Electronic Structure of Elemental Plutonium: A Dynamical Mean Field Perspective (DMFT)

Electronic Structure of Elemental Plutonium: A Dynamical Mean Field Perspective (DMFT)

Gabriel KotliarPhysics Department and

Center for Materials TheoryRutgers University

MRS Boston 2003

Collaborators, References

S. Savrasov and G. Kotliar PRL 84 3670 (2000). S.Savrasov G. Kotliar and E. Abrahams, Nature 410,793 (2001).X. Dai,S. Savrasov, G. Kotliar,A. Migliori, H. Ledbetter, E. Abrahams Science, Vol300, 954 (2003). S. Murthy Rutgers Ph.D Thesis (2004).

Introduction: basic questions and alternative theories.Dynamical Mean Field Theory (DMFT). Alpha and Delta Pu. Both phases are strongly correlated and differ in only in the distribution of one electron spectral weight. Delta and Epsilon Pu differ dramatically in their phonon spectra. Epsilon Pu is strongly anharmonic.Outlook.

Support: Dynamical mean field method NSF. Actinides DOE Basic Energy Sciences

THE STATE UNIVERSITY OF NEW JERSEY

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Pu phases Small amounts of Ga stabilize the phase (A. Lawson LANL) Los Alamos Science,26, (2000).


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Americium under pressure (Lindbaum et. al. PRB 2003)


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More conventional electronic structure approaches

o DFT in the LDA or GGA is a well established tool for the calculation of ground state properties.

o Many studies (Freeman, Koelling 1972)APW methods, ASA and FP-LMTO Soderlind et. al 1990, Kollar et.al 1997, Boettger et.al 1998, Wills et.al. 1999) give

o an equilibrium volume of the an equilibrium volume of the phasephaseIs 30% Is 30% lower than experimentlower than experiment

o This is the largest discrepancy ever known in DFT based calculations.


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DFT Studies LSDA predicts magnetic long range (Solovyev

et.al.) Experimentally Pu is not magnetic. If one treats the f electrons as part of the core

LDA overestimates the volume by 30% DFT in GGA predicts correctly the volume of the

phase of Pu, when full potential LMTO (Soderlind Eriksson and Wills) is used. This is usually taken as an indication that Pu is a weakly correlated system

Alternative approach to delta Pu, Wills et. al. (5f)4

core+ 1f(5f)in conduction band. [SIC-LDA]


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Shear anisotropy fcc Pu (GPa) C’=(C11-C12)/2 = 4.78

C44= 33.59

C44/C’ ~ 7 Largest shear anisotropy in any element!

LDA Calculations (Bouchet et. al. ) C’= -48


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Dynamical Mean Field Theory (Reviews) A. Georges, G. Kotliar, W. Krauth and M.

Rozenberg Rev. Mod. Phys. 68,13 (1996)] Lichtenstein Katsnelson and Kotliar cond-

mat-0211076 K. Held et.al. , Psi-k Newsletter 56 (April

2003).


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Dynamical Mean Field Theory Basic idea: reduce the quantum many body problem to a

one site or a cluster of sites, in a medium of non interacting electrons obeying a self consistency condition.[A. Georges and GK Phys. Rev. B 45, 6497, 1992].

Atom in a medium = Quantum impurity model. Solid in a frequency dependent potential. Basic idea: instead of using functionals of the density,

use more sensitive functionals of the one electron spectral function. [density of states for adding or removing particles in a solid, measured in photoemission] [GK R. Chitra GKPhys. Rev. B62, 12715 (2000). and S. Savrasov cond-matt 0308053]. Allows computation of total energy AND one electron spectra.


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Dynamical Mean Field Theory


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Approximations Electronic structure. LMTO’s ASA , LMTO full

potential. Crystal field splitting in the self energies is

neglected. W(r,r’) (w) replaced by U on the f electrons. 4 ev.No multiplet splittings. Non perturbative treatment of spin orbit coupling. Approximate Impurity Solver. Interpolative

Perturbation Theory and Hubbard I.


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W (ev) vs (a.u. 27.2 ev) N.Zein G. Kotliar and S. Savrasov

iw


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What is the dominant atomic configuration?

