CASINO manual - Theory of Condensed Matter

Population-control catastrophes should not occur under normal circumstances. The following have
been found to lead to catastrophic behaviour: (i) the use of a trial wave function that does not satisfy
the electron–nucleus cusp conditions, e.g., when a non-cusp-corrected Gaussian basis is used; (ii) the
use of certain nonlocal pseudopotentials in the absence of the T -move scheme [20] (set use tmove
to T in order to use the T -move scheme); (iii) the severe truncation of localized orbitals, especially
when a smooth truncation scheme is used; (iv) the use of an inadequate basis set to represent the
orbitals; (v) pathological trial wave functions, resulting from optimizations that have gone awry (use
jastrow plot to examine the behaviour of u(r ij ): it should increase monotonically to 0). All of these
circumstances should be avoided if possible.
Note finally that, in large systems, the likelihood of population explosions may be reduced if you
use (unreweighted) variance minimization rather than energy minimization to optimize the wave
function. The reason for this is related to the fact that energy minimisation doesn’t care much about
what happens near nodes, since those regions do not contribute much to the energy expectation
value. However, the divergent local energies there make a big difference to the stability of the DMC
algorithm. If you’re using backflow then you have to use energy minimisation, but then you’re unlikely
to be looking at a big system.
13.10 Reducing the computational expense of DMC equilibration
It is always necessary to discard the results of propagating the configuration population over the first
few correlation periods (the equilibration phase) of a DMC simulation. Suppose the target population
is large. Because the number of statistics-accumulation steps required to achieve a given error bar
is relatively small in this case, equilibration can be a large fraction of the total computational effort.
Indeed, for some calculations, DMC equilibration accounts for as much as half of the total CPU time.
By contrast, if one uses a small population and runs for a large number of correlation periods until
one has generated the required amount of data, equilibration is a small fraction of the total run time.
Unfortunately, when running on a large number of cores, the configuration population is necessarily
large, since each core must have at least one configuration on average.
Here we suggest an alternative approach for equilibrating a DMC calculation with a large target
population P . Instead of using VMC to generate P initial configurations, we suggest that VMC be used
to generate a small number of configurations q, which are then used in a preliminary DMC calculation
to generate P configurations. In this preliminary DMC calculation with q configurations, equilibration
is followed by (P/q)T corr time steps of statistics accumulation, where T corr is the correlation period
in terms of time steps, during which a total of P configurations are written out. Following this, a P -
configuration DMC statistics-accumulation calculation is performed. Note that q must be sufficiently
large that population-control bias is negligible in the preliminary DMC calculation.
For example, if one wanted to use a population of 10,000 configurations in a DMC calculation, one could
carry out a preliminary DMC calculation with 1,000 configurations, until 10,000 independent configurations
were generated, then use these as the initial population for the main 10,000-configuration
run (which would not need to be equilibrated). The computational effort of equilibration would be
reduced nearly tenfold.
If the data generated in the preliminary DMC calculation were discarded altogether, this approach
would still lead to a substantial reduction in the computational effort, because the DMC equilibration
would be performed with a small population. However, one can further reduce the waste of computational
effort: the data generated in the statistics accumulation phase of the preliminary DMC
calculation can be averaged with the data generated in the main DMC calculation. Since some of
the configurations generated in the preliminary calculation are reused in the initial population of the
main DMC calculation, some care must be taken to ensure that the error bar is not underestimated.
The most effective way of combining the data from the preliminary and main calculations is still being
investigated, and it is left up to the user to combine the data ‘by hand’ for the time being.
The drawbacks of the proposed method are (i) having an extra stage to the calculation; (ii) the fact
that the preliminary DMC calculation would probably have to be run on a relatively small number of
processors compared with the main calculation; and (iii) the fact that population-control bias might
be a problem if the population q in the preliminary calculation is very small.
In order to perform a preliminary DMC calculation (i.e., to write out configurations during DMC
statistics accumulation), the dmc nconf prelim keyword should be set to the total number of configurations
to be written out (summed over all processors). The configurations are written out once
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