The CFSE is illustrated in figure . This shows the total energy calculated using VMC for diamond-structure silicon using a finite simulation cell with periodic boundary conditions plotted for simulation cells containing multiples, n=2,3,4,5 of primitive lattice cells, which corresponds to 16,54,128 and 250 atoms, respectively. For each of the calculations, a full Hartree-Fock-Jastrow-Chi wavefunction as in Eq.() was used where the and u functions were optimised using the variance minimisation techniques described in chapter . Six stars of vectors were used to describe the function and 22 variational parameters were used in the u function. The single particle orbitals used to construct the Slater determinant have -points chosen from a reciprocal space grid that is offset from the origin by sampling, where the are the primitive reciprocal lattice vectors of the supercell, i.e. L-point sampling. This choice of sampling almost totally removes the IPFSE, leaving the CFSE as the dominant finite size effect. A norm-conserving, non-local pseudopotential was used to describe the silicon cores. The extensions to the variance minimisation scheme introduced in chapter to deal with non-local pseudopotentials were experimented with in both the simple ``fixed non-local'' and ``full optimisation'' forms. The two resulting wavefunctions produced variational energies that were indistinguishable at the level of the statistical noise. This suggests that in silicon the ``fixed non-local'' approximation in the variance minimisation scheme is sufficient.
Figure shows the total energy per atom asymptotically approaching a value that can be taken as the energy per atom in the bulk solid. The CFSE can then be defined as the difference in the total energy (once any IPFSE have been removed) at a specific system size and this bulk value.
Figure: Total energy per atom calculated using VMC as a function of system size. The statistical error bars are smaller than the size of the symbols.
The motivation for reducing the CFSE is twofold. Firstly, if the desired level of accuracy in a specific calculation can be achieved by performing that calculation on a smaller system as a result of reducing the CFSE, then the computational benefits will be large as the computational time scales as approximately the third power of the number of electrons in the system. In Figure , the number of electrons scales as the cube of the system size, n and hence the total computational time scales as the ninth power of the x-axis.
Secondly, there are some problems in which even if one is able to perform the calculation on a large system size this still fails to reduce the CFSE. A standard problem in electronic structure theory is to calculate the energy required to create a point defect. This is done by subtracting the energy of the perfect crystal from that of a large simulation cell containing a single defect. Taking the example of the n=3 (54 atom) simulation cell of silicon, we find (see figure .) that the CFSE error in the VMC energy of the whole simulation cell is -5 eV. This is much larger than the energies of interest, which are often tenths of an eV per simulation cell. Moreover, it has been observed in previous work [46, 33] that these CFSE are approximately inversely proportional to the number of atoms in the simulation cell and hence the CFSE for the whole cell is almost independent of N. Of course there will be a cancellation between the CFSEs in the perfect and defective solids, which will become more complete as the size of the simulation cell increases, so that eventually the energy difference will converge. However, we must expect that the incomplete cancellation of errors for finite simulation cells will lead to a significant uncertainty in the defect energy. Therefore, the only guaranteed way of improving the accuracy of such calculations is to significantly reduce the CFSE at all system sizes.