becke_numerical_grid

This module contains all quantities needed to build Becke’s grid used in general for DFT integration. Note that it can be used for whatever integration in R^3 as long as the functions to be integrated are mostly concentrated near the atomic regions.

This grid is built as the reunion of a spherical grid around each atom. Each spherical grid contains a certain number of radial and angular points. No pruning is done on the angular part of the grid.

The main keyword for that module is:

  • becke_numerical_grid grid_type_sgn which controls the precision of the grid according the standard SG-n grids. This keyword controls the two providers n_points_integration_angular n_points_radial_grid.

The main providers of that module are:

  • n_points_integration_angular which is the number of angular integration points. WARNING: it obeys to specific rules so it cannot be any integer number. Some of the possible values are [ 50 | 74 | 170 | 194 | 266 | 302 | 590 | 1202 | 2030 | 5810 ] for instance. See angular.f for more details.
  • n_points_radial_grid which is the number of radial angular points. This can be any strictly positive integer. Nevertheless, a minimum of 50 is in general necessary.
  • final_grid_points which are the (x,y,z) coordinates of the grid points.
  • final_weight_at_r_vector which are the weights at each grid point

For a simple example of how to use the grid, see example.irp.f.

The spherical integration uses Lebedev-Laikov grids, which was used from the code distributed through CCL (http://www.ccl.net/). See next section for explanations and citation policies.

This subroutine is part of a set of subroutines that generate
Lebedev grids [1-6] for integration on a sphere. The original
C-code [1] was kindly provided by Dr. Dmitri N. Laikov and
translated into fortran by Dr. Christoph van Wuellen.
This subroutine was translated using a C to fortran77 conversion
tool written by Dr. Christoph van Wuellen.

Users of this code are asked to include reference [1] in their
publications, and in the user- and programmers-manuals
describing their codes.

This code was distributed through CCL (http://www.ccl.net/).

[1] V.I. Lebedev, and D.N. Laikov
    "A quadrature formula for the sphere of the 131st
     algebraic order of accuracy"
    Doklady Mathematics, Vol. 59, No. 3, 1999, pp. 477-481.

[2] V.I. Lebedev
    "A quadrature formula for the sphere of 59th algebraic
     order of accuracy"
    Russian Acad. Sci. Dokl. Math., Vol. 50, 1995, pp. 283-286.

[3] V.I. Lebedev, and A.L. Skorokhodov
    "Quadrature formulas of orders 41, 47, and 53 for the sphere"
    Russian Acad. Sci. Dokl. Math., Vol. 45, 1992, pp. 587-592.

[4] V.I. Lebedev
    "Spherical quadrature formulas exact to orders 25-29"
    Siberian Mathematical Journal, Vol. 18, 1977, pp. 99-107.

[5] V.I. Lebedev
    "Quadratures on a sphere"
    Computational Mathematics and Mathematical Physics, Vol. 16,
    1976, pp. 10-24.

[6] V.I. Lebedev
    "Values of the nodes and weights of ninth to seventeenth
     order Gauss-Markov quadrature formulae invariant under the
     octahedron group with inversion"
    Computational Mathematics and Mathematical Physics, Vol. 15,
    1975, pp. 44-51.

EZFIO parameters

grid_type_sgn

Type of grid used for the Becke’s numerical grid. Can be, by increasing accuracy: [ 0 | 1 | 2 | 3 ]

Default: 2

Providers

alpha_knowles

File : becke_numerical_grid/integration_radial.irp.f

double precision, allocatable   :: alpha_knowles        (100)

Recommended values for the alpha parameters according to the paper of Knowles (JCP, 104, 1996) as a function of the nuclear charge

Needed by:

angular_quadrature_points

File : becke_numerical_grid/grid_becke.irp.f

double precision, allocatable   :: angular_quadrature_points    (n_points_integration_angular,3)
double precision, allocatable   :: weights_angular_points       (n_points_integration_angular)

weights and grid points for the integration on the angular variables on the unit sphere centered on (0,0,0) According to the LEBEDEV scheme

Needs:

Needed by:

dr_radial_integral

File : becke_numerical_grid/grid_becke.irp.f

double precision, allocatable   :: grid_points_radial   (n_points_radial_grid)
double precision        :: dr_radial_integral

points in [0,1] to map the radial integral [0,infty]

Needs:

Needed by:

final_grid_points

File : becke_numerical_grid/grid_becke_vector.irp.f

   double precision, allocatable   :: final_grid_points    (3,n_points_final_grid)
   double precision, allocatable   :: final_weight_at_r_vector     (n_points_final_grid)
   integer, allocatable    :: index_final_points   (3,n_points_final_grid)
   integer, allocatable    :: index_final_points_reverse   (n_points_integration_angular,n_points_radial_grid,nucl_num)


final_grid_points(1:3,j) = (/ x, y, z /) of the jth grid point

final_weight_at_r_vector(i) = Total weight function of the ith grid point which contains the Lebedev, Voronoi and radial weights contributions

index_final_points(1:3,i) = gives the angular, radial and atomic indices associated to the ith grid point

index_final_points_reverse(i,j,k) = index of the grid point having i as angular, j as radial and l as atomic indices

Needs:

  • nucl_num

Needed by:

final_weight_at_r

File : becke_numerical_grid/grid_becke.irp.f

double precision, allocatable   :: final_weight_at_r    (n_points_integration_angular,n_points_radial_grid,nucl_num)

Total weight on each grid point which takes into account all Lebedev, Voronoi and radial weights.

Needs:

Needed by:

final_weight_at_r_vector

File : becke_numerical_grid/grid_becke_vector.irp.f

   double precision, allocatable   :: final_grid_points    (3,n_points_final_grid)
   double precision, allocatable   :: final_weight_at_r_vector     (n_points_final_grid)
   integer, allocatable    :: index_final_points   (3,n_points_final_grid)
   integer, allocatable    :: index_final_points_reverse   (n_points_integration_angular,n_points_radial_grid,nucl_num)


final_grid_points(1:3,j) = (/ x, y, z /) of the jth grid point

final_weight_at_r_vector(i) = Total weight function of the ith grid point which contains the Lebedev, Voronoi and radial weights contributions

index_final_points(1:3,i) = gives the angular, radial and atomic indices associated to the ith grid point

index_final_points_reverse(i,j,k) = index of the grid point having i as angular, j as radial and l as atomic indices

Needs:

  • nucl_num

Needed by:

grid_points_per_atom

File : becke_numerical_grid/grid_becke.irp.f

double precision, allocatable   :: grid_points_per_atom (3,n_points_integration_angular,n_points_radial_grid,nucl_num)

x,y,z coordinates of grid points used for integration in 3d space

Needs:

Needed by:

grid_points_radial

File : becke_numerical_grid/grid_becke.irp.f

double precision, allocatable   :: grid_points_radial   (n_points_radial_grid)
double precision        :: dr_radial_integral

points in [0,1] to map the radial integral [0,infty]

Needs:

Needed by:

index_final_points

File : becke_numerical_grid/grid_becke_vector.irp.f

   double precision, allocatable   :: final_grid_points    (3,n_points_final_grid)
   double precision, allocatable   :: final_weight_at_r_vector     (n_points_final_grid)
   integer, allocatable    :: index_final_points   (3,n_points_final_grid)
   integer, allocatable    :: index_final_points_reverse   (n_points_integration_angular,n_points_radial_grid,nucl_num)


final_grid_points(1:3,j) = (/ x, y, z /) of the jth grid point

final_weight_at_r_vector(i) = Total weight function of the ith grid point which contains the Lebedev, Voronoi and radial weights contributions

index_final_points(1:3,i) = gives the angular, radial and atomic indices associated to the ith grid point

index_final_points_reverse(i,j,k) = index of the grid point having i as angular, j as radial and l as atomic indices

Needs:

