Introduction to density-functional theory

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Introduction to density-functional theory

Julien Toulouse Laboratoire de Chimie Th´eorique

Sorbonne Universit´ e and CNRS, Paris, France

European Summerschool in Quantum Chemistry (ESQC) September 2019, Sicily, Italy

www.lct.jussieu.fr/pagesperso/toulouse/presentations/presentation_esqc_19.pdf

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Why and how learning density-functional theory?

Density-functional theory (DFT) is:

apractical electronic-structure computational method, widely used in quantum chemistry and condensed-matter physics;

anexact and elegant reformulation of the quantum many-body problem, which has led to new ways of thinking in the field.

Classical books:

R. G. Parr and W. Yang,Density-Functional Theory of Atoms and Molecules, Oxford University Press, 1989.

R. M. Dreizler and E. K. U. Gross,Density Functional Theory: An Approach to the Quantum Many-Body Problem, Springer-Verlag, 1990.

W. Koch and M. C. Holthausen,A Chemist’s Guide To Density Functional Theory, Wiley-VCH, 2001.

My lecture notes:

http://www.lct.jussieu.fr/pagesperso/toulouse/enseignement/introduction_dft.pdf

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Outline

1 Basic density-functional theory The quantum many-electron problem The universal density functional The Kohn-Sham method

2 More advanced topics in density-functional theory

Exact expressions for the exchange and correlation functionals Fractional electron numbers and frontier orbital energies

3 Usual approximations for the exchange-correlation energy The local-density approximation

Semilocal approximations

Hybrid and double-hybrid approximations

Range-separated hybrid and double-hybrid approximations Semiempirical dispersion corrections

4 Additional topics in density-functional theory Time-dependent density-functional theory

Some less usual orbital-dependent exchange-correlation functionals Uniform coordinate scaling

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Outline

1

Basic density-functional theory

The quantum many-electron problem The universal density functional

The Hohenberg-Kohn theorem Levy’s constrained-search formulation

The Kohn-Sham method

Decomposition of the universal functional The Kohn-Sham equations

Practical calculations in an atomic basis Extension to spin density-functional theory

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The quantum many-electron problem

We consider aN-electron systemin the Born-Oppenheimer and non-relativistic approximations.

Theelectronic Hamiltonianin the position representation is, in atomic units, H(r1,r2, ...,rN) =−1

2 XN

i=1

∇²r_i+1 2

XN

i=1

XN

j=1 i6=j

1

|ri−rj|+ XN

i=1

v_ne(ri)

wherevne(ri) =−P

αZα/|ri−Rα|is the nuclei-electron interaction potential.

Stationary states satisfy thetime-independent Schr¨odinger equation H(r1,r2, ...,rN)Ψ(x1,x2, ...,xN) =EΨ(x1,x2, ...,xN) where Ψ(x1,x2, ...,xN) is a wave function written with space-spin coordinates xi = (ri, σi) (withri ∈R³andσi ∈ {↑,↓}) which is antisymmetric with respect to the exchange of two coordinates, andE is the associated energy.

UsingDirac notations(representation-independent formalism):

Hˆ|Ψi=E|Ψi where Hˆ = ˆT + ˆWee+ ˆVne

These operators can be conveniently expressed in (real-space) second quantization.

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Wave-function variational principle

Theground-state electronic energyE0can be expressed with thewave-function variational principle

E0= min

Ψ hΨ|Hˆ|Ψi

where the minimization is over allN-electron antisymmetric wave functions Ψ, normalized to unityhΨ|Ψi= 1.

Remark: IfH does not bind N electrons, then the minimum does not exist but the ground-stateˆ energy can still be defined as an infimum, i.e. E0= inf

ΨhΨ|H|Ψi.ˆ

DFT is based on a reformulation of this variational theorem in terms of the one-electron densitydefined as

n(r) =N Z

· · · Z

|Ψ(x,x2, ...,xN)|²dσdx2...dxN

which is normalized to the electron number,R

n(r)dr=N.

Remark: Integration over a spin coordinateσmeans a sum over the two values ofσ, i.e.

Rdσ=P

σ=↑,↓.

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Outline

1

Basic density-functional theory

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The Hohenberg-Kohn theorem

Consider an electronic system with an arbitrary external local potentialv(r) in place of vne(r).

The corresponding ground-state wave function Ψ (or one of them if there are several) can be obtained by solving the Schr¨odinger equation, from which the associated ground-state densityn(r) can be deduced. Therefore, one has the mapping:

v(r)−−−−−→n(r)

In 1964, Hohenberg and Kohn showed that this mapping can be inverted, i.e. the ground-state densityn(r) determines the potentialv(r) up to an arbitrary additive constant:

n(r)−−−−−−−−−→Hohenberg-Kohn v(r) + const

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Proof of the Hohenberg-Kohn theorem (1/2)

This is a two-step proof by contradiction.

