE OF CONTENTS


iii

List of Figures









v

LIST OF FIGURES


vi

List of Tables







vii













EQ Equilibrium

FM Ferromagnetic







GS Ground state



HF Hartree-Fock

HS Hilbert-Schmidt







ix


MC Monte-Carlo


NO Natural orbitals



PM Paramagnetic







SC Self-consistent




SW Spin waves



x

1 Introduction







1135

1 INTRODUCTION











2135

1 INTRODUCTION




3135

4135



Section Description


2223






H = T + W + Vext (21)


intdx =

sumσ



T =

intdx lim

xprimerarrx

(minusnabla

2

2


Vext =



W =



w(x xprime) =1




D =sum


i

wi = 1 (26)



(27)

5135

Prerequisites

T

T1

DN


T

TradicN

ΓN




N =




(29)


AHS =

radicTrA+A (210)


T1 =A | AHS le 1

(211)






γ(x xprime) =sum

i

niφlowasti (x

prime)φi(x) (213)

6135

Prerequisites


Φi(r) =

(φi1(r)

φi2(r)

) (214)


γ(r rprime) =sum

i

niΦdaggeri (r


=

(γ11(r r



prime)

)=sum

i

ni

(φlowasti1(r



prime)φi2(r)

)(216)


Φi1(r) =

(φi1(r)

0

)Φi2(r) =

(0

φi2(r)

)(217)



sum

i

niσφlowastiσ(r





sumi niφ

lowasti (x



i

ni = N (220)





(222)

7135

Prerequisites




radicN

TradicN =

A | AHS le

radicN

(223)





n(x) = γ(x x) (224)


nσ(r) = γσσ(r r) (225)





rs =

(3

4πn

)13

(226)

8135

Prerequisites


kF =(3π2n

) 13 =

(9

4π

) 13

rminus1s (227)

εF =k2F2

(228)

TF =εFkB

(229)



(230)









Enuc [n(r)] =



Vext σσprime =



9135

10135




Section Description








D = |Ψgs〉〈Ψgs| (31)


E = Ek[γ] + Vext[γ] +W [Γ(2)] (32)

Ek[γ] =

intdx lim

xprimerarrx

(minusnabla

2

2

)γ(xprime x) (33)

Vext[γ] =



W [Γ(2)] =




2TrDΨ+(x4)Ψ

+(x3)Ψ+(x1)Ψ

+(x2) (36)

11135

ZT-RDMFT


γ(x xprime) =2

N minus 1

intdx2Γ

(2)(x x2xprime x2) (37)



W [Γ(2)] =W [γ] (38)





prime) (310)






12135

ZT-RDMFT




W [γ] = EH [γ] + Ex[γ] + Ec[γ] (311)

EH [γ] =1

2


Ex[γ] = minus1

2








13135

ZT-RDMFT


Exc[γ] = Ex[γ] + Ec[γ] (315)

Exc[γ] = minus1

2

sum

ij

f(ni nj)



prime) (316)











14135

ZT-RDMFT



intd3RεHEG

c [γ(kR)] (317)



γ(kR) =















15135

ZT-RDMFT

10010101001

minus030

minus020

minus010

0

minus005

minus015

minus025

rs(au)

e c(Ha)


MLα (α = 055)




33 BOW functional




16135

ZT-RDMFT

0

02

04

06

08

10

0 02 04 06 08 10

ninj

f

fHF






2

sum

ij

fBOW (ni nj α)



prime) (319)








17135

ZT-RDMFT

00 05 10 15 2000

02

04

06

08

10

kkF

n(kk

F)

rs = 05

rs = 1

GZBOW

GZBOW

00 05 10 15 2000

02

04

06

08

10

kkF

n(kk

F)

rs = 5

rs = 10

GZBOW

GZBOW


10010101001

minus08

minus07

minus06

minus05

minus04

minus03

minus02

minus01

0


MLα (α = 06)

rs(au)

e c(Ha)




18135

ZT-RDMFT










E(nuarr ndarr) =1







19135

ZT-RDMFT

001 01 1 10 100minus010

minus008

minus006

minus004

minus002

0

rs(au)

∆F(Ha)

ξ00204060810


001 01 1 10 100

0

002

004

006

008

010

rs(au)

∆P(Ha)

ξ00204060810


0

01

02

03

04

001 01 1 10 100rs(au)

δF

00204

060810


minus08

minus06

minus04

minus02

0

001 01 1 10 100rs(au)

δP

00204

060810





δF =∆F

Wc(328)

δP =∆P

Wc (329)



20135

ZT-RDMFT

3D

ξ 00 01 02 03 04 05 06 07 08 09 10

α 056 056 056 057 057 057 058 059 061 063 066

BOW 061 061 061 062 062 063 063 064 065 067 069

2D

ξ 00 01 02 03 04 05 06 07 08 09 10

α 063 063 063 063 064 064 065 066 067 07 074

BOW 066 066 066 067 067 068 069 071 073 076 080





2

sum

ijσ


csum

ij

((niuarrnjdarr)

αU



21135

ZT-RDMFT

αP αU c

BOW-TIE 070 20 019









22135




Section Description







23135

FT-RDMFT


H


∆Nmicro

∆E β



H =

infinoplus

n=0

Shotimesn (41)










24135

FT-RDMFT

F [D]

D

downwards convex

upwards convex

42a Convexity

F [D]

