Study of Electron Correlation in Helium-Like Systems Using an Exactly Soluble Model

TLDR

The exact solution of a two‑electron system with Coulombic electron–electron repulsion and harmonic oscillator electron–nucleus attraction yields a wave function χ₁χ₂ f(r₁₂) built from one‑particle spherical harmonic oscillator orbitals. The model is employed to gain insight into electron correlation in real two‑electron atoms and to evaluate common approximations. A variational trial function f(r₁₂)=1+α r₁₂, with α optimized, reproduces energies and wave functions close to the exact solution. The exact f(r₁₂) exhibits only slight curvature, and the model’s predicted correlation energy and average electron–electron angle are somewhat lower than those observed in actual helium‑like systems.

Abstract

The exact solution of a two-electron system in which the electron-electron interaction is Coulombic but the electron-nuclear attraction is a harmonic oscillator potential gives the following wave function: ${\ensuremath{\chi}}_{1}{\ensuremath{\chi}}_{2}f({r}_{12})$, where the $\ensuremath{\chi}'\mathrm{s}$ are one-particle spherical harmonic oscillator wave functions. The exact $f({r}_{12})$ has only a small curvature in ${r}_{12}$, the distance between the two electrons. This exactly soluble model is used to gain some insight into electron correlation in actual two electron atoms and into the usual approximations. A variational trial function, $f({r}_{12})=1+\ensuremath{\alpha}{r}_{12}$, where $\ensuremath{\alpha}$ is a variable parameter, gives energies and even wave functions close to the exact solution. The correlation energy determined from an analytic Hartree-Fock solution and the average angle between the two electrons are somewhat less than in actual helium-like systems.