Monte Carlo Calculation of the Born-Oppenheimer Potential Between Two Helium Atoms

Abstract:
The results of the calculations of extremely accurate wave functions for the ground state of two helium atoms, including energies obtained from these wave functions, are presented herein. These energies provide a variational upper bound to the Born-Oppenheimer potential curve for this system. The necessary expectation values were calculated by biased Monte Carlo techniques at seven internuclear distances. The energy obtained from the trial wave function at the potential minimum is -11.6149685 ± 0.0000030 Ry giving a well depth of -7.10 ± 0.30 x 10^-5 Ry at the nuclear separation distance of 5.6 Bohr radii (ä). It is estimated that this energy is above the energy of the exact wave function by no more than 1.8 x 10^-6 Ry. The extremely small Monte Carlo standard deviation (a) of 3.0 x 10~^ Ry was made possible through a combination of the three factors: 1. Evaluation of the integrands for many (over 10 ) Monte Carlo points. For the seven internuclear distances this took a total of about 50 hours of CPU time on an Amdahl 47 0/V6.
2. Monte Carlo methods (which allowed for analytic removal of all singularities) for finding good weight function.
3. The extremely accurate wave functions reported herein.
These wave functions, in fact, were found by minimizing, rather than the energy () , the standard deviation in this energy (a) which is zero for a perfect wave function. This enabled us to optimize the set of values for the 2 9 variational parameters by using very few Monte Carlo points and, therefore, made this step financially feasible.
Monte Carlo evaluation of the integrals allows total freedom to choose a natural and concise expansion for the wave functions. The wave functions used combine Schwartz's 189-term Hylleraas-type atomic wave function with molecular terms containing dipole-dipole, dipole-quadrupole, and further terms in the expansion of the interatomic potential energy.
The Born-Oppenheimer potential curve found in this work is in rough agreement with the experimental results of Burgmans, Farrar, and Lee (BFL) . The greatest departure is at the nuclear separation distance of 5.6 ab, where the potential found is 1.3a below the BFL result of -6.70 Ry. Therefore the upper bound found herein should be considered to be in agreement with the BFL potential curve with just a hint that the exact curve is deeper than the BFL curve.
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