My PhD thesis - Condensed Matter Theory - Imperial College London

CHAPTER 4.
ERRORS IN QMC SIMULATIONS
Here M is the block length. If M is large enough, then successive values of Ōj will
be uncorrelated; this occurs if M is greater than the correlation length of the data.
The new error estimate is given by equation (4.25) with O i replaced by Ōi and N
replaced by the number of blocks.
If M is not sufficiently large then the blocking method gives an error estimate
which is too small. Thus a way of determining the correct block length is to increase
M until the calculated error reaches a plateau; this is then the best error estimate.
This technique may not be possible if the correlation length is very long or there are
insufficient sample points.
In both VMC and DMC, the correlation length depends on the time step. If
the time step is too small, then it takes many steps before a configuration changes
significantly; however, if it is too large, then too many steps are rejected, and the
time required to generate the next uncorrelated configuration is also large.
4.4 Surface calculations
There are several additional challenges which must must be met when dealing with
surfaces, which will be detailed in the rest of this chapter.
4.4.1 System geometry
In order to study surfaces, it is usual to simulate slab systems (with two surfaces);
using only a single surface creates problems with the boundary conditions.
The aim is to model a slab with infinite extent in two dimensions, but the simulation
cell must be finite; it is therefore normal to apply periodic boundary conditions
to the cell. However, rather than applying these conditions in two dimensions, most
surface calculations use fully three-dimensional periodicity. In addition, it is conventional
to use the 3D version of the Ewald interaction; thus the system actually
being simulated is a stack of slabs, as shown in figure 4.3.
In density-functional and other mean-field calculations, this is not important.
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