In this chapter a way to reduce the finite size errors in quantum many-body simulations is presented. A key step in this process is the use of a new periodic interaction instead of the conventionally used Ewald interaction. This interaction more accurately describes the interactions in the infinite system than the Ewald potential, which is designed to model the artificial array of supercells. The new ``Model Periodic Coulomb'' (MPC) interaction also has the additional advantage that the residual finite size effects are reasonably well described by standard DFT calculations. The accuracy can be further increased by using an extrapolation procedure, but the extrapolation corrections are considerably reduced and can therefore be evaluated using a smaller range of system sizes.

In section 6.3 the Hamiltonian within periodic boundary conditions is described. In section 6.4 various finite size correction and extrapolation procedures that have previously been used are presented. In section 6.6 the ``independent particle finite size effects'' review, showing how k-space sampling techniques in many-body theory are related to those used in mean-field theories. The MPC interaction for reducing finite size effects in periodic systems is introduced in section 6.7. It is shown that it can be applied to all the Coulomb interactions. The MPC interaction is tested within HF theory (section 6.8.1), VMC (section 6.8.2), and DMC (section 6.8.3).

In section 6.9 the finite size errors present in calculations of excitation energies are discussed. HF and VMC calculations of excitation energies using the MPC interaction are presented in sections 6.9.1 and section 6.9.2. VMC results for the ``optical absorption'' and ``photoemission'' gaps are presented, the latter being the first such calculations for a three-dimensional periodic system. Calculations with up to 512 electrons demonstrate that the finite size errors in the ``optical absorption'' and ``photoemission'' gaps are similar and that the finite size effects are quite small even for a 64-electron simulation cell. Conclusions are given in section 6.10.