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Density Functional Theory (DFT) in a nutshell.

Élise Dumont

ENS Lyon

Élise Dumont Lecture M2 (RFCT, 1st week and ATOSIM)� 25th Oct. 2010


Outline

Why resorting to DFT ?From a 3N-D to a 3D view : RDM theory...Fundations : Hohenberg-Kohn and Kohn-Sham theoremsClimbing Jacob's Ladder : LD(S)A, GGA, meta-GGA, hybrid andACPitfalls of DFTDFT shortcomingsMost recent developments...Further reading



Why resorting to DFT ?

0. Key advantages of DFT over post HF-methods � both treating (moreor less accurately) the dynamical electron correlation :

Cost O(N3) � also the need for large basis sets is reduced...

Conceptutal DFT (density easier to visualize and manipulate... )

Ability to treat excited states (DNA photostability)

AIMD (because, after all, molecules do vibrate)

DFT is �rstly justi�ed by a computational bottleneck.→ "Get a correlated calculation at the price of a simple HF calculation".

For a (rather) recent review, for non-specialists : R. A. Friesner, "Ab initio

quantum chemistry: Methodology and applications", PNAS, 2005, 102, 19,

6648�6653.



Why resorting to DFT ?

An alternative→ RI : a neat & universal mathematical trick!

Key idea: approximate calculations of many (ia|jb)...

1 Use a projector I =∑

K |K 〉〈K |2 Auxiliary basis expansion (ABE): |jb〉 =

∑K CK

jb |K 〉3 Fit of CK

jb coe�cients... (density �tting DF)

(ia|jb) =∑K ,L

(ia|L)(L|K )−1(K |jb)

Main advantages:

Error ≈ 60 µH/atom, mostly from cores

Small prefactor (4) → speed-up by 10 or 100 ! Still O(N5)

Resolution of Coulomb operator...

Fairly universal: RI-HF, DFT, MP2, CC ...



From a 3N-D to a 3D view : RDM theory...

1. Reduced density matrices (RDM) theory

Key idea : Ψ(~ri ) fully describes a N particles system (SE), but is amany-complicated object... (1 ≤ i ≤ N, a priori 4N ) Let's write downequations for the ground-state total energy, as functional of the totalelectronic density ρ(~r) � or more generally a reduced density matrix.

E [Ψ]→ E [ρ] (1)

The RDM theory allows us to condensate Ψ(~ri ) into a more handyquantity, e.g. a molecular descriptor. The �rst one we shall de�ne is anoverall density matrix (DM) :

γN(~r1, ~r2, . . . , ~rN , ~r ′1, ~r ′

2, . . . , ~r ′N) = Ψ(~ri ) ∗Ψ?(~r ′j ) (2)




By integrating over a subset of ~ri (p+1,. . .,N) , one de�nes a reduced

DM, in a p-particles con�guration space :

γp(~r1, ~r2, . . . , ~rp, ~r ′1, ~r ′

2, . . . , ~r ′p) = CN

p∫. . .

∫γN(~r1, ~r2, . . . , ~rN , ~r ′

1, ~r ′

2, . . . , ~r ′N)d ~rp+1 . . . d ~rN(3)

Assuming the electronic dance (condensed into a single number, i.e. thedynamic correlation energy) can be expressed when considering a one- ortwo-electron informations, one will mostly refer to:

1 the one-particle DM

γ1(~r1, ~r ′1) = N

∫. . .

∫γN(~r1, ~r2, . . . , ~rN , ~r ′

1, ~r ′

2, . . . , ~r ′N)d ~r2 . . . d ~rN

2 the two-particles DMγ2(~r1, ~r2, ~r ′

1, ~r ′

2)




From 1-RDM to density...

By restricting the one-particle RDM to its diagonal terms, one gets theprobability to �nd a particule (an electron here) in an elementary volumed ~r1 centered on ~r1.

γ1(~r1, ~r1) = N

∫. . .

∫|Ψ(~r1, ~r2, . . . , ~rp)|2d ~r2 . . . d ~rN = ρ1(~r1) (4)

At the end of this "distillation" one has expressed a well-celebratedone-electron quantity . . . the electron density in ~r1, which veri�es∫

ρ1(~r1)d ~r1 = N.