D








γ(x xprime) = Tr

eminusβ(T+V+W )

ZΨ+(xprime)Ψ(x)

(48)



(49)



T (see Section 23)





25135

FT-RDMFT





F [Dk] (412)













S[A B] = supλ

((1λ)


))(419)




S[Ak Bk] (420)

26135

FT-RDMFT


TrA = supP

TrP A (421)

TrP A le PTrA (422)






S[A B] = supP λ

((1λ)tr


))(425)


S[Ak Bk] (426)





Ω[Dk] (427)


S[Dk] (428)






27135

FT-RDMFT

Deq

Deq

Deq

γeq

γeq neq

neq

neq

+ =rArr










U =


βlnZ

Z prime (430)





minus 1

βlnZ

Z prime = u11 = u22 = u11 + u22 (431)



28135

FT-RDMFT







eq) (433)


leads to



eq) (435)
















Ω[γeq] =







Ωeq = minDisinDN

Ω[D] (443)

29135

FT-RDMFT


Ωeq = minγisinΓN

infDisinDNrarrγ

Ω[D[γ]] (444)


Ωeq = minγisinΓN

(F[γ] +





TrD(T + W + 1β ln D) (446)


Ω[γ] = F[γ] +




Ωeq = minγisinγN

Ω[γ] (448)



δF [γ]

δγ(xprime x)

∣∣∣∣γeq




30135

FT-RDMFT





H(1)s =

sum

i

εi|φi〉〈φi| (450)


γeq(x xprime) =

sum

i

niφlowasti (x

prime)φi(x) (451)


ni =1



εi minus micro =1

βln

(1minus nini

) (453)



veff (x xprime) =

sum

ij






31135

FT-RDMFT


Ωk[γ] =

intdxprime lim

xrarrxprime

(minusnabla

2

2

)γ(x xprime) (456)

N [γ] =

intdxγ(x x) (457)

Ωext[γ] =



ΩH [γ] =1

2


Ωx[γ] = minus1

2


S0[γ] = minussum

i







Ω[γ] =

intdxprime lim

xrarrxprime

(minusnabla

2

2

)γ(x xprime) +

intdxvext(x x


1

2


2


1βsum

i



δΩ[γ]



0 =

intdxprimeφi(x

prime)δΩ[γ]

δγ(xprime x)(465)

=

(minusnabla

2

2

)φi(x) +

intdxprimevext(x x



prime)+



32135

FT-RDMFT





Ω[γ] = F [γ] +
















L sub DN )]


F [γ] = TrD(H0 + 1β ln D) (470)


33135

FT-RDMFT


limkrarrinfin

TrDk(H0 + 1β ln Dk) = F [γ] (471)












Mf =









34135

FT-RDMFT



infinsum

i=M+1









TrD(H0 + 1β ln D) = F [γ] (479)






F [γk] (480)


F [γ] le TrD(H0 + 1β ln D) (481)


F [γk] (482)

35135

FT-RDMFT

433 Convexity









infγisinΓN

(F [γ] +

intv0γ

)= F [γ0] +

intv0γ0 (484)






36135

FT-RDMFT








F [γ] +

intvγ ge F [γ0] +

intvγ0 (487)

infγisinΓN

(F [γ] +

intvγ

)ge F [γ0] +

intvγ0 (488)



F [γ0] +

intv0γ0 = inf

γisinΓN

(F [γ] +

intv0γ

)(489)






37135

FT-RDMFT





intv0γ



1 γk rarr γ




∆0 = F [γ0] +

intv0γ0 minus inf

γisinΓN

F [γ] +

intv0γ

(492)








38135

FT-RDMFT




Dc =sum

α


α

wαN = 1 (495)


F [D] = TrD(H + 1β ln D) (496)


Dceq =

eminusβH

TreminusβH (497)



TrD(T + W + 1β ln D) (498)


F [γ] = Fc[γ] + Vext[γ] (499)



F [γ] = Ω[γ] + microN [γ] (4100)

39135

FT-RDMFT

T

T1

DN

DV

DNL


T

TradicN

ΓN

ΓV








DNL




40135

FT-RDMFT







1minus1larrrarr ΓN




solved by

leads to

solved by

leads to

solved by

enables

used for

allows


41135

42135



Section Description









Ω0[D0] lt Ω0[Dw] (52)








43135

Exact properties


Sc[γ] lt 0


Ωc[γ] lt 0

(58)

(59)

(510)




Hλ = T + V0 + λW (511)




)(512)

= minDrarrγ


)minus Tr

DV λ[γ]

) (513)





)minus Tr

D[γ]V λ[γ]

(514)


dΩλ[γ]

dλ= Tr

Dλ[γ]

(W +

d

dλV λ[γ]

)minus d

dλTrD[γ]V λ[γ]

(515)


Ω[γ] = Ω0[γ] +

int 1

0

dλTrDλ[γ]W

(516)


Ωc[γ] =

int 1

0

dλ

λWλ

c [γ] (517)



44135

Exact properties

minus4 minus2 2 4

minus02

minus01

01

02

z

Ψ210(0 0 z)

51a Wavefunction

minus4 minus2 2 4

001

002

003

004

0z

ρ210(0 0 z)

51b Density







ψλ(r1 rN ) = 〈r1 rN |U(λ)|ψ〉 (518)

