CASINO manual - Theory of Condensed Matter

apidly (typically thousands of times per second); furthermore the CPU time required is independent
of the system size; and (ii) the LSF along any line in parameter space is a simple quartic polynomial,
so that the exact, global minimum of the LSF along that line can be computed analytically.
The method has two drawbacks: (i) only linear Jastrow parameters can be optimized in this fashion;
and (ii) the number of quartic coefficients to be evaluated and stored in memory grows as the fourth
power of the number of parameters to be optimized.
Detailed information about the varmin-linjas method can be found in Ref. [53].
25.2.2 Using the varmin-linjas method
A casino variance-minimization calculation using the varmin-linjas method is carried out in exactly
the same way as an ordinary variance-minimization calculation except that:
1. The opt method keyword in input should be set to varmin linjas.
2. If desired, the user may change the method used to minimize the LSF with respect to the
set of parameters by using the vm linjas method keyword. This can take the values: ‘CG’
(conjugate gradients); ‘SD’ (steepest descents); ‘GN’ (Gauss-Newton); ‘MC’ (Monte Carlo line
minimization); ‘LM’ (simple line minimization); ‘CG MC’ (alternate conjugate gradients and
Monte Carlo line minimization); ‘BFGS’ (Broyden-Fletcher-Goldfarb-Shanno); ‘BFGS MC’ (alternate
BFGS and Monte Carlo line minimization); or ‘GN MC’ (alternate Gauss-Newton and
Monte Carlo line minimization). If the vm linjas method keyword is omitted then the BFGS
method will be used by default. If you experience difficulty optimizing a large set of parameters
then the Gauss-Newton method is worth trying. The BFGS method seems to be the most
efficient method in general, however.
3. If desired, the user can change both the maximum number of iterations and the number of
line minimizations to be performed by means of the vm linjas its keyword. If this keyword is
omitted, or it is given a negative value, then a default number of iterations will be performed.
The cutoff lengths in the Jastrow factor are important variational parameters, and some attempt
to optimize them should always be made. It is recommended that a (relatively cheap) calculation
using the standard variance-minimization method should be carried out in order to optimize the
cutoff lengths, followed by an accurate optimization of the linear parameters using the varmin-linjas
method. For some systems, good values of the cutoff lengths can be supplied immediately (for example,
in periodic systems at high density with small simulation cells, the cutoff length L u should be set equal
to the radius of the sphere inscribed in the WS cell of the simulation cell), and one can make use of
the varmin-linjas method straight away.
25.3 Energy minimization
25.3.1 Motivation
Although it was originally motivated by difficulties in optimizing the variational energy, variance minimization
has proven capable of reliably producing very high quality trial wave functions. Nevertheless,
there are several reasons why optimizing the energy may still be desirable:
• Since trial wave functions generally cannot exactly represent an eigenstate, the energy and
variance minima do not coincide. Energy minimization should therefore produce lower VMC
energies. This might in turn yield lower DMC energies, although it is not clear how improvements
in the energy (or variance) relate to improvements in the nodal surface. (In practice, lower VMC
energies usually lead to lower DMC energies.)
• Energy-optimized wave functions have been shown to give better estimates of other expectation
values [55, 56, 57].
• It is known that the variance of the DMC wave function is proportional to the difference between
the ground-state energy and the VMC energy [58, 59].
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