CASINO manual - Theory of Condensed Matter
CASINO manual - Theory of Condensed Matter
CASINO manual - Theory of Condensed Matter
Create successful ePaper yourself
Turn your PDF publications into a flip-book with our unique Google optimized e-Paper software.
19.2 Evaluating the nonlocal pseudopotential energy<br />
The action <strong>of</strong> the nonlocal pseudopotential on the wave function can be written as a sum <strong>of</strong> contributions<br />
from each electron and each angular momentum channel. The contribution to the local energy<br />
made by the nonlocal pseudopotential is<br />
V nl = Ψ −1 ˆVnl Ψ<br />
= ∑ i<br />
Ψ −1 ˆV<br />
ps<br />
nl,i Ψ = ∑ i<br />
V nl,i , (103)<br />
where for simplicity we consider the case <strong>of</strong> a single atom placed at the origin. V nl,i may be written<br />
as [15]<br />
V nl,i = ∑ V ps<br />
nl,l (r i) 2l + 1 ∫<br />
P l [cos(θ<br />
4π<br />
i)] ′ Ψ(r 1, . . . , r i−1 , r ′ i , r i+1, . . . , r N )<br />
Ψ(r 1 , . . . , r i−1 , r i , r i+1 , . . . , r N ) dΩ r ′ , (104)<br />
i<br />
l<br />
where P l denotes a Legendre polynomial.<br />
casino currently performs the nonlocal projections for l = 0, 1, 2 only. The integral over the surface <strong>of</strong><br />
the sphere in Eq. (104) is evaluated numerically. The r ′ dependence <strong>of</strong> the many-body wave function<br />
is expected to have predominantly the angular momentum character <strong>of</strong> the orbitals in the Slater<br />
part <strong>of</strong> the wave function. A suitable integration scheme is therefore to use a quadrature rule that<br />
integrates products <strong>of</strong> spherical harmonics exactly up to some maximum value l max . The quadrature<br />
grids currently available are listed in Table 1. To avoid bias the orientation <strong>of</strong> the axes is chosen<br />
randomly each time such an integral is evaluated.<br />
non local grid l max N p<br />
1 0 1<br />
2 2 4<br />
3 3 6<br />
4 5 12<br />
5 5 18<br />
6 7 26<br />
7 11 50<br />
Table 1: Quadrature grids for the nonlocal integration. non local grid (also known as nlrule) is the label<br />
for the rule, l max is the maximum value <strong>of</strong> l which is integrated exactly and N p is the number <strong>of</strong> points in the<br />
grid.<br />
Within a VMC calculation it is <strong>of</strong>ten possible to use a low-order quadrature rule because the error<br />
cancels over the run, but higher accuracy is required for wave-function optimization and DMC calculations,<br />
which are biased by errors in the nonlocal integration. In principle the nonlocal energy<br />
should be summed over all the ionic cores and all electrons in the system. However, since the nonlocal<br />
potential <strong>of</strong> each ion is short ranged, one need only sum over the few atoms nearest to each electron.<br />
Exact sampling <strong>of</strong> the nonlocal energy with DMC is problematic and we use the localization approximation<br />
in which the nonlocal operator acts on the trial wave function in exactly the same way as in<br />
VMC. The error introduced by this approximation is proportional to (Ψ − Ψ 0 ) 2 [33], where Ψ 0 is the<br />
exact wave function.<br />
19.3 The core-polarization potential energy<br />
Core-polarization potentials (CPPs) account for the polarization <strong>of</strong> the pseudo-ion cores by the fields<br />
<strong>of</strong> the other charged particles in the system. The polarization <strong>of</strong> the pseudo-ion cores by the fields <strong>of</strong><br />
the valence electrons is a many-body effect which includes some <strong>of</strong> the core-valence correlation energy<br />
[34]. In the CPP approximation the polarization <strong>of</strong> a particular core is determined by the electric<br />
field at the nucleus. The electric field acting on a given ion core at R I due to the other ion cores at<br />
R J and the electrons at r i is<br />
F I = − ∑ J≠I<br />
Z J<br />
R JI<br />
|R JI | 3 + ∑ i<br />
r iI<br />
|r iI | 3 , (105)<br />
139