CASINO manual - Theory of Condensed Matter

where L is a cutoff length, C is the truncation order and Θ denotes the Heaviside function.
Notice that the above can be easily extended to inhomogeneities other than atoms by introducing any
other coordinates appearing in the potential energy into the expressions. This is not implemented in
casino yet.
23.1.1 The η function
Three forms of the homogeneous backflow function have been tested with casino. Following extensive
tests, no advantages were found from using the rational and Gaussian forms, and only the polynomial
form remains in the code,
N
∑ η
η ij = f(r ij ; L η ) c k rij k , (175)
where N η is the expansion order and {c l } are the optimizable coefficients. We expect this simple form
to be flexible and efficient. The natural polynomial basis is used to improve the speed of evaluation,
and rounding errors are not likely to appear for N η + C < 20 24 .
k=0
23.1.2 The µ function
We use the following polynomial expansion for µ,
N
∑ µ,I
µ iI = f(r iI ; L µ,I )
k=0
d k,I r k iI , (176)
where L µ,I is the cutoff length for ion I, N µ,I is the expansion order and {d k,I } are the optimizable
coefficients.
23.1.3 The Φ and Θ functions
Our choice for Φ jI
i
and Θ jI
i
are the following power expansions,
N
∑ eN,I
Φ jI
i = f(r iI ; L Φ,I )f(r jI ; L Φ,I )
k=0
N
∑ eN,I
Θ jI
i = f(r iI ; L Φ,I )f(r jI ; L Φ,I )
k=0
N
∑ eN,I
l=0
N
∑ eN,I
l=0
N
∑ ee,I
m=0
N
∑ ee,I
m=0
ϕ klm,I r k iIr l jIr m ij , (177)
θ klm,I r k iIr l jIr m ij , (178)
where N eN,I and N ee,I are the expansion orders, L Φ,I are the truncation lengths and ϕ klm,I and θ klm,I
are optimizable coefficients.
23.2 Constraints on the backflow parameters
In SJ wave functions it is common practice to impose the electron–electron cusp conditions on the
parameters in the Jastrow factor and the electron–nucleus ones on the orbitals in the Slater determinant.
Backflow can modify the cusp conditions; we have constrained the backflow parameters so that
they do not.
When AE atoms are present, it can be shown that the electron–nucleus cusp conditions cannot be
fulfilled unless there is no homogeneous backflow term present. However, this issue can be bypassed
by smoothly truncating η(r ij ) around such nuclei. This is automatically done by casino.
An additional set of constraints can be added in order to satisfy the relation
ξ i ({r j }) = ∇ i Y ({r j }) , (179)
where Y ({r j }) is an object called backflow potential, which appears in the derivation of backflow of
Ref. [47]. This equation is already satisfied by both the electron–electron and the electron–nucleus
24 It is estimated that numerical problems arise in the evaluation of polynomials beyond order ∼ 20 when using
double-precision arithmetics. More complicated polynomial forms (e.g., Chebyshev polynomials) should be used to be
able to exceed this limit.
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