CASINO manual - Theory of Condensed Matter

where u k has the periodicity of the primitive lattice. The function φ ∗ k
is a Bloch function with wave
vector −k. Therefore we can make two real orbitals from φ k and φ ∗ k
as follows:
φ + (r) = 1 √
2
[φ k (r) + φ ∗ k(r)] ,
φ − (r) =
1
√
2i
[φ k (r) − φ ∗ k(r)] . (59)
The orbitals φ + and φ − are orthogonal if φ k and φ ∗ k = φ −k are orthogonal, which is true unless
k − (−k) = G p , i.e., their wave vectors differ by a reciprocal lattice vector of the primitive lattice. In
this case φ + and φ − are linearly dependent and we must use only one of them. Therefore the scheme
is:
Case 1. If k ≠ G p
2
Case 2. If k = G p
2
use φ + and φ − .
use φ + or φ − . (60)
In the second case, if one of φ + or φ − is zero then obviously one must use the other one.
It may happen that we have in our k-point grid the vectors k and −k which are not related by a
reciprocal lattice vector of the primitive lattice. We may wish to occupy only one of the orbitals from
these k points. We can then form a real orbital as cos(θ)φ + + sin(θ)φ − , where θ is a phase angle
between zero and 2π. If both k and −k orbitals are supplied, casino chooses one, which in general
is sufficient to generate all linearly independent orbitals.
Note that, if complex wf is set to T, then the Slater wave function is complex, and casino makes
no attempt to construct real orbitals. See Sec. 28 for information about the use of complex wave
functions under twisted boundary conditions.
16 Cusp corrections for Gaussian orbitals
Gaussian basis sets are unable to describe the cusps in the single-particle orbitals at the nuclei that
would be present in the exact single-particle orbitals, because the Gaussian basis functions have zero
gradient at the nuclei on which they are centred. This leads to divergences in the local energy at the
nuclei, which should be removed. This can either be done using the Jastrow factor (which is generally
a poor method) or by using the cusp correction scheme described here [31].
In this scheme the molecular orbitals are modified so that each of them obeys the cusp condition at
each nucleus. This ensures that the local energy remains finite whenever an electron is in the vicinity
of a nucleus, although it generally has a discontinuity at the nucleus.
16.1 Electron–nucleus cusp corrections
The Kato cusp condition [32] applied to an electron at r i and a nucleus of charge Z at the origin is
( ∂〈Ψ〉
= −Z〈Ψ〉 ri=0 , (61)
∂r i
)r i=0
where 〈Ψ〉 is the spherical average of the many-body wave function about r i = 0. For a determinant of
orbitals to obey the Kato cusp condition at the nuclei it is sufficient for every orbital to obey Eq. (61)
at every nucleus. We need only correct the orbitals which are nonzero at a particular nucleus because
the others already obey Eq. (61). This is sufficient to guarantee that the local energy is finite at the
nucleus provided at least one orbital is nonzero there. In the unlikely case that all of the orbitals
are zero at the nucleus then the probability of an electron being at the nucleus is zero and it is not
important whether Ψ obeys the cusp condition.
An orbital, ψ, expanded in a Gaussian basis set can be written as
ψ = φ + η , (62)
where φ is the part of the orbital arising from the s-type Gaussian functions centred on the nucleus
in question (which, for convenience is at r = 0), and η is the rest of the orbital. The spherical average
of ψ about r = 0 is given by
〈ψ〉 = φ + 〈η〉 . (63)
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