MOLPRO

17 THE DENSITY FUNCTIONAL PROGRAM 112
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 = −α log e (1 − x m r
) , (2)
with 0 ≤ x ≤ 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. α is taken to
be scale times the recommended value for α 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
(1 + x)
r = α
(1 − x) , (3)
using points in −1 ≤ x ≤ +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 α 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 =
α
( ) 2
log e 2 (1 + x)0.6 log e , (4)
1 − x
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.
17.3.3 Angular integration grid (ANGULAR)
ANGULAR,method,acca,crowd
LMIN,l0 min ,l1 min ,l2 min ,l3
min
LMAX,l0 max ,l1 max ,l2 max ,l3
max
Specify the details of the angular quadrature scheme. The default choice for method is LEBEDEV
(ie. as in A. D. Becke, J. Chem. Phys. 88 (1988) 2547) which provides angular grids of octahedral
symmetry. The alternative choice for method is LEGENDRE which gives Gauss-Legendre
quadrature in θ and simple quadrature in φ, as defined by C. W. Murray, N. C. Handy and G. J.
Laming, Mol. Phys. 78 (1993) 997.
Each type of grid specifies a family of which the various members are characterized by a single
quantum number l; spherical harmonics up to degree l are integrated exactly. lmin i and
lmax i ,i = 0,1,2,3 specify allowed ranges of l for hydrogen/helium, first row, second row, and
other elements respectively. For the Lebedev grids, if the value of l is not one of the set implemented
in MOLPRO (3, 5, 7, 9, 11, 13, 15, 17, 19, 23, 29, 41, 47, 53), then l is increased to give