One can already intuite that it will be hard to infer a correct correlation

functional... (How to describe a two-electron dance starting with a one-electron

averaged information... It is possible: we don't know how.) � see slide xx.




From 2-RDM to electron pair probability

Similarly, one may privilege the use of a two-particule DM.It is associated to a diagonal part :

γ2(~r1, ~r2, ~r1, ~r2) = ρ2(~r1, ~r2) = CN2

∫. . .

∫|Ψ(~r1, ~r2, . . . , ~rp)|2d ~r3 . . . d ~rN

which gives the probability P2(~r1, ~r2) � modulo a division by 2.

This is the starting point quantity used by Baerends and coworkers to derive a

DMDF theory. Gill and coworkers chose a Wigner intracule, and relative

position ~u and momenta ~v . We will point out later the intrinsic advantages of

such a choice... and why the two of them abandon DFT !




2. Expression of the energy � implying ρ2

For a given system, E can be expressed as a function of ~r1, ~r ′1and ~r2 :

H =N∑i

[−12∇2

i −∑A

ZA

riA

]+

∑i<j

1rij

(5)

E =

∫ [−12∇2γ1(~r1, ~r ′

1)

]r1=r ′1

d ~r1−∑A

∫ZA

r1Aρ1(~r1)d ~r1+

∫ ∫ρ2(~r1, ~r2)

r12d ~r1d ~r2

(6)

Löwdin decomposition into Coulomb and exchange interactions :∫ρ2(~r1, ~r2)

r12d ~r1d ~r2 =

12

∫ ∫ρ1(~r1)ρ1(~r2)

r12−12

∫ ∫γ1(~r1, ~r2)γ1(~r2, ~r1)

r12d ~r1d ~r2

(7)

To be meaningful, a distribution must be N-representable i.e. verify :1 Antisymmetry by permutation of two electrons2 Normation to N




3. What we do know on electron density...

Conceptually a much more simple object than Ψ(~ri )

1 De�ned in all points, strictly positive and normed. ρ1(~r1)d ~r1 = N

2 Far from nuclei : limr1→∞

[ρ1(~r1)] = 0

3 At the nuclei ZA, a cusp

4 The derivative of ρ close to the cusp is proportionnal to ZA.

limri,A→0

∂

∂riAρ1(riA) = −2ZAρ(0)

→ Wilson's "proof" to consolidate early works in the 1930s. A moredirect statement ?



Fundations : Hohenberg-Kohn and Kohn-Sham theorems

Functional → a mathematical object that maps a function into a number.

Following Wilson's argument, one can write down :

E0 = E [N,Vext ] = E [N,ZA,RA] = E [ρ1(~r1)] (8)

Since 1927, one has long waited for a more formal proof of this verysimple and intuitive picture...→ Hohenberg-Kohn (HK) : there exists a unique form of the potential

which, is used self-consistently, actually yields the exact Schrödinger

energy E.

In a more formal way, two theorems.




Existence : First Hohenberg-Kohn Theorem (HK-I, 1964)

"The electron density ρ(~r) determinated the external potential."

ρ(~r)→ H → Ψ

"The external potential Vext(r) is (to within a constant) a unique functional of

ρ(r); since in turn Vext(r) �xes the Hamiltonian, we see that the full many

particle ground state is a unique functional of ρ(r). "

Demonstration...

Holds for the ground-state only !!




E0[ρ0] =

∫ρ0(~r)Vext(~r) + T [ρ0] + Eee [ρ0] (9)

=

∫ρ0(~r)Vext(~r) + T [ρ0] + J[~r ] + Enonclassic [~r ] (10)

=

∫ρ0(~r)Vext(~r) + FHK [ρ0] (11)

FHK is a unique functional, the same for all N-electron systems but itsanalytic expression (as a function of ~r) remains unknown. The �rst termis system-dependent... and is also unknown.




A variational principle : HK-II

If ρ(~r) is the exact density, then E[ρ(~r)] is minimal and equal to the exact

energy.

This is true for the exact functional only, that we don't know. Then,unlike in the HF formalism, one can go down below Eexact .