= λ3N2ψ(λr1 λrN ) (519)


45135

Exact properties


〈ψλ|Oλ|ψλ〉 = 〈ψ|O|ψ〉 (520)


Tλ =1

λ2T (521)

Wλ =1

λW (522)

Nλ = N (523)


T [γλ] = λ2Ωk[γ] (524)

ΩH [γλ] = λΩH [γ] (525)

Ωx[γλ] = λΩx[γ] (526)

S0[γλ] = S0[γ] (527)





get


λ[γλ]) (529)

TrD 1λ[γλ](W 1

λ+ τ ln D 1


λ+ τ ln D[γ]) (530)











(535)

(536)

46135

Exact properties





(537)

(538)

(539)


0 =d

dλ

(Ω[D 1

λ[γλ]]

)∣∣∣λ=1

(540)


0 =d

dλTr

D 1

λ[γλ]

(1

λW 1

λ+ τ ln(D 1

λ[γλ]

)∣∣∣∣λ=1

(541)


W [γ] =d



Wc[γλ] = λd

dλΩc[γλ] (543)


Ωc[γλ] =

int λ

0

dmicro



Sc[γλ] = λ2(part

partλ

1

λΩc[γλ]

)(545)


d

dλτSc[γλ] = λ

d

dλ

(1

λWc[γλ]

) (546)


d

dλΩc[γλ] =

1

λWc[γλ] lt 0

d

dλWc[γλ] lt

d

dλΩc[γλ] lt 0

d

dλSc[γλ] lt 0

(547)

(548)

(549)

47135

Exact properties

Ωc[γλ]

λ1

minusτSc[γ]

0


Ωc[γλ]




d2


1

λ2Wc[γλ] +

1

λ

d


1

λ2Wc[γλ] +

1

λ

d

dλΩc[γλ] (550)

d2








Dλ(τ w)[γ] = D(λ2τ λw)[γλ] (553)

48135

Exact properties




λ2Sc(τλ

2 wλ)[γ]


(554)

(555)

(556)


Ωc(τ w)[γ] =

int 1

0

dλ

λWc(τ λw)[γ] (557)

Ωc(τ w)[γ] =

int 1

0








49135

50135



Section Description










prime) =sumi εiφ

lowasti (x



i

φi(x)φlowasti (x


ni ν gt νprime




i

niφlowasti (x

prime)φi(x) (62)

51135


W + Veff




ni =1


εi =1

βln

(1minus nini

)+ micro (64)




52135




Ω[γ] = Ω0[γ] +

int 1

0

dλTrDλ[γ]W

(65)




〈λW 〉λ =1

2

intdxdxprime lim

νprimerarrν+





〈λW 〉λ =1

2

intdxdxprime lim

νprimerarrν+




1

2

intd3xd3xprime


)γ(x xprime) (68)






〈λW 〉unifλ =1

2

Σlowast minus

(610)


53135


λw(r rprime)

vλ(x xprime)






veff = v(1)eff + v

(2)eff + (611)



0

0+

=

0

0+

+

0

0+

Σlowast (612)


0 =

0

0+

Σlowast (613)

54135


+ + + +

+ + + + + +

+ + + +

+ + + + + +



(1)

0

0+

= minus

0

0+

minus

0+

0

(614)

55135


intdydyprime

int β

0


= minusintdydyprime

int β

0


int β

0



v(1)eff (x x



= minus minus (617)



potential v(2)eff

(2)

0

0+

= minus

0+

0

minus

0+

0

(618)


(2)

0

0+

= minus

0

0+

M (619)


intdydyprime

int β

0


= minusintdydyprime

int β

0


56135



v(2)eff =

sumij φi(x)φ

lowastj (x

prime)int β



int β

0eν


where

Mij(ν νprime) =








M =

sumij φi(x)φ

lowastj (x

prime)int β



int β

0eν








Ω(1)xc [γ] = + (625)


Ωxc[γ] =1

2


2


57135



Ω(2)xc [γ] = + + + (627)







Ω[γ]

γΓV ΓNΓV ΓV

(γV EV

) (γN EN

)


Ω[γ]

γΓN ΓV

(γV EV

) (γN EN

)



58135






prime) (628)




bull 2n+1 G0 lines






c [γ] (631)



c [γ] = nΩ(n)c [γλ] (632)


c [γ] = (nminus 1)Ω(n)c [γλ] (633)









part




part


59135





prime) (640)






c (τ w)[γ] (643)



c (τ w)[γ] (644)





60135



Section Description










61135

Numerical treatment

γ

F [γ]

γ0γ1γ2γ3

Ω(0)effΩ

(1)effΩ

(2)eff

Ω








62135

Numerical treatment






δΩeff [γ]

δγ(x xprime)=

δΩ[γ]

δγ(x xprime) (73)




(1

β0minus 1

β


where


δΩHxc[γ]

δγ(x xprime)(75)


δγ(x xprime) (76)


δnkδγ(xprime x)


δφk(y)

δγ(xprime x)=sum

l 6=k



δφlowastk(y)

δγ(xprime x)=sum

l 6=k




gij =




heff ij = δij

(partΩ[γ]

partni+ micro0 +

σiβ0

)+


intdy

(δΩ[γ]

δφi(y)φj(y)minus

δΩ[γ]


) (711)


σi =partS0[γ]

partni= ln

(1minus nini

) (712)

