part v: generalized kohn-sham dft - université de genève · 2016-04-22 · a. d. becke...
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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
Exchange-correlation holeAdiabatic Connection
Hybrid functionals
Table of contents
1 Exchange-correlation hole
2 Adiabatic Connection
3 Hybrid functionals
Tomasz A. Wesolowski, Universite de Geneve Part V: Generalized Kohn-Sham DFT
Exchange-correlation holeAdiabatic Connection
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.
Tomasz A. Wesolowski, Universite de Geneve Part V: Generalized Kohn-Sham DFT
Exchange-correlation holeAdiabatic Connection
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
Tomasz A. Wesolowski, Universite de Geneve Part V: Generalized Kohn-Sham DFT
Exchange-correlation holeAdiabatic Connection
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))
Tomasz A. Wesolowski, Universite de Geneve Part V: Generalized Kohn-Sham DFT
Exchange-correlation holeAdiabatic Connection
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
Tomasz A. Wesolowski, Universite de Geneve Part V: Generalized Kohn-Sham DFT
Exchange-correlation holeAdiabatic Connection
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]).
Tomasz A. Wesolowski, Universite de Geneve Part V: Generalized Kohn-Sham DFT
Exchange-correlation holeAdiabatic Connection
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)
Tomasz A. Wesolowski, Universite de Geneve Part V: Generalized Kohn-Sham DFT
Exchange-correlation holeAdiabatic Connection
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
Tomasz A. Wesolowski, Universite de Geneve Part V: Generalized Kohn-Sham DFT
Exchange-correlation holeAdiabatic Connection
Hybrid functionals
Examples of ρxc(r, r′)
Data from Ref. a
aO.Gritsenko and E.J. Baerends,
J.Phys.Chem. A, 101 (1997) 53xx
Tomasz A. Wesolowski, Universite de Geneve Part V: Generalized Kohn-Sham DFT
Exchange-correlation holeAdiabatic Connection
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]
Tomasz A. Wesolowski, Universite de Geneve Part V: Generalized Kohn-Sham DFT
Exchange-correlation holeAdiabatic Connection
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.
Tomasz A. Wesolowski, Universite de Geneve Part V: Generalized Kohn-Sham DFT
Exchange-correlation holeAdiabatic Connection
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
Exchange-correlation holeAdiabatic Connection
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
Tomasz A. Wesolowski, Universite de Geneve Part V: Generalized Kohn-Sham DFT
Exchange-correlation holeAdiabatic Connection
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
Tomasz A. Wesolowski, Universite de Geneve Part V: Generalized Kohn-Sham DFT
Exchange-correlation holeAdiabatic Connection
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
Tomasz A. Wesolowski, Universite de Geneve Part V: Generalized Kohn-Sham DFT
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