In turn, there is a variational criterion to estimate the quality of a guessdensity ρ(~r) ...




Analogy : DFT, HF and the self-consistent �eld

Hartree kinetic energy � averaged �eld :

EH28

T = −12

n∑i

∫Ψi (~r)∇2Ψi (~r)d~r (12)

ETF27T =

310

(6π2)2/3

∫ρ5/3

α (~r)d~r (13)

Not very clear at this stage... TF27 is very beautiful � see its (elegant)derivation as detailled at the end of the following Ref. 1, but raisesissues... (not chemically acceptable keeping in mind the viriel theorem).≈ 10% smaller than H28.

Two main options : introduce a dependence on ∇ρ(~r) (von Weizsacker,1935) or KS (much later, 1965), from which one sees the parrallelbetween DFT and HF frameworks. In the latter, one gives up the sole useof ρ(~r)... hence the following taxonomy.




Taken from Ref. 1 : The Encyclopedia of Computational Chemistry,

1998, "DFT, HF and the self-consistent �eld" by P. M. W. Gill




De�nition of a �ctitious system : Kohn-Sham theory.

• The total energy E , each of its components and every single propertyof a system are functionals of the density.

Analytic expressions of T[ρ] : beautiful but a pathologic chemicalinaccuracy.

Kohn-Sham : there exists a way to approach T [ρ], hence E [ρ].

Let's de�ne a �ctitious system of N fermions with no mutual interactions(they respect the antisymmetry principle) � pseudo-electrons with nospin, neither charge. One de�nes Kohn-Sham orbitals, that exactlyrepresent the system. These φi are used as auxiliary variables to tacklethe problem.

This system is exactly represented by a Slater determinant of the Nspin-orbitals, and one can de�ne...




Where KS orbitals come into play...

... a N-representable density : ρ(~r) =∑N

i φ2

i (~r)

and one writes down the kinetic energy, again for this �ctitious system :

Ts [ρ] =∑

< φi | −12∇2

i |φi > (14)

The later experiences an external potential Vs(~r). KS orbitals φi mustmimimize the energy of this �ctitious system, respecting theorthonormality. They are solution of the following Euler-Lagrange system(one introduces an operator FKS) � because of the constraint∫

ρ(~r)d~r = N. [−12∇2 + Vs

]φi = εiφi

FKS .C = S .C .E




These are known as Kohn-Sham equations.[−12∇2 −

M∑A

ZA

|r − rA|+

∫ρ(~r ′)

|r − r ′|d~r ′ +

∂EXC (ρ(~r)

∂ρ(~r)

]φi (~r) = εiφi (~r)

(15)

In perfect line with the Hartree-Fock equations :−12∇2 −

M∑A

ZA

|r − rA|+

∑j

(Jij − Kij)

φi (~r) = εiφi (~r) (16)

where J is a local operator, unlike K . A key di�erence : VXC includesexchange and correlation components... at the price of a "simple" HFcalculation :)

One stands with the equality : Vs = V +∫ ρ(~r2)

r12d ~r2 + VXC




A short reminder in terms of E...

The energy associated to the �ctitious system is expressed as �independent particles :

E [ρ] = Ts [ρ] +

∫Vs(~r)ρ(~r)d~r (17)

... to be compared to the real system energy � where ee stands for theinter-electronic repulsion.

E [ρ] = T [ρ] + Eee [ρ] +

∫V (~r)ρ(~r)d~r (18)

The key idea is to build up the �ctitious system such that both thedensity ρ(~r) and the energy E [ρ] of the pseudo-particles system areidentical to density and energy of the real system.

Ts [ρ] +

∫Vs(~r)ρ(~r)d~r = T [ρ] + Eee [ρ] +

∫V (~r)ρ(~r)d~r (19)




What's beyound EXC ?

