part v: generalized kohn-sham dft - université de genève · 2016-04-22 · a. d. becke...

Exchange-correlation holeAdiabatic Connection

Hybrid functionals

Part V: Generalized Kohn-Sham DFT

Tomasz A. Wesolowski, Universite de Geneve

EPFL, Spring 2016

Tomasz A. Wesolowski, Universite de Geneve Part V: Generalized Kohn-Sham DFT


Hybrid functionals

Table of contents

1 Exchange-correlation hole

2 Adiabatic Connection

3 Hybrid functionals



Hybrid functionals

Reduced density matrices

The wave-function (Ψ), defined in the Schrodinger equation, provides the completedescription of a given system. There exists, however, an alternative formulation ofquantum mechanics which uses reduced density matrices.Two types of density matrices are especially useful in quantum chemistry:

One-particle density matrix γ1(x′1, x1):

γ1(x′1, x1) = N

∫· ·∫

Ψ(x′1, x2, ··, xN)Ψ∗(x1, x2, ··, xN)dx2 · ·dxN

Two-particle density matrix γ2(x′1, x′2, x1, x2):

γ2(x′1, x′2, x1, x2) =

N(N − 1)

2

∫· ·∫

Ψ(x′1, x′2, ··, xN)Ψ∗(x1, x2, ··, xN)dx3 · ·dxN

where xi denotes the spin (si ) and the space (ri ) coordinates.



Hybrid functionals

Some useful relations involving reduced density matrices γ1, γ2

γ1 and γ2 are Hermitian:

γ1(x′1, x1) = γ∗1 (x1, x′1)

γ2(x′1, x′2, x1, x2) = γ∗2 (x1, x2, x

′1, x′2)

γ1 is normalized: ∫γ1(x1, x1)dx1 = N

Once γ2 is known also γ1 is known:

γ1(x′1, x1) =2

N − 1

∫γ2(x′1, x2, x1, x2)dx2

γ1 and γ2 are positively defined:

γ1(x1, x1) ≥ 0

γ2(x1, x2, x1, x2) ≥ 0



Hybrid functionals

Spin-less density matrices

One-particle spin-less density matrix:

ρ1(r′1, r1) = N

∫· ·∫

Ψ(r′1, s1, x2, ··, xN )Ψ∗(r1, s1, x2, ··, xN )ds1dx2 · ·dxN

Two-particle spin-less density matrix:

ρ2(r′1, r′2, r1, r2) =N(N − 1)

2

∫· ·∫

Ψ(r′1, s1, r′2, s2, ··, xN )Ψ∗(r1, s1, r2, s2, ··, xN )ds1ds2dx3 · · dxN

The diagonal element of the one-particle spin-less density matrix equals to the electron density:

ρ1(r1, r1) = ρ(r1)

The diagonal elements of the two-particle spin-less density matrix ρ2(r1, r2, r1, r2) ≡ ρ2(r1, r2) are conventionallyexpressed by means of the pair-correlation function h(r1, r2):

ρ2(r1, r2) =1

2ρ(r1)ρ(r2) (1 + h(r1, r2))



Hybrid functionals

Exercise:

Show that:

Vee ≡∫

Ψ∗(r1, ··, rN )

∣∣∣∣∣∣N∑i

N∑i<j

1∣∣ri − rj∣∣∣∣∣∣∣∣Ψ(r1, ··, rN )dr1, ··, drN =

∫ ∫1

r12

ρ2(r1, r2)dr1dr2



Hybrid functionals

Exchange-correlation hole: definition

Exchange correlation hole is defined as:

ρxc (r1, r2) = ρ(r2)h(r1, r2)

where, h(r1, r2) is the pair-correlation function:

ρ2(r1, r2) =1

2ρ(r1)ρ(r2) (1 + h(r1, r2))

Since:

ρxc (r1, r2) =

(2ρ2(r1, r2)

ρ(r1)− ρ(r2)

)

ρxc (r1, r2) is called also exchange-correlation charge.Using ρxc (r1, r2) we obtain:

Vee ≡ 〈Ψ |vee |Ψ〉 =

∫ ∫1

r12

ρ2(r1, r2)dr1dr2 = J[ρ] +1

2

∫ ∫1

r12

ρ(r1)ρxc (r1, r2)dr1dr2

Note that Vee − J[ρ] is one of the components of the exchange-correlation energy in the Kohn-Sham formulation

of DFT (the difference is just T [ρ]− Ts [ρ]).



