CASINO manual - Theory of Condensed Matter

22.3 The RPA form of the u term
An alternative to the u term described above is the common RPA form, which in casino has been
implemented as
u RPA
ij = − A ( [1 − exp − r )]
ij
, (158)
r ij F
in three dimensions, where A and F are variational parameters. In two dimensions, the equivalent
expression is
u RPA
ij = −√ A ( [1 − exp − r √ )]
ij
rij 2F − rij
. (159)
F
The usual polynomial cutoff function is applied to these terms. This alters the cusp conditions, which
are, in three dimensions,
A(F ) = 2L uΓ ij F 2
L u + 2CF , (160)
and in two dimensions,
A(F ) = 3L uΓ ij F 3/2
L u + 3CF , (161)
where Γ ij = 1/4 if i and j are electrons of the same spin, and Γ ij = 1/2 if their spins are different. If i
and j are particles other than electrons, the value of Γ ij will also depend on their masses and charges.
Note that in two-dimensional bilayer systems, where electrons of different spins are on separate layers,
the anti-parallel cusp conditions are lifted, and the antiparallel A is a free parameter.
The RPA u term is rarely used in practice, as the polynomial u term generally gives lower variational
energies.
22.4 The p and q terms in the Jastrow factor
The p term takes the cuspless form
p(r ij ) = ∑ A
∑
a A cos(G A · r ij ) , (162)
G + A
where the {G A } 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 G A is included in the sum, −G A is excluded. The p term is important if the finite-size correction
to the kinetic energy is to be calculated (see Sec. 29).
For systems with inversion symmetry the q term takes the cuspless form
q(r i ) = ∑ ∑
b B cos(G B · r i ), (163)
B
G + B
where the {G B } 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 G B is included in the sum, −G B is excluded. The q term is rarely of use in practice.
22.5 The three-body W term
The W term is given by
W =
N∑
¯δ jk¯δjl¯δkl s lj · s lk , (164)
j,k,l
s jk = w jk r jk , (165)
where w jk = w(r jk ) is a suitably parametrized function of the distance between electrons j and k,
r jk = r j − r k , and the symbol ¯δ jk (no-delta of j, k) is short-hand for 1 − δ jk .
The core function w ij is parametrized in casino as
w ij = w(r ij ) = f C (r ij ; L w )
148
n∑
c l rij l . (166)
l=0