Snapshots of the f electron Dominant configuration:(5f)5

Naïve view Lz=-3,-2,-1,0,1 ML=-5 B

S=5/2 Ms=5 B Mtot=0 More realistic calculations,Mtot 0 is quenched by crystalFields and Kondo effect. Moment

lives at high q Contrast Am:(5f)6


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Pu: DMFT total energy vs volume Savrasov Kotliar and Abrahams Nature 410,793 (2001)


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Double well structure and Pu Qualitative explanation

of negative thermal expansion


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LDA+DMFT calculations for fcc Americium S. Murthy and G. K(2003)


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Dynamical Mean Field View of Pu(Savrasov Kotliar and Abrahams, Nature 2001)

Delta and Alpha Pu are both strongly correlated, the DMFT mean field free energy has a double well structure, for the same value of U. One where the f electron is a bit more localized (delta) than in the other (alpha).

Is the natural consequence of earlier studies of the Mott transition phase diagram once electronic structure is about to vary.


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Alpha and delta Pu Photoemission Spectra DMFT(Savrasov et.al.) EXP (Arko Joyce Morales Wills Jashley PRB 62, 1773 (2000))


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Pu and delta Pu differ electronically by the distribution of spectral weight in the resonance and the Hubbard band.

U/W is not so different in alpha and delta The specific heat of delta Pu, is only twice as

big as that of alpha Pu. The susceptibility of alpha Pu is in fact larger

than that of delta Pu. The resistivity of alpha Pu is comparable to

that of delta Pu and near the Mott limit.


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Anomalous Resistivity

PRL 91,061401 (2003)


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Pu is NOT MAGNETIC, alpha and delta have comparable susceptibility and specifi heat.


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Important Physics Proximity to the Mott Transition.

Redistribution of spectral weight.

Simultaneous description of band physics and atomic physics.

All captured by DMFT in the approximations used.!


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Pu phases Small amounts of Ga stabilize the phase (A. Lawson LANL) Los Alamos Science,26, (2000).


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The delta –epsilon transition The high temperature phase, (epsilon) is body centered

cubic, and has a smaller volume than the (fcc) delta phase.

What drives this phase transition?

LDA+DMFT functional computes total energies opens the way to the computation of phonon frequencies in correlated materials (S. Savrasov and G. Kotliar 2002). Combine linear response and DMFT.


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Phonon freq (THz) vs q in delta Pu X. Dai et. al. Science vol 300, 953, 2003


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Expts’ Wong et. al. Science 301. 1078 (2003) Theory Dai et. al. Science 300, 953, (2003)


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Comparison of theory and experiment.

Good agreement over the majority of the Brillouin zone, is significant. The phonon frequencies depend on the forces acting on the atoms as a result of their displacement. Ability to compute forces, is a first step to derive potentials, and do molecular dynamics.

Discrepancies along (111) are significant. Role of temperature ? Improve the impurity solver ? Non local corrections, and deviations from DMFT.


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Elastic constants theory (LDA+DMFT with a Hubbard1 solver, Dai et. al. and experiments,( Letbetter and Moment ). Large c44/c’ ratio.

C11 (GPa) C44 (GPa) C12 (GPa) C'(GPa) Theory 34.56 33.03 26.81 3.88 Expt 36.28 33.59 26.73 4.78


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The delta –epsilon transition The high temperature phase, (epsilon) is body

centered cubic, and has a smaller volume than the (fcc) delta phase.

What drives this phase transition?

Having a functional, that computes total energies opens the way to the computation of phonon frequencies in correlated materials (S. Savrasov and G. Kotliar 2002)


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Epsilon Plutonium.


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Phonon entropy drives the epsilon delta phase transition

Epsilon is slightly more delocalized than delta, has SMALLER volume and lies at HIGHER energy than delta at T=0. But it has a much larger phonon entropy than delta.

At the phase transition the volume shrinks but the phonon entropy increases.

Estimates of the phase transition following Drumont and G. Ackland et. al. PRB.65, 184104 (2002); (and neglecting electronic entropy). TC ~ 600 K.


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Transverse Phonon along (0,1,1) in epsilon Pu in self consistent Born approximation.


