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arXiv:physics/0506109

Expressions for the Exchange Correlation Potential and Exchange Correlation Functional of Kohn--Sham Density Functional Theory

James P. Finley

Published Jun 12, 2005• 0 citations
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Abstract

The State--Specific Kohn--Sham Density Functional Theory [arXiv:physics/0506037] is used to derive the Kohn-Sham exchange-correlation potential $\vxc$ and exchange-correlation energy $\Eco$ as explicit functionals of $v_s$ and $ϕ_1$, where $v_s$ is the local, one-body potential from the Kohn--Sham equations, and $ϕ_1$ is the spinless one-particle density matrix from the Kohn--Sham noninteracting state, say $|ϕ_1\ran$. In other words, $|ϕ_1\ran$ is the ground state eigenfunction of the noninteracting Schrödinger equation with the one-body potential $v_s$. For simplicity, we only consider noninteracting states that are closed-shell states and interacting states that are nondegenerate, singlet ground-states.

DOI
10.48550/arxiv.physics/0506109

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The State--Specific Kohn--Sham Density Functional Theory [arXiv:physics/0506037] is used to derive the Kohn-Sham exchange-correlation potential \vxc\vxc\vxc and exchange-correlation energy \Eco\Eco\Eco as explicit functionals of vsv_svs​ and ϕ1ϕ_1ϕ1​, where vsv_svs​ is the local, one-body potential from the Kohn--Sham equations, and ϕ1ϕ_1ϕ1​ is the spinless one-particle density matrix from the Kohn--Sham noninteracting state, say ∣ϕ1\ran|ϕ_1\ran∣ϕ1​\ran. In other words, ∣ϕ1\ran|ϕ_1\ran∣ϕ1​\ran is the ground state eigenfunction of the noninteracting Schrödinger equation with the one-body potential vsv_svs​. For simplicity, we only consider noninteracting states that are closed-shell states and interacting states that are nondegenerate, singlet ground-states.
Using an approach based on many body perturbation theory, the correlation energy $\cEco$ is expressed as an explicit functional of ρ1ρ_1ρ1​, vvv, and vsv_svs​, where ρ1ρ_1ρ1​ is the one-particle density matrix from the noninteracting, or reference, determinantal-state; vvv is the external potential from the interacting, or target, state; vsv_svs​ is the (kernel of the) external potential from the noninteracting determinantal-state. In other words we have $\cEco[ρ_1,v,v_s]$. Anther possibility is the following explicit functional: $\cEco[ρ_1,v_{\text{co}},v_s]$, where vcov_{\text{co}}vco​ is the (kernel of the) correlation potential from the noninteracting Hamiltonian. The proposed method can, in principle, be used to compute $\cEco$ in a very accurate and efficient manner, since, like the Kohn--Sham approach, there are no virtual orbitals to consider. However, in contrast to the Kohn--Sham approach, $\cEco$ is a known, explicit functional that can be approximated in a systematic manner. For simplicity, we only consider noninteracting closed-shell states and target states that are nondegenerate, singlet ground-states; so, in that case, ρ1ρ_1ρ1​ denotes the spin-less one-particle density matrix from the determinantal reference state.
Following a recent work [Gál, Phys. Rev. A64, 062503 (2001)], a simple derivation of the density-functional correction of the Hartree–Fock equations, the Hartree–Fock–Kohn–Sham equations, is presented, completing an integrated view of quantum mechanical theories, in which the Kohn–Sham equations, the Hartree–Fock–Kohn–Sham equations and the ground-state Schrödinger equation formally stem from a common ground: density-functional theory, through its Euler equation for the ground-state density. Along similar lines, the Kohn–Sham formulation of the Hartree–Fock approach is also considered. Further, it is pointed out that the exchange energy of density-functional theory built from the Kohn–Sham orbitals can be given by degree-two homogeneous N-particle density functionals (N = 1, 2, …), forming a sequence of degree-two homogeneous exchange-energy density functionals, the first element of which is minus the classical Coulomb-repulsion energy functional.
Aiming to combine density functional theory (DFT) and wave-function theory, we study a mapping from the many-body interacting system to an effectively interacting Kohn-Sham system instead of a noninteracting Kohn-Sham system. Because a ground state of effectively interacting systems requires having a solution for the correlated many-body wave functions, this provides a natural framework to many-body wave-function theories such as the configuration interaction and the coupled-cluster method in the formal theoretical framework of DFT. Employing simple one-dimensional two-electron systems---namely, the one-dimensional helium atom, the hydrogen molecule, and the heteronuclear diatomic molecule---we investigate properties of many-body wave functions and exact exchange-correlation potentials of effectively interacting Kohn-Sham systems. As a result, we find that the asymptotic behavior of the exact exchange-correlation potential can be controlled by optimizing that of the effective interaction. Furthermore, the typical features of the exact noninteracting Kohn-Sham system, namely, a spiky feature and a step feature in the exchange-correlation potential for the molecular dissociation limit, can be suppressed by a proper choice of the effective interaction. These findings open a possibility to construct numerically robust and efficient exchange-correlation potentials and functionals based on the effectively interacting Kohn-Sham scheme.
We use the exact strong-interaction limit of the Hohenberg-Kohn energy density functional to approximate the exchange-correlation energy of the restricted Kohn-Sham scheme. Our approximation corresponds to a highly nonlocal density functional whose functional derivative can be easily constructed, thus transforming exactly, in a physically transparent way, an important part of the electron-electron interaction into an effective local one-body potential. We test our approach on quasi-one-dimensional systems, showing that it captures essential features of strong correlation that restricted Kohn-Sham calculations using the currently available approximations cannot describe.
From the local force equation of quantum mechanics and by expanding the interacting wave function in terms of the noninteracting Kohn-Sham wave function, we derive differential equations defining the local Hartree-exchange and the local correlation potential of density functional theory. The derived equations apply to time-dependent as well as to time-independent problems. Approximations to the correlation part of the interacting wave function result in approximations to the effective potential which take the Hartree-exchange forces into account exactly. Under some assumptions, we find explicit expressions for the local effective potential. We analyze the local Hartree-exchange-only approximation for the ground and the first two excited states of an exactly solvable one-dimensional model system of helium and compare the results to an optimized effective potential approach.