Consider two local potentials differing by more than an additive constant:

v₁(r)6=v₂(r) + const We have two Hamiltonians:

Hˆ1= ˆT + ˆWee+ ˆV1with a ground state ˆH1|Ψ1i=E1|Ψ1iand ground-state densityn1(r) Hˆ₂= ˆT + ˆW_ee+ ˆV₂with a ground state ˆH₂|Ψ2i=E₂|Ψ2iand ground-state densityn₂(r) 1 We first show that Ψ16= Ψ2:

Assume Ψ1= Ψ2= Ψ. Then we have:

( ˆH₁−Hˆ₂)|Ψi= ( ˆV₁−Vˆ₂)|Ψi= (E1−E₂)|Ψi or, in position representation,

XN i=1

[v1(ri)−v2(ri)]Ψ(x1,x2, ...,xN) = (E1−E2)Ψ(x1,x2, ...,xN)

If Ψ(x1,x2, ...,xN)6= 0 for all (r1,r2, ...,rN) and at least one fixed set of (σ1, σ2, ..., σN), which is “true almost everywhere for reasonably well behaved potentials”, then it implies thatv1(r)−v2(r) = const, in contradiction with the initial hypothesis.

=⇒Intermediate conclusion: two local potentials differing by more than an

additive constant cannot share the same ground-state wave function. _9/81

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Proof of the Hohenberg-Kohn theorem (2/2)

2 We now show thann16=n2:

Assumen₁=n₂=n. Then, by the variational theorem, we have:

E1=hΨ1|Hˆ1|Ψ1i<hΨ2|Hˆ1|Ψ2i=hΨ2|Hˆ2+ ˆV1−Vˆ2|Ψ2i=E2+ Z

[v1(r)−v2(r)]n(r)dr The strict inequality comes from the fact that Ψ2cannot be a ground-state wave function of ˆH1, as shown in the first step of the proof.

Symmetrically, by exchanging the role of system 1 and 2, we have the strict inequality E2<E1+

Z

[v2(r)−v1(r)]n(r)dr Adding the two inequalities gives the inconsistent result

E1+E2<E1+E2

=⇒Conclusion: there cannot exist two local potentials differing by more than an additive constant which have the same ground-state density.

Remark: This proof does not assume non-degenerate ground states (contrary to the original Hohenberg-Kohn proof).

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The universal density functional and the variational property

The Hohenberg-Kohn theorem can be summarized as

n(r)−−−−−→v(r)−−−−−→Hˆ −−−−−→everything

v is a functional of the ground-state densityn, i.e. v[n], and all other quantities as well.

In particular, aground-state wave functionΨ for a given potentialv(r) is a functional ofn, denoted by Ψ[n]. Hohenberg and Kohn defined theuniversal density functional

F[n] =hΨ[n]|Tˆ+ ˆW_ee|Ψ[n]i and thetotal electronic energy functional

E[n] =F[n] + Z

vne(r)n(r)dr for the specific external potentialvne(r) of the system considered.

Hohenberg and Kohn showed that we have avariational propertygiving the exact ground-state energy

E0= min

n

F[n] +

Z

vne(r)n(r)dr

where the minimization is overN-electron densitiesnthat are ground-state densities associated with some local potential (referred to asv-representable densities). The minimum is reached for an exact ground-state densityn0(r) of the potentialvne(r).

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Levy’s constrained-search formulation

In 1979 Levy, and later Lieb, proposed to redefine theuniversal density functionalas F[n] = min

Ψ→nhΨ|Tˆ+ ˆW_ee|Ψi=hΨ[n]|Tˆ+ ˆW_ee|Ψ[n]i

where “Ψ→n” means that the minimization is done over normalized antisymmetric wave functions Ψ which yield the fixed densityn.

Remark: This definition of F[n]does not require the existence of a local potential associated with the density: it is defined on the larger set of N-electron densities coming from an antisymmetric wave function (referred to as N-representable densities).

Thevariational propertyis easily obtained using the constrained-search formulation:

E₀= min

Ψ hΨ|Tˆ+ ˆW_ee+ ˆV_ne|Ψi

= min

n min

Ψ→nhΨ|Tˆ + ˆWee+ ˆVne|Ψi

= min

n

Ψ→nminhΨ|Tˆ+ ˆW_ee|Ψi+ Z

v_ne(r)n(r)dr

= min

n

F[n] +

Z

v_ne(r)n(r)dr

×Ψ1 ×Ψ2

×Ψ3

×Ψ4

×Ψ5

×Ψ6

n1

n2 n3

Hence, in DFT, we replace “min

Ψ ” by “min

n ” which is atremendous simplification!

However,F[n] =T[n] +Wee[n] isvery difficult to approximate, in particular the kinetic energy partT[n].

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Outline

1

Basic density-functional theory

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Kohn-Sham (KS) method: decomposition of the universal functional

In 1965, Kohn and Sham proposed to decomposeF[n] as F[n] =Ts[n] +EHxc[n]

Ts[n] is thenon-interacting kinetic-energy functional:

T_s[n] = min

Φ→nhΦ|Tˆ|Φi=hΦ[n]|Tˆ|Φ[n]i

where “Φ→n” means that the minimization is done over normalizedsingle-determinant wave functions Φ which yield the fixed densityn. The minimizing single-determinant wave function is called theKS wave functionand is denoted by Φ[n].

Remark: Introducing a single-determinant wave function is notan approximation because any N-representable density n can be obtained from a single-determinant wave function.

The remaining functionalEHxc[n] is called theHartree-exchange-correlation functional.

