CASINO manual - Theory of Condensed Matter

It may be activated by setting vmc sampling to ‘HF optimum’, together with values for E 0 and ɛ
via the keywords vmc optimum e0 and vmc optimum ew, as above.
26.2.4 Efficient sampling
We may reduce computational expense further at the cost of an increase in fixed-sample-size error,
and still obtain a net increase in performance when compared to standard sampling. This is achieved
by avoiding the evaluation of the Jastrow factor, backflow function, multideterminant expansion and
local energy that appears in the previous distributions and using the form
[ ( ) 2 ) ] 2
P (R) = λ D ↑ 1 (R)D↓ 1 (R) +
(D ↑ 2 (R)D↓ 2 (R) . (235)
This is an arbitrary choice that is not the mod-square of any fermionic trial wave function, is not
an approximation to any physical quantity, and does not possess a nodal surface. This choice is not
unique, but has been specifically chosen such that the LLN and CLT are valid and for the increase
in fixed-sample-size error to be manageable (note that the validity of the CLT relies on the presence
of two terms in this expansion, and dropping the second term results in an invalid estimate.) As
implemented we do not sum the mod-square of the first two determinants, but the mod-square of the
first two configuration state functions (CSFs), in order to avoid naturally the coincidence of nodal
surfaces for the two terms through symmetry.
For such a choice we achieve a net gain in computational efficiency even though the fixed-sample-size
error is greater, since more samples are accessible for a given computational budget. For example,
to achieve a given accuracy in all-electron atomic carbon calculations, standard sampling takes ×25
longer than efficient sampling [69].
Efficient sampling can be activated by setting vmc sampling to ‘efficient’. The first two CSFs are
included in the sum as above.
26.2.5 Optimization
Optimization proceeds as for standard sampling, with the same control keywords and with variance
and energy minimization defined using estimates of the same integral expressions using the different
sampling distributions provided above. The only significant change arises for variance minimization.
It is often not clear whether to interpret the optimized quantity in terms of a Monte Carlo estimate
or as a least-squares fitting with random sample points, hence many interpretations arise that are
equivalent for standard sampling but differ for general sampling. In addition, many of the variations
of variance minimization available are specifically designed to prevent the failure of the CLT during
optimization, something that does not occur for any of the alternative sampling methods presented
here. As it stands the code minimizes only one quantity in variance minimization: the Monte Carlo
estimate of the actual error in the estimated energy, σ2
¯
E
. Note that this has not been tested, and a
wide range of other options may perform better, such as minimizing an estimate of the optimum or
standard error.
27 Use of localized orbitals and bases in CASINO
27.1 Theoretical background
27.1.1 Introduction
The rate-determining step in practical QMC calculations is the evaluation of the orbitals in the Slater
part of the trial wave function after each electron has moved 27 . We explain how the amount of CPU
time spent evaluating the orbitals after each electron move can be made essentially independent of
system size.
27 The time spent updating the cofactor matrices will in principle dominate in the limit of large system size, but this
limit is not usually reached in practice. Note that, throughout this section, we assume that electron-by-electron QMC
algorithms are used.
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