Minimizers for open-shell, spin-polarised Kohn-Sham equations for non-relativistic and quasi-relativistic molecular systems
type: Item
instance of: scholarly article
title: Minimizers for open-shell, spin-polarised Kohn-Sham equations for non-relativistic and quasi-relativistic molecular systems
zbMATH Open document ID: 1362.35277
published in: Methods and Applications of Analysis
publication date: 2017-01-03T00:00:00Z
review text: This interesting paper is devoted to the open-shell, spin-polarized Kohn-Sham models for non-relativistic and quasi-relativistic \(N\)-electron Coulomb systems. Note that nowadays the Density Functional Theory (DFT) of \textit{W. Kohn} and \textit{L. J. Sham} [``Self-consistent equations including exchange and correlation effects'', Phys. Rev. A, 140, No. 4, 1133--1138 (1965; \url{doi:10.1103/PhysRev.140.A1133})] is one of the most applicable to the electronic structure calculations in modern quantum chemistry [\textit{A. Szabo} and \textit{N. S. Ostlund}, Modern quantum chemistry: an introduction to advanced electronic structure theory. New York, NY: Palgrave Macmillan (1982)], and also in the condensed matter theory. Here it is established existence of a ground state (minimizer) to the non-relativistic Kohn-Sham problem as well as to the quasi-relativistic one, that is shown for the spin-polarized molecular systems with \(N^{\uparrow }\) electrons (with spin up), and \(N^{\downarrow }\) electrons (with spin down). The open-shell, spin-polarized Kohn-Sham models for non-relativistic and quasi-relativistic \(N\)-electron Coulomb systems are characterized with the properties that the kinetic energy of the electrons is given by either the non-relativistic operator \(-\triangle_{x_n}\) or the quasi-relativistic operator \(\sqrt{-\alpha^{-2}\triangle_{x_n}+\alpha^{-4}}-\alpha^{-2}\). Here \(\alpha \) is the Sommerfeld's fine structure constant. For the standard and extended Kohn-Sham models in the local density approximation the existence of a ground state would be of interest. Note that it is required that the total charge \(Z_{\text{tot}}\) of \(K\) nuclei must be greater than \(N - 1\). Also it is considered the quasi-relativistic setting, where it is required that \(Z_{\text{tot}}\) to be smaller than a critical charge \(Z_c = 2\alpha^{-1}\pi^{-1}\). To show the main result it is required the total electron density \(\rho \) to be defined, and the exchange-correlati…
reviewed by: Dimitar A. Kolev
Mathematics Subject Classification ID: 35Q55
Mathematics Subject Classification ID: 35R09
Mathematics Subject Classification ID: 35J60
Mathematics Subject Classification ID: 47J10
Mathematics Subject Classification ID: 58Z05
Mathematics Subject Classification ID: 81V55
zbMATH DE Number: 6670052
zbMATH Keywords: open-shell
zbMATH Keywords: spin-polarised Kohn-Sham equations
zbMATH Keywords: ground state
zbMATH Keywords: variational methods
zbMATH Keywords: concentration-compactness
author: Carlos Argáez
author: M. Melgaard
MaRDI profile type: Publication
full work available at URL: maa.2016.v23.n3.a4
OpenAlex ID: W2549283994
DOI: 10.4310/MAA.2016.V23.N3.A4
description: scientific article; zbMATH DE number 6670052