CASINO manual - Theory of Condensed Matter

Jastrow factor from each cycle of the variance-minimization process. When casino has finished, type
envmc. The output should look something like this (again, I ran the calculation on 12 cores):
ENVMC v0.60: Script to extract VMC energies from CASINO output files.
Usage: envmc [-kei] [-ti] [-fisq] [-pe] [-vee] [-vei] [-vnl] [-nc]
[-vr] [-rel] [-ct] [-nf ] [files]
File: ./out
(energies in au/particle, variances in au)
VMC #1: E = -0.03892(8) ; var = 0.191(5) (correlation.out.0)
VMC #2: E = -0.04640(5) ; var = 0.0601(9) (correlation.out.1)
VMC #3: E = -0.05335(1) ; var = 0.0056(1) (correlation.out.2)
Total CASINO CPU time ::: 189.9371 seconds
We see that optimizing the parameters in this way lowers the variance and total energy significantly.
You should use the output of envmc to choose which correlation.out file you want to use in subsequent
calculations, e.g., for DMC. In general, one should choose the correlation.out file that gives
the lowest variational energy. Therefore, in the example above, correlation.out.2 should be copied
to correlation.data for use in subsequent DMC calculations.
To get an even better wave function with varmin one could try additional things, such as optimizing
the cutoffs, or using more parameters, or having different functional forms between different spin types
to optimize the Jastrow still further. Feel free to try this.
For this example we have only used the RPA form of the electron–electron term u in the Jastrow
factor, along with a three-body W -term. If we were optimizing a wave function for a real system with
atoms then we would include atom-centred electron–nucleus χ terms (one for each type of atom), and
possibly electron–electron–nucleus f terms (again, one for each atom type). There are two additional
types of term, p and q, which are plane-wave extensions of u and χ, respectively, for periodic systems.
The p term is known to improve answers for periodic systems, but the q term is rarely used. A
three-electron homogeneous term h can be used too: see Sec. 7.4.2.
If you are only interested in optimizing linear parameters in casino’s Jastrow factor (i.e., all Jastrow
parameters except cutoffs, RPA parameters and parameters in the three-body terms), then the
‘varmin-linjas’ method should be used, as it is considerably faster (see Sec. 25.2). The input files
should be set up exactly as for an ordinary variance minimization, except that the opt method keyword
should be set to ‘varmin linjas’. Unfortunately you can’t use this on the example above, because
it only contains non-linear terms, but feel free to experiment with some of the other examples.
6.3.2 How to do an energy-minimization calculation
casino also includes a different optimization method, in which the variational energy is minimized.
Full details of the method and its usage are given in Sec. 25.3.
Energy minimization is done in a very similar manner to standard variance minimization, and is
selected by setting opt method to ‘emin’. As before, the process consists of cycles, each comprising a
VMC configuration-generation stage followed by an optimization stage. The number of configurations
which must be generated per cycle (vmc nconfig write) is usually similar to the number required
for variance minimization.
There are some differences in the capabilities and behaviour of energy and variance minimization.
Whereas variance minimization typically achieves all of its improvement to the wave function in one
or two cycles, energy minimization will often require up to ten cycles to converge. The exception to
this is that when optimizing only determinant coefficients, energy minimization should converge in
at most two cycles (see Sec. 25.3 for an explanation of this). Energy-minimization cycles are usually
faster than variance-minimization cycles, so that the overall time to convergence is similar for both
methods. Energy minimization is also more sensitive than variance minimization to the presence of
optimizable parameters which have very little effect on the wave function (it is often best to avoid
including such parameters). More importantly, energy minimization has some difficulty in optimizing
cutoff parameters (in the Jastrow, backflow, or orbital functions). Lastly, if the Jastrow u term is
present and being optimized, starting from zeroed parameters, it is possible for the energy to increase
after the first cycle of energy minimization. Subsequent cycles should lower the energy as usual. This
behaviour is explained in Sec. 25.3.
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