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Kohn-Sham (KS) method: variational principle

Theexact ground-state energycan then be expressed as E0= min

n

F[n] +

Z

vne(r)n(r)dr

= min

n

Φ→nminhΦ|Tˆ|Φi+EHxc[n] + Z

vne(r)n(r)dr

= min

n min

Φ→n

nhΦ|Tˆ+ ˆVne|Φi+EHxc[nΦ]o

= min

Φ

nhΦ|Tˆ + ˆV_ne|Φi+E_Hxc[nΦ]o

and the minimizing single-determinant KS wave function gives an exact ground-state densityn0(r).

Hence, in KS DFT, we replace “min

Ψ ” by “min

Φ ” which is still atremendous

simplification! The advantage of KS DFT over pure DFT is that a major part of the kinetic energy is treated explicitly with the single-determinant wave function Φ.

KS DFT is similar to Hartree-Fock (HF) EHF= min

Φ hΦ|Tˆ+ ˆVne+ ˆWee|Φi

but in KS DFT the exact ground-state energy and density are in principle obtained!

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Kohn-Sham (KS) method: the Hartree-exchange-correlation functional

E_Hxc[n] is decomposed as

EHxc[n] =EH[n] +Exc[n]

E_H[n] is theHartree energy functional EH[n] =1

2

x n(r1)n(r2)

|r1−r2| dr1dr2

representing the classical electrostatic repulsion energy for the charge distributionn(r) and which is calculated exactly.

Exc[n] is theexchange-correlation energy functionalthat remains to approximate.

This functional is often decomposed as

E_xc[n] =E_x[n] +E_c[n]

whereE_x[n] is theexchange energy functional

Ex[n] =hΦ[n]|Wˆee|Φ[n]i −EH[n]

andEc[n] is thecorrelation energy functional

Ec[n] =hΨ[n]|Tˆ+ ˆWee|Ψ[n]i − hΦ[n]|Tˆ+ ˆWee|Φ[n]i=Tc[n] +Uc[n]

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The Kohn-Sham equations (1/2)

The single determinant Φ is constructed from a set ofNorthonormal occupied spin-orbitalsψi(x) =φi(r)δσi,σ.

Thetotal energyto be minimized is E[{φi}] =

XN i=1

Z φ^∗_i(r)

−1

2∇²+v_ne(r)

φi(r)dr+E_Hxc[n]

and thedensityis

n(r) = XN

i=1

|φi(r)|²

For minimizing over the orbitals{φi}with the constraint of keeping the orbitals orthonormalized, we introduce theLagrangian

L[{φi}] =E[{φi}]− XN

i=1

εi

Z

φ^∗_i(r)φi(r)dr−1

whereεi is the Lagrange multiplier associated with the normalization condition of φi(r).

The Lagrangian must bestationarywith respect to variations of the orbitalsφi(r) δL

δφ^∗_i(r) = 0

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Interlude: Review on functional calculus

For afunctionalF[f] of the functionf(x), an infinitesimal variationδf off leads to an infinitesimal variation ofF which can be expressed as

δF[f] =

Z δF[f] δf(x)δf(x)dx

This defines thefunctional derivativeofF[f] with respectf(x): δF[f] δf(x) Remark: For a function F(f1,f2, ...)of several variables f1, f2, ..., we have

dF=X

i

∂F

∂f_i df_i

δF[f]/δf(x)is the analog of∂F/∂fi for the case of an infinite continuous number of variables.

For a functionalF[f] of a functionf[g](x) which is itself a functional of another functiong(x), we have thechain rule

δF δg(x) =

Z δF δf(x^′)

δf(x^′) δg(x)dx^′

Remark: It is the analog of the chain rule for a function F(f1,f2, ...)of several variables f_j(g1,g2, ...)which are themselves functions of other variables g1,g2, ...

∂F

∂gi

=X

j

∂F

∂fj

∂gi

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The Kohn-Sham equations (2/2)

We find for the functional derivative of the Lagrangian 0 = δL

δφ^∗_i(r) =

−1

2∇²+vne(r)

φi(r) +δEHxc[n]

δφ^∗_i(r) −εiφi(r) We calculate the termδEHxc[n]/δφ^∗_i(r) using the chain rule

δEHxc[n]

δφ^∗_i(r) =

Z δEHxc[n]

δn(r^′) δn(r^′)

δφ^∗_i(r)dr^′=vHxc(r)φi(r)

where we have usedδn(r^′)/δφ^∗_i(r) =φi(r)δ(r−r^′) and we have introduced theHartree-exchange-correlation potentialvHxc(r)

v_Hxc(r) =δEHxc[n]

δn(r) which is itself a functional of the density.

We arrive at theKS equations

−1

2∇²+vne(r) +v_Hxc(r)

φi(r) =εiφi(r)

The orbitalsφi(r) are called the KS orbitals andεi are the KS orbital energies.

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The Kohn-Sham equations and the Kohn-Sham Hamiltonian

The KS orbitals are eigenfunctions of the KS one-electron Hamiltonian hs(r) =−1

2∇²+vs(r) where vs(r) =vne(r) +vHxc(r) is the KS potential.

Mathematically, the KS equations are a set of coupled self-consistent equations since the potentialv_Hxc(r) depends on all the occupied orbitals{φi(r)}through the density.

Physically,hs(r) defines the KS system which is a system ofNnon-interacting electrons in an effective external potentialv_s(r) ensuring that its ground-state densityn(r) is the same as the exact ground-state densityn0(r) of the physical system ofNinteracting electrons.

The KS equations also defines virtual KS orbitals{φa(r)}which together with the occupied KS orbitals form a complete basis sinceh_s(r) is a self-adjoint operator.

