CASINO manual - Theory of Condensed Matter
CASINO manual - Theory of Condensed Matter
CASINO manual - Theory of Condensed Matter
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where u k has the periodicity <strong>of</strong> the primitive lattice. The function φ ∗ k<br />
is a Bloch function with wave<br />
vector −k. Therefore we can make two real orbitals from φ k and φ ∗ k<br />
as follows:<br />
φ + (r) = 1 √<br />
2<br />
[φ k (r) + φ ∗ k(r)] ,<br />
φ − (r) =<br />
1<br />
√<br />
2i<br />
[φ k (r) − φ ∗ k(r)] . (59)<br />
The orbitals φ + and φ − are orthogonal if φ k and φ ∗ k = φ −k are orthogonal, which is true unless<br />
k − (−k) = G p , i.e., their wave vectors differ by a reciprocal lattice vector <strong>of</strong> the primitive lattice. In<br />
this case φ + and φ − are linearly dependent and we must use only one <strong>of</strong> them. Therefore the scheme<br />
is:<br />
Case 1. If k ≠ G p<br />
2<br />
Case 2. If k = G p<br />
2<br />
use φ + and φ − .<br />
use φ + or φ − . (60)<br />
In the second case, if one <strong>of</strong> φ + or φ − is zero then obviously one must use the other one.<br />
It may happen that we have in our k-point grid the vectors k and −k which are not related by a<br />
reciprocal lattice vector <strong>of</strong> the primitive lattice. We may wish to occupy only one <strong>of</strong> the orbitals from<br />
these k points. We can then form a real orbital as cos(θ)φ + + sin(θ)φ − , where θ is a phase angle<br />
between zero and 2π. If both k and −k orbitals are supplied, casino chooses one, which in general<br />
is sufficient to generate all linearly independent orbitals.<br />
Note that, if complex wf is set to T, then the Slater wave function is complex, and casino makes<br />
no attempt to construct real orbitals. See Sec. 28 for information about the use <strong>of</strong> complex wave<br />
functions under twisted boundary conditions.<br />
16 Cusp corrections for Gaussian orbitals<br />
Gaussian basis sets are unable to describe the cusps in the single-particle orbitals at the nuclei that<br />
would be present in the exact single-particle orbitals, because the Gaussian basis functions have zero<br />
gradient at the nuclei on which they are centred. This leads to divergences in the local energy at the<br />
nuclei, which should be removed. This can either be done using the Jastrow factor (which is generally<br />
a poor method) or by using the cusp correction scheme described here [31].<br />
In this scheme the molecular orbitals are modified so that each <strong>of</strong> them obeys the cusp condition at<br />
each nucleus. This ensures that the local energy remains finite whenever an electron is in the vicinity<br />
<strong>of</strong> a nucleus, although it generally has a discontinuity at the nucleus.<br />
16.1 Electron–nucleus cusp corrections<br />
The Kato cusp condition [32] applied to an electron at r i and a nucleus <strong>of</strong> charge Z at the origin is<br />
( ∂〈Ψ〉<br />
= −Z〈Ψ〉 ri=0 , (61)<br />
∂r i<br />
)r i=0<br />
where 〈Ψ〉 is the spherical average <strong>of</strong> the many-body wave function about r i = 0. For a determinant <strong>of</strong><br />
orbitals to obey the Kato cusp condition at the nuclei it is sufficient for every orbital to obey Eq. (61)<br />
at every nucleus. We need only correct the orbitals which are nonzero at a particular nucleus because<br />
the others already obey Eq. (61). This is sufficient to guarantee that the local energy is finite at the<br />
nucleus provided at least one orbital is nonzero there. In the unlikely case that all <strong>of</strong> the orbitals<br />
are zero at the nucleus then the probability <strong>of</strong> an electron being at the nucleus is zero and it is not<br />
important whether Ψ obeys the cusp condition.<br />
An orbital, ψ, expanded in a Gaussian basis set can be written as<br />
ψ = φ + η , (62)<br />
where φ is the part <strong>of</strong> the orbital arising from the s-type Gaussian functions centred on the nucleus<br />
in question (which, for convenience is at r = 0), and η is the rest <strong>of</strong> the orbital. The spherical average<br />
<strong>of</strong> ψ about r = 0 is given by<br />
〈ψ〉 = φ + 〈η〉 . (63)<br />
133