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arXiv:0708.0576 · International Journal of Modern Physics B

TREATMENTS OF THE EXCHANGE ENERGY IN DENSITY-FUNCTIONAL THEORY

TAMÁS GÁL

Published Jun 9, 2008• 0 citations
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Abstract

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.

DOI
10.1142/s0217979208039344

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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.
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.
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.
By introducing a set of auxiliary equations representing a many-body system, we have derived an extension of the Kohn-Sham scheme for the density functional theory. These equations consist of a Kohn-Sham-type equation determining single-particle orbitals and an eigen-value equation for an effective many-body problem. A variational method similar to the Kohn-Sham technique was utilized to derive effective interactions as well as effective potentials without artificial substitution of a Hubbard-type interaction and a mean-field correction in the energy functional. The second equation is described by an effective many-body Hamiltonian with both 2-body interactions and mean-field terms. Rigorous formulation of the extended Kohn-Sham equation is also given in accordance with the Hadjisavvas-Theophilou formulation. Our formulation can be interpreted as a way to define models of the strongly correlated electron systems, e.g. the Hubbard model
This note is intended for expanding the details on the derivation and properties of density functional theory, in hope to make them more systematic, better motivated, and step-by-step for readers new to the domain. The note starts with basic concepts in quantum mechanics, then takes the step towards many-body systems using the tools of second quantization and Fock space, with some highlights on properties of the Coulomb system. Given these general technical preparations, the Hartree-Fock method is naturally unrolled, with expressions for various cases and quantities. Density functional theory is then presented, with a motivating reasoning of the Kohn-Sham formulation, and a connection and comparison of the expressions with the Hartree-Fock method. Lieb's celebrated 1983 paper on functional analysis on density functionals is also summarized.
It is known in density-functional theory that the noninteracting kinetic-energy density functional ${T}_{s}[\ensuremath{\rho}]$ is not first-degree homogeneous in density scaling. However, it is shown here that, for every particle number N, there is an N-particle noninteracting kinetic-energy density functional ${T}_{N}[\ensuremath{\rho}],$ that is, a density functional that gives the noninteracting kinetic energy for N-particle densities, which is of first-degree homogeneity in the density $\ensuremath{\rho}(\stackrel{\ensuremath{\rightharpoonup}}{r}).$ This gives a powerful tool, a strong requirement, for constructing such functionals. A systematic procedure to obtain the real part of ${T}_{N}[\ensuremath{\rho}],$ the full ${T}_{N}[\ensuremath{\rho}]$ in one-dimension, for each N is also proposed. It is pointed out, further, that in the Euler-Lagrange equations that determine the one-particle orbitals that define ${T}_{s}[\ensuremath{\rho}],$ the Lagrange multiplier that forces the orbitals to yield $\ensuremath{\rho}(\stackrel{\ensuremath{\rightharpoonup}}{r})$ is not other than the first derivative of ${T}_{s}[\ensuremath{\rho}],$ $\ensuremath{\delta}{T}_{s}[\ensuremath{\rho}]/\ensuremath{\delta}\ensuremath{\rho}(\stackrel{\ensuremath{\rightharpoonup}}{r}),$ which yields a natural derivation of the Kohn-Sham equations. Utilizing the same idea, it is shown for ground states how the Schr\"odinger equation can be derived from the basics of density-functional theory as well.