22.3.2 Radial integration grid (RADIAL)

RADIAL,method,$m_r$,scale, $n_0,n_1,n_2,n_3$

Specify the details of the radial quadrature scheme. Four different radial schemes are available, specified by method = EM, BECKE, AHLRICHS or LOG, with the latter being the default.

EM is the Euler-Maclaurin scheme defined by C. W. Murray, N. C. Handy and G. J. Laming, Mol. Phys. 78 (1993) 997. $m_r$, for which the default value is 2, is defined in equation (6) of the above as

$$r = \alpha {x^{m_r}\over (1-x)^{m_r}}$$ (1)

whilst scale (default value 1) multiplied by the Bragg-Slater radius of the atom gives the scaling parameter $\alpha$.

LOG is the scheme described by M. E. Mura and P. J. Knowles, J. Chem. Phys. 104 (1996) 9848. It is based on the transformation

$$r = - \alpha \log_e (1-x^{m_r})\; ,$$ (2)

with $0\le x \le 1$ and simple Gauss quadrature in $x$-space. The recommended value of $m_r$ is 3 for molecular systems, giving rise to the Log3 grid; $m_r$=4 is more efficient for atoms. $\alpha$ is taken to be scale times the recommended value for $\alpha$ given by Mura and Knowles, and scale defaults to 1.

BECKE is as defined by A. D. Becke, J. Chem. Phys. 88 (1988) 2547. It is based on the transformation

$$r = \alpha {(1+x)\over (1-x)} \; ,$$ (3)

using points in $-1\le x \le +1$ and standard Gauss-Chebyshev quadrature of the second kind for the $x$-space quadrature. Becke chose his scaling parameters to be half the Bragg-Slater radius except for hydrogen, for which the whole Bragg-Slater radius was used, and setting scale to a value other than 1 allows a different $\alpha$ to be used. $m_r$ is not necessary for this radial scheme.

AHLRICHS is the radial scheme defined by O. Treutler and R. Ahlrichs, J. Chem. Phys. 102 (1995) 346. It is based on the transformation their M4 mapping

$$r= {\alpha \over \log_e 2} (1+x)^{0.6} \log_e\left( {2\over 1-x}\right)\; ,$$ (4)

with using standard Gauss-Chebyshev quadrature of the second kind for the $x$-space integration. $m_r$ is not necessary for this radial scheme.

$n_0$, $n_1$, $n_2$, $n_3$ are the degrees of quadrature $n_r$ (see equation (3) of Murray et al.), for hydrogen/helium, first row, second row, and other elements respectively.

accr as given by the THR command specifies a target accuracy; the number of radial points is chosen according to a model, instead of using an explicit $n_i$. The stricter of $n_i$, accr is used, unless either is zero, in which case it is ignored.