CASINO manual - Theory of Condensed Matter

25.3.5 Matrix manipulation (level-shifting)
We can rewrite Eq. (210) as a simple eigenproblem:
(S + S T ) −1 (H + H T )a = Ea. (227)
If the (S + S T ) −1 (H + H T ) matrix had the form
⎛
⎞
λ 0 0 0 0
0 λ 1 0 0
⎜
⎝
.
0 0 ..
⎟
0 ⎠ , λ 0 < λ i ∀ i = 1 . . . p, (228)
0 0 0 λ p
then there would be no change at all in the variable parameters α. It is possible to bring any
(S + S T ) −1 (H + H T ) matrix qualitatively closer to this ‘no-change’ condition using ‘level-shifting’
[65, 66], in which a positive real constant L is added to all the diagonal elements of the matrix except
the first:
⎛
⎞ ⎛
⎞
λ 0 • • • λ 0 • • •
• λ 1 • •
⎜
⎝
.
• • ..
⎟
• ⎠ → • λ 1 + L • •
⎜
⎝
.
• • ..
⎟
• ⎠ . (229)
• • • λ p • • • λ p + L
The approximation in Eq. (208) fails when the parameter changes are too large. Level-shifting can
be used to reduce the size of the parameter changes, stabilizing the optimization. In casino, the
best value for the constant L is determined automatically by performing a line minimization of the
VMC energy estimate with respect to L. (Where possible, correlated sampling is used, both to reduce
computational effort and to increase the statistical precision of energy differences.) As noted by
Umrigar et al. [60, 61], level-shifting can equally well be applied to the H + H T matrix.
25.3.6 Using energy minimization
Energy minimization is used very similarly to standard variance minimization. After selecting it by
setting opt method to ‘emin’, the user must choose the number of optimization cycles (opt cycles),
and the number of optimization configurations per cycle (vmc nconfig write). As for variance
minimization, vmc decorr period should be chosen so as to generate approximately uncorrelated
configurations, and the number of configurations required is usually similar to the number required
for variance minimization.
The keyword emin min energy is unique to energy minimization. During level-shifting (described
above), some of the candidate wave functions may be so poor that they cause numerical errors resulting
in an energy estimate which appears to be very low, but is actually spurious. To protect against
this, any energy estimate which is lower than emin min energy is rejected as invalid. During
diagonalization, eigenvalues are also thrown away if they are below emin min energy. The user
may set emin min energy to an energy below the ground-state energy if it is know, else casino will
calculate an estimate at each cycle based on the energy and variance of the initial set of local energies,
which works well in practice.
Two other keywords affect the stabilization procedures (semi-orthogonalization and level-shifting) but
they are mainly intended for the use of developers and should rarely, if ever, be changed by users.
emin xi value controls the value of ξ in Eq. (223), but the default value of 0.5 performs well in all
tests.
As mentioned in Sec. 25.3.3, parameters with respect to which the energy is a shallow function may not
allow energy minimization to converge accurately to the best of its precision. Therefore casino fixes
parameters internally flagged as ‘shallow’ for the second half of the optimization cycles if using energy
minimization; the precise behaviour can be controlled using the input keyword opt noctf cycles.
Finally, a brief list of possibilities to investigate if an energy minimization run appears to diverge or
otherwise behave badly (in no particular order):
• Are enough configurations being used? (Try increasing vmc nconfig write.)
• Is the reported VMC correlation time significantly greater than 1? (If so, increase
vmc decorr period.)
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