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Negative thermal expansion of Pu revisited. The distortion described by C'

is very soft, nearly like a liquid, . C' measures the rigidity against the volume conserving tetragonal deformation. This is in fact the deformation from fcc towards a bcc along a Bain path. Previous LDA+ U study [Bouchet et. al. ] and our DMFT study show that the total energy difference between phase and phases is quite small and is around 1000K. Soft behavior along the Bain path. Pu can sample the bcc structure, which has lower volume by the thermal fluctuation along Bain path.

d

d

e


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Insights into the anomalous properties of Pu

Physical anomalies, are the result of the unique position of Pu in the periodic table, where the f electrons are near a localization delocalization transition. The Mott transition across the actinide series [ B. Johansson Phil Mag. 30,469 (1974)] concept has finally been worked out! .We learned how to think about this unusual situation using DMFT, Weiss fields, local spectral functions etc.


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Conclusions DMFT produces non magnetic state, around a fluctuating

(5f)^5 configuration with correct volume the qualitative features of the photoemission spectra, quasiparticle resonance and Hubbard band, and a double minima structure in the E vs V curve.

Correlated view of the alpha and delta phases of Pu. Interplay of correlations and electron phonon

interactions account for delta-epsilon transition. Anomalous phonons in epsilon Pu.

Calculations can be refined, include multiplets, better impurity solvers, frequency dependent U’s, electronic entropy. User friendly interfaces.


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Experiments and Theory are Needed to test the different pictures of the elctronic structure of PU

Model of Wills et al. : 4 (5f) electrons are core-like and 1 is delocalized. DMFT picture: all the 5 (5f) electrons are equivalent, they are localized over short time scales and itinerant over long time scales resulting in Hubbard band and quasiparticle resonance in the spectra.

Both pictures require strong correlations in the delta phase but how to differentiate between them experimentally ?

Alpha phase. Resonant Photoemission (J. Tobin et. al. ) Probe unoccupied states. Upper Hubbard band, BIS.

Optics. X ray absortion. Etc.. Fermi Surface Probes. Is Luttinger theorem obeyed ?


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Open questions ? Is the softening along the 110 direction in delta

Pu, temperature dependent ? Is the discrepancy between theory and

experiments the result of not including the resonance in the phonon calculation or the result of not including non local corrections ?


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J. Tobin et. al. PHYSICAL REVIEW B 68, 155109 ,2003


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A. Arko et. al. PRB 15. (2000), 1773.


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Acknowledgements: Development of DMFT

Collaborators: E. Abrahams,V. Anisimov, R. Chitra, V. Dobrosavlevic, X. Dai, D. Fisher, A. Georges, K. Haule H. Kajueter, W.Krauth, E. Lange, A. Lichtenstein, G. Moeller, Y. Motome, G. Palsson, A. Poteryaev, M. Rozenberg, S. Savrasov, Q. Si, V. Udovenko, I. Yang, X.Y. Zhang

Support: NSF DMR 0096462

Support: Instrumentation. NSF DMR-0116068

Work on Fe and Ni: ONR4-2650

Work on Pu: DOE DE-FG02-99ER45761 and LANL subcontract No. 03737-001-02


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Mott transition in layered organic conductors S Lefebvre et al. cond-mat/0004455, Phys. Rev. Lett. 85, 5420 (2000)


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More recent work, organics, Limelette et. al. PRL 91,061401 (2003)


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Magnetic moment L=5, S=5/2, J=5/2, Mtot=Ms=B gJ =.7 B

Crystal fields

GGA+U estimate (Savrasov and Kotliar 2000) ML=-3.9 Mtot=1.1

This bit is quenched by Kondo effect of spd electrons [ DMFT treatment]

Experimental consequence: neutrons large magnetic field induced form factor (G. Lander).