63135

Numerical treatment









Ω[n1 n2] =α1

2(n1 minus 05)2 +

α2

2(n2 minus 05)2 (713)

= Ω1[n1] + Ω2[n2] (714)









64135

Numerical treatment

0

2

4

6

8

10

0 02 04 06 08 1

n1

Ω[n

1]

Ω1[n1]Ωeff1[n1]


0

005

010

015

020

0 02 04 06 08 1

n2

Ω[n

2]

Ω2[n2]Ωeff2[n2]




ni =1


εi minus micro =1

βiln

(1minus nini

) (720)


Ω[γ] =sum

i


1


)(721)

=sum

i

Ωi[ni βi] (722)


βi = ηpart2S0[γ]

partn2i

part2Ω[γ]partn2i

(723)

= η1

ni(1minus ni)

(part2Ω[γ]

partn2i

)minus1

(724)


βi =η

αi

1

ni(1minus ni) (725)


65135

Numerical treatment

0

2

4

6

8

10

0 02 04 06 08 1

n1

Ω[n

1]

Ω1[n1]

Ωeff1[n1]


0

005

010

015

020

0 02 04 06 08 1

n2

Ω[n

2]

Ω2[n2]

Ωeff2[n2]








j 6=i

heff ji



nprimei =1


66135

Numerical treatment


i n(k)i φ


2) fromn(k)i

and

φ(k)i

calculate H

(k)eff by Eq (711)


ε(k+1)i

and

φ(k+1)i (x)

4) constructn(k+1)i


(k+1)0 so that

sumi n

(k+1)i = N

5) fromφ(k+1)i (x)

and

n(k+1)i





nprimei =ni


(partΩpartni

minus∆micro

) (729)



= βini(ni minus 1)

(partΩ[γ]

partniminus∆(0)

micro

) (731)




∆(0)micro =



(732)

67135

Numerical treatment




heff ij (734)


∆Ω =

intdxdxprime

δΩ[γ]


=sum

ij

δΩ[γ]

δγij∆γji (736)

=sum

i

δnipartΩ[γ]

partni︸︷︷︸

∆Ω1

+sum

i6=j


|heff ij |2

︸︷︷︸∆Ω2

(737)




∆Ω1 =sum

i

δnipartΩ[γ]

partni(738)

=sum

i

βini(ni minus 1)

(partΩ[γ]

partniminus∆(0)

micro

)partΩ[γ]

partni(739)

=sum

i

βini(ni minus 1)

(partΩ[γ]

partni

)2

minus


partni

)2


(740)


∆Ω1 =

sum

j

βjnj(nj minus 1)

sum

i

ci

(partΩ[γ]

partni

)2

minus(sum

k

ckpartΩ[γ]

partnk

)2 (741)

=

sum

j

βjnj(nj minus 1)

sum

i

ci

(partΩ[γ]

partniminussum

k

ckpartΩ[γ]

partnk

)2

(742)


∆Ω1 le 0 (743)



∆Ω2 =sum

i6=j


|heff ij |2 (744)

68135

Numerical treatment


∆Ω2 =sum

i6=j

ni minus nj1βi

ln(

1minusni

ni

)minus 1

βjln(

1minusnj

nj

) |heff ij |2 (745)


∆Ω2 = β0sum

i6=j

ni minus njln(


) |heff ij |2 (746)


∆Ω2 le 0 (747)








69135

Numerical treatment



to the ONs

χ2n =

1

Nunpinned

Nunpinnedsum

i

(partΩ

partniminus micro

)2

(748)

micro =1

Nunpinned

Nunpinnedsum

i

partΩ

partni(749)



χ2φ =

1

N minus 1


i ε2i

(750)





Eαxc[γ] = minus

1

2

sum

ij

nαi nαj



prime) (751)






70135

Numerical treatment

minus794

minus793

minus792

minus791

minus790

minus789

minus788

0 5 10 15 20

it

E(Ha)

τ = 20β0 = 30η = 10


1eminus25

1eminus20

1eminus15

1eminus10

1eminus05

1

0 20 40 60 80 100

it

χ2 n

τ = 20β0 = 30η = 10


minus808

minus806

minus804

minus802

minus800

0 5 10 15 20

it

E(Ha)

τ = 20β0 = 30η = 10


1eminus30

1eminus25

1eminus20

1eminus15

1eminus10

1eminus05

1

0 20 40 60 80 100

it

χ2 n τ = 20

β0 = 30η = 10


minus812

minus810

minus808

minus806

minus804

minus802

0 5 10 15 20

it

E(Ha)

τ = 10β0 = 30η = 10


1eminus30

1eminus25

1eminus20

1eminus15

1eminus10

1eminus05

1

0 20 40 60 80 100

it

χ2 n

τ = 10β0 = 30η = 10



71135

Numerical treatment





cycle


72135

Numerical treatment

minus79445

minus79440

minus79435

minus79430

minus79425

0 100 200 300 400 500

it

E(Ha)

τ = 10

β0 = 20


1eminus24

1eminus20

1eminus16

1eminus12

1eminus08

1eminus04

0 100 200 300 400 500

it

χ2 φ

τ = 10β0 = 20


minus8085

minus8084

minus8083

minus8082

minus8081

minus8080

0 100 200 300 400 500

it

E(Ha)