VXC is the 'bin' of DFT... In the exchange�correlation energy, onegathers :

1 Fermi correlation for electrons of same spin

2 Coulomb between electrons of opposite spin

3 self-interaction correction

4 di�erence of kinetic energy between virtual and real systems

E [ρ] =

∫V (~r)ρ(~r) + Ts [ρ] + J[ρ] + T [ρ]− Ts [ρ] + Eee [ρ]− J[ρ] (20)

EXC [ρ] = T [ρ]− Ts [ρ]− Vee [ρ]− J[ρ] (21)

Nota It is a complete non-sense to consider separately X and C energies !One really relies on an inner cancellation of errors. Second order for T ...




Three e�ects of exchange�correlation

Fermi hole : the non-independance of motion arising from the Pauliexclusion particle (particles of same spin).One has exactly :

Eee =

∫ ∫ρ2(~r1, ~r2)

r12d ~r1d ~r2 (22)

It is important to note that this implies ρ2 and we will have toapproximate it as

ρ2(~r1, ~r2)

r12≈ 1

2ρ1(~r1)ρ1(~r2)

r12(23)

which is correct at the limit r →∞ but is not when r12 → 0.

Integrating ρ2 over the entire space leads to N(N-1) against N2 for therho1 product, hence a correction factor for self-interaction.




Coulomb hole : purely electrostatic (opposite spin)

A cusp for ρ2 (this time σ1 6= σ2 that is also not reproduced by theoversimpli�ed ρ1 product

this leads us to de�ne hXC as an exchange�correlation hole

To enforce a correct description of the two holes, one has to enforce apurely-ρ1 description and to modify ρ1(~r2).

2ρ2(~r1, ~r2) = ρ1(~r1)[ρ1(~r2) + hXC (~r1, ~r2)] (24)

hXC is a negative quantity, the so-called exchange�correlation hole. Itaccounts for three e�ects (Fermi, Coulomb and the self-interaction error): physically speaking, it corresponds to the density deformation inducedin r2 by an electron placed in r1.

One splits hXC into two components depending on the spin, one for theexchange (X) and one for the correlation (C).

hXC = hX + hC (25)




Interlude : physical interpretation of the εi

Di�erent than within the HF framework

→ "Janak theorem" instead of Koopman's one.

∂E

∂ni= εi (26)

Electroa�nity : -AE ≈ εLUMO and Ionization potential -IP = ≈ εHOMO

Anyway, too qualitative and barely used.




A fourfold taxonomy

Reminder: one likes to partition the total energy as :

E = ET + EV + EJ + EX + EC (27)

whose respective magnitudes are very di�erent.

Depending on whether ET / EXC are expressed as a function of Ψ or ρ...

1 Hartree-Fock based theory, where both come from Ψi .

2 Adiabatic connection (AC) theories : ET ← ρ and EXC ← Ψi ,ρ

3 Kohn-Sham theories : ET ← Ψi and EXC ← ρ

4 Pure density functional theories : both from ρ




Adiabatic connexion (AC) � 1976,1977

The remaining di�erence to be taken into account is the di�erence ofkinetic energy T − TS ... A formal way to build EXC is to consider acontinuum of �ctitious systems. described by :

Hλ = T + V λext + λVee Vee =

∑i<j

1|r − r ′|

with a coupling parameter λ between the �ctitious system (λ=0) and thereal one (λ=1). Since HK holds for each value of λ, one can choose λsuch that ρλ(~r) = ρλ=1 = ρ. Let's introduce φλ and pose :

UλXC =< φλ[ρ]|Vee |φλ[ρ] > −1

2

∫ ∫ρ(~r1, ~r2)

|~r1 − ~r2|d ~r1d ~r1 (28)

for a XC potential energy. Then EXC =∫UλXCdλ




Still limitations of validity...

1 for λ→ 0, exchange-only limit. Less accuracy : exact exchange isneeded !

2 for λ→ 1 � fully interacting system, correlation is crucial and EEwill not work !

In between the two worlds → hybrid functionals...

Climbing Jacob's Ladder : LD(S)A, GGA, meta-GGA, hybrid and �nally(?) ACFD (�uctuation-dissipation) and RPA (random phaseapproximation).



Climbing Jacob's Ladder : LD(S)A, GGA, meta-GGA, hybrid and AC

1. LDA and jellium (= uniform electron gas)

Two underlying hypothesis : X�C separability and uniform electron gas.