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LDA+DMFT functional2 *log[ / 2 ( ) ( )]

( ) ( ) ( ) ( )

1 ( ) ( ')( ) ( ) ' [ ]2 | ' |

[ ]

R R

n

n KS

KS n ni

LDAext xc

DCR

Tr i V r r

V r r dr Tr i G i

r rV r r dr drdr Er r

G

a b ba

w

w c c

r w w

r rr r

- +Ñ - - S -- S +

+ + +-F - F

åòò òå

Sum of local 2PI graphs with local U matrix and local G

1[ ] ( 1)2DC G Un nF = - ( )0( ) i

ababi

n T G i ew

w += å

KS ab [ ( ) G V ( ) ]LDA DMFT a br r


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LDA+DMFT and LDA+U • Static limit of the LDA+DMFT functional , • with atom HF reduces to the LDA+U functional

of Anisimov Andersen and Zaanen.

Crude approximation. Reasonable in ordered Mott insulators. Short time picture of the systems.

• Total energy in DMFT can be approximated by LDA+U with an effective U . Extra screening processes in DMFT produce smaller Ueff.

• ULDA+U < UDMFT

®


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LDA+DMFT References Anisimov Poteryaev Korotin Anhokin and Kotliar J. Phys.

Cond. Mat. 35, 7359 (1997). Lichtenstein and Katsenelson PRB (1998).Reviews: Kotliar, Savrasov, in Kotliar, Savrasov, in New Theoretical approaches New Theoretical approaches

to strongly correlated systemsto strongly correlated systems, Edited by A. Tsvelik, , Edited by A. Tsvelik, Kluwer Publishers, (2001).Kluwer Publishers, (2001).

Held Nekrasov Blumer Anisimov and Vollhardt et.al. Int. Held Nekrasov Blumer Anisimov and Vollhardt et.al. Int. Jour. of Mod PhysB15, 2611 (2001).Jour. of Mod PhysB15, 2611 (2001).

A. Lichtenstein M. Katsnelson and G. Kotliar (2002)


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X.Zhang M. Rozenberg G. Kotliar (PRL 1993)

Concepts : three peak structure and transfer of spectral weigth. Evolution at T=0 half filling full frustration


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Evolution of the Spectral Function with Temperature

Anomalous transfer of spectral weight connected to the proximity to the Ising Mott endpoint (Kotliar Lange and Rozenberg Phys. Rev. Lett. 84, 5180 (2000)


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Generalized phase diagram

T

U/WStructure, bands,

orbitals


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Minimum of the melting point Divergence of the compressibility at the Mott

transition endpoint. Rapid variation of the density of the solid as

a function of pressure, in the localization delocalization crossover region.

Slow variation of the volume as a function of pressure in the liquid phase


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Minimum in melting curve and divergence of the compressibility at the Mott endpoint

( )dT Vdp S

Vsol

Vliq


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Cerium


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Comments on LDA+DMFT• Static limit of the LDA+DMFT functional , with

= HF reduces to LDA+U• Gives the local spectra and the total energy

simultaneously, treating QP and H bands on the same footing.

• Luttinger theorem is obeyed.• Functional formulation is essential for

computations of total energies, opens the way to phonon calculations.


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Spectral Density Functional The exact functional can be built in perturbation

theory in the interaction (well defined diagrammatic rules )The functional can also be constructed from the atomic limit, but no explicit expression exists.

DFT is useful because good approximations to the exact density functional DFT(r)] exist, e.g. LDA, GGA

A useful approximation to the exact functional can be constructed, the DMFT +LDA functional.


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DMFT: Effective Action point of view.R. Chitra and G. Kotliar Phys Rev. B.(2000), (2001).

Identify observable, A. Construct an exact functional of <A>=a, [a] which is stationary at the physical value of a.

Example, density in DFT theory. (Fukuda et. al.) When a is local, it gives an exact mapping onto a local

problem, defines a Weiss field. The method is useful when practical and accurate

approximations to the exact functional exist. Example: LDA, GGA, in DFT.

It is useful to introduce a Lagrange multiplier conjugate to a, [a,

It gives as a byproduct a additional lattice information.


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Interface DMFT with electronic structure.

