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arXiv:1909.02510 · Physical Review A

Accurate real-time evolution of electron densities and ground-state properties from generalized Kohn-Sham theory

M. J. P. Hodgson, Jack Wetherell

Published Mar 2, 2020• 3 citations
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Abstract

The exact static and time-dependent Kohn-Sham (KS) exchange-correlation potential is extremely challenging to approximate as it is a local multiplicative potential that depends on the electron density everywhere in the system. The KS approach can be generalized by allowing part of the potential to be spatially nonlocal. We take this nonlocal part to be that of unrestricted Hartree-Fock theory. The additional local correlation potential in principle ensures that the single-particle density exactly equals the many-body density. In our case, the local correlation potential is predominantly nearsighted in its dependence on the density and hence an (adiabatic) local-density approximation to this potential yields accurate ground-state properties and real-time densities for one-dimensional test systems.

DOI
10.1103/physreva.101.032502

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The exact static and time-dependent Kohn-Sham (KS) exchange-correlation potential is extremely challenging to approximate as it is a local multiplicative potential that depends on the electron density everywhere in the system. The KS approach can be generalized by allowing part of the potential to be spatially nonlocal. We take this nonlocal part to be that of unrestricted Hartree-Fock theory. The additional local correlation potential in principle ensures that the single-particle density exactly equals the many-body density. In our case, the local correlation potential is predominantly nearsighted in its dependence on the density and hence an (adiabatic) local-density approximation to this potential yields accurate ground-state properties and real-time densities for one-dimensional test systems.
We obtain the exact Kohn-Sham potentials VKS{V}_{\mathrm{KS}}VKS​ of time-dependent density-functional theory for one-dimensional Hubbard chains, driven by a dc external field, using the time-dependent electron density and current density obtained from exact many-body time evolution. The exact Vxc{V}_{\mathrm{xc}}Vxc​ is compared to the adiabatically exact Vxcad{V}_{\mathrm{xc}}^{\mathrm{ad}}Vxcad​ and the ``instantaneous ground state'' Vxcigs{V}_{\mathrm{xc}}^{\mathrm{igs}}Vxcigs​. The latter is shown to work effectively in some cases when the former fails. Approximations for the exchange-correlation potential Vxc{V}_{\mathrm{xc}}Vxc​ and its gradient, based on the local density and on the local current density, are also considered and both physical quantities are observed to be far outside the reach of any possible local approximation. Insight into the respective roles of ground-state and excited-state correlation in the time-dependent system, as reflected in the potentials, is provided by the pair correlation function.
Given the time-evolution of an electron charge density, the local potential in Kohn-Sham time-dependent density functional theory (KS-TDDFT) can be modeled as a sum of instantaneous and dynamic contributions by assuming a certain form of the time-dependent Kohn-Sham (TD-KS) orbitals. The instantaneous part is obtained numerically using methods from ground-state density functional theory (DFT) and the dynamic part is expressed in terms of a velocity potential that depends on the electron current density. The suggested form of the TD-KS orbitals satisfies several known constraints (orthonormality, N-representability, J-representability), and the domain of validity is shown to depend on the evolution of the instantaneous quantities. Through this decomposition, we can relate time-dependent and ground -state V-representability. The resulting potentials are shown numerically to approximate the exact time-dependent Kohn-Sham potentials for a set of 3 non-singlet two-particle systems (a Kohn-mode, a Coulomb explosion, and a double quantum well) where the exact solutions and reference densities are known or obtained through configuration interaction (CI) approaches.
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.
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.
We calculate the exact Kohn-Sham potential that describes, within time-dependent density-functional theory, the propagation of an electron quasiparticle wave packet of nonzero crystal momentum added to a ground-state model semiconductor. The potential is observed to have a highly nonlocal functional dependence on the charge density, in both space and time, giving rise to features entirely lacking in local or adiabatic approximations. The dependence of the nonequilibrium part of the Kohn-Sham electric field on the local current and charge density is identified as a key element of the correct Kohn-Sham functional.