τ = 10

β0 = 10


1eminus13

1eminus11

1eminus09

1eminus07

1eminus05

0 100 200 300 400 500

it

χ2 φ

τ = 10

β0 = 10


minus81285

minus81280

minus81275

minus81270

minus81265

minus81260

minus81255

0 100 200 300 400 500

it

E(Ha) τ = 10

β0 = 10


1eminus13

1eminus11

1eminus09

1eminus07

1eminus05

0 100 200 300 400 500

it

χ2 φ

τ = 10

β0 = 10



73135

Numerical treatment

minus81284

minus81282

minus81280

minus81278

minus81276

minus81274

minus81272

0 100 200 300 400 500

it

E(Ha)

β0 = 10β0 = 20β0 = 50


1eminus13

1eminus11

1eminus09

1eminus07

1eminus05

0 100 200 300 400 500

it

χ2 φ

β0 = 10

β0 = 20

β0 = 50











74135

8 Applications


Section Description








75135

Applications








76135

Applications

minus40

minus50

minus45

5

10

15

20

0

02

08

06

04

10

rs(au)

ξF(10minus

2Ha)





S[D] =sum

i

wi lnwi (82)




77135

Applications










FMPP

PM

r s(au)

T (K)

rc (rTc Tc)

TF


78135

Applications





n =

intd3k



Ωkσ =1

n

intd3k

(2π)3k2

2nσ(k) (84)

Ωextσ =1

n

intd3k


nσ =1

n

intd3k

(2π)3nσ(k) (86)

S0σ = minus 1

n

intd3k


F =sum

σ








s We therefore set



ΩSW = grminus1

s (1minus ξ2) (811)

79135

Applications

FM

PP

PM

0 20 40 6010 30 50

60

80

100

120

140

40

160

200

180

r s(au)

T (K)

rc(rc Tc)

TF



83 FT-MBPT




n(k) =1


ε(k) =k2

2(814)


t =T

TF k =

k

kF(815)

80135

Applications



3 (816)


nσ(k) =1


(817)


2

3tminus

32 =

int infin

0

dx

radicx

1 + exminusα(818)


Fj(x) =1

Γ(j + 1)

int infin

0

dzzj

1 + ezminusx(819)


nσ =3

8π

12 t

32F 1

2(ασ) (820)

Ekσ =9

32π

12 k2F t

52F 3

2(ασ) (821)

microσ =1

2k2F tασ (822)

S0σ =

(15

16π

12 t

32F 3


)(823)


4

3πminus 1

2 tminus 3

2σ = F 1

2(ασ) (824)


831 Exchange-only


F = F0 +

int 1

0

dλ

λ

︸︷︷︸rarrΩx

(825)



Ωxσ = minus 1

n

intd3k

(2π)3d3q


81135

Applications

0

02

04

06

08

1

0 1 2 3 4 5

t

I x(t)

84a Exchange

0

05

1

15

2

25

3

0 1 2 3 4 5

t

I 2b(t)




W (k) =4π

k2 (828)



3


Ix(t) =

intdx

intdy

x

1 + ex2

tminusα(t)

y

1 + ey2

tminusα(t)

ln(x+ y)2

(xminus y)2 (830)




82135

Applications

0 2000 4000 6000 8000 100000

2

4

6

8

10

12

14

16

18

20

PP

PM

FMr s(au)

T (K)

rc(rTc Tc)

TF


832 Exchange + RPA


F = F0 +

int 1

0

dλ

λ

︸︷︷︸

rarrΩx

+

︸︷︷︸rarrΩr

(831)



Ωr =1

n

1

2β

intd3q

(2π)3

infinsum

a=minusinfin


) (833)


νa =2πa

β (834)

83135

Applications

0 2 4 6 8 10

minus05

minus04

minus03

minus02

minus01

00 2 4 6 8 10

minus025

minus020

minus015

minus010

0 2 4 6 8 10

minus009

minus008

minus007

minus006

minus005

minus004

0 2 4 6 8 10

minus0011

minus0010

minus0009

minus0008

tt

Ωr(t)

Ωr(t) Ω

r (t)Ω

r (t)

rs = 025 rs = 114

rs = 540 rs = 117




d3k




Ωr =k2F t

8π2

intdqq2

infinsum

a=minusinfin


)(836)


(2π)2kFσ

q

intdk

k


ln

((2πatσ)

2 + (q2 + 2kq)2


) (837)



84135

Applications

minus1eminus3

minus5eminus4

1eminus3

5eminus4

0

1 10 100

δ(r sξ)

rs(au)

ξ = 00

ξ = 02

ξ = 04

ξ = 06

ξ = 08

ξ = 10


0 50 100 200150 250

10

15

20

25

30

35

40

45

50

55

60

PP

PM

FM

300 350

r s(au)

T (K)

rc(rTc Tc)

TF




85135

Applications



F = F0 +

int 1

0

dλ

λ

︸︷︷︸

rarrΩx

+


+

︸︷︷︸rarrΩr

(838)



Ω2bσ =1

n2minus10πminus9

intd3k

intd3p

intd3q





2Ω2bσ(0)I2b(tσ) (841)

I2b(t) = minus3


intd3k

intd3p

intd3q


k2p2k middot p(842)

Ω2bσ(0) =

(ln 2

6minus 3

4π2ζ(3)

) (843)



86135

Applications

0 4000 8000 120000

4

8

12

16

20

24

28

32

36

40

16000FM

PP

PMr s(au)

