A Direct Monte Carlo Evaluation of the Derivative of the Born-Oppenheimer Potential between Two Ions in Liquid Sodium

Monte Carlo methods for finding trial wavefunctions by minimizing the Monte Carlo standard deviation in the expectation value of the Hamiltonian are investigated for a periodically repeated system of 2 sodium nuclei and 22 electrons. Methods are described for using the biased selection technique to find reasonable numbers of statistically independent configurations in the regions where the wavefunction is sizable. A technique for making a direct Monte Carlo estimate of the derivative of the Born-Oppenheimer (BO) potential is described which numerically eliminates the part of the standard deviation in this estimate coming from the fluctuations within the ion cores. Finally, the resulting bound to the BO potential given by a rather simple trial wavefunction for the system at 1/14 the normal liquid density is presented.