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The determination of the Dirac density matrix of the d-dimensional harmonic oscillator for an arbitrary number of closed shells
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| Abstract |
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| 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). |
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English
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Journal of physics: A: mathematical and general. - London, 1968 - 2006 | |
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London : 2002
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0305-4470
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35:24(2002), p. 4985-4997
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000176858400004
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