Monte carlo calculation of the born-oppenheimer potential between two helium atoms using hylleraas-type electronic wave functions

Abstract

The fact that Monte Carlo estimates for the expectation value of the Hamiltonian have no standard deviation for exact eigenfunctions of the Hamiltonian has been used to construct electronic wave functions accounting for the total correlation energy of two helium atoms separated by distances larger than 5aB to within 5 × 10−4 Ry. The biased selection Monte Carlo method used is an extension of information sampling, which enables the analytical elimination of all possible singularities in Vψ and also allows the use of the Hartree-Fock atomic wave function to select the points. The trial wave functions are an antisymmetric combination of Hylleraas-type atomic wave functions multiplying an exponential containing dipole-dipole and dipole-quadrupole terms in a form involving 18 parameters. The standard deviations in the energy values were estimated using 200 well-chosen sample points. Best wave functions were found by keeping the points fixed and varying the parameters in the wave functions until the minimum deviations were obtained. The resulting expectation values and standard deviations were then estimated using 6400 points (˜3 min of Amdahl 460 V time). The range of energy values from this step were in agreement with experiment to within their standard deviations of less than 5 × 10−4 Ry. The wave functions for two separation distances were tested much more severely to make sure that the above results were not due to a failure to sample enough odd points. A 10-hr run was made in which the expectation values were evaluated at 6.6 and 9aB separation and also the difference between these values. The runs using 6400 points were found to be valid, and in addition it was discovered that the energy bounds given by these wave functions are in agreement with experiment to within standard deviations of ˜3 × 10−5 Ry and that their difference (1.3 ± 1.4) × 10−5 Ry differs from the experimental 3.2 by about 1.4 standard deviations.