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arXiv:0812.1877 · Physical Review A

Kohn-Sham scheme for frequency-dependent linear response

Ryan Requist, Oleg Pankratov

Published Mar 5, 2009• 0 citations
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Abstract

We study the Kohn-Sham scheme for the calculation of the steady-state linear response $\ensuremath{\lambda}{n}_{\ensuremath{\omega}}^{(1)}(\mathbf{r})\text{cos}\text{ }\ensuremath{\omega}t$ to a harmonic perturbation $\ensuremath{\lambda}{v}^{(1)}(\mathbf{r})\text{cos}\text{ }\ensuremath{\omega}t$ that is turned on adiabatically. Although in general the exact exchange-correlation potential ${v}_{\text{xc}}(\mathbf{r},t)$ cannot be expressed as the functional derivative of a universal functional due to the so-called causality paradox, we show that for a harmonic perturbation the exchange-correlation part of the first-order Kohn-Sham potential $\ensuremath{\lambda}{v}_{s}^{(1)}(\mathbf{r})\text{cos}\text{ }\ensuremath{\omega}t$ is given by ${v}_{\text{xc}}^{(1)}(\mathbf{r})=\ensuremath{\delta}{K}_{\text{xc}}^{(2)}/\ensuremath{\delta}{n}_{\ensuremath{\omega}}^{(1)}(\mathbf{r})$. ${K}_{\text{xc}}^{(2)}$ is the exchange-correlation part of the second-order quasienergy ${K}_{v}^{(2)}$. The Frenkel variation principle implies a stationary principle for ${K}_{v}^{(2)}[{n}_{\ensuremath{\omega}}^{(1)}]$. We also find an analogous stationary principle and Kohn-Sham scheme in the time-dependent extension of one-matrix functional theory, in which the basic variable is the one-matrix (one-body-reduced density matrix).

