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The condition that the f term has no electron–nucleus cusps is
∂f
= 0,
∂riI riI =0
(90)
rij =rjI
which gives
ee
NfeN
I Nf I
XX
m+n
(Cγ0mnI − Lf I γ1mnI )(−Lf I )C−1 rjI
(rjI − Lf I )C = 0,
(91)
m=0 n=0
ee
for all rjI . It is therefore required that, ∀ k 0 ∈ {0, . . . , NfeN
I + Nf I },
X
(Cγ0mnI − Lf I γ1mnI ) = 0.
(92)
m,n : m+n=k0
10.14.3
The p and q terms in the new Jastrow factor
The p term takes the cuspless form
p(rij ) =
X
aA
X
cos(GA · rij ) ,
(93)
G+
A
A
where the {GA } are the reciprocal lattice vectors of the simulation cell belonging to the Ath star of
vectors that are equivalent under the full symmetry group of the Bravais lattice, and “+” means that,
if GA is included in the sum, −GA is excluded.
For systems with inversion symmetry the q term takes the cuspless form
X X
cos(GB · ri ),
q(ri ) =
bB
(94)
G+
B
B
where the {GB } are the reciprocal lattice vectors of the primitive unit cell belonging to the Bth star
of vectors that are equivalent under the space-group symmetry of the crystal, and the “+” means
that, if GB is included in the sum, −GB is excluded.
10.14.4
Old Jastrow factor
The old redundant Jastrow factor jasfun.data used by CASINO before Easter 2004 is described in
this section (as it is still supported) can be written as follows (We denote the distance of electron i
from ion I by riI and the distance between electrons i and j by rij ).
J= −
X
[u0 (rij ) + S1 (rij )] −
i>j,j
−
i
XX
i>j,j
XX
S2 (riI )
I
[S3 (riI , rjI ) + S4 (riI , rij ) + S4 (rjI , rij ) + S5 (riI , rjI , rij )] .
(95)
I
The one-electron term S2 is allowed to depend on the electron spin, while the other terms which
all involve two electrons are allowed to depend on the relative spins. At the moment the terms for
two up-spin electrons are the same as for two down-spin electrons. In three dimensions, the function
u0 is taken to be
!
2
r i
rij
A h
ij
1 − exp −
exp − 2 ,
(96)
u0 (rij ) =
rij
F
L0
√
√
where F is chosen so that the cusp conditions are obeyed, i.e., F↑↑ = 2A and F↑↓ = A. L0 is
chosen so that u0 is effectively zero at the shortest distance to the surface of the Wigner-Seitz cell,
LWS . We normally take L0 ≈ 0.3LWS , and A is either set to the inverse of the plasma energy or
obtained by an optimization procedure. In two-dimensional electron and electron-hole gases, u0 is set
to
!
r 2
rij
A
rij
rij
u0 (rij ) = √
1 − exp −
−
exp − 2 ,
(97)
rij
2F
F
L0
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