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Physical Review A

Local Hartree-exchange and correlation potential defined by local force equations

Michael Ruggenthaler, D. Bauer

Published Nov 5, 2009• 22 citations
Publisher

Abstract

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.

DOI
10.1103/physreva.80.052502

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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.
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.
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 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 $|ϕ_1\ran$. In other words, $|ϕ_1\ran$ 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.
The exact static and time-dependent Kohn-Sham (KS) exchange-correlation potential is extremely challenging to approximate as it is a local multiplicative potential that depends on the electron density everywhere in the system. The KS approach can be generalized by allowing part of the potential to be spatially nonlocal. We take this nonlocal part to be that of unrestricted Hartree-Fock theory. The additional local correlation potential in principle ensures that the single-particle density exactly equals the many-body density. In our case, the local correlation potential is predominantly nearsighted in its dependence on the density and hence an (adiabatic) local-density approximation to this potential yields accurate ground-state properties and real-time densities for one-dimensional test systems.
We formulate the inverse Kohn–Sham equations based on the density equation theory. The present formulation relies on neither action functionals nor force balance equation, but is constructed by comparing the first-order density equation for the electron interacting real system and that for the electron non-interacting Kohn–Sham system. The derived inverse Kohn–Sham integral equation requires information of an exact ground-state electron density to calculate an exact Kohn–Sham potential for a ground state and it requires an exact time-dependent electron density and a time-derivative of the exact time-dependent first-order density matrix to calculate an exact time-dependent Kohn–Sham potential. A possible algorithm to construct the Kohn–Sham potential for a ground state is suggested as well as a solution algorithm for the time-dependent inverse Kohn–Sham problem without resorting to any approximations. These two solution algorithms, in combination, suggest a way to perform an exact time-dependent density functional calculation as a predictive method.
We obtain the exact Kohn-Sham potentials VKS{V}_{\mathrm{KS}}VKS​ of time-dependent density-functional theory for one-dimensional Hubbard chains, driven by a dc external field, using the time-dependent electron density and current density obtained from exact many-body time evolution. The exact Vxc{V}_{\mathrm{xc}}Vxc​ is compared to the adiabatically exact Vxcad{V}_{\mathrm{xc}}^{\mathrm{ad}}Vxcad​ and the ``instantaneous ground state'' Vxcigs{V}_{\mathrm{xc}}^{\mathrm{igs}}Vxcigs​. The latter is shown to work effectively in some cases when the former fails. Approximations for the exchange-correlation potential Vxc{V}_{\mathrm{xc}}Vxc​ and its gradient, based on the local density and on the local current density, are also considered and both physical quantities are observed to be far outside the reach of any possible local approximation. Insight into the respective roles of ground-state and excited-state correlation in the time-dependent system, as reflected in the potentials, is provided by the pair correlation function.