1 TF27 for kinetic energy

2 Dirac exchange (D30) functional. D30 yields energies that are roughly

10% smaller than F30. Also, it only partly removes the spurious

self-interactions.

ED30

X = −32

(34π

)1/3 ∫ρ4/3

α (~r)d~r (29)

3 Vosko-Wilk-Nusair correlation (VWN) functional. VWN is a

complicated expression (not reported here), proposed in 1980 from QMC

data, which overestimates Ec by a factor of ca. 2 (!) when applied to

atoms and molecules.

Also Xα Slater 1951.




1bis. Exchange functionals

Sham-Kleinman exchange functional (within 3% from F30):

ESK71

X = ED30

XC −5

(36π)5/3

∫ρ4/3

α x2αd~r

Has the "merit" to be an ab initio functional... but in the realm of DFT,a parametrized functional has been widely adopted.

EB88X = ED30

X − b

∫ρ4/3

α

x2α1 + 6βxαsinh−1xα

d~r (30)

dampled x2 behavior � b=0.042 �tted.




2. GGA 1 Physical idea : introduces a dependence on ∇ρ(~r)L(S)DA results are often improved wrt. HF (deviation of +36 vs. +72kcal/mol along the G2 test). It also provides a satisfactory asymptoticbehavior... but a noticeable tendency to overbinding.

Atoms and molecules are intrinsically far from a universal characterassumed by jellium. Let introduce a gradient correction (generalizedgradient approximation, GGA) to improve agreement.

EGGAXC [ρ] =

∫f [ρ(~r),∇ρ(~r)]ρ(~r)d~r (31)

Von Weizsacker kinetic functional (1935, within 1% from H28) :

EW35

T = ETF27T +

18

∫ρ5/3

α x2αd~r

EGGAX = ELDA

X +

∫F (s)ρ4/3(~r)d~r with s(~r) =

|∇ρ(~r)|ρ4/3(~r)

(32)

where s is a reduced gradient density.1Let us mention in passing the existence of GEA and GWA approaches




2bis. Some GGA (exchange) functionals... and beyond

EXC [ρ] = EXC (ρ,∇ρ) (33)

Two main "pragmatic" options :

1 Fit an expression of F (s) on rare gas energies → B88 (Axel Becke)

2 Develop as a rational function of (even) power of s → P86 (JohnPerdrew)

Following the same idea... let's climb another step in the so-known DFTJacob's ladder ! Again, two strategies...

1 introducing an explicit dependence on ∇2ρ

2 introducing the kinetic energy of the electrons (meta-GGA, e.g.M06-L, TPSS)




3. Hybrid functionals.

Such functionals still su�er from chronic de�ciences : problem withlocalized electrons...Correlation is a very short-range phenomenom, and the non-localcounterpart only comes from exchange (→ range-separated hybridfunctionals, see supra).

Wait... one knows an exact expression of the exchange energy. Let'sreplace EGGA

X by EKSX or EHF

X ! But, doing so, one a�ects therelocalization of hXC ... 2

Half-and-half functional : EHHXC = 1

2EKSX + 1

2ELDAXC

Examples : B3LYP, B3PW91, PBE0, . . .

EB3LYPXC = aEHF

X + (1− a)ELDAX + bEGGA

X + cEGGAC + (1− c)EVWN

C

→ optimal value of exchange exact ?2For a non-conventional (?) point-of-view : "Obituary: Density Functional Theory

(1927-1993)", P. M. W. Gill. Aust. J. Chem., 2001, 54, 661.Élise Dumont Lecture M2 (RFCT, 1st week and ATOSIM)� 25th Oct. 2010



EA, IP, geometries, dipolar moments of good quality.

Need GGA for dissociation barriers.




A dream ? Jacob's Ladder

i.e. begin able to systematically improve DFT performance.



Pitfalls of DFT

1 Self repulsion, interaction-error : 2c-3e systems.

2 H+2dissociation.

3 Weak interactions and dispersion.

4 Protobranching : a mid-range correlation ?



Pitfalls of DFT

• Self repulsionOne electron repulses itself in DFT...