Hybrid functionals

Properties of ρxc(r, r′)

The exact exchange-correlation hole (ρxc) satisfies the sum rule:∫ρxc(r1, r2)dr2 = −1

ExerciseDerive the above relation.

1) Show that:

ρ1(r′1, r1) =2

N − 1

∫ρ2(r′1, r2, r1, r2)dr2

2) Use the definition of ρxc(r1, r2)



Hybrid functionals

Properties of ρxc(r, r′) cnt.

The separate components of ρxc(r, r′):

ρxc(r, r′) = ρx(r, r′) + ρc(r, r′)

are called: ρx(r, r′)) is the Fermi hole and ρc(r, r′) is the Coulomb hole.

In the constrained search definition the partitioning of the total hole is unique.The functions Ψ[ρ] and Ψs [ρ] are available and can be used to evaluateρxc(r, r′) (Ψ[ρ]) and ρx(r, r′) (Ψs [ρ]).

The exact holes satisfy the following conditions:

ρx(r, r′) ≤ 0∫ρx(r, r′)dr′ = −1∫ρc(r, r′)dr′ = 0



Hybrid functionals

Examples of ρxc(r, r′)

Data from Ref. a

aO.Gritsenko and E.J. Baerends,

J.Phys.Chem. A, 101 (1997) 53xx



Hybrid functionals

Artificial system with modified Coulomb interactions (1)

The coupling parameter λ determines the strength of the electron-electron repulsion.For λ = 1 the entire electron-electron repulsion is switched on, but all other values0 ≤ λ < 1 correspond to an artificial refernce system. Kohn-Sham system is aparticular case λ = 0).

The functional F [ρ], defined using the constrained search for the Kohn-Sham theorycan be generalized for 0 < Fλ[ρ] ≤ 1 in a straightforward manner:

Fλ[ρ] = minΨ→ρ

⟨Ψ∣∣∣T + λVee

∣∣∣Ψ⟩

=⟨

Ψλρ

∣∣∣T + λVee

∣∣∣Ψλρ

⟩where Ψλρ is the N-electron wave function that minimizes

⟨T + λVee

⟩and yields the

density ρ.

Properties of Fλ[ρ]

Fλ=0[ρ] = Ts [ρ]

Fλ=1[ρ] = T [ρ] + Vee [ρ]



Hybrid functionals

Artificial system with modified Coulomb interactions (2)

From Ψλρ , we get the following λ dependent quantities:

The diagonal elements of two-particle spinless density matrix:

ρλ2 (r1, r2) =

N(N − 1)

2

∫· ·∫

Ψλρ (r1, ds1, r2, ds2, ··, xN )Ψλ∗ρ (r1, ds1, r2, ds2, ··, xN )ds1, ds2, dx3 · · dxN

Pair-correlation function:

ρλ2 (r1, r2) =1

2ρ(r1)ρ(r2)

(1 + hλ(r1, r2)

)Exchange-correlation hole:

ρλxc (r1, r2) = ρ(r2)hλ(r1, r2)

Expectation value of the electron-electron repulsion:

⟨Ψλρ |vee |Ψλρ

⟩=

∫ ∫1

r12ρλ2 (r1, r2)dr1dr2 = J[ρ] +

1

2

∫ ∫1

r12ρ(r1)ρλxc (r1, r2)dr1dr2

Note that the functional Ts [ρ] does not depend on λ.



