CASINO manual - Theory of Condensed Matter

• Is emin min energy too high? This may be the case if casino rejects what appear to be
perfectly good energies.
• Is energy minimization being used to optimize cutoff parameters? It is recommended to fix them
at some point to allow the rest of the parameters to fully converge.
26 Alternative sampling strategies
26.1 Summary
In VMC we directly use Monte Carlo sampling to estimate expectation values defined as the quotient
of two integrals over 3N dimensional space, such as the total energy expectation value,
E tot = 〈Ψ|Ĥ|Ψ〉/〈Ψ|Ψ〉. Estimation of integrals using Monte Carlo methods can generally be performed
using samples drawn from any of a wide range of distributions. The usual VMC choice of
P (R) = λ|Ψ 2 (R)| (with λ an unknown and un-required normalization prefactor) is an analytically
convenient choice, but is not in any sense optimal [67]. This leaves open the possibility of choosing a
different sampling distribution, such as one that is optimal in the sense of providing the minimum statistical
error for a given number of samples, or one that is maximally efficient in the sense of providing
a given accuracy for less computational cost than other choices.
Unfortunately there are some strong limitations on the distribution functions that we can use. Firstly,
we require the ‘estimate’ to be an estimate, that is, to converge to the true value with increasing sample
size due to the law of large numbers (LLN) being valid. Secondly, we usually require meaningful error
bars to be available for a finite sample size estimate, that is that the central limit theorem (CLT)
is valid in some form. The validity of either of these conditions must be arrived at analytically, and
if they are not true then the conventional formulae for the sample mean and standard error provide
numerical results that are unrelated to the estimate and error. Examples of when this occurs are
provided by force estimates [68] and parameter optimization [69].
In what follows, estimates constructed using an arbitrary distribution are described. Following this,
three alternative choices of sampling distribution are summarized. The usual VMC choice is referred
to as ‘standard sampling’, whereas the alternative methods are referred to as ‘alternative sampling’.
26.2 Alternative sampling
For estimating the Hamiltonian expectation value using an arbitrary sampling distribution, P (R), the
estimate is composed of the quotient of two sample means
∑ w(R)EL (R)
Ē tot = ∑ , (230)
w(R)
where w(R) = |Ψ 2 |/P (R). In the limit of large sample size, this is a value drawn from a normal
distribution with mean E tot and variance
σE 2 = 1 ∫
|Ψ 2 (R)|/w(R)d 3 R ∫ w(R)|Ψ 2 (R)| 2 (E L (R) − E tot ) 2 d 3 R
[∫
N
|Ψ2 (R)| 2 d 3 R ] 2
, (231)
whose estimate is
σ¯
E 2 = N
∑ ( w(R)
2
E L (R) − Ētot) 2
N − 1 [ ∑ w(R)] 2 (232)
as long as the bivariate CLT is valid. For standard sampling as described below, the random error is
normal for energy estimates, but is not normal for most other estimates such as the energy surface
implicit in optimization. For the three alternative sampling strategies, the random error is normal for
most estimates, including the energy surface implicit in optimization.
Note that the estimates given above are not the sample mean and standard error of any set of independent
identically distributed random variables. In the input file this form of alternative sampling
is activated by setting the vmc sampling keyword to one of the values described below.
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