CASINO manual - Theory of Condensed Matter

estricts the flexibility of the algorithm. In practice we define small regions around each node where
the effective one-electron local energies are not taken into account during the minimization, and from
which the cusp correction radius is excluded.
The Gaussian cusp correction is activated through the input keyword cusp correction—this has an
effect only if the system contains at least one all-electron atom. In periodic systems it has proved
difficult to implement this scheme efficiently, and while it works perfectly well the performance of
the code is significantly affected. Check the timings. More information about the cusp correction of
each orbital at each nucleus can be produced with the keyword cusp info which can be useful in fine
tuning. Clearly this can produce a lot of output, so beware.
17 General-purpose cusp corrections
A scheme for modifying real, Gaussian orbitals so that they satisfy the Kato cusp conditions is
described in Sec. 16 and Ref. [31]. An extension of this scheme can be used to enforce the Kato cusp
conditions on complex orbitals expanded in any smooth basis set. Both cusp-correction schemes are
present in casino. We refer to the former as the Gaussian cusp-correction scheme and the latter as
the general-purpose (GP) cusp-correction scheme.
For simplicity, we consider only the case of a single nucleus of charge Z at the origin in the following
discussion.
In the Gaussian cusp-correction scheme, the s-type basis functions are replaced by radial functions
in the vicinity of the bare nucleus. These functions satisfy the cusp conditions and make the singleparticle
local energy resemble an ‘ideal’ curve [31]. In the GP scheme, instead of replacing part of the
orbital, we add a spherically symmetric function of constant phase to the orbital. The function added
to uncorrected orbital ψ(r) is
∆ψ(r) = exp(iθ 0 )∆φ(r) = exp(iθ 0 ) [C + exp [p(r)] − φ(r)] Θ(r c − r), (78)
where Θ is the Heaviside function, C is a real constant, r c is a cutoff length, θ 0 = arg[ψ(0)],
( ∫ )
exp(−iθ0 )
φ(r) = Re
ψ(r) dΩ , (79)
4π
and
p(r) = α 0 + α 1 r + α 2 r 2 + α 3 r 3 + α 4 r 4 , (80)
where the {α} are real constants to be determined. In practice φ(r) is calculated by cubic spline
interpolation: the spherical averaging of the uncorrected orbital is performed at the outset on a grid
of radial points. C is chosen so that φ(r) − C is positive everywhere within the Bohr radius of the
nucleus.
The uncorrected orbital may be written as
ψ(r) = exp(iθ 0 )φ(r) + η(r), (81)
where η(r) consists of the l > 0 spherical harmonic components of ψ(r), together with the phasedependence
of the l = 0 component. Note that exp(iθ 0 )φ(0) = ψ(0), and hence η(0) = 0. Let
These are real, spherically symmetric functions.
˜φ(r) = φ(r) + ∆φ(r). (82)
We may now apply the scheme of Ma et al. to determine {α} and r c , with exp(iθ 0 )φ and exp(iθ 0 ) ˜φ
playing the roles of the uncorrected and corrected s-type Gaussian functions centred on the nucleus
in question [31]. The constant phase exp(iθ 0 ) cancels out of the equations determining the {α} (Eqs.
(9)–(13) in Ref. [31]), so the determination of the {α} and r c is exactly as described in Ma et al.,
except that we do not need to modify Z at the nuclei when more than one atom is present, because
η(0) = 0.
Each orbital is corrected at each all-electron ion in the simulation cell. If twisted boundary conditions
are used then the phase of the uncorrected orbital at the nucleus in simulation cell R s is exp[i(θ 0 +
k s · R s )], where k s is the simulation-cell Bloch vector. [So we can evaluate the cusp correction using
minimum-image distances, then multiply by exp(ik s · R s ), where R s is the difference of the actual
and minimum-image positions of the electron relative to the ion.]
To use the scheme, set use gpcc to T. Note that cusp correction (which activates the Gaussian
cusp correction) must be set to F.
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