[-](rirs|radrule) rule-number
This keyword (Radial Integration Rule Selection) specifies the
rule to be used for numerical integration of the radial function.
The general methodology to calculate the integral of a function
f(r) over the interval [0, ∞] spanned by the radial
coordinate r, is to map the semi-infinite interval [0, ∞] of the
variable r to a finite interval [a, b] of the variable
q by a suitable variable transformation r = r(q),
and then to use a quadrature formula defined on this interval.
Therefore, each value of the radial-integration-rule selector will combine
a specific quadrature rule (qr)
with a suitable transformation of the radial coordinate
(trc):
radrule = qr + trc |
Available quadrature rules and the associated qr-values are reported in Table 1 and are described in section "Methods of Numerical Integration" of this manual.
quadrature rule | [a, b] | qr-value | (qr + trc)-value |
---|---|---|---|
Gauss-Legendre | [−1, +1] | 10 | 15−17 |
Gauss-Chebyshev of the second kind | [−1, +1] | 20 | 25−27 |
Gauss-Gill | [0, 1] | 30 | 31−34, 37 |
Gauss-Laguerre | [0, ∞] | 40 | 47 |
Generalized Gauss-Laguerre | [0, ∞] | 50 | 57 |
Gauss-Hermite | [−∞, +∞] | 60 | 67 |
Euler-Maclaurin | [a, b] | 70 | 71−77 |
The available mappings of the radial coordinate r ∈ [r_{0}, r_{∞}] into a new coordinate q ∈ [a, b] and the associated trc-values are given in Table 2 and are described in section "Radial integral" of this manual.
r ∈ [r_{0}, r_{∞}] ⇔ q ∈ [a, b] | [a, b] | trc-value | (qr + trc)-value |
---|---|---|---|
MultiExp | [0, 1] | 1 | 31, 71 |
Knowles | [0, 1] | 2 | 32, 72 |
Handy | [0, 1] | 3 | 33, 73 |
Handy (modified) | [a, b] | 4 | 34, 74 |
Becke | [−1, +1] | 5 | 15, 25, 75 |
Ahlrichs | [−1, +1] | 6 | 16, 26, 76 |
linear | [a, b] | 7 | 17, 27, 37, 47, 57, 67, 77 |
Because of the restrictions on the interval [a, b], only a few combinations of qr- and trc-values are allowed, and they are reported in the last columns of Tables 1 and 2.