Electron correlation calibrated at the large dimension limit

Abstract

Correlation energies (CEs) for two-electron atom ground states have recently been obtained to good approximation from a simple perturbation treatment using 1/D as the expansion parameter, with D the dimensionality of space. In hydrogenic units, the CE varies almost linearly with 1/D between limits at D→1 and D→∞ which are exactly calculable. However, for D→∞ the CE is only about 35% smaller than the ‘‘true-world’’ value at D=3. This is in striking contrast to the analogous error in the mean field approximation of statistical mechanics, which vanishes for sufficiently large D. Here we show that the CE for D→∞ can be made to vanish by modifying the Hartree–Fock (HF) variational wave function. A separable form is retained but a factor Θ(θ) is included, with θ the angle between the electron–nucleus radii r1 and r2. Likewise, the error in the HF value for the first derivative of the energy with respect to 1/D can be made to vanish by employing a suitable choice of coordinates in separate factors of the wave function. The choice is determined by the vibrational normal modes of the electrons about the rigid configuration attained in the D→∞ limit. We estimate that these improvements in the HF wave function at large D will reduce the CE for D=3 by about a factor of 10 or more for any two-electron atom. We also relate our results to those obtained with hyperspherical coordinates and show that the large-D limit accounts for the success enjoyed by the hyperspherical approximation at D=3. These findings offer prospects for reducing CEs for multielectron systems by exploiting dimensional calibration of the HF wave function.