Title
The determination of the Dirac density matrix of the d-dimensional harmonic oscillator for an arbitrary number of closed shells The determination of the Dirac density matrix of the d-dimensional harmonic oscillator for an arbitrary number of closed shells
Author
Faculty/Department
Faculty of Sciences. Physics
Publication type
article
Publication
London ,
Subject
Physics
Source (journal)
Journal of physics: A: mathematical and general. - London, 1968 - 2006
Volume/pages
35(2002) :24 , p. 4985-4997
ISSN
0305-4470
ISI
000176858400004
Carrier
E
Target language
English (eng)
Full text (Publishers DOI)
Affiliation
University of Antwerp
Abstract
In 1959, March and Young (Nucl. Phys. 12 237) rewrote the equation of motion for the Dirac density matrix gamma(x, x(0)) in terms of sum and difference variables. Here, gamma((r) over right arrow, (r) over right arrow (0)) for the d-dimensional isotropic harmonic oscillator for an arbitrary number of closed shells is shown to satisfy, using the variables \(r) over bar + (r) over bar (0)\/2 and \(r) over right arrow - (r) over right arrow (0)\/2, a generalized partial differential equation embracing the March-Young equation for d = 1. As applications, we take in turn the cases d = 1, 2, 3 and 4, and obtain both the density matrix gamma((r) over right arrow,(r) over right arrow (0)) and the diagonal density rho(r) = gamma((r) over right arrow, (r) over right arrow (0))\((r) over right arrow0=(r) over right arrow), this diagonal element already being known to satisfy a third-order linear homogeneous differential equation for 1 through 3. Some comments are finally made on the d-dimensional kinetic energy density, which is important for first-principles density functional theory in allowing one to bypass one-particle Schrodinger equations (the so-called Slater-Kohn-Sham equations).
E-info
https://repository.uantwerpen.be/docman/iruaauth/020c5b/b5d5991.pdf
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