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.. _becke_numerical_grid:
.. program :: becke_numerical_grid
.. default-role :: option
====================
becke_numerical_grid
====================
This module contains all quantities needed to build the 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.
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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 modue is:
* :option: `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`
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The main providers of that module are:
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* `n_points_integration_angular` :file: `angular.f` for more details.
* `n_points_radial_grid`
* `final_grid_points`
* `final_weight_at_r_vector`
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For a simple example of how to use the grid, see :file: `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.
.. code-block :: text
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
----------------
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.. option :: grid_type_sgn
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Type of grid used for the Becke's numerical grid. Can be, by increasing accuracy: [ 0 | 1 | 2 | 3 ]
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Default: 2
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Providers
---------
.. c:var :: alpha_knowles
.. code :: text
double precision, allocatable :: alpha_knowles (100)
File: :file: `integration_radial.irp.f`
Recommended values for the alpha parameters according to the paper of Knowles (JCP, 104, 1996) as a function of the nuclear charge
.. c:var :: angular_quadrature_points
.. code :: text
double precision, allocatable :: angular_quadrature_points (n_points_integration_angular,3)
double precision, allocatable :: weights_angular_points (n_points_integration_angular)
File: :file: `grid_becke.irp.f`
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
.. c:var :: dr_radial_integral
.. code :: text
double precision, allocatable :: grid_points_radial (n_points_radial_grid)
double precision :: dr_radial_integral
File: :file: `grid_becke.irp.f`
points in [0,1] to map the radial integral [0,\infty]
.. c:var :: final_grid_points
.. code :: text
double precision, allocatable :: final_grid_points (3,n_points_final_grid)
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double precision, allocatable :: final_weight_at_r_vector (n_points_final_grid)
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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)
File: :file: `grid_becke_vector.irp.f`
final_grid_points(1:3,j) = (/ x, y, z /) of the jth grid point
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final_weight_at_r_vector(i) = Total weight function of the ith grid point which contains the Lebedev, Voronoi and radial weights contributions
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index_final_points(1:3,i) = gives the angular, radial and atomic indices associated to the ith grid point
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index_final_points_reverse(i,j,k) = index of the grid point having i as angular, j as radial and l as atomic indices
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.. c:var :: final_weight_at_r
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.. code :: text
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double precision, allocatable :: final_weight_at_r (n_points_integration_angular,n_points_radial_grid,nucl_num)
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File: :file: `grid_becke.irp.f`
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Total weight on each grid point which takes into account all Lebedev, Voronoi and radial weights.
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.. c:var :: final_weight_at_r_vector
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.. code :: text
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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)
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File: :file: `grid_becke_vector.irp.f`
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final_grid_points(1:3,j) = (/ x, y, z /) of the jth grid point
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final_weight_at_r_vector(i) = Total weight function of the ith grid point which contains the Lebedev, Voronoi and radial weights contributions
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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
.. c:var :: final_weight_functions_at_final_grid_points
.. code :: text
double precision, allocatable :: final_grid_points (3,n_points_final_grid)
double precision, allocatable :: final_weight_functions_at_final_grid_points (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)
File: :file: `grid_becke_vector.irp.f`
final_grid_points(1:3,j) = (/ x, y, z /) of the jth grid point
final_weight_functions_at_final_grid_points(i) = Total weight function of the ith grid point which contains the Lebedev, Voronoi and radial weights contributions
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index_final_points(1:3,i) = gives the angular, radial and atomic indices associated to the ith grid point
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index_final_points_reverse(i,j,k) = index of the grid point having i as angular, j as radial and l as atomic indices
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.. c:var :: final_weight_functions_at_grid_points
.. code :: text
double precision, allocatable :: final_weight_functions_at_grid_points (n_points_integration_angular,n_points_radial_grid,nucl_num)
File: :file: `grid_becke.irp.f`
Total weight on each grid point which takes into account all Lebedev, Voronoi and radial weights.