  • nucl_num

Needed by:

index_final_points_reverse

File : becke_numerical_grid/grid_becke_vector.irp.f

   double precision, allocatable   :: final_grid_points    (3,n_points_final_grid)
   double precision, allocatable   :: final_weight_at_r_vector     (n_points_final_grid)
   integer, allocatable    :: index_final_points   (3,n_points_final_grid)
   integer, allocatable    :: index_final_points_reverse   (n_points_integration_angular,n_points_radial_grid,nucl_num)


final_grid_points(1:3,j) = (/ x, y, z /) of the jth grid point

final_weight_at_r_vector(i) = Total weight function of the ith grid point which contains the Lebedev, Voronoi and radial weights contributions

index_final_points(1:3,i) = gives the angular, radial and atomic indices associated to the ith grid point

index_final_points_reverse(i,j,k) = index of the grid point having i as angular, j as radial and l as atomic indices

Needs:

  • nucl_num

Needed by:

m_knowles

File : becke_numerical_grid/grid_becke.irp.f

integer :: m_knowles

value of the “m” parameter in the equation (7) of the paper of Knowles (JCP, 104, 1996)

Needed by:

n_points_final_grid

File : becke_numerical_grid/grid_becke_vector.irp.f

integer :: n_points_final_grid

Number of points which are non zero

Needs:

  • nucl_num

Needed by:

n_points_grid_per_atom

File : becke_numerical_grid/grid_becke.irp.f

integer :: n_points_grid_per_atom

Number of grid points per atom

Needs:

n_points_integration_angular

File : becke_numerical_grid/grid_becke.irp.f

integer :: n_points_radial_grid
integer :: n_points_integration_angular

n_points_radial_grid = number of radial grid points per atom

n_points_integration_angular = number of angular grid points per atom

These numbers are automatically set by setting the grid_type_sgn parameter

Needs:

  • grid_type_sgn

Needed by:

n_points_radial_grid

File : becke_numerical_grid/grid_becke.irp.f

integer :: n_points_radial_grid
integer :: n_points_integration_angular

n_points_radial_grid = number of radial grid points per atom

n_points_integration_angular = number of angular grid points per atom

These numbers are automatically set by setting the grid_type_sgn parameter

Needs:

  • grid_type_sgn

Needed by:

weight_at_r

File : becke_numerical_grid/grid_becke.irp.f

double precision, allocatable   :: weight_at_r  (n_points_integration_angular,n_points_radial_grid,nucl_num)

Weight function at grid points : w_n(r) according to the equation (22) of Becke original paper (JCP, 88, 1988)

The “n” discrete variable represents the nucleis which in this array is represented by the last dimension and the points are labelled by the other dimensions.

Needs:

Needed by:

weights_angular_points

File : becke_numerical_grid/grid_becke.irp.f

double precision, allocatable   :: angular_quadrature_points    (n_points_integration_angular,3)
double precision, allocatable   :: weights_angular_points       (n_points_integration_angular)

weights and grid points for the integration on the angular variables on the unit sphere centered on (0,0,0) According to the LEBEDEV scheme

Needs:

Needed by:

Subroutines / functions

cell_function_becke:()

File : becke_numerical_grid/step_function_becke.irp.f

double precision function cell_function_becke(r,atom_number)

atom_number :: atom on which the cell function of Becke (1988, JCP,88(4)) r(1:3) :: x,y,z coordinantes of the current point

Needs:

  • nucl_num
derivative_knowles_function:()

File : becke_numerical_grid/integration_radial.irp.f

double precision function derivative_knowles_function(alpha,m,x)

Derivative of the function proposed by Knowles (JCP, 104, 1996) for distributing the radial points

example_becke_numerical_grid:()

File : becke_numerical_grid/example.irp.f

subroutine example_becke_numerical_grid

subroutine that illustrates the main features available in becke_numerical_grid

Needs:

f_function_becke:()

File : becke_numerical_grid/step_function_becke.irp.f

double precision function f_function_becke(x)
knowles_function:()

File : becke_numerical_grid/integration_radial.irp.f

double precision function knowles_function(alpha,m,x)

Function proposed by Knowles (JCP, 104, 1996) for distributing the radial points : the Log “m” function ( equation (7) in the paper )

step_function_becke:()

File : becke_numerical_grid/step_function_becke.irp.f

double precision function step_function_becke(x)

Step function of the Becke paper (1988, JCP,88(4))