T (K)

TF

(rTc Tc)rc




87135

Applications

minus15

minus10

minus5

0

5

minus8

minus6

minus4

minus2

0 05 10 15

minus7

minus6

minus5

minus4

minus3

0 05 10 15 20

3

4

5

6

7

F0(10minus

2Ha) Ω

x(10 minus

2Ha)

Ωr(10minus

2Ha) Ω

2b (10 minus

2Ha)

tt

ξ = 0ξ = 05ξ = 08ξ = 10


84 FT-RDMFT



F [γ] = F0[γ] +

int 1

0

dλ

λ

︸︷︷︸rarrΩx

(844)



88135

Applications




σi

nσi

int

Vi

d3k





Ωk[nσi] =sum

σi

nσiti (848)

S0[nσi] = minussum

σi


Ωx[nσi] = minus1

2

sum

σij

nσinσjKij (850)


ti =1

n

int

Vi

d3k

(2π)3k2

2(851)

Kij =1

n

int

Vi

dk31(2π)3

int

Vj

dk32(2π)3

4π


ωi =1

n

int

Vi

d3k

(2π)3 (853)







89135

Applications

4 6 8 10 12 14 160

02

04

06

08

10

T=0T=5000T=8000T=10000

ξ min

rs(au)


05

10

15

20

0

0 02 04 06 08

minus4

0 02 04 06 08 1

minus3

0

3

6

1

2

3

minus3

minus2

minus1

ξξ

Ωk(10minus

2Ha) Ω

x(10 minus

2Ha)S

∆F(10 minus

3Ha)

t = 0t = 10t = 16t = 22



∆1minus0 = Ω(ξ = 1)minus Ω(ξ = 0) (854)


90135

Applications

0

1

2

3

4

0 1 2 0 1 2 3

minus8

minus6

minus4

minus2

minus1

0

1

2

3

4

minus3

minus2

minus1

tt

Ωk(10minus

2Ha) Ω

x(10 minus

2Ha)

SF(10 minus

2Ha)

FD ξ = 0

FD ξ = 1

HF ξ = 0

HF ξ = 1

FD ∆1minus0

HF ∆1minus0










91135

Applications





92135

Applications




φ1k(r) =

(cos(Θk

2

)eminusiqmiddotr2

sin(Θk

2

)eiqmiddotr2

)eikmiddotrradicV

(855)

φ2k(r) =

(minus sin

(Θk

2

)eminusiqmiddotr2

cos(Θk

2

)eiqmiddotr2




intd3k

(2π)3

(n1k cos

2

(Θk

2

)+ n2k sin

2

(Θk

2



intd3k

(2π)31




intd3k

(2π)31




intd3k

(2π)3

(n1k sin

2

(Θk

2

)+ n2k cos

2

(Θk

2



Ωx[γ] = minus1

2

sum

σ1σ2




Ωx[γ] = minus1

2n

intd3k1(2π)3

d3k2(2π)3

4π

|k1 minus k2|2[sum

b

(nbk1nbk2


2

)+ (n1k1

n2k2+ n2k1

n1k2) sin2

(Θk1minusΘk2

2

)] (862)


93135

Applications


F [q nbi Θi] =sum

bi

nbiti +q2

8minus q

sum

i


1βsum

bi


)ωiminus

1

2

sum

bij

(nbinbj) cos2

(Θi minusΘj

2

)Kij minus

sum

ij

(n1in2j) sin2

(Θi minusΘj

2

)Kij (863)


Qi =1

2n

int

Vi

d3k

(2π)3kz (864)


m(r) = minus


B

(865)


A =1

2

intd3k


B =1

2

intd3k





(868)

Θkρplusmn|kz| =π

2(1∓ akρ|kz|) (869)


i n1i and N2 =sum

i n2i

ξPSS =N1 minusN2

N1 +N2(870)






94135

Applications

kρkρkρkρ

kzkzkzkz

qkF = 0 qkF = 10 qkF = 18 qkF = 2

kF kF kF kF

-kF -kF -kF -kF

q2

q2

q2

- q2- q2 - q2


5

6

7

minus112

minus105

minus98

minus91

0 05 1 15 2 25

0

0 05 1 15 2 25 3

minus49

minus48

minus47

minus46

5

10

15

qkFqkF

Ωk(10minus

2Ha)

Ωx(10 minus

2Ha)

S(10minus

2)

F(10 minus

2Ha)

T = 0KT = 2500KT = 4000KT = 5000K








95135

Applications

0

1

2

3

4

5

1 15 2 25minus494

minus493

minus492

minus491

minus490

minus489

qkF

F(10minus

2Ha)

A(10minus

4au)

T = 4700K t = 0244T = 4800K t = 0249T = 4900K t = 0254T = 5000K t = 0260






96135

Applications

0 1000 2000 3000 4000 50000

1

2

3

4

T (K)

Aopt(10minus

4au)

rs = 55rs = 50rs = 45rs = 40







rc(au) 40 45 50 55

Tc(K) 2700 3400 4100 5000


97135

Applications

0 2000 4000 6000 8000 100000

2

4

6

8

10

12

14

16

18

20

PSS

FMPP

PM

r s(au)

T (K)

rc(rTc Tc)

