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DMC Results

Although it would be computationally prohibitive to repeat all the calculations of figure gif 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, tex2html_wrap_inline7873 , where tex2html_wrap_inline5833 is a normalised wavefunction. If a similar procedure is carried out for a functional including the electron-electron interaction of Eq. (gif), the electron-electron interaction operator in the resulting Schrödinger-like equation is


Again, as in Eq. (gif), we chose to use the LDA charge density, tex2html_wrap_inline7877 , as the input density, tex2html_wrap_inline7879 , to the second term in the Hamiltonian. The total electron-electron energy, tex2html_wrap_inline7881 , is then the expectation value of tex2html_wrap_inline7883 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 tex2html_wrap_inline6593 -points chosen on a reciprocal space grid centred at the origin, i.e. tex2html_wrap_inline6701 -point sampling. This sampling was chosen as DMC calculations with tex2html_wrap_inline6701 -point sampling are required in the next chapter as well.

The first DMC calculation was performed with the standard Hamiltonian as described in chapter gif. The second calculation used the new DMC Hamiltonian from Eq.(gif). The same changes were made to the DMC algorithm (described in chapter gif) as were made to the VMC algorithm in section gif 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 gif 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 gif were performed using tex2html_wrap_inline6701 -point sampling to facilitate the comparison.


Ewald Interaction New Interaction
(eV per atom) (eV per atom)
VMC -106.88 tex2html_wrap_inline7561 0.03 -106.70 tex2html_wrap_inline7561 0.03
DMC -107.41 tex2html_wrap_inline7561 0.03 -107.30 tex2html_wrap_inline7561 0.03
tex2html_wrap_inline7903 0.53 tex2html_wrap_inline7561 0.04 0.60 tex2html_wrap_inline7561 0.04
Table: Comparison of VMC and DMC results using the Ewald and New interactions. All calculations use tex2html_wrap_inline6701 -point sampling of the one-electron wavefunctions. DMC results obtained using the energy expression of Eq.(gif).


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.

next up previous contents
Next: Variance Minimisation with the Up: Results for Silicon Previous: HF Results

Andrew Williamson
Tue Nov 19 17:11:34 GMT 1996