CASINO manual - Theory of Condensed Matter

energy. In the limit of perfect sampling, the reweighted variance is equal to the actual variance, and
is therefore independent of the configuration distribution, so that the optimized parameters would
not change over successive cycles of reweighted variance minimization. This is not the case for unreweighted
variance minimization; nevertheless, by carrying out a number of cycles, a ‘self-consistent’
parameter set may be obtained.
In casino the minimization of the variance is carried out by the routine nl2sno in module nl2sol,
which performs an unconstrained minimization (without requiring derivatives) of a sum of m squares
of functions which contain n variables, where m ≥ n. (Information on the minimization routine can
be found in Ref. [52]).
Before carrying out this process, the user must decide how they wish to parametrize the trial wave
function, and with how many parameters. They must also decide on the number of configurations to
be used in the optimization. These choices are system specific, and depend on the level of accuracy
to which the user wishes to work.
A systematic approach to deciding on an appropriate number of variational parameters is to start by
optimizing a few parameters, then to add more and re-optimize, and so on, until the decrease in energy
resulting from the inclusion of additional parameters is small compared with the energy difference that
the user wishes to resolve. Note that the error bars on the VMC energy must be smaller still, so that
the user can make accurate judgements about the energy differences.
It is clearly desirable for the VMC-generated configurations to be completely uncorrelated. This can
be achieved by giving vmc decorr period a large value (e.g., 10). Reblocking VMC energies in
a preliminary VMC run will allow the user to determine the correlation period for VMC energies,
which in turn suggests a suitable value for vmc decorr period. It is also essential that the VMC
configuration-generation run is fully equilibrated. Since VMC equilibration is usually computationally
inexpensive, this should be straightforward enough. The utility plot hist can be used to verify that
the VMC energies have equilibrated.
The user may choose whether to optimize the Jastrow factor, determinant-expansion coefficients,
or the pairing parameter and orbital coefficients, by setting the opt jastrow, opt detcoeff and
opt orbitals flags as appropriate. For most applications, it is only necessary to optimize the Jastrow
factor. If only linear parameters in casino’s Jastrow factor are to be optimized then the ‘varmin-linjas’
method should be used: see Sec. 25.2.
The user may choose between the reweighted or unreweighted variance-minimization algorithms by
choosing the vm reweight keyword to be T or F, respectively. As shown in Ref. [53], the unreweighted
variance-minimization has several desirable properties, which often make it a more useful technique
than reweighted variance minimization: (i) the unreweighted algorithm is numerically more stable; (ii)
in general the unreweighted variance has a simple functional form and only a single minimum in the
space of linear Jastrow parameters; (iii) for a large number of model systems it can be demonstrated
that the wave functions generated by unreweighted variance minimization iterated to self-consistency
have a lower variational energy than wave functions optimized by reweighted variance minimization.
If reweighted variance minimization is performed then it is possible to limit the values that the weights
can take, in an attempt to improve the stability. The vm w max and vm w min parameters can
be used to specify the maximum and minimum values that the weights can take.
When optimizing parameters that affect the nodal surface, the local energies of configurations may
diverge as the nodal surface is moved. The affected configurations will then have a disproportionate
effect on the value of the unreweighted variance. It is therefore desirable to remove configurations
from the optimization procedure when their local energies deviate substantially from the mean local
energy. This can be achieved by introducing an effective weight for each configuration, which is a
function of the deviation of the local energy from the mean local energy,
⎧
⎪⎨
1
∣ EL − Ē∣ ∣
[
/σEL < T
( ) ] 2
f(E L ) =
|EL
⎪⎩ exp −
−Ē|/σ E L −T ∣∣EL
W
− Ē∣ ∣ , (201)
/σEL > T
where T is a user-defined threshold, W is a user-defined filter width, E L is the local energy of a
configuration, Ē is the average energy, and σ EL is the square root of the unreweighted variance. To
activate the configuration-filtering scheme, turn vm filter to T in input; the T parameter corresponds
to the keyword vm filter thres and W corresponds to vm filter width. The default values of T = 4
and W = 2 are found to work well in most cases.
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