Hybrid functionals

Averaged exchange-correlation hole: ρxc(r1, r2)

Definitions of ρ2(r1, r2) and ρxc (r1, r2):

ρ2(r1, r2) =

∫ 1

0ρλ2 (r1, r2)dλ

ρxc (r1, r2) = ρ(r2)−ρ2(r1, r2)

ρ(r1)

From the definition of Exc [ρ]1

Exc [ρ] = Vee [ρ]− J[ρ] + T [ρ]− Ts [ρ] = F1[ρ]− F0[ρ]− J[ρ] =

∫ 1

0dλ

∂Fλ[ρ]

∂λ− J[ρ]

The Hellman-Feynman theorem ∂Fλ[ρ]∂λ

=< Ψλ∣∣∣Vee

∣∣∣Ψλ > and

the relation⟨Ψλρ |vee |Ψλρ

⟩= J[ρ] + 1

2

∫ ∫1r12ρ(r1)ρλxc (r1, r2)dr1dr2 yield:

Exc [ρ] =

∫ 1

0dλ < Ψλ

∣∣∣Vee

∣∣∣Ψλ > −J[ρ] =

∫ ∫1

r12ρ(r1)ρxc (r1, r2)dr1dr2

1A silent assumption: ρ is v -representableTomasz A. Wesolowski, Universite de Geneve Part V: Generalized Kohn-Sham DFT


Hybrid functionals

Adiabatic-connection formula for Exc [ρ]

Exc [ρ] =

∫ ∫1

r12ρ(r1)ρxc (r1, r2)dr1dr2 (1)

Eq. 1 opens new possibilities to develop better approximations for Exc [ρ] 2 and also

provides also the theoretical justification for hybrid schemes, in which the Kohn-Sham

orbitals are used to evaluate (part of) the exchange energy 3.

2Gorling and Levy J.Chem.Phys. 106 (1997) 2675; W. Yang J.Chem.Phys. 109 (1998) 10107

3A. D. Becke J.Chem.Phys., 98 (1993)1372



Hybrid functionals

Adiabatic-connection formula for Exc [{φi}]

In the scheme proposed by Becke 4 (’half-and-half’ approach), the integral∫ 10 (ρλxc (r1, r2)dλ is approximated by the average of two terms:

∫ 1

0(ρλxc (r1, r2)dλ ≈

1

2

(ρ1xc (r1, r2) + ρ0

xc (r1, r2))

The numerical value of the term corresponding λ = 0 can be evaluated exactly usingthe known the Kohn-Sham orbitals - it is equal to the exchange energy calculatedusing Kohn-Sham orbitals.

The other term, corresponding to λ = 1 is not known. It is known to good accuracy

(LDA or GGA for instance). Becke showed that for many properties of organic

molecules the introduction of the explicit dependence of the exchange-correlation

energy on the Kohn-Sham orbitals improves the obtained results.

4A. D. Becke J.Chem.Phys., 98 (1993)1372



Hybrid functionals

Becke three parameter approximation for the exchange energy

Currently, the hybrid functional known as B3 (Becke’s three parameter exchangefunctional) 5 is one of the most commonly used functional (especially in organicchemistry). This functional is frequently combined with either LYP, or PW91correlation functional. In the original publication the PW91 density functional wasused for correlation. Most of current applications use LYP for this purpose. TheB3LYP functional reads:

EB3LYPxc [{φi}] = (1− a)ELSDA

x [ρ] + aE′exact′x [{φi}] + b(EB88

X [ρ]− ELSDAx [ρ])

+ EVWNC [ρ] + c(ELYP

c − EVWNc [ρ])

where a, b, c are parameters fitted to thermochemical data in a learning set ofmolecules (a ≈ 0.2, b ≈ 0.7, c ≈ 0.8).

5A.D. Becke J.Chem.Phys., 98 (1993) 5648


part v: generalized kohn-sham dft - université de genève · 2016-04-22 · a. d. becke...

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