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The Hartree-exchange-correlation potential

Remark: The existence of the functional derivative vHxc(r) =δEHxc[n]/δn(r)has been assumed.

This is in fact not true for all densities but only for vs-representable densities, i.e. densities that are the ground-state densities of a non-interacting system with some local potential.

Remark: The KS potential is defined only up to an additive constant. For finite systems, we choose the constant so that the potential vanishes at infinity: lim_|r|→∞vs(r) = 0

Following the decomposition ofEHxc[n], the potentialvHxc(r) is also decomposed as vHxc(r) =vH(r) +vxc(r)

with theHartree potential vH(r) = δEH[n]

δn(r) =

Z n(r^′)

|r−r^′|dr^′ and theexchange-correlation potential v_xc(r) =δExc[n]/δn(r) The potentialv_xc(r) can be decomposed as v_xc(r) =v_x(r) +v_c(r)

with theexchange potential vx(r) =δEx[n]/δn(r) and thecorrelation potential vc(r) =δEc[n]/δn(r)

Remark: Contrary to Hartree-Fock, the KS exchange potential is local.

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Practical calculations in an atomic basis (1/3)

We consider abasis ofM atom-centered functions{χν(r)}, e.g. GTO basis functions.

The orbitals are expanded as

φi(r) = XM

ν=1

c_νi χν(r)

Inserting this expansion in the KS equations hs(r)φi(r) =εiφi(r)

and multiplying on the left byχ^∗µ(r) and integrating overr, we arrive at the familiarSCF generalized eigenvalue equation

XM ν=1

Fµνcνi =εi

XM ν=1

Sµνcνi

whereFµν =R

χ^∗_µ(r)hs(r)χν(r)drare the elements of the KS Fock matrix and Sµν =R

χ^∗_µ(r)χν(r)drare the elements of the overlap matrix.

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Practical calculations in an atomic basis (2/3)

The Fock matrix is calculated as Fµν=hµν+Jµν+Vxc,µν

hµν are the one-electron integrals: hµν = Z

χ^∗µ(r)

−1

2∇²+vne(r)

χν(r)dr Jµν is the Hartree potential matrix:

J_µν= Z

χ^∗_µ(r)vH(r)χν(r)dr= XM

λ=1

XM

γ=1

P_γλ(χµχν|χλχγ)

where (χµχν|χλχγ) =x χ^∗_µ(r1)χν(r1)χ^∗_λ(r2)χγ(r2)

|r1−r2| dr1dr2are the two-electron integrals (in chemists’ notation) andP_γλ=

XN

i=1

c_γic_λi^∗ is the density matrix.

Vxc,µν is the exchange-correlation potential matrix: Vxc,µν= Z

χ^∗_µ(r)vxc(r)χν(r)dr The total electronic energy is calculated as

E= XM µ=1

XM ν=1

Pνµhµν+1 2

XM µ=1

XM ν=1

PνµJµν+Exc

The density is calculated as n(r) = XM

γ=1

XM

λ=1

P_γλχγ(r)χ^∗λ(r)

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Practical calculations in an atomic basis (3/3)

In the simplest approximation, the exchange-correlation energy functional has a local form

E_xc^local= Z

f(n(r))dr

wheref(n(r)) has a complicated nonlinear dependence on the densityn(r).

For example, in the local-density approximation (LDA), the exchange energy is E_x^LDA=cx

Z

n(r)^4/3dr wherecxis a constant, and the exchange potential is

v_x^LDA(r) =4

3cxn(r)^1/3

Therefore, the integrals cannot be calculated analytically, but are instead evaluated by numerical integration on a grid

V_xc,µν≈X

k

w_kχ^∗_µ(rk)vxc(rk)χν(rk) and E_xc^local≈X

k

w_kf(n(rk)) whererkandwk are quadrature points and weights. For polyatomic molecules, the multicenter numerical integration scheme of Becke (1988) is generally used.

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Extension to spin density-functional theory (1/2)

For dealing with an external magnetic field, DFT has been extended from the total density tospin-resolved densities(von Barth and Hedin, 1972)

nσ(r) =N Z

· · · Z

|Ψ(rσ,x2, ...,xN)|²dx2...dxN with σ=↑or↓ Without external magnetic fields, this isin principle not necessary, even for open-shell

systems. However, in practice, the dependence on the spin densities allows one to constructmore accurate approximate exchange-correlation functionalsand is thereforealmost always used for open-shell systems.

Theuniversal density functionalis now defined as F[n↑,n↓] = min

Ψ→n_↑,n↓hΨ|Tˆ+ ˆW_ee|Ψi

where the search is over normalized antisymmetric wave functions Ψ which yield fixed spin densities integrating to the numbers ofσ-spin electrons, i.e. R

n_σ(r)dr=N_σ. AKS methodis obtained by decomposingF[n↑,n↓] as

F[n↑,n↓] =Ts[n↑,n↓] +E_H[n] +Exc[n↑,n↓]

whereTs[n↑,n↓] is defined with a constrained search overspin-unrestrictedSlater determinants Φ

Ts[n↑,n↓] = min

Φ→n_↑,n_↓hΦ|Tˆ|Φi

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Extension to spin density-functional theory (2/2)

Writing the spatial orbitals of the spin-unrestricted determinant asφiσ(r) (with indices explicitly including spin now), we have now the spin-dependent KS equations

