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[52]. 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[3].
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.