Consistency of the Local Density Approximation and Generalized Quantum Corrections for Time Dependent Closed Quantum Systems

TL;DR

This paper investigates the mathematical foundations of time-dependent density functional theory (TDDFT), establishing existence and uniqueness of solutions for models using local density approximations and quantum corrections in quantum systems.

Contribution

It provides the first rigorous proof of existence and uniqueness of solutions for TDDFT models with local density approximations and various quantum corrections.

Findings

Proved existence of weak solutions for a broad class of quantum corrections.

Established uniqueness of solutions on arbitrary time intervals.

Included models with time-history and ionic Coulomb potentials.

Abstract

Time dependent quantum systems are the subject of intense inquiry, in mathematics, science, and engineering, particularly at the atomic and molecular levels. In 1984, Runge and Gross introduced time dependent density functional theory (TDDFT), a noninteracting electron model, which predicts charge exactly. An exchange-correlation potential is included in the Hamiltonian to enforce this property. We have previously investigated such systems on bounded domains for Kohn-Sham potentials by use of evolution operators and fixed point theorems. In this article, motivated by use in the physics community, we consider local density approximations (LDA) for building the exchange-correlation potential, as part of a set of quantum corrections. Existence and uniqueness of solutions are established separately within a framework for general quantum corrections, including time-history potentials and ionic Coulomb potentials, in addition to general LDA potentials. In summary, we are able to demonstrate a unique weak solution, on an arbitrary time interval, for a general class of quantum corrections, including those typically used in numerical simulations of the model.