Although it would be computationally prohibitive to repeat all the
calculations of figure within DMC to confirm that
the new interaction behaves in a similar way, it is possible to compare
DMC calculations at the smallest system size, *n*=2, to see the effect
on the diffusion algorithm of switching to the new electron-electron
interaction.

To perform a DMC calculation, one requires not only an energy expression but also an expression for the Hamiltonian of the system being studied. The Schrödinger equation may be ``derived'' by minimising an energy functional, , where is a normalised wavefunction. If a similar procedure is carried out for a functional including the electron-electron interaction of Eq. (), the electron-electron interaction operator in the resulting Schrödinger-like equation is

Again, as in Eq. (), we chose to use the LDA charge density, , as the input density, , to the second term in the Hamiltonian. The total electron-electron energy, , is then the expectation value of minus a double counting term for the electrostatic interactions;

This double counting term can itself be accumulated during the DMC calculation and then subtracted off at the end of the simulation as it is a fixed constant that will not affect the diffusion algorithm.

Two DMC calculations were performed on the *n*=2 system of
diamond-structure silicon to compare the effect of the new interaction
in DMC and VMC. The DMC calculations were performed using a Slater
determinant of single-particle orbitals with -points chosen
on a reciprocal space grid centred at the origin, i.e. -point
sampling. This sampling was chosen as DMC calculations with
-point sampling are required in the next chapter as well.

The first DMC calculation was performed with the standard Hamiltonian as described in chapter . The second calculation used the new DMC Hamiltonian from Eq.(). The same changes were made to the DMC algorithm (described in chapter ) as were made to the VMC algorithm in section to implement the new electron-electron interaction. The new DMC calculation exhibited the same stability in the population of walkers as the original Hamiltonian. It required a similar number of steps to diffuse to a state where energies could be accumulated and the intrinsic variance of the energy over the run was also very similar. Table shows a comparison of the total energies obtained in VMC and DMC using the Ewald and new electron-electron interactions. The VMC results in table were performed using -point sampling to facilitate the comparison.

The results show that the reduction in energy obtained by performing a DMC calculation rather than a VMC calculation is similar for the two electron-electron interactions. This suggests that, as expected, the finite size effects in DMC broadly follow those in VMC and that using the new electron-electron interaction yields a similar improvement in DMC calculations to VMC calculations.

Tue Nov 19 17:11:34 GMT 1996