Derive model Hamiltonians, solve by DMFT(or cluster extensions). Total energy? Full many body aproach, treat light electrons by

GW or screened HF, heavy electrons by DMFT [E-DMFT frequency dependent interactionsGK and S. Savrasov, P.Sun and GK cond-matt 0205522]

Treat correlated electrons with DMFT and light electrons with DFT (LDA, GGA +DMFT)


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Spectral Density Functional : effective action construction

Introduce local orbitals, R(r-R), and local GF G(R,R)(i ) =

The exact free energy can be expressed as a functional of the local Greens function and of the density by introducing sources for (r) and G and performing a Legendre transformation, (r),G(R,R)(i)]

' ( )* ( , ')( ) ( ')R Rdr dr r G r r i r


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LDA+DMFT approximate functional The light, SP (or SPD) electrons are extended,

well described by LDA The heavy, D (or F) electrons are localized,treat by

DMFT. LDA already contains an average interaction of the

heavy electrons, substract this out by shifting the heavy level (double counting term)

The U matrix can be estimated from first principles (Gunnarson and Anisimov, McMahan et.al. Hybertsen et.al) of viewed as parameters


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LDA+DMFT-outer loop relax

G0 G

Im puritySolver

S .C .C .

0( ) ( , , ) i

i

r T G r r i e ww

r w += å

2| ( ) | ( )k xc k LMTOV H ka ac r c- Ñ + =

DMFT

U

Edc

0( , , )HHi

HHi

n T G r r i e w

w

w += å

ff &


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Outer loop relax

0( ) ( , , ) i

i

r T G r r i e ww

r w += å

2| ( ) | ( )k xc k LMTOV H ka ac r c- Ñ + =

U

Edc

0( , , )HHi

HHi

n T G r r i e w

w

w += å

ff &

Impurity Solver

SCC

G,G0

DMFTLDA+U

Imp. Solver: Hartree-Fock


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References Long range Coulomb interactios, E-DMFT. R.

Chitra and G. Kotliar Combining E-DMFT and GW, GW-U , G. Kotliar

and S. Savrasov Implementation of E-DMFT , GW at the model

level. P Sun and G. Kotliar. Also S. Biermann et. al.


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Dynamical Mean Field Theory(DMFT)Review: A. Georges G. Kotliar W. Krauth M. Rozenberg. Rev Mod Phys 68,1 (1996)

Local approximation (Treglia and Ducastelle PRB 21,3729), local self energy, as in CPA.

Exact the limit defined by Metzner and Vollhardt prl 62,324(1989) inifinite.

Mean field approach to many body systems, maps lattice model onto a quantum impurity model (e.g. Anderson impurity model )in a self consistent medium for which powerful theoretical methods exist. (A. Georges and G. Kotliar prb45,6479 (1992).


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Technical details Multiorbital situation and several atoms per

unit cell considerably increase the size of the space H (of heavy electrons).

QMC scales as [N(N-1)/2]^3 N dimension of H

Fast interpolation schemes (Slave Boson at low frequency, Roth method at high frequency, + 1st mode coupling correction), match at intermediate frequencies. (Savrasov et.al 2001)


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LDA+U bands. (Savrasov GK ,PRL 2000). Similar work Bouchet et. al. 2000


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Energy vs Volume


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S. Murthy Rutgers Ph.D ThesisP vs V for fcc Am


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Lda vs Exp Spectra


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Comparaison with LDA+U


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Am photoemission spectra. Expt (Negele ) DMFT Theory (S. Murthy)


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Plutonium and The Mott PhenomenaEvolution of the electronic structure between the atomic limit and the

band limit in an open shell situation.The “”in between regime” is ubiquitous central them in strongly

correlated systems, gives rise to interesting physics. Example Mott transition across the actinide series [ B. Johansson Phil Mag. 30,469 (1974)]

Revisit the problem using a new insights and new techniques from the solution of the Mott transition problem within dynamical mean field theory in the model Hamiltonian context.

Use the ideas and concepts that resulted from this development to give physical qualitative insights into real materials.

Turn the technology developed to solve simple models into a practical quantitative electronic structure method .


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Phases of Pu (A. Lawson LANL) Los Alamos Science 26, (2000)

Summary

LDA

LDA+U LRO

DMFT

Spectra Method E vs V


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DMFT: elastic constants


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Phases of Pu (A. Lawson LANL) Los Alamos Science 26, (2000)

Electronic Structure of Elemental Plutonium: A Dynamical Mean Field Perspective (DMFT)

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