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Physical Review A

Exact relations of the quasienergy functional and the exchange-correlation potential from the Floquet formulation of time-dependent density functional theory

Dmitry A. Telnov, Shih‐I Chu

Published Dec 13, 2000• 11 citations
Publisher

Abstract

In the framework of the Floquet formulation of time-dependent density functional theory we present several exact relations involving different parts of the quasienergy functional. These relations hold when the exact densities and exchange-correlation energy functional are employed. They can be used as useful constraints and tests when searching for the approximate forms of the time-dependent exchange-correlation functionals. The general results are illustrated on an exactly soluble model, Hooke's atom in a linearly polarized monochromatic laser field.

DOI
10.1103/physreva.63.012514

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In the framework of the Floquet formulation of time-dependent density functional theory we present several exact relations involving different parts of the quasienergy functional. These relations hold when the exact densities and exchange-correlation energy functional are employed. They can be used as useful constraints and tests when searching for the approximate forms of the time-dependent exchange-correlation functionals. The general results are illustrated on an exactly soluble model, Hooke's atom in a linearly polarized monochromatic laser field.
The exact time-dependent correlation energy in time-dependent density-functional theory is shown to sometimes become positive, which is impossible with most present TDDFT approximations. Both the correlation potential and energy can be strongly nonlocal in time. A new inequality is derived for the time-dependent exchange-correlation energy. The correlation energy appears to scale to a constant function of scaled time in the high-density limit. In the linear response regime, the correlation energy is shown to become purely adiabatic, but the correlation potential is generally nonadiabatic. The usefulness of the virial theorem as a test of numerical accuracy is demonstrated. All results are found or inspired by exact numerical solution of a simple model system (Hooke’s atom), and inversion of the corresponding Kohn–Sham equations.
We illustrate the scope of time-dependent density-functional theory for strongly correlated (lattice) models out of equilibrium. Using the exact many-body time evolution, we reverse engineer the exact exchange correlation (xc) potential v_(xc) for small Hubbard chains exposed to time-dependent fields. We introduce an adiabatic local density approximation to v_(xc) for the 1D Hubbard model and compare it to exact results, to gain insight about approximate xc potentials. Finally, we provide some remarks on the v-representability for the 1D Hubbard model.
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.
We illustrate the potential of Time Dependent Density Functional Theory (TDDFT) for describing the nonequilibrium behavior of strongly correlated (lattice) models. Starting from an exact time-evolution of the many-body wavefunction, we determine, via reverse engineering, the exact exchange correlation (xc) potential vxcv_{xc}vxc​ for small Hubbard chains of different lengths and electron fillings and exposed to time-dependent perturbations. We compare some of the exact results to those of adiabatic-like treatments, in order to extract some of the properties that approximate xc potentials should have. Finally, we provide details of work in progress and future directions.
A decomposition of the exact exchange-correlation potential of time-dependent density functional theory into an interaction component and a kinetic component offers a new starting point for non-adiabatic approximations. The components are expressed in terms of the exchange-correlation hole and the difference between the one-body density matrix of the interacting and Kohn-Sham systems, which must be approximated in terms of quantities accessible from the Kohn-Sham evolution. We explore several preliminary approximations, evaluate their fulfillment of known exact conditions, and test their performance on simple model systems for which available exact solutions indicate the significance of going beyond the adiabatic approximation.