CASINO manual - Theory of Condensed Matter

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START SET 1
Spherical harmonic l,m
0 0
Expansion order N_h
3
Spin dep (0->uu=dd=ud; 1->uu=dd/=ud; 2->uu/=dd/=ud)
-1
Cutoff (a.u.) ; Optimizable (0=NO; 1=YES)
0.d0 1
Parameter values ; Optimizable (0=NO; 1=YES)
END SET 1
END H TERM
where use is made of spin-triplet dependencies as defined in the custom striplet dep input block.
Notes:
• There are 6 types of term available: u (isotropic electron–electron terms), χ (isotropic
electron–nucleus terms), f (isotropic electron–electron–nucleus terms), p (plane-wave expansions
in electron–electron separation), q (plane-wave expansions in electron position) and W /H
(isotropic, homogeneous three-electron term). All of these terms are optional: e.g., omitting all
the lines from ‘START Q TERM’ to ‘END Q TERM’ will give a Jastrow factor with no q terms.
In many cases (particularly in the presence of pseudopotentials) only u and χ terms are needed,
and this is good since higher-order terms such as f-functions can become very expensive in large
systems.
• The ‘truncation order’ should be either 2 or 3. If it is of value C then the Jastrow factor is
C times differentiable everywhere; it must therefore be at least 2 for the kinetic-energy integrand
to be well-defined. If it is 3 then the local energy is continuous in configuration space
(assuming the orbitals are smooth). This is not strictly required, and leads to the loss of some
variational freedom, but it makes the numerical optimization of cutoff lengths easier. In general,
we recommend setting C = 3.
• In a future release, the u and χ terms may be made anisotropic; there are place-holders for this
in the Jastrow data set. At present, however, both must be isotropic. There should only be one
set of u terms, and the spherical harmonic l and m values should therefore be set to 0.
• The ‘expansion order’ of u determines the number of parameters. Typically it is given a value
between 4 and 8.
• The ‘spin-dep params’ line allows the user to specify whether the same u-parameters are to
be used for parallel- and antiparallel-spin electron-pairs. If the value is set to 0 then the same
parameters are used for parallel and antiparallel pairs (this option should not be used in general);
if the value is set to 1 then different parameters are used for parallel and antiparallel pairs; if
it is set to 2 then different parameters are used for up-spin, down-spin and opposite-spin pairs:
this is useful for spin-polarized systems. 12
• The cutoff length is given a default value if it is set to 0. Note that it is possible to optimize
the cutoff length using variance minimization. This can be useful, because the choice of cutoff
length has a significant effect on the optimized energy and variance. Unfortunately, optimizing
the cutoff length is numerically difficult: variance minimization will take many more iterations
to converge if the cutoff is optimizable. Setting the truncation order to 3 can help somewhat.
• It is possible to specify exactly which of the expansion coefficients (‘parameter values’) are
optimizable and which are not. One does not need to specify all of the expansion coefficients:
any that are not listed are assumed to be zero. If all of the parameter values in all of the
Jastrow terms are blank, as is the case in the example given above, then the following default
behaviour will be used: if one is studying a homogeneous system, or a 3D-periodic system
or is using energy minimisation, a simple default for u will be chosen that satisfies the Kato
12 When particles other than electrons are present in the systems, introduced via the particles block input keyword,
these definitions change a little. Information on the particle and particle-pair splittings are given near the beginning of
the out file.
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