A formally exact expression for the interaction density response function $\ensuremath{\chi}(\mathbf{r},{\mathbf{r}}^{\ensuremath{'}},\ensuremath{\omega})$ exists in terms of (i) its noninteracting counterpart ${\ensuremath{\chi}}_{0}(\mathbf{r},{\mathbf{r}}^{\ensuremath{'}},\ensuremath{\omega})$ and (ii) an exchange
(x)-correlation
(c) kernel ${f}_{xc}(\mathbf{r},{\mathbf{r}}^{\ensuremath{'}},\ensuremath{\omega})$. In the absence of a first-principles theory for the $\ensuremath{\omega}$ dependence of
fxc, the adiabatic approximation is most frequently made in this term, to construct a workable time-dependent density-functional theory. In the present study, a proposal is put forward to avoid the adiabatic approximation by working in the exchange-only limit in which
fxc is set equal to ${f}_{x}(\mathbf{r},{\mathbf{r}}^{\ensuremath{'}},\ensuremath{\omega})$. We then refer to a result for the exchange energy given by Pines and Nozieres to motivate the assumption that ${f}_{x}={f}_{x}[\mathrm{Im}\phantom{\rule{0.2em}{0ex}}{\ensuremath{\chi}}_{0}(\mathbf{r},{\mathbf{r}}^{\ensuremath{'}},\ensuremath{\omega})]$. The essential proposal here is therefore that the integral equation to be solved for the interacting density response function $\ensuremath{\chi}$ is in the exchange-only case characterized entirely by the noninteracting response function ${\ensuremath{\chi}}_{0}$.
The general expressions for the number density and current are first considered under the action of an external time-dependent perturbing potential of arbitrary strength and cast in terms of the single-particle lesser Green’s function. The expansion of the number density up to linear order in the perturbing potential is then considered, yielding the density–density correlation function of linear-response theory. A connection is also considered with the temperature correlation function of the Matsubara formalism via an analytic continuation in frequency space.
A new approach for the calculation of the linear response of both the ground and thermal-equilibrium states of many-electron systems is proposed via an integral formulation of the density-functional theory. Based on the path-integral representation of the one-particle Green's function, the frequency-dependent linear-response function for the Kohn-Sham noninteracting system is expressed as an explicit functional of the Kohn-Sham local potential, instead of the summation over the single-electron orbitals. Thus the dynamic linear density response can be determined as the solution of self-consistent equations which only need as input the total electron density of the ground or the equilibrium state. Exchange and correlation effects are incorporated. Because orbitals are not employed, the formulation provides the possibility for calculations of linear-response properties, such as dynamic polarizability and photoabsorption cross sections of systems with very many electrons. The present formulation can also be applied to general fermions.
We have obtained an exact analytical solution for a system of two particles moving under the action of a central harmonic-oscillator potential and the mutual Coulomb interaction augmented by a potential linear in the interparticle coordinate. The expressions for the single-particle density, the exchange-correlation energy quantities, and the correlation functions are thereby derived. Numerical results on the exchange-correlation contributions are presented using plots of various local energy densities as well as the corresponding global energy quantities for varying strengths of the oscillator potential.
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
vs and
ϕ1, where
vs is the local, one-body potential from the Kohn--Sham equations, and
ϕ1 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
vs. For simplicity, we only consider noninteracting states that are closed-shell states and interacting states that are nondegenerate, singlet ground-states.