TPSSc

TF




εσ(k) =k2

2minus 2

πfd(kkFσ) (871)


f3(x) =1

2+

1minus x24x

log

∣∣∣∣1 + x

1minus x

∣∣∣∣ (872)


f2(x) =

E(x) x lt 1

x(E(1x

)minus(1minus 1

x2

)K(1x

)) x ge 1



partF

partni= 0 =

partΩk

partniminus 1

β

partS0

partni+partΩx

partni (873)

98135

Applications

minus4

minus3

minus2

minus1

0

1

2

3

4

0 05 1 15 2 25

kkF

ε σ(kk

F)ε F

k2200105

12510100


minus8

minus6

minus4

minus2

0

2

4

0 05 1 15 2 25

kkF

ε σ(kk

F)ε F

k220

0105

12510100

820b Two dimensions




partni+partΩx

partni (874)


1d where d is



e0b(k) =

(k plusmn q

2

)2

2minus q2

8=

1



99135

Applications

minus4

minus2

0

2

4

6

8

10

0 05 1 15 2

kkF

ε σ(kk

F)ε F

ni

T = 0

FM


minus4

minus2

0

2

4

6

8

10

0 05 1 15 2

kkF

ε(kk

F)ε F

091787









Ωc[γ] = ΩDFTc [n] (876)



nminγrarrn

F [γ] (877)

= minn



) (878)


100135

Applications





FTrarr0= Ωk[n


c (n) (879)



101135

Applications

0 1000 1500 2000 2500 30000

20

40

60

80

100

120

140

160

180

200

FM

PP

PM

500

T (K)

r s(au)

TF

(rTc Tc)rc




0 20 40 60 80 100 120 140

15

20

25

30

35

40

45

FM

PP

PM

T (K)

r s(au)

(rTc Tc)

rc


102135

Applications







Ωαxc[nσi] =

sum

σij









103135

Applications

α 1 9 8 7 6 55

rc(au) 556 545 539 528 470 asymp5Tc(K) 10500 9400 8500 7200 6500 4600

rTc (au) 710 725 775 825 900 950


0 05 1 15 2 250

02

04

06

08

1

kkf

nk

HF T = 0HFα = 08α = 0565ni







α 1 095 09 085

TPSSc (K) 4500 3000 2000 500


104135

Applications

0 1000 2000 3000 4000 5000

10

20

30

40

50

60

70

80

FM

PP

PM

T (K)

r s(au)

TF

(rTc Tc)

rc




Ω+c [γ] = minus

1

2





Ω+c [γ] = minus

1

2n

intd3k1(2π)3

d3k2(2π)3

4π


cos2(Θk1

2

)+ n2k1

sin2(Θk1

2

))(n2k2

cos2(Θk2

2

)+ n1k2

sin2(Θk2

2

))(883)


105135

Applications





ε(mk) =k2

2m(884)


n(tmk) =1

1 + ek2

mtminusα(mt)

= n(mt 1k) (885)


Ωk(tm) = Ωk(mt 1) (886)


N(tm) = N(mt 1) (888)

S0(tm) = S0(mt 1) (889)

Ωx(tm) = Ωx(mt 1) (890)



Ω2bσ(tm) = mΩ2bσ(mt 1) (892)



106135

Applications



WRPA(tm) =W





εσ(k) = ε0σ(k) =k2

2(895)



partn(k)




partn(k)

partε(k)

1 +



partε(k)partkjsum

i qipartε(k)partki

(897)

107135

Applications



intd3k

(2π)3nσ(k)(1minus nσ(k)) +O(q2) (898)


Ωr =1

β(2π)minus2




q gt q


r (8100)

Ω0r =

1

β(2π)minus2

int q

0

dqq2(W (q)χ(q)) (8101)

=4π

β(2π)minus2

int radicβ

0

dqχ(q)) (8102)

Ωinfinr = minus 1

β(2π)minus2

int infin

radicβ

dqq2(W (q)χ(q))2

2 (8103)


behaves like


κ ge 2 (8105)


χκ[n(k)](q) =sum

σ

χκσ[n(k)](q) (8106)


β

1 + (qkF )κ

intd3k

(2π)3nσ(k)(1minus nσ(k)) (8107)

= minus β


) (8108)

where

〈n2σ〉 =int

d3k

(2π)3n2σ(k) (8109)






108135

Applications

01 1 10 100

0

minus002

minus004

minus006

minus008

minus010

minus012

rs(au)

e c(Ha)

PWCAκ = 29κ = 27κ = 25κ = 23



0

02

04

06

08

10

0 05 10 15 20 25 30

kkF

n(kk

F)

rs = 1 GZ

rs = 5 GZ

rs = 10 GZ

rs = 1 κ

rs = 5 κ

rs = 10 κ


KAPPA-functional

ξ 00 01 02 03 04 05 06 07 08 09 10

κ 29 29 29 30 30 30 31 31 31 32 32


109135

Applications

κ c

KAPPA-TIE 32 024


0 200 400 600 800 1000

10

20

30

40

50

60

70

80

01200

FM

PP

PM

T (K)

r s(au)

TF

(rTc Tc)





1 + q2k2F

intd3k







110135

Applications







111135

Applications

(1) (2) (3) (4) (5) (6) rc


FT-MBPT




FT-RDMFT








112135

A APPENDIX




γ(x xprime) =sum

i


=sum

i

niφlowasti (x

prime)φi(x) (A2)


|ψi〉 =sum

l

cij |χj〉 (A3)



