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.  and Williamsonet al.  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.