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Physical Review Letters

Harmonic-Potential Theorem: Implications for Approximate Many-Body Theories

John F. Dobson

Published Oct 17, 1994• 321 citations
Publisher

Abstract

We consider interacting particles in an external harmonic potential. We extend the $B=0$ case of the generalized Kohn theorem, giving a "harmonic-potential theorem" (HPT), demonstrating rigid, arbitrary-amplitude, time-oscillatory Schr\"odinger transport of a many-body eigenfunction. We show analytically that the time-dependent local-density approximation (TDLDA) satisfies the HPT exactly. Other approximations, such as linearized TDLDA with frequency-dependent exchange correlation kernel and certain inhomogeneous hydrodynamic formalisms, do not. A simple modification permits such explicitly frequency-dependent local theories to satisfy the HPT, however.

DOI
10.1103/physrevlett.73.2244

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We consider interacting particles in an external harmonic potential. We extend the B=0B=0B=0 case of the generalized Kohn theorem, giving a "harmonic-potential theorem" (HPT), demonstrating rigid, arbitrary-amplitude, time-oscillatory Schr"odinger transport of a many-body eigenfunction. We show analytically that the time-dependent local-density approximation (TDLDA) satisfies the HPT exactly. Other approximations, such as linearized TDLDA with frequency-dependent exchange correlation kernel and certain inhomogeneous hydrodynamic formalisms, do not. A simple modification permits such explicitly frequency-dependent local theories to satisfy the HPT, however.
One of the few exact results for the description of the time evolution of an inhomogeneous, interacting many-particle system is given by the harmonic potential theorem (HPT). The relevance of this theorem is that it sets a tight constraint on time-dependent many-body approximations. In this contribution, we show that the original formulation of the HPT is valid also for the case of spin-, velocity-, and density-dependent interactions. This result is completely general and relevant, among the rest, for nuclear structure theory both in the case of ab initio and of more phenomenological approaches. As an example, we report on a numerical implementation by testing the small-amplitude limit of the time-dependent Hartree-Fock-also known as the random phase approximation-for the translational frequencies of a neutron system trapped in a harmonic potential.
The Hohenberg-Kohn theorem is extended to the case that the external potential is nonlocal. It is shown that, in this more general case, a nondegenerate ground-state wave function is a universal functional of the one-particle density kernel $\ensuremath{\mu}(x,{x}^{\ensuremath{'}})$, but probably not of the particle density $n(\stackrel{\ensuremath{\rightarrow}}{\mathrm{r}})=\ensuremath{\Sigma}{s}^{}\ensuremath{\mu}(\stackrel{\ensuremath{\rightarrow}}{\mathrm{r}}s,\stackrel{\ensuremath{\rightarrow}}{\mathrm{r}}s)$. The variational equations for the local and nonlocal cases are compared. The former must be replaced by a variational equation for an equivalent system of noninteracting particles, following a prescription of Kohn and Sham, in order to obtain a Schr\"odinger-like form, and contains only local potentials. The latter may be obtained directly in Schr\"odinger-like form, but the exchange-correlation potential is nonlocal. If the nonlocal pseudo-Hamiltonian exists [i.e., if the functional derivative $\frac{\ensuremath{\delta}E}{\ensuremath{\delta}\ensuremath{\mu}(x,{x}^{\ensuremath{'}})}$ exists for a nondegenerate ground-state density kernel], then the eigenfunctions of the pseudo-Hamiltonian are natural spin orbitals, and all partially occupied orbitals ($0<〈{\ensuremath{\varphi}}_{i}|\ensuremath{\mu}|{\ensuremath{\varphi}}_{i}〉<1$) belong to the same degenerate eigenvalue of the pseudo-Hamiltonian. Finally, it is shown, as a corollary of Coleman's theorem for NNN-representable density kernels, that any finite non-negative differentiable function is an NNN-representable particle density.
The Floquet theorem allows one to reformulate periodic time-dependent problems such as the interaction of a many-body system with a laser field in terms of time-independent, field-dressed states, also known as Floquet states. If this was possible for density-functional theory as well, one could reduce in such cases time-dependent density-functional theory to a time-independent Floquet density-functional theory. We analyze under which conditions the Floquet theorem is applicable in a density-functional framework. By employing numerical ab initio solutions of the interacting time-dependent Schr\"odinger equation with time-periodic external potentials we show that the exact effective potential in the corresponding Kohn-Sham equation is not unconditionally periodic. Whenever several Floquet states in the interacting system are involved in a physical process the corresponding Hartree-exchange-correlation potential is not periodic with the external frequency only. Using an analytically solvable example we demonstrate that, in general, the periodicity of the time-dependent Kohn-Sham Hamiltonian cannot be restored by choosing a different initial state. Only if the external periodic potential is sufficiently weak such that the initial state of the interacting system evolves adiabatically to a single, field-dressed state, the resulting Kohn-Sham system admits the application of the Floquet theorem.
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.
We propose a memory form of exchange-correlation potential vXC(r,t){v}_{\mathrm{XC}}(\mathbf{r},t)vXC​(r,t) for time-dependent interacting many-particle systems. Unlike previous memory-XC potentials, our vXC{v}_{\mathrm{XC}}vXC​ is not limited to the linear response regime. The proposed form of vXC{v}_{\mathrm{XC}}vXC​ is a generalized local-density approximation chosen so as to satisfy the nonlinear harmonic potential theorem and Newton's third law. For the case of the inhomogeneous electron gas, we give an explicit prescription for vXC{v}_{\mathrm{XC}}vXC​ based solely on an existing parametrization of the linear XC response kernel ${f}_{\mathrm{XC}}^{\mathrm{hom}}(n,\ensuremath{\omega})$ of the uniform gas. Application to quantum wells seems promising.
Approximations for the ground-state exchange-correlation potential of density-functional theory have reached a high level of sophistication. By contrast, time- or frequency-dependent exchange-correlation potentials are still being treated in a local approximation. Here we propose a novel approximation scheme, which effectively brings the power of the generalized gradient approximation (GGA) and meta-GGA to time-dependent density-functional theory. The theory should allow a more accurate treatment of strongly inhomogeneous electronic systems (e.g. molecular junctions) while remaining essentially exact for slowly varying densities and slowly varying external potentials.