−1

2∇²+vne(r) +vH(r) +vxc,σ(r)

φiσ(r) =εiσφiσ(r) with the spin-dependent exchange-correlation potentials

vxc,σ(r) = δExc[n↑,n↓] δnσ(r) and the spin densities

nσ(r) =

Nσ

X

i=1

|φiσ(r)|²

The spin-dependent exchange functionalE_x[n↑,n↓] can be obtained from the spin-independent exchange functionalEx[n] with thespin-scaling relation

Ex[n↑,n↓] = 1

2(Ex[2n↑] +Ex[2n↓])

Therefore, any approximation for the spin-independent exchange functionalEx[n] can be easily extended to an approximation for the spin-dependent exchange functional Ex[n↑,n↓]. Unfortunately, there is no such relation for the correlation functional.

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Outline

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Outline

2

The exchange-correlation hole

Thepair densityassociated with the wave function Ψ[n] is n₂(r1,r2) =N(N−1)

Z

· · · Z

|Ψ[n](x1,x2, ...,xN)|²dσ1dσ2dx3...dxN

which is a functional of the density. It is normalized to the number of electron pairs:

s n₂(r1,r2)dr1dr2=N(N−1). It is proportional to the probability density of finding two electrons at positions (r1,r2) with all the other electrons anywhere.

It can be used to express the electron-electron interaction energy hΨ[n]|Wˆee|Ψ[n]i= 1

2

x n2(r1,r2)

|r1−r2|dr1dr2

Mirroring the decomposition ofE_Hxc[n], the pair density can be decomposed as n2(r1,r2) =n(r1)n(r2) +n2,xc(r1,r2)

| {z }

independent electrons

| {z }

exchange and correlation effects

We also introduce theexchange-correlation holenxc(r1,r2) by n_2,xc(r1,r2) =n(r1)nxc(r1,r2)

It can be interpreted as the modification due to exchange and correlation effects of the conditional probability of finding an electron atr2knowing that one has been found atr1. We have the exact constraints: n_xc(r1,r2)≥ −n(r2) and R

n_xc(r1,r2)dr2=−1

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The exchange hole

Similarly, we define theKS pair densityassociated with the KS single determinant Φ[n]

n2,KS(r1,r2) =N(N−1) Z

· · · Z

|Φ[n](x1,x2, ...,xN)|²dσ1dσ2dx3...dxN

It can be decomposed as

n2,KS(r1,r2) =n(r1)n(r2) +n2,x(r1,r2)

and we introduce theexchange holenx(r1,r2) by n2,x(r1,r2) =n(r1)nx(r1,r2)

n_x(r1,r2) 0r1

r2

which satisfies the exact constraints:

nx(r1,r2)≥ −n(r2) and R

nx(r1,r2)dr2=−1 and nx(r1,r2)≤0 The exchange energy functional is the electrostatic interaction energy between an

electron and its exchange hole:

Ex[n] =1 2

x n(r1)nx(r1,r2)

|r1−r2| dr1dr2= Z

n(r1)εx[n](r1)dr1

whereεx[n](r1) is the exchange energy per particle. In approximate exchange density functionals, the quantityεx[n](r1) is usually what is approximated.

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The correlation hole

Thecorrelation holeis defined as the difference n_c(r1,r2) =n_xc(r1,r2)−n_x(r1,r2)

nc(r1,r2)

0r1 r2

and satisfies the sum rule Z

n_c(r1,r2)dr2= 0

which implies that the correlation hole has negative and positive contributions.

The potential contribution to the correlation energy can be written in terms of the correlation hole

U_c[n] =1 2

x n(r1)nc(r1,r2)

|r1−r2| dr1dr2

But in order to express the total correlation energyE_c[n] =T_c[n] +U_c[n] in a similar form, we need to introduce the adiabatic-connection formalism.

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The adiabatic connection (1/3)

The idea of theadiabatic connectionis to have a continuous path between the non-interacting KS system and the physical system while keeping the ground-state density constant.

For this, we introduce a Hamiltonian depending on acoupling constantλwhich switches on the electron-electron interaction

Hˆ^λ= ˆT+λWˆ_ee+ ˆV^λ

where ˆV^λis the external local potential imposing that the ground-state density is the same as the ground-state density of the physical system for allλ, i.e. n^λ(r) =n₀(r),∀λ.

By varyingλ, we connect the KS non-interacting system (λ= 0) to the physical interacting system (λ= 1):

Hˆ^λ=0

| {z }

KS non-interacting system

0≤λ≤1

←−−−−−−−−−→ Hˆ^λ=1

| {z }

Physical interacting system

We define a universal functional for each value of the parameterλ F^λ[n] = min

Ψ→nhΨ|Tˆ+λWˆee|Ψi=hΨ^λ[n]|Tˆ+λWˆee|Ψ^λ[n]i

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The adiabatic connection (2/3)

The functionalF^λ[n] can be decomposed as

F^λ[n] =T_s[n] +E_H^λ[n] +E_x^λ[n] +E_c^λ[n]

E_H^λ[n] andE_x^λ[n] are the Hartree and exchange functionals associated with the interactionλWˆee and are simply linear inλ

E_H^λ[n] =λEH[n] and E_x^λ[n] =λEx[n]

