CASINO manual - Theory of Condensed Matter

where n is the order of the expansion, c l are the expansion coefficients and f C (r ij ; L w ) = (r ij − L w ) C
is the usual cutoff function.
23 Backflow transformations
The most widely used form of the trial wave function Ψ T is the Slater-Jastrow (SJ) form
Ψ T = e J Ψ S , (167)
where the Jastrow correlation factor e J is an optimizable function of the particle coordinates, J =
J({r i }), and the Slater part Ψ S is in general a multideterminant expansion,
Ψ S =
∑
N D N S
c k
k=1
∏
D (jk) , (168)
j=1
D (jk) = D (jk) ({r i }) , (169)
where {c k } are the expansion coefficients, and D (jk) is the Slater determinant of the one-electron
orbitals of electrons of spin j in the kth term of such expansion.
Backflow corrections in QMC are capable of introducing further correlations in Ψ T by substituting
the coordinates in the Slater determinants by a set of collective coordinates x i ({r j }), given by
x i = r i + ξ i ({r j }) , (170)
where ξ i is the backflow displacement of particle i, which depends on the configuration of the system
{r j }, and contains optimizable parameters that can be fed into a standard method like variance
minimization.
While the use of a Jastrow factor does not modify the nodal surface of the wave function, backflow
transformations do change it.
23.1 The generalized backflow transformation
The form of the backflow displacement ξ i in homogeneous systems has traditionally been taken as
[46]
N∑
ξ (e−e)
i = η ij r ij , (171)
j≠i
where η ij = η(r ij ) is an appropriate function of interparticle distance. Equation (171) can be regarded
as the most general two-body coordinate transformation for a homogeneous system.
Notice that the above expression implicitly assumes that there is a set of preferred directions in the
system, given by the electron–electron vectors {r ij }. In a system with nuclei a new set of preferred
directions is introduced, the electron–nucleus vectors {r iI }. Following the same idea, one is led to
introduce an electron–nucleus contribution to ξ i , of the form:
N∑
ion
ξ (e−N)
i = µ iI r iI , (172)
where µ iI = µ(r iI ). However, this is a one-electron term; to be consistent with the order of the η term,
it is necessary to introduce an inhomogeneous two-electron term (i.e., an electron–electron–nucleus
term), which would be given by
where Φ jI
i
ξ (e−e−N)
i =
N∑
N∑
ion
j≠i
= Φ(r iI , r jI , r ij ) and Θ jI
i = Θ(r iI , r jI , r ij ).
I
I
(
Φ
jI
i r ij + Θ jI
i r iI)
, (173)
These functions must be cut off at given lengths for efficiency. We use a simple truncation function,
( ) C L − r
f(r; L) =
Θ(L − r) , (174)
L
149