My PhD thesis - Condensed Matter Theory - Imperial College London

CHAPTER 4.
ERRORS IN QMC SIMULATIONS
Figure 4.1: Illustrating the lattice generated by the periodic repeat of the simulation cell. Note
that any movement of the particle in the simulation cell is copied by all the images.
The periodic repeat of the simulation cell leads to two distinct kinds of error,
known as the independent-particle and Coulomb finite-size errors.
4.1.1 The independent-particle finite-size effect
The independent-particle finite-size effect is a result of replacing the smooth density
of states of an infinite material with a set of discrete energy levels, as is inevitable
when moving to a finite system. Figure 4.2 illustrates this point; it shows the kinetic
energy per electron of a non-interacting electron gas as a function of the size of the
box in which it is contained. The oscillations evident in figure 4.2 are a result of shell
filling. Each effective one-electron wave function is associated with a wave vector;
the wave vectors are grouped in shells of equivalent magnitude (and correspondingly
equivalent one-particle kinetic energy).
This effect may be described more completely in terms of k-point sampling. In
this context, two cases will be discussed: the periodic boundary conditions may be
imposed on the wave function or (more generally) on the Hamiltonian operator. The
more general case will be described first. In real solids, there is usually an under-
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