My PhD thesis - Condensed Matter Theory - Imperial College London

Chapter 4
Errors in QMC simulations
This chapter will describe the most significant errors inherent in extended-system
QMC simulations. Two of these — statistical noise and the fixed-node error —
have already been mentioned in the previous chapter; when attempting to simulate
extended systems, finite-size errors also become important. 1
4.1 Finite-size errors
The QMC methods described in Chapter 3 apply to finite systems. To investigate the
properties of materials which have infinite extent in at least one spatial dimension,
some kind of scheme for extrapolation is required.
In practice, such materials are studied by placing a finite set of particles in a
simulation cell, which is then subjected to periodic boundary conditions. This defines
a lattice (see figure 4.1). To complicate things, the particles must also undergo
interactions with their periodically-repeated images; the best way to take account
of this is not obvious. In this report, the particles will be exclusively electrons,
interacting via the Coulomb force, the long-ranged nature of which is a key part of
the problem.
1 The use of pseudopotentials introduces additional errors in the simulation of real atoms, but
these will not be discussed in this work.
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