Most simulations of extended quantum systems are performed using finite simulation cells. This introduces ``finite size errors'' which are one of the major problems limiting the application of accurate many-body techniques to extended systems. The standard method of reducing the finite size errors is to apply periodic boundary conditions, but important finite size errors often remain. Minimising and quantifying the size of these errors is essential where high accuracy is required.

In this chapter, developments of the theory of finite size effects in
quantum many-body simulations subject to periodic boundary conditions
are presented. The motivation is to understand and reduce the finite
size effects encountered in quantum Monte Carlo simulations, although
the techniques described are of wide generality and can be readily
applied to other many-body electronic structure methods. The work
contained in this chapter is an extension of earlier work by Fraser
*et al.* [86] and Williamson*et
al.* [87] A more thorough analysis of the finite
size effects in presented, including application to excitation
energies. The theory is applied to large silicon supercells of up to
250 atoms (1000 valence electrons) in DFT, HF, VMC and DMC. These
calculations are computationally very demanding.

By more accurately modelling the infinite system, the techniques developed in this chapter enable far greater accuracy to be obtained at lower computational cost than would otherwise be possible.