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.. c:var :: grid_points_per_atom
.. code :: text
double precision, allocatable :: grid_points_per_atom (3,n_points_integration_angular,n_points_radial_grid,nucl_num)
File: :file: `grid_becke.irp.f`
x,y,z coordinates of grid points used for integration in 3d space
.. c:var :: grid_points_radial
.. code :: text
double precision, allocatable :: grid_points_radial (n_points_radial_grid)
double precision :: dr_radial_integral
File: :file: `grid_becke.irp.f`
points in [0,1] to map the radial integral [0,\infty]
.. c:var :: index_final_points
.. code :: text
double precision, allocatable :: final_grid_points (3,n_points_final_grid)
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double precision, allocatable :: final_weight_at_r_vector (n_points_final_grid)
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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)
File: :file: `grid_becke_vector.irp.f`
final_grid_points(1:3,j) = (/ x, y, z /) of the jth grid point
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final_weight_at_r_vector(i) = Total weight function of the ith grid point which contains the Lebedev, Voronoi and radial weights contributions
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index_final_points(1:3,i) = gives the angular, radial and atomic indices associated to the ith grid point
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index_final_points_reverse(i,j,k) = index of the grid point having i as angular, j as radial and l as atomic indices
.. c:var :: index_final_points_reverse
.. code :: text
double precision, allocatable :: final_grid_points (3,n_points_final_grid)
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double precision, allocatable :: final_weight_at_r_vector (n_points_final_grid)
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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)
File: :file: `grid_becke_vector.irp.f`
final_grid_points(1:3,j) = (/ x, y, z /) of the jth grid point
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final_weight_at_r_vector(i) = Total weight function of the ith grid point which contains the Lebedev, Voronoi and radial weights contributions
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index_final_points(1:3,i) = gives the angular, radial and atomic indices associated to the ith grid point
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index_final_points_reverse(i,j,k) = index of the grid point having i as angular, j as radial and l as atomic indices
.. c:var :: m_knowles
.. code :: text
integer :: m_knowles
File: :file: `grid_becke.irp.f`
value of the "m" parameter in the equation (7) of the paper of Knowles (JCP, 104, 1996)
.. c:var :: n_points_final_grid
.. code :: text
integer :: n_points_final_grid
File: :file: `grid_becke_vector.irp.f`
Number of points which are non zero
.. c:var :: n_points_grid_per_atom
.. code :: text
integer :: n_points_grid_per_atom
File: :file: `grid_becke.irp.f`
Number of grid points per atom
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.. c:var :: n_points_integration_angular
.. code :: text
integer :: n_points_radial_grid
integer :: n_points_integration_angular
File: :file: `grid_becke.irp.f`
n_points_radial_grid = number of radial grid points per atom
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n_points_integration_angular = number of angular grid points per atom
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These numbers are automatically set by setting the grid_type_sgn parameter
.. c:var :: n_points_radial_grid
.. code :: text
integer :: n_points_radial_grid
integer :: n_points_integration_angular
File: :file: `grid_becke.irp.f`
n_points_radial_grid = number of radial grid points per atom
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n_points_integration_angular = number of angular grid points per atom
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These numbers are automatically set by setting the grid_type_sgn parameter
.. c:var :: weight_at_r
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.. code :: text
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double precision, allocatable :: weight_at_r (n_points_integration_angular,n_points_radial_grid,nucl_num)
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File: :file: `grid_becke.irp.f`
Weight function at grid points : w_n(r) according to the equation (22) of Becke original paper (JCP, 88, 1988)
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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.
.. c:var :: weight_functions_at_grid_points
.. code :: text
double precision, allocatable :: weight_functions_at_grid_points (n_points_integration_angular,n_points_radial_grid,nucl_num)
File: :file: `grid_becke.irp.f`
Weight function at grid points : w_n(r) according to the equation (22) of Becke original paper (JCP, 88, 1988)
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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.
.. c:var :: weights_angular_points
.. code :: text
double precision, allocatable :: angular_quadrature_points (n_points_integration_angular,3)
double precision, allocatable :: weights_angular_points (n_points_integration_angular)
File: :file: `grid_becke.irp.f`
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
Subroutines / functions
-----------------------
.. c:function :: cell_function_becke
.. code :: text
double precision function cell_function_becke(r,atom_number)
File: :file: `step_function_becke.irp.f`
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
.. c:function :: derivative_knowles_function
.. code :: text
double precision function derivative_knowles_function(alpha,m,x)
File: :file: `integration_radial.irp.f`
Derivative of the function proposed by Knowles (JCP, 104, 1996) for distributing the radial points
.. c:function :: example_becke_numerical_grid
.. code :: text
subroutine example_becke_numerical_grid
File: :file: `example.irp.f`
subroutine that illustrates the main features available in becke_numerical_grid
.. c:function :: f_function_becke
.. code :: text
double precision function f_function_becke(x)
File: :file: `step_function_becke.irp.f`
.. c:function :: knowles_function
.. code :: text
double precision function knowles_function(alpha,m,x)
File: :file: `integration_radial.irp.f`
Function proposed by Knowles (JCP, 104, 1996) for distributing the radial points : the Log "m" function ( equation (7) in the paper )
.. c:function :: step_function_becke
.. code :: text
double precision function step_function_becke(x)
File: :file: `step_function_becke.irp.f`
Step function of the Becke paper (1988, JCP,88(4))