LDA → does not tend to 1r � like in HF, yet one has a too rapidlydecaying asymptotic e(−λr). Improved by GGA functionals... but hybrido�er a better solution towards Rydberg states, three-electron two-centerbonding, charge transfer → of special importance for excited states !

• Dispersion

Inter-molecular correlation is non-local, hence badly reproduced by anyfunctional → empirical dispersion proposed by Grimme notably.Also source of errors for weak interactions.



Pitfalls of DFT

Some simple organic systems...

Protobranch = 1,3-alkyl-alkyl interaction (ca. 2.8 kcal/mol)

Protobranching is a very useful & versatile concept:→ ring strain, (hyper)conjugation, aromaticity, isomerisation energies...

Grimme & Schleyer: stabilization from mid-range electron correlation...→ seriously underestimated by HF and most DFT levels.

1P. van Ragué Schleyer, Chem. Eur. J. 2007, 13, 7731-7744.Élise Dumont Lecture M2 (RFCT, 1st week and ATOSIM)� 25th Oct. 2010


Most recent developments...

Recently enough in the DFT community.

1 TD-DFT

2 Doubly Hybrid functionals

3 Long-range separated hybrids → optimal value of the attenuationparameter µ ?

4 Double hybrid

5 Truhlar functionals...

6 Multi-reference systems (broken symmetry)

7 Intracules, cumulants and DM functional theory...




Time-dependent density functional theory (TD-DFT).

Allows to consider excited states (vertically, but also to look for conicalintersections)

Runge-Gross theoremLinear response

Ref. : A. Dreuw and M. Head-Gordon, Chem. Rev., 2005, 105,4009�4037, "Single-Reference ab Initio Methods for the Calculation of

Excited States of Large Molecules"




A double-hybrid functional B2PLYP-D1, S. Grimme, Munster

A physically-based decomposition

EXC = EX + EC

EXC = (1− ax)EX + aXEHF + EC

EXC = (1− ax)EX + aXEHF

+ (1− aC )EC + aCEPT2

⊕Edisp = −s6∑j<i

Cij6

R6

ij

fdmp(Rij)

aC → "mid-range" EC (≈ 2�5 Å)

6= lengths for electron correlation

1Schwabe and Grimme, Phys. Chem. Chem. Phys. 2007, 9, 3397-3406.2Y. Zhang, X. Xu and W. A. Goddard III, PNAS, 2009, 106, 4963�68.




Wigner intracule W(u,v) for the berryllium atom.




Truhlar M-0x functionals

M062 functionals of Don Truhlar, UMN, 2006 Schrodinger MedalAn hybrid meta-GGA functional

- How many parameters? I don't count them!

Derivatives: M06-2X, M06-HF, M06-L

- Why several functionals? A charpenter needs several saws!

2Zhao and Truhlar, Acc. Chem. Res. 2008, 41, 157-167.Élise Dumont Lecture M2 (RFCT, 1st week and ATOSIM)� 25th Oct. 2010



In practice...

1 Always the need to calibrate (adapt ?) a functional against post HFor experimental data (or refer to an existing benchmark)

2 the grid issue...

3 the Coulomb problem

4 certainly other points...




The Coulomb problem

EJ expression involves a double sum of integrals over space, whichimplies a prohibitive (!) O(N2) behavior.

→ bottleneck tackled by Pople et al. in the 60s.

Currently, one relies on FMM (or KWIK) algorithms.




Towards a gridless DFT ?

Functionals lead to cumbersome expressions that one cannot solve inclosed form.

E = f (ρ(~r), x(~r)) ≈Ngrid∑i

wi f (ρ(~ri ), x(~ri )) (34)

Choosing a quadrature for an e�cient grid... 'fuzzy' Voronoidecomposition into atomic contributions in modern functionals asproposed by Becke.

Let's keep in mind that a DFT calculation is also dependent on thechoice of the grid → grid=ultrafine in G09. (Possible source ofdiscrepancy between di�erent QM packages).



Further reading

Further references

In addition to the ones listes above :

"A Chemist's Guide to Density Functional Theory", 2nd edition, Wiley-,W. Koch and M. C. Holthausen.

"14 easy lessons in DFT", IJQC, 2010, 110, 2801�07, Perdew andRuzsinsky.



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