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.
|Ewald Interaction||New Interaction|
|(eV per atom)||(eV per atom)|
|VMC||-106.88 0.03||-106.70 0.03|
|DMC||-107.41 0.03||-107.30 0.03|
|0.53 0.04||0.60 0.04|
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.