Generalized Kohn Sham theory

Generalized Kohn-Sham theory (GKS) is an extension of Kohn–Sham (KS) density functional theory (DFT)^[1][2] and Hartree–Fock (HF) theory.^{[3][4][5][6][7]} It is used to give a rigorous basis for Hybrid functionals. GKS theory was introduced by Seidl, Görling, Vogl, Majewski and Levy in 1996^[8] and 1997^[9].

Formalism of GKS theory ^[8][10]

Instead of the regular Levy constrained search formulation of DFT

${\displaystyle E[v]=\min _{\rho (r)\rightarrow N}{\bigg \{}F[\rho ]+\int drv(r)\rho (r){\bigg \}}}$

where

${\displaystyle F[\rho ]=\min _{\Psi \rightarrow \rho (r)}\langle \Psi |{\hat {T}}+{\hat {V}}_{ee}|\Psi \rangle }$

is the Hohenberg-Kohn (HK) functional, one decomposes the electronic energy further into a HK-type functional ${\textstyle F^{S}[\rho ]}$ of a given subsystem denoted ${\textstyle S}$ and a remainder functional ${\textstyle R^{S}[\rho ]}$ as

${\displaystyle F[\rho ]=F^{S}[\rho ]+R^{S}[\rho ]}$

This leads to the GKS variational principle

${\displaystyle {\begin{aligned}E[v]=\min _{\rho (r)\rightarrow N}{\bigg \{}F^{S}[\rho ]+R^{S}[\rho ]+\int drv(r)\rho (r){\bigg \}}=\min _{\rho (r)\rightarrow N}{\bigg \{}\min _{\Phi \rightarrow \rho (r)}S[\Phi ]+R^{S}[\rho ]+\int drv(r)\rho (r){\bigg \}}=\\=\min _{\Phi \rightarrow N}{\bigg \{}S[\Phi ]+R^{S}[\rho [\Phi ]]+\int drv(r)\rho [\Phi ](r){\bigg \}}=\min _{\{\phi _{i}\}\rightarrow N}{\bigg \{}S[\{\phi _{i}\}]+R^{S}[\rho [\{\phi _{i}\}]]+\int drv(r)\rho [\{\phi _{i}\}](r){\bigg \}}\end{aligned}}}$

where the HK-type functional of the subsystem is defined as

${\displaystyle F^{S}[\rho ]\equiv \min _{\Phi \rightarrow \rho (r)}S[\Phi ]=\min _{\{\phi _{i}\}\rightarrow N}S[\{\phi _{i}\}]}$

and the energy of the subsystem is given as

${\displaystyle E^{S}[v_{\text{eff}};\{\phi _{i}\}]=\min _{\rho (r)\rightarrow N}{\bigg \{}F^{S}[\rho ]+\int drv_{\text{eff}}(r)\rho (r){\bigg \}}=S[\{\phi _{i}\}]+\int drv_{\text{eff}}(r)\rho (r)}$

Here the effective potential is the sum of the external potential and remainder potential

${\displaystyle v_{\text{eff}}(r)=v(r)+\underbrace {\frac {\delta R^{S}[\rho ]}{\delta \rho (r)}} _{v_{R}(r)}}$

Minimizing the energy of the subsystem under constrained, that the set of orbitals is orthonormalized, one obtains the GKS equations

${\displaystyle {\hat {O}}^{S}[\{\phi _{i}\}]\phi _{j}+v_{\text{eff}}\phi _{j}={\hat {O}}^{S}[\{\phi _{i}\}]\phi _{j}+v\phi _{j}+v_{R}\phi _{j}=\varepsilon \phi _{j}}$

The operator ${\textstyle {\hat {O}}^{S}[\{\phi _{i}\}]}$ is invariant under unitary orbital transformations. The choice of the Slater determinant functional ${\textstyle S[\Phi ]=S[\{\phi _{i}\}]}$ distinguished the different GKS shemes.

Kohn-Sham (KS)

Choosing the Slater determinant functional as the kinetic energy of a Slater determinant

${\displaystyle {\begin{aligned}S[\Phi ]\equiv T[\Phi ]\Rightarrow {\begin{cases}R^{s}[\rho ]=E_{\text{Hxc}}[\rho ]\\v_{R}=v_{\text{Hxc}}\end{cases}}\Rightarrow v_{\text{eff}}=v+v_{\text{Hxc}}=v_{S},\\O^{S}=-{\frac {1}{2}}\Delta \Rightarrow O^{S}+v_{\text{eff}}=-{\frac {1}{2}}\Delta +v_{S}=h_{\text{KS}}\end{aligned}}}$

one obtains the Kohn–Sham (KS) equations.

Hartree-Fock-Kohn-Sham (HF-KS)

Choosing the Slater determinant functional as the kinetic energy and electron-electron interaction of a Slater determinant

${\displaystyle {\begin{aligned}S[\Phi ]\equiv T[\Phi ]+E_{\text{Hx}}[\Phi ]\Rightarrow {\begin{cases}R^{s}[\rho ]=E_{\text{c}}^{\text{HF}}[\rho ]\\v_{R}=v_{\text{c}}^{\text{HF}}\end{cases}}\Rightarrow v_{\text{eff}}=v+v_{\text{c}}^{\text{HF}},\\O^{S}=-{\frac {1}{2}}\Delta +v_{\text{H}}+v_{\text{x}}^{\text{NL}}\Rightarrow O^{s}+v_{\text{eff}}=-{\frac {1}{2}}\Delta +v+v_{\text{H}}+v_{\text{x}}^{\text{NL}}+v_{\text{c}}^{\text{HF}}=h_{\text{HF-KS}}\end{aligned}}}$

one obtains the Hartree-Fock-Kohn-Sham (HF-KS) equations.

Hybrid formalism: GKS along the adiabatic connection (AC)^[9]

Choosing the Slater determinant functional as the kinetic energy and electron-electron interaction of a Slater determinant scaled along the adiabatic connection (AC)

${\displaystyle {\begin{aligned}S[\Phi ]\equiv T[\Phi ]+\alpha E_{\text{Hx}}[\Phi ]\Rightarrow {\begin{cases}R^{s}[\rho ]=\alpha E_{\text{c}}^{\text{HF}}+(1-\alpha )E_{\text{Hxc}}[\rho ]\\v_{R}=\alpha v_{\text{c}}^{\text{HF}}+(1-\alpha )v_{\text{Hxc}}\end{cases}}\\\Rightarrow v_{\text{eff}}=v+\alpha v_{\text{c}}^{\text{HF}}+(1-\alpha )v_{\text{Hxc}},O^{S}=-{\frac {1}{2}}\Delta +\alpha v_{\text{H}}+\alpha v_{\text{x}}^{\text{NL}}\\\Rightarrow O^{S}+v_{\text{eff}}=\\-{\frac {1}{2}}\Delta +v+\underbrace {\alpha v_{\text{H}}+(1-\alpha )v_{\text{H}}} _{v_{\text{H}}}+\alpha v_{\text{x}}^{\text{NL}}+\alpha v_{\text{c}}^{\text{HF}}+(1-\alpha )v_{\text{x}}+(1-\alpha )v_{\text{c}}\in [h_{\text{KS}},h_{\text{HF-KS}}]\end{aligned}}}$

one obtains the global (double) hybrid equations.

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