D =sum

ijk



γ(x xprime) =sum

i

sum

jk


φlowasti (x

prime)φi(x) (A6)


γ(x xprime) =sum

i

sum

jk


φlowasti (x

prime)φi(x) (A7)


P (A) =sum

B

P (A|B)P (B) (A8)


γ(x xprime) =sum

i

P (φi)φlowasti (x

prime)φi(x) (A9)


113135

Appendix



ni =1




γ(x xprime) =sum

i

niφlowasti (x

prime)φi(x) (A11)


ni = c+i ci (A12)


written as

ni = TrDni =sum

e



Ψe =sum

α

ceαΦα (A14)


sum

e

|ceα|2 =sum

α

|ceα|2 = 1 (A15)


ni =sum

e

we

sum

αβ



ni =sum

α

(sum

e

we|ceα|2)

︸︷︷︸fα


(A17)



fα gt 0 (A18)sum

α

fα = 1 (A19)

114135

Appendix


0 lt ni lt 1 (A20)




H = T + V + W (A21)

γgs(x xprime) =

sum

i

niφlowasti (x

prime)φi(x) (A22)


E[γ] = 〈Ψ[γ]|H|Ψ[γ]〉 (A23)



partE[γ]

partni

∣∣∣∣γgs

=

ai gt micro ni = 0

micro 0 lt ni lt 1

bi lt micro ni = 1

(A24)



δE[γ]

δγ(x xprime)

∣∣∣∣γgs

= microδN [γ]

δγ(x xprime)

∣∣∣∣γgs




δEu[γ]

δγ(x xprime)

∣∣∣∣γgs




115135

Appendix





Eα[γ] = E[γ] +



partEα[γ]

partni

∣∣∣∣γgs

=

ai + uδiα ni = 0


bi + uδiα ni = 1

(A30)





Z=sum

i



H =



γ =


116135

Appendix


γ(x xprime) =sum

i


=sum

i

niφlowasti (x

prime)φi(x) (A35)


Ψ(x) =sum

i

ciφi(x) (A36)


γ(x xprime) = Tr


ZΨ+(xprime)Ψ(x)

(A37)

=1

Z

sum

i


sum

n1n2



=1

Z

sum

i


sum

n1n2


=1

Z

sum

i


prod

j

sum

nj


〈ni|e


(A40)


γ(x xprime) =1

Z

sum

i


prod

j


1




(A42)

=sum

n1n2

〈n1 n2 |prod

i


=prod

i

sum

ni


=prod

i



γ(x xprime) =sum

i

niφlowasti (x

prime)φi(x) (A46)

ni =1


117135

Appendix


εi minus micro =1

βln

(1minus nini

) (A48)


Ω = minus 1

βlnZ (A49)


Ω0 = minus 1

β

sum

i


)(A50)

=1

β

sum

i

ln(1minus ni) (A51)


S =part

partTΩ

∣∣∣∣micro

(A52)


part

partTni

∣∣∣∣micro

=1

Tni(1minus ni) ln

(1minus nini

)(A53)


S0 = minussum

i


A5 Feynman rules







0dτ




118135

Appendix



Ω2b =1

n2minus10πminus9

intd3k

intd3p

intd3q



=1

n2minus6πminus7

intd3k

intd3p

intd3q




krarr q (A57)


q rarr k (A59)

Ω2b = minus1

n2minus6πminus7

intd3k

intd3p

intd3q


k2p2k middot p(A60)


Ω2b = minus1

n2minus2πminus5

int infin

0

dk

int infin

0

dp

int infin

0

dq

int π

0

dθp

int π

0

dθq

int π2

minusπ2

dφq




F (kp q θp) =

int π

0


int π2

minusπ2



G(k p q) = q2fq

int π2

0

dθp tan(π2+ θp

)(F(k p q

π

2+ θq

)minus F

(k p q

π

2minus θq

)) (A63)


Ω2b = minus1

n2minus2πminus5

int infin

minusinfindk

int infin

minusinfindp

int infin

0

dqG(ek ep q) (A64)






Ω2b = minus1

n2minus2πminus5

int infin

0

dxx2int π

0

dθx sin(θx)

int π2

minusπ2


119135

120135

B Bibliography

References






3865





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10106312987202


0304-8853(79)90333-0


PhysRevB5514975

121135

Bibliography



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dxdoiorg10106311906203

122135

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123135

Bibliography



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124135

Bibliography




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org101098rspa20021027



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866


057

125135

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126135

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127135

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128135

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softwaregsl




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19664730108


129135

Bibliography







PhysRev139A796

130135








131135

132135

D Publications










gas Exchange only









133135

134135

E Acknowledgements











Tim

135135

Title

Abstract

Table of Contents

List of Figures

List of Tables


Introduction










BOW functional


Summary and outlook













Convexity



Summary and outlook






Summary and outlook






Scaling behaviour

Summary and outlook

Numerical treatment



Temperature tensor





Sample calculations


Full minimization

Summary and outlook

Applications



FT-MBPT

Exchange-only

Exchange + RPA


FT-RDMFT









Summary and outlook

APPENDIX





Feynman rules


Bibliography


Publications

Acknowledgements

Reduced Density Matrix Functional Theory at Finite ...

Documents

Transcript of Reduced Density Matrix Functional Theory at Finite ...