CASINO manual - Theory of Condensed Matter

19.2 Evaluating the nonlocal pseudopotential energy
The action of the nonlocal pseudopotential on the wave function can be written as a sum of contributions
from each electron and each angular momentum channel. The contribution to the local energy
made by the nonlocal pseudopotential is
V nl = Ψ −1 ˆVnl Ψ
= ∑ i
Ψ −1 ˆV
ps
nl,i Ψ = ∑ i
V nl,i , (103)
where for simplicity we consider the case of a single atom placed at the origin. V nl,i may be written
as [15]
V nl,i = ∑ V ps
nl,l (r i) 2l + 1 ∫
P l [cos(θ
4π
i)] ′ Ψ(r 1, . . . , r i−1 , r ′ i , r i+1, . . . , r N )
Ψ(r 1 , . . . , r i−1 , r i , r i+1 , . . . , r N ) dΩ r ′ , (104)
i
l
where P l denotes a Legendre polynomial.
casino currently performs the nonlocal projections for l = 0, 1, 2 only. The integral over the surface of
the sphere in Eq. (104) is evaluated numerically. The r ′ dependence of the many-body wave function
is expected to have predominantly the angular momentum character of the orbitals in the Slater
part of the wave function. A suitable integration scheme is therefore to use a quadrature rule that
integrates products of spherical harmonics exactly up to some maximum value l max . The quadrature
grids currently available are listed in Table 1. To avoid bias the orientation of the axes is chosen
randomly each time such an integral is evaluated.
non local grid l max N p
1 0 1
2 2 4
3 3 6
4 5 12
5 5 18
6 7 26
7 11 50
Table 1: Quadrature grids for the nonlocal integration. non local grid (also known as nlrule) is the label
for the rule, l max is the maximum value of l which is integrated exactly and N p is the number of points in the
grid.
Within a VMC calculation it is often possible to use a low-order quadrature rule because the error
cancels over the run, but higher accuracy is required for wave-function optimization and DMC calculations,
which are biased by errors in the nonlocal integration. In principle the nonlocal energy
should be summed over all the ionic cores and all electrons in the system. However, since the nonlocal
potential of each ion is short ranged, one need only sum over the few atoms nearest to each electron.
Exact sampling of the nonlocal energy with DMC is problematic and we use the localization approximation
in which the nonlocal operator acts on the trial wave function in exactly the same way as in
VMC. The error introduced by this approximation is proportional to (Ψ − Ψ 0 ) 2 [33], where Ψ 0 is the
exact wave function.
19.3 The core-polarization potential energy
Core-polarization potentials (CPPs) account for the polarization of the pseudo-ion cores by the fields
of the other charged particles in the system. The polarization of the pseudo-ion cores by the fields of
the valence electrons is a many-body effect which includes some of the core-valence correlation energy
[34]. In the CPP approximation the polarization of a particular core is determined by the electric
field at the nucleus. The electric field acting on a given ion core at R I due to the other ion cores at
R J and the electrons at r i is
F I = − ∑ J≠I
Z J
R JI
|R JI | 3 + ∑ i
r iI
|r iI | 3 , (105)
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