The correlation functionalE_c^λ[n] is nonlinear inλ

E_c^λ[n] =hΨ^λ[n]|Tˆ+λWˆ_ee|Ψ^λ[n]i − hΦ[n]|Tˆ+λWˆ_ee|Φ[n]i We can get rid of ˆT by taking the derivative with respect toλand using the

Hellmann-Feynman theorem for the wave function Ψ^λ[n]

∂E_c^λ[n]

∂λ =hΨ^λ[n]|Wˆ_ee|Ψ^λ[n]i − hΦ[n]|Wˆ_ee|Φ[n]i

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The adiabatic connection (3/3)

Reintegrating overλfrom 0 to 1, and usingE_c^λ=1[n] =E_c[n] andE_c^λ=0[n] = 0, we arrive at theadiabatic-connection formula

Ec[n] = Z1

0

dλhΨ^λ[n]|Wˆee|Ψ^λ[n]i − hΦ[n]|Wˆee|Φ[n]i

Introducing the correlation holen^λ_c(r1,r2) associated with the wave function Ψ^λ[n], the adiabatic-connection formula can also be written as

Ec[n] = 1 2

Z 1 0

dλx n(r1)n_c^λ(r1,r2)

|r1−r2| dr1dr2

Introducing theλ-integrated correlation hole ¯nc(r1,r2) =R1

0 dλn^λ_c(r1,r2), we finally write Ec[n] =1

2

x n(r1)¯n_c(r1,r2)

|r1−r2| dr1dr2= Z

n(r1)εc[n](r1)dr1

whereεc[n](r1) is the correlation energy per particle, which is the quantity usually approximated in practice.

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Outline

2

Quantum mechanics with fractional electron numbers (1/2)

The ground-state energy of a system with afractional number of electrons N =N−1 +f (whereNis an integer and 0≤f ≤1) can be defined as

E₀^N−1+f = min

ˆΓ

Trh ˆΓ

Tˆ+ ˆWee+ ˆVne

i

where the minimization is overensemble density matricesΓ of the formˆ ˆΓ = (1−f)|Ψ^N−1ihΨ^N−1|+f |Ψ^NihΨ^N|

wheref is fixed, and Ψ^N−1and Ψ^Nare (N−1)- andN-electron normalized wave functions to be varied.

The minimizing ensemble density matrix is

Γˆ0= (1−f)|Ψ^N−1₀ ihΨ^N−1₀ |+f |Ψ^N0ihΨ^N0|

where Ψ^N−1₀ and Ψ^N₀ are the (N−1)- andN-electron ground-state wave functions.

The ground-state energy islinearinf between the integer numbersN−1 andN E₀^N−1+f = (1−f)E₀^N−1+f E₀^N

whereE₀^N−1andE₀^N are the (N−1)- andN-electron ground-state energies.

Similarly, between the integer electron numbersN andN+ 1, we have E₀^N+f = (1−f)E₀^N+f E₀^N+1

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Quantum mechanics with fractional electron numbers (2/2)

The ground-state energy is acontinuous piecewise linear functionof the fractional electron numberN.

E₀^N

N−1 N N+ 1 N

E₀^N+1 E₀^N E₀^N−¹

−I_N

−AN

The derivative ofE₀^N with respect toN defines the electronic chemical potential

µ=∂E0^N

∂N

Taking the derivative with respect toN corresponds to taking the derivative with respect tof, we find forN−1<N <N

∂E₀^N

∂N

N−1<N<N

=E₀^N−E₀^N−1=−IN

whereIN is theionization energyof theNelectron system.

Similarly forN<N <N+ 1 ∂E₀^N

∂N

N<N<N+1

=E₀^N+1−E₀^N=−AN

whereA_N is theelectron affinityof theNelectron system.

The electronic chemical potentialµhas thus adiscontinuityat the integer electron numberN. So, the plot ofE₀^N with respect toN is made of a series of straight lines between integer electron numbers, with derivative discontinuities at each integer. 37/81

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DFT with fractional electron numbers (1/2)

The universal density functionalF[n] is extended to densities integrating to a fractional electron number,R

n(r)dr=N =N−1 +f, as F[n] = min

ˆΓ→n

Trh Γˆ

Tˆ+ ˆWee

i

As usual, to set up a KS method,F[n] is decomposed as F[n] =Ts[n] +EHxc[n]

where Ts[n] = min

Γˆs→n

Tr[ˆΓsTˆ] is the KS non-interacting kinetic-energy functional and the minimization is over ensemble non-interacting density matrices ˆΓsof the form

ˆΓs= (1−f)|Φ^N−1,fihΦ^N−1,f|+f |Φ^N,fihΦ^N,f|

where Φ^N−1,f and Φ^N,f are (N−1)- andN-electron single-determinant wave functions, constructed from acommon set of orbitals{φi}depending on the fixedf.

The exact ground-state energy can then be expressed as E₀^N−1+f = min

ˆΓs

nTrh ˆΓs

Tˆ + ˆV_nei

+E_Hxc[nˆΓs]o

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DFT with fractional electron numbers (2/2)

The total energy can be written in terms of orbital occupation numbersn_i E^N−1+f =

XN i=1

n_i Z

φ^∗_i(r)

−1

2∇²+v_ne(r)

φi(r)dr+E_Hxc[n]

with the densityn(r) =PN

i=1ni|φi(r)|²and the occupation numbersni = 1 fori≤N−1 andnN=f for the HOMO (ignoring degeneracy for simplicity).

