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The Journal of Chemical Physics

Exact local density method for linear harmonic oscillator

G. P. Lawes, N. H. March

Published Jul 15, 1979• 60 citations
Publisher

Abstract

Using an exact form for the particle density for N particles moving independently in a linear harmonic oscillator potential, a differential equation is derived for the particle density. This is then shown to be exactly that which is obtained from a local density assumption for the kinetic energy density t, in the form −1/2Σψi(δ2/δx2) ψi. The form of the kinetic energy density functional is then discussed in this example. The implications for wider classes of potential are briefly referred to.

DOI
10.1063/1.438398

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Using an exact form for the particle density for N particles moving independently in a linear harmonic oscillator potential, a differential equation is derived for the particle density. This is then shown to be exactly that which is obtained from a local density assumption for the kinetic energy density t, in the form −1/2Σψi(δ2/δx2) ψi. The form of the kinetic energy density functional is then discussed in this example. The implications for wider classes of potential are briefly referred to.
Using an exact form for the particle density for N particles moving independently in a linear harmonic oscillator potential, a differential equation is derived for the particle density. This is then shown to be exactly that which is obtained from a local density assumption for the kinetic energy density t, in the form −1/2Σψi(δ2/δx2) ψi. The form of the kinetic energy density functional is then discussed in this example. The implications for wider classes of potential are briefly referred to.
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From the explicit solution of a two-level problem previously obtained, it is proved that the single-particle kinetic energy density functional ts{t}_{s}ts​[\ensuremath{\rho}] is the density \ensuremath{\rho} times a function solely of the ratio \ensuremath{\rho}'/\ensuremath{\rho}, with \ensuremath{\rho}' the first derivative of \ensuremath{\rho}. For an n-level system with a harmonic-oscillator potential, the form of ts{t}_{s}ts​[\ensuremath{\rho}] is explicitly obtained. Again ts{t}_{s}ts​[\ensuremath{\rho}] involves only \ensuremath{\rho} and \ensuremath{\rho}'; no higher derivatives appear.
We investigate the particle and kinetic-energy densities for a system of N fermions confined in a potential V (r). In an earlier paper (Brack and Murthy 2003 J. Phys. A: Math. Gen. 36 1111), some exact and asymptotic relations involving the particle density and the kinetic-energy density locally, i.e. at any given point r, were derived for isotropic harmonic oscillators in arbitrary dimensions. In this paper, we show that these local virial theorems (LVTs) also hold exactly for linear potentials in arbitrary dimensions and for the one-dimensional box. We also investigate the validity of these LVTs when they are applied to arbitrary smooth potentials. We formulate generalized LVTs that are suggested by a semiclassical theory which relates the density oscillations to the closed non-periodic orbits of the classical system. We test the validity of these generalized theorems numerically for various local potentials. Although formally they are only valid asymptotically for large particle numbers N, we show that practically they are surprisingly accurate also for moderate values of N.