DOI
10.1103/physreva.79.032502

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We study the Kohn-Sham scheme for the calculation of the steady-state linear response \ensuremathλn\ensuremathω(1)(r)cos \ensuremathωt\ensuremath{\lambda}{n}_{\ensuremath{\omega}}^{(1)}(\mathbf{r})\text{cos}\text{ }\ensuremath{\omega}t\ensuremathλn\ensuremathω(1)​(r)cos \ensuremathωt to a harmonic perturbation \ensuremathλv(1)(r)cos \ensuremathωt\ensuremath{\lambda}{v}^{(1)}(\mathbf{r})\text{cos}\text{ }\ensuremath{\omega}t\ensuremathλv(1)(r)cos \ensuremathωt that is turned on adiabatically. Although in general the exact exchange-correlation potential vxc(r,t){v}_{\text{xc}}(\mathbf{r},t)vxc​(r,t) cannot be expressed as the functional derivative of a universal functional due to the so-called causality paradox, we show that for a harmonic perturbation the exchange-correlation part of the first-order Kohn-Sham potential \ensuremathλvs(1)(r)cos \ensuremathωt\ensuremath{\lambda}{v}_{s}^{(1)}(\mathbf{r})\text{cos}\text{ }\ensuremath{\omega}t\ensuremathλvs(1)​(r)cos \ensuremathωt is given by vxc(1)(r)=\ensuremathδKxc(2)/\ensuremathδn\ensuremathω(1)(r){v}_{\text{xc}}^{(1)}(\mathbf{r})=\ensuremath{\delta}{K}_{\text{xc}}^{(2)}/\ensuremath{\delta}{n}_{\ensuremath{\omega}}^{(1)}(\mathbf{r})vxc(1)​(r)=\ensuremathδKxc(2)​/\ensuremathδn\ensuremathω(1)​(r). Kxc(2){K}_{\text{xc}}^{(2)}Kxc(2)​ is the exchange-correlation part of the second-order quasienergy Kv(2){K}_{v}^{(2)}Kv(2)​. The Frenkel variation principle implies a stationary principle for Kv(2)[n\ensuremathω(1)]{K}_{v}^{(2)}[{n}_{\ensuremath{\omega}}^{(1)}]Kv(2)​[n\ensuremathω(1)​]. We also find an analogous stationary principle and Kohn-Sham scheme in the time-dependent extension of one-matrix functional theory, in which the basic variable is the one-matrix (one-body-reduced density matrix).
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.
The frequency-dependent exchange-correlation potential, which appears in the usual Kohn-Sham formulation of a time-dependent linear response problem, is a strongly nonlocal functional of the density, so that a consistent local density approximation generally does not exist. This problem can be avoided by choosing the current density as the basic variable in a generalized Kohn-Sham theory. This theory admits a local approximation which, for fixed frequency, is exact in the limit of slowly varying densities and perturbing potentials.
In the context of inhomogeneous one-dimensional finite systems, recent numerical advances [Phys. Rev. B 103, 125155 (2021)] allow us to compute the exact coupling-constant dependent exchange-correlation kernel ${f}_{\text{xc}}^{\ensuremath{\lambda}}(x,{x}^{\ensuremath{'}},\ensuremath{\omega})$ within linear response time-dependent density-functional theory. This permits an improved understanding of ground-state total energies derived from the adiabatic-connection fluctuation-dissipation theorem (ACFDT). We consider both one-shot and self-consistent ACFDT calculations, and demonstrate that chemical accuracy is reliably preserved when the frequency dependence in the exact functional ${f}_{\text{xc}}[n](\ensuremath{\omega}=0)$ is neglected. This performance is understood on the grounds that the exact fxc[n]{f}_{\text{xc}}[n]fxc​[n] varies slowly over the most relevant $\ensuremath{\omega}$ range (but not in general), and hence the spatial structure in ${f}_{\text{xc}}[n](\ensuremath{\omega}=0)$ is able to largely remedy the principal issue in the present context: self-interaction (examined from the perspective of the exchange-correlation hole). Moreover, we find that the implicit orbitals contained within a self-consistent ACFDT calculation utilizing the adiabatic exact kernel ${f}_{\text{xc}}[n](\ensuremath{\omega}=0)$ are remarkably similar to the exact Kohn-Sham orbitals, thus further establishing that the majority of the physics required to capture the ground-state total energy resides in the spatial dependence of fxc[n]{f}_{\text{xc}}[n]fxc​[n] at $\ensuremath{\omega}=0$.
The Kohn-Sham approach to time-dependent density-functional theory (TDDFT) can be formulated, in principle, exactly by invoking the force-balance equation for the density, which leads to an explicit expression for the exchange-correlation potential as an implicit density functional. It is shown that this suggests a reformulation of TDDFT in terms of the second time derivative of the density, rather than the density itself. The result is a time-local Kohn-Sham scheme of second-order in time whose causal structure is more transparent than that of the usual Kohn-Sham formalism. The scheme can be used to construct new approximations at the exchange-only level and beyond, and it offers a straightforward definition of the exact adiabatic approximation.
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.
This chapter introduces the time-dependent Kohn-Sham scheme. The first section shows how an interacting time-dependent many-body system can be replaced by an auxiliary noninteracting system which reproduces the exact time-dependent density. This noninteracting system is called the Kohn-Sham system. It features an effective single-particle potential, consisting of external, Hartree, and exchange-correlation potential. The Kohn-Sham formalism is then generalized to spin-dependent systems. The exchange-correlation potential needs to be approximated in practice. Of particular importance is the adiabatic approximation, which uses any static exchange-correlation potential and evaluates it with time-dependent densities. The last two sections deal with practical issues related to the numerical implementation of the time-dependent Kohn-Sham scheme, namely, the meaning of self-consistency and unitary time propagation. The Crank-Nicholson algorithm is introduced, and important practical concepts such as the predictor-corrector scheme and absorbing boundary conditions are discussed.