εi

The orbitals satisfy standard-looking KS equations

−1

2∇²+vs(r)

φi(r) =εiφi(r) with vs(r) =vne(r) +δEHxc[n]

δn(r) with the important difference that we can now fix the arbitrary constant invs(r).

This is because we can now allow variations ofn(r) changingN, i.e. R

δn(r)dr6= 0 δEHxc[n] =

Z δEHxc[n]

δn(r) + const

δn(r)dr

making the constant no longer arbitrary. This unambiguously fixes the values of the KS orbital energiesεi.

Janak’s theorem(1978): After optimizing the orbitals with fixed occupation numbers, we have

∂E^N

∂ni

=εi for occupied orbitals

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The HOMO energy and the ionization energy

For clarity in the discussion, we will now explicitly indicate the dependence on the electron numberN.

Janak’s theorem applied to the HOMO forN =N−δwhereδ→0⁺gives ∂E₀^N

∂N

N−δ

=ε^N−δ_H ≡ε^NH

whereε^N_H is the HOMO energy of theN-electron system (defined as the left side of discontinuity).

This implies theKS HOMO energy is the opposite of the exact ionization energy ε^N_H=−IN

Combining this result with the known asymptotic behavior of the exact ground-state density (for finite systems)

n^N(r)_r→∞∼ e⁻²√

2INr

it can be shown that it implies that the KS potentialv_s^N(r)≡v_s^N−δ(r) (defined as the limit from the left side) vanishes asymptotically

r→∞lim v_s^N(r) = 0

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The LUMO energy, the electron affinity, the derivative discontinuity (1/2)

Janak’s theorem applied to the HOMO but now forN =N+δwhereδ→0⁺ gives ∂E₀^N

∂N

N+δ

=ε^N+δ_H =−AN

whereε^N+δ_H is the HOMO energy from the right side of the discontinuity.

Remark: ∂E₀^N/∂N is constant for all N<N<N+ 1, so:ε^N+δ_H =ε^N+1−δ_H ≡ε^N+1_H One may think thatε^N+δ_H is equal to the LUMO energy of theN-electron systemε^N_L

ε^N+δ_H =^? ε^N_L ≡ε^N−δ_L but this is WRONG!

εi

ε^N−_L ^δ=^?

N−δ

ε^N+_H ^δ

N+δ Let us compareε^N+δ_H andε^N−δ_L :

ε^N+δ_H = Z

φ^N+δ_H (r)^∗

−1

2∇²+v_s^N+δ(r)

φ^N+δ_H (r)dr

ε^N−δ_L = Z

φ^N−δ_L (r)^∗

−1

2∇²+v_s^N−δ(r)

φ^N−δ_L (r)dr

The density is continuous at the integerN, i.e. n^N+δ(r) =n^N−δ(r), but this only imposes thatv_s^N+δ(r) andv_s^N−δ(r) be equal up to anadditive constant(according to

the Hohenberg-Kohn theorem). _41/81

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The LUMO energy, the electron affinity, the derivative discontinuity (2/2)

Indeed, it turns out thatv_s^N+δ(r) andv_s^N−δ(r) do differ by a uniform constant ∆^N_xc v_s^N+δ(r)−v_s^N−δ(r) = ∆^Nxc

The orbitals are continuous at the integerN, soφ^N+δ_H (r) =φ^N−δ_L (r), and we find ε^N+δ_H =ε^N−δ_L + ∆^N_xc

In conclusion, theKS LUMO energy is not the opposite of the exact electron affinity ε^N_L =−AN−∆^N_xc

due to the discontinuity∆^Nxcin the KS potential.

Such a discontinuity can only come from the exchange-correlation part of the potential v_xc^N(r) sincevne(r) is independent fromN and the Hartree potential

v_H^N(r) =R

n^N(r^′)/|r−r^′|dr^′ is a continuous function ofN. So, we have

∆^N_xc=v_xc^N+δ(r)−v_xc^N−δ(r) =

δExc[n]

δn(r)

N+δ

−

δExc[n]

δn(r)

N−δ

i.e. ∆^N_xcis thederivative discontinuityin the exchange-correlation energy functional E_xc[n].

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Kohn-Sham frontier orbital energies: Graphical summary

εi

−IN

−AN

ε^N−δ_H ε^N−δ_L

∆^N_xc

ε^N+δ_H−1

ε^N+δ_H ε^N+1_H =−A_N=−I_N+1

N≡N−δ N+δ N+ 1

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Fundamental gap

Thefundamental gapof theN-electron system is defined as E_gap^N =IN−AN

In KS DFT, it thus be expressed as

E_gap^N =ε^N_L −ε^N_H+ ∆^N_xc

| {z }

KS gap

Sothe KS gap is not equal to the exact fundamental gapof the system, the difference coming from the derivative discontinuity ∆^N_xc.

The derivative discontinuity ∆^Nxccan represent an important contribution to the fundamental gap. In the special case of open-shell systems, we haveε^N_L =ε^N_H, and thus if the fundamental gap of an open-shell system is not zero (Mott insulator), it is entirely given by ∆^N_xc.

