Consistency of local density approximations and quantum corrections for time-dependent quantum systems

Joseph W. Jerome*

*Corresponding author for this work

Research output: Contribution to journalArticle

1 Citation (Scopus)

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, a non-interacting 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 usage 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 corrections and ionic Coulomb potentials, in addition to 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.

Original languageEnglish (US)
JournalApplicable Analysis
DOIs
StatePublished - Jan 1 2019

Fingerprint

Local density approximation
Quantum Systems
Hamiltonians
Approximation
Time-dependent Density Functional Theory
Density functional theory
Coulomb Potential
Physics
Evolution Operator
Existence and Uniqueness of Solutions
Gross
Weak Solution
Fixed point theorem
Electrons
Bounded Domain
Computer simulation
Charge
Electron
Engineering
Predict

Keywords

  • 35Q41
  • 81Q05
  • Siegfried Carl
  • Time-dependent quantum systems
  • local density approximation
  • quantum corrections
  • smoothing
  • time-history

ASJC Scopus subject areas

  • Analysis
  • Applied Mathematics

Cite this

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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, a non-interacting 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 usage 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 corrections and ionic Coulomb potentials, in addition to 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.",
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