An orbital-dependent correlation energy functional Ec to be accompanied by the exact exchange energy functional Ex is proposed for applications of density-functional theory (DFT). The present Ec comprises spin-antiparallel and spin-parallel contributions, Ecσ−σ and Ecσσ. Ecσ−σ is a modification of the spin-antiparallel component of the Hartree energy functional with a factor of gσ−σ (r, r′)−1 and Ecσσ a modification of the spin-parallel component of the same energy functional with gcσσ(r, r′) where gσ−σ(r, r′) (or gcσσ(r, r′)) is the spin-antiparallel (or the correlational part of the spin-parallel) coupling-constant-averaged pair correlation function. The present orbital-dependent gσ−σ(r, r′) and gcσσ(r, r′) fulfill the symmetric property, the Pauli principle and the sum rules. In the limit of uniform density the two correlation functions are reduced to the very accurate analogues of the electron liquid that involve long-, intermediate-, and short-range correlations as well as their exchange counterparts. It is stressed that the correlation energy functional Ec in DFT should by its very nature be defined as a functional only of occupied Kohn-Sham orbitals and occupied Kohn-Sham energies for the purpose of employing the optimized potential method (OPM) to evaluate the correlation potential υc(r). The present scheme for Ec, if applied to finite systems after making a suitable change in the treatment of long-range correlation, can give the correct asymptotic form of υc(r) of order r−4 for large r as well as the van der Waals potential.
An orbital-dependent correlation energy functional Ec to be accompanied by the exact exchange energy functional Ex is proposed for applications of density-functional theory (DFT). The present Ec comprises spin-antiparallel and spin-parallel contributions, Ecσ−σ and Ecσσ. Ecσ−σ is a modification of the spin-antiparallel component of the Hartree energy functional with a factor of gσ−σ (r, r′)−1 and Ecσσ a modification of the spin-parallel component of the same energy functional with gcσσ(r, r′) where gσ−σ(r, r′) (or gcσσ(r, r′)) is the spin-antiparallel (or the correlational part of the spin-parallel) coupling-constant-averaged pair correlation function. The present orbital-dependent gσ−σ(r, r′) and gcσσ(r, r′) fulfill the symmetric property, the Pauli principle and the sum rules. In the limit of uniform density the two correlation functions are reduced to the very accurate analogues of the electron liquid that involve long-, intermediate-, and short-range correlations as well as their exchange counterparts. It is stressed that the correlation energy functional Ec in DFT should by its very nature be defined as a functional only of occupied Kohn-Sham orbitals and occupied Kohn-Sham energies for the purpose of employing the optimized potential method (OPM) to evaluate the correlation potential υc(r). The present scheme for Ec, if applied to finite systems after making a suitable change in the treatment of long-range correlation, can give the correct asymptotic form of υc(r) of order r−4 for large r as well as the van der Waals potential.
Abstract Density functional theory (DFT) is usually formulated in terms of the electron density as a function of position n(r). Here we discuss an alternative formulation of DFT in terms of the orbital occupation numbers {n α } associated with a local-orbital orthonormal basis set {ϕ α }. First, we discuss how the building blocks of DFT, namely the Hohenberg–Kohn theorems, the Levy–Lieb approach and the Kohn–Sham method, can be adapted for a description in terms of {n α }. In particular, the total energy is now a function of {n α }, E[{n α }], and a Kohn–Sham-like Hamiltonian is derived introducing the effects of the electron–electron interactions via effective potentials, { V α eff = ∂ E e e [ { n β } ] / ∂ n α } . In a second step we consider the Hartree and exchange energies and discuss how to describe them, in the spirit of a DFT approach, in terms of the orbital occupation numbers. In this contribution special attention is paid to the description of the (intra-atomic) correlation energy and corresponding correlation potentials {V corr,α }. For this purpose, a model system is analyzed in detail, whereby an atomic Hamiltonian interacts with the environment via a simplified model; the use of this model allows us to obtain the correlation energy and potentials (in terms of {n α }) for different cases corresponding to low, intermediate and high electron correlations.
It is known that the separation of electrons into spinons and chargons, the spin-charge separation, plays a decisive role when describing one-dimensional (1D) strongly correlated systems [ Phys. Rev. B 2012 , 86 , 075132 ]. In this paper, within the density-functional theory (DFT) formalism, we extend the investigation by considering a model for the third electron fractionalization: the separation into spinons, chargons and orbitons, the last associated with the electronic orbital degree of freedom. Specifically, we deal with two exact constraints of exchange-correlation (XC) density-functionals: (i) the constancy of the highest occupied (HO) Kohn-Sham (KS) eigenvalues upon fractional electron numbers and (ii) their discontinuities at integers. By means of 1D discrete Hubbard chains and 1D H2 molecules in the continuum, we find that spin-charge separation yields almost constant HO KS eigenvalues, whereas the spin-orbital counterpart can be decisive when describing derivative discontinuities of XC potentials at strong correlations.
Within density-functional theory a time-dependent perturbation theory is derived that yields exact orbital-dependent expressions for the time-dependent exchange-correlation potential in the form of a perturbation series. This leads to an exact time-dependent Kohn-Sham scheme based on perturbation theory. In first order an ``exchange-only'' Kohn-Sham procedure is obtained, that is a generalization of a recently presented time-dependent optimized potential method. The exact, in higher order, but quite complicated expressions for the exchange-correlation potential may serve as a starting point for the construction of approximate exchange-correlation potentials. An adiabatic connection or coupling constant path between a time-dependent physical system and its corresponding Kohn-Sham system is introduced.
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.
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.