11.1.3 grid_set_radial: set parameters for radial grids

      subroutine grid_set_radial (record,rqu,mr,scale,idftnr,accr)
      implicit double precision (a-h,o-z)
      character *(*) rqu
      integer idftnr(0:*)

Specify the details of the radial quadrature scheme. Four different radial schemes are available, specified by rqu = EM, BECKE, AHLRICHS or LOG, with subsidiary parameters scale,mr$=m_r$ specifying additional information.

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

$$r = \alpha {x^{m_r}\over (1-x)^{m_r}}$$ (3)

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

$$r = - \alpha \log_e (1-x^{m_r})\; ,$$ (4)

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

$$r = \alpha {(1+x)\over (1-x)} \; ,$$ (5)

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

$$r= {\alpha \over \log_e 2} (1+x)^{0.6} \log_e\left( {2\over 1-x}\right)\; ,$$ (6)

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

idftnr 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 specifies a target accuracy; the number of radial points is chosen according to a model, instead of using idftnr etc. Implementation: the stricter of idftnr, accr is used, unless either is zero, in which case it is ignored.