CASINO manual - Theory of Condensed Matter

where R JI = R J − R I and r iI = r i − R I . The CPP energy is then
V CPP = − 1 ∑
α I F I · F I , (106)
2
where α I is the dipole polarizability of core I.
I
Equation (105) assumes a classical description, which is valid when the valence electrons are far from
the core. When a valence electron penetrates the core, the classical result is a very poor approximation,
diverging at the nucleus. To remove this unphysical behaviour each contribution to the electric field
in Eq. (105) is multiplied by a cutoff function f(r iI /¯r I ), which tends to unity at large r iI . A further
possible modification is to allow the one-electron term in Eq. (106), which takes the form −α I /(2r 4 iI ),
to depend on the angular momentum component, l, so that ¯r I in the cutoff function is replaced by
¯r lI . With these modifications the CPP energy operator becomes
V CPP = − 1 2
∑
I
α I
⎡
⎣ ∑ i
1
r 4 iI
∑
f
l
−2 ∑ i
(
riI
∑
J≠I
¯r lI
) 2
ˆPl + ∑ i
∑
j≠i
( )
r iI · R JI riI
riI 3 f
R3 JI
¯r I
( ) ( )
r iI · r jI riI rjI
riI 3 f f
r3 jI
¯r I ¯r I
Z J +
⎛
⎝ ∑ J≠I
R JI
R 3 JI
Z J
⎞
⎠2 ⎤ ⎥ ⎦ , (107)
where ˆP l is the projector onto the lth angular momentum component of the ith electron with respect
to the Ith ion.
We use the cutoff function [35, 34],
I
f (x) =
For efficient evaluation, Eq. (107) is written as
V CPP = − 1 ∑
α I |¯F I | 2 + 1 ∑ ∑
α I
2
2
1
[
∑lmx
f
where
¯F I = − ∑ J≠I
Z J
R JI
|R JI | 3 + ∑ i
I
(
1 − e −x2) 2
. (108)
i
r 4 iI
l=0
( ) 2 ( ) ] 2 riI riI
− f ˆP l , (109)
¯r I ¯r lI
r iI
|r iI | 3 f (
riI
¯r I
)
, (110)
and the maximum angular momentum is lmx = 2. In our approach the cutoff parameter for all
angular momenta l > 2 is ¯r I , which is slightly different from Shirley and Martin [34] who use ¯r 2I .
Equation (109) contains 5 parameters for each ion, α I , ¯r 0I , ¯r 1I , ¯r 2I and ¯r I , whose values are entered at
the end of the xx pp.data file, see Sec. 7.5. Suitable values of the parameters are given in the paper
by Shirley and Martin [34]. If ¯r 0I = ¯r 1I = ¯r 2I = ¯r I , the second term in Eq. (109) is zero and it is not
calculated. The second term in Eq. (109) is short-ranged because f(x) → 1 at large x. This term is
calculated in real space.
The second term in Eq. (109) is added to the pseudopotential and the core radii lcutofftol and
nlcutofftol are determined from the resulting potential. The electric field evaluation is activated by
the presence of core-polarization terms in the pseudopotential files; they are not calculated by default
since they may be expensive, especially when periodic boundary conditions are used.
In periodic boundary conditions the electric fields are evaluated directly from the analytic first derivatives
of the Ewald potential: see Sec. 19.4. Calculations using CPPs may be 5–10% slower than ones
without CPPs in periodic systems.
Note: the evaluation of the first derivatives of the periodic potential in 1D polymers has not yet been
implemented, and thus the core-polarization energy cannot be evaluated in such systems.
19.4 Evaluation of infinite Coulomb sums
19.4.1 3D Ewald interaction
In three dimensionally periodic systems, the periodic potential of a neutralized lattice of point charges
may be evaluated using the Ewald method [36, 37]. Consider the periodic charge density consisting
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