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Physical Review A

Two electrons in an external oscillator potential: Particular analytic solutions of a Coulomb correlation problem

M. Taut

Published Nov 1, 1993• 430 citations
Publisher

Abstract

The problem of the Schr\"odinger equation for two electrons (interacting with Coulomb potentials) in an external harmonic-oscillator potential is revisited and shown to be solvable analytically for a particular, denumerably infinite set of oscillator frequencies. Solutions are given for ground and excited states in the singlet and triplet spin configurations.

DOI
10.1103/physreva.48.3561

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The problem of the Schr"odinger equation for two electrons (interacting with Coulomb potentials) in an external harmonic-oscillator potential is revisited and shown to be solvable analytically for a particular, denumerably infinite set of oscillator frequencies. Solutions are given for ground and excited states in the singlet and triplet spin configurations.
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We have obtained an exact analytical solution for a system of two particles moving under the action of a central harmonic-oscillator potential and the mutual Coulomb interaction augmented by a potential linear in the interparticle coordinate. The expressions for the single-particle density, the exchange-correlation energy quantities, and the correlation functions are thereby derived. Numerical results on the exchange-correlation contributions are presented using plots of various local energy densities as well as the corresponding global energy quantities for varying strengths of the oscillator potential.