As well as scaling with the fifth or sixth power of the atomic number of the atomic species being studied, QMC calculations also scale as the third power of the number of the electrons in the system. This is due to the process of updating the Slater determinant of one-electron orbitals after moving each electron (see appendix ). Therefore, even after the introduction of a pseudopotential to reduce the effective atomic number of the ionic cores, it is still important to keep the number of electrons in the system as small as possible.
One method of simulating a solid is to construct a cluster of atoms and then investigate the properties of the cluster as the number of atoms increases. As the size of the cluster increases, the collective behaviour of the atoms within the cluster should asymptotically approach those of the bulk solid. In practice, it turns out that the number of atoms that can be simulated in a QMC calculation is so small that any cluster constructed from such a small number of atoms would be completely dominated by surface effects and would not be able to reproduce the properties of atoms deep within the bulk of a true solid.
An alternative approach to simulating solids is the use of supercells. Here one constructs a supercell containing relatively few atoms and electrons and then repeats the supercell throughout all space using periodic (or toroidal) boundary conditions. These boundary conditions mean that the supercell is wrapped around on itself and as an electron moves out of one side of the supercell it immediately moves back in through the opposite side. The advantage of using such a supercell is that there are no longer any ``surface electrons'' and hence the problems of the cluster method are removed. However, the supercell method itself still suffers from very significant finite size effects. These are due to the absence of long wavelength fluctuations in the charge density. For a simulation cell of linear dimension, L, the periodicity will remove any electron density waves with wavelength greater than L. One would expect this omission to be especially important in materials where long range effects are dominant such as superconductors containing Cooper pairs of electrons separated by many lattice constants. In these cases, the only cure for the finite size effects is to increase the size of the simulation cell being studied. Finite size effects are also present in the simplest systems such as the HEG. Methods of dealing with these finite size effects are discussed in chapter .
In supercell calculations, the standard choice of supercell is an integer multiple of primitive unit cells. In the following work, we will refer to the size of supercell by an integer, n, where n=2 refers to a supercell consisting of a 2x2x2 array of primitive unit cells. This is illustrated in figure .
Figure: Illustration of different supercell sizes.