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Outline

45/81

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Outline

3

Usual approximations for the exchange-correlation energy

The local-density approximation Semilocal approximations

The gradient-expansion approximation Generalized-gradient approximations Meta-generalized-gradient approximations

Hybrid approximations Double-hybrid approximations

Range-separated hybrid and double-hybrid approximations

Range-separated hybrid approximations Range-separated double-hybrid approximations

Semiempirical dispersion corrections

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The local-density approximation

In thelocal-density approximation(LDA), introduced by Kohn and Sham (1965), the exchange-correlation functional is approximated as

E_xc^LDA[n] = Z

n(r)ε^unifxc (n(r))dr

whereε^unif_xc (n) is the exchange-correlation energy per particle of the infiniteuniform electron gas (UEG)with the densityn.

The exchange energy per particle of the UEG can be calculated analytically ε^unif_x (n) =cxn^1/3 Dirac (1930) and Slater (1951)

For the correlation energy per particleε^unif_c (n) of the UEG, there are some parametrized functions ofnfitted to QMC data and imposing the high- and low-density expansions (using the Wigner-Seitz radiusr_s= (3/(4πn))^1/3)

ε^unif_c =

rs→0Alnrs+B+C rslnrs+O(rs) high-density limit or weak-correlation limit ε^unif_c =

rs→∞

a r_s + b

r_s^3/2+O 1

r_s²

low-density limit or strong-correlation limit The two most used parametrizations are VWN and PW92. Generalization to spin densitiesε^unif_c (n↑,n↓) is sometimes referred to as local-spin-density (LSD) approximation.

47/81

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Outline

3

Usual approximations for the exchange-correlation energy

The local-density approximation Semilocal approximations

The gradient-expansion approximation Generalized-gradient approximations Meta-generalized-gradient approximations

Hybrid approximations Double-hybrid approximations

Range-separated hybrid and double-hybrid approximations

Range-separated hybrid approximations Range-separated double-hybrid approximations

Semiempirical dispersion corrections

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The gradient-expansion approximation

The next logical step beyond the LDA is thegradient-expansion approximation(GEA) which consists in a systematic expansion ofExc[n] in terms of the gradients ofn(r).

To derive the GEA, one starts from the uniform electron gas, introduce a weak and slowly-varying external potentialv(r), and expand the exchange-correlation energy in terms of the gradients of the density. Alternatively, one can perform a semiclassical expansion of the exactExc[n].

At second order, the GEA has the form E_xc^GEA[n] =E_xc^LDA[n] +

Z

C_xc(n(r))n(r)^4/3

∇n(r) n(r)^4/3

2

dr whereCxc(n) =Cx+Cc(n) are known coefficients.

We use thereduced density gradient|∇n|/n^4/3which is a dimensionless quantity.

The GEA should improve over the LDA provided that the reduced density gradient is small. Unfortunately, for real molecular systems, the reduced density gradient can be large in some regions of space, and the GEA turns out to be a worse approximation than the LDA.

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Generalized-gradient approximations (1/4)

The failure of the GEA lead to the development of generalized-gradient approximations (GGAs), started in the 1980s, of the generic form

E_xc^GGA[n] = Z

f(n(r),∇n(r))dr

The GGAs provide a big improvement over LDA for molecular systems.

The GGAs are often calledsemilocalapproximations, which means that they involve a single integral onrusing “semilocal information” through∇n(r).

For simplicity, we consider here only the spin-independent form, but in practice GGA functionals are more generally formulated in terms of spin densities and their gradients

E_xc^GGA[n] = Z

f(n↑(r),n↓(r),∇n↑(r),∇n↓(r))dr

(Too) Many GGA functionals have been proposed. We will review some of the most widely used ones.

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Generalized-gradient approximations (2/4)

Becke 88 (B88 or B) exchange functional E_x^B[n] =E_x^LDA[n] +

Z

n(r)^4/3f

|∇n(r)| n(r)^4/3

dr

Functionf chosen so as to satisfy the exact asymptotic behavior of the exchange energy per particle:

εx(r)_r_→∞∼ −1 2r

It contains an empirical parameter fitted to Hartree-Fock exchange energies of rare-gas atoms.

Lee-Yang-Parr (LYP) correlation functional(1988)

One of the rare functionals not constructed starting from LDA.

It originates from the Colle-Salvetti (1975) correlation-energy approximation depending on the curvature of Hartree-Fock hole and containing four parameters fitted to Helium data.

LYP introduced a further approximation to retain dependence on onlyn,∇n,∇²n.

The density Laplacian∇²ncan be exactly eliminated by an integration by parts.

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Generalized-gradient approximations (3/4)

Perdew-Wang 91 (PW91) exchange-correlation functional

It is based on a model of exchange and correlation holes from which we express the exchange and correlation energies per particle:

εx(r1) = 1 2

Z nx(r1,r2)

|r1−r2|dr2 and εc(r1) = 1 2

Z ¯nc(r1,r2)

|r1−r2|dr2

It starts from the GEA model of these holes and removes the unrealistic long-range parts of these holes to restore important conditions satisfied by the LDA.

For the GEA exchange hole: the spurious positive parts are removed to enforce nx(r1,r2)≤0 and a cutoff in|r1−r2|is applied to enforceR

nx(r1,r2)dr2=−1.

For the GEA correlation hole: a cutoff is applied to enforceR

¯

nc(r1,r2)dr2= 0.

The exchange and correlation energies per particle calculated from these numerical holes are then fitted to functions ofnand|∇n|chosen to satisfy a number of exact conditions.