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399 lines
12 KiB
ReStructuredText
399 lines
12 KiB
ReStructuredText
.. _becke_numerical_grid:
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.. program:: becke_numerical_grid
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.. default-role:: option
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====================
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Becke Numerical Grid
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====================
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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.
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For a simple example of how to use the grid, see :file:`example.irp.f`.
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The spherical integration uses Lebedev-Laikov grids, which was used from the code distributed through CCL (http://www.ccl.net/).
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See next section for explanations and citation policies.
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.. code-block:: text
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This subroutine is part of a set of subroutines that generate
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Lebedev grids [1-6] for integration on a sphere. The original
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C-code [1] was kindly provided by Dr. Dmitri N. Laikov and
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translated into fortran by Dr. Christoph van Wuellen.
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This subroutine was translated using a C to fortran77 conversion
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tool written by Dr. Christoph van Wuellen.
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Users of this code are asked to include reference [1] in their
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publications, and in the user- and programmers-manuals
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describing their codes.
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This code was distributed through CCL (http://www.ccl.net/).
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[1] V.I. Lebedev, and D.N. Laikov
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"A quadrature formula for the sphere of the 131st
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algebraic order of accuracy"
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Doklady Mathematics, Vol. 59, No. 3, 1999, pp. 477-481.
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[2] V.I. Lebedev
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"A quadrature formula for the sphere of 59th algebraic
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order of accuracy"
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Russian Acad. Sci. Dokl. Math., Vol. 50, 1995, pp. 283-286.
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[3] V.I. Lebedev, and A.L. Skorokhodov
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"Quadrature formulas of orders 41, 47, and 53 for the sphere"
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Russian Acad. Sci. Dokl. Math., Vol. 45, 1992, pp. 587-592.
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[4] V.I. Lebedev
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"Spherical quadrature formulas exact to orders 25-29"
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Siberian Mathematical Journal, Vol. 18, 1977, pp. 99-107.
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[5] V.I. Lebedev
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"Quadratures on a sphere"
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Computational Mathematics and Mathematical Physics, Vol. 16,
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1976, pp. 10-24.
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[6] V.I. Lebedev
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"Values of the nodes and weights of ninth to seventeenth
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order Gauss-Markov quadrature formulae invariant under the
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octahedron group with inversion"
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Computational Mathematics and Mathematical Physics, Vol. 15,
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1975, pp. 44-51.
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EZFIO parameters
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----------------
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.. option:: n_points_integration_angular
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Number of angular points per atom for 3d numerical integration, needed for DFT for example [ 50 | 74 | 266 | 590 | 1202 | 2030 | 5810 ]
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Default: 590
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.. option:: n_points_radial_grid
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Number of radial points per atom for 3d numerical integration, needed for DFT for example
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Default: 60
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Providers
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---------
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.. c:var:: alpha_knowles
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.. code:: text
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double precision, allocatable :: alpha_knowles (100)
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File: :file:`integration_radial.irp.f`
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Recommended values for the alpha parameters according to the paper of Knowles (JCP, 104, 1996) as a function of the nuclear charge
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.. c:var:: angular_quadrature_points
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.. code:: text
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double precision, allocatable :: angular_quadrature_points (n_points_integration_angular,3)
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double precision, allocatable :: weights_angular_points (n_points_integration_angular)
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File: :file:`grid_becke.irp.f`
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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
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.. c:var:: dr_radial_integral
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.. code:: text
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double precision, allocatable :: grid_points_radial (n_points_radial_grid)
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double precision :: dr_radial_integral
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File: :file:`grid_becke.irp.f`
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points in [0,1] to map the radial integral [0,\infty]
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.. c:var:: final_grid_points
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.. code:: text
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double precision, allocatable :: final_grid_points (3,n_points_final_grid)
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double precision, allocatable :: final_weight_functions_at_final_grid_points (n_points_final_grid)
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integer, allocatable :: index_final_points (3,n_points_final_grid)
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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_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_final_grid_points
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.. code:: text
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double precision, allocatable :: final_grid_points (3,n_points_final_grid)
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double precision, allocatable :: final_weight_functions_at_final_grid_points (n_points_final_grid)
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integer, allocatable :: index_final_points (3,n_points_final_grid)
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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_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
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.. code:: text
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double precision, allocatable :: final_weight_functions_at_grid_points (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:: grid_points_per_atom
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.. code:: text
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double precision, allocatable :: grid_points_per_atom (3,n_points_integration_angular,n_points_radial_grid,nucl_num)
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File: :file:`grid_becke.irp.f`
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x,y,z coordinates of grid points used for integration in 3d space
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.. c:var:: grid_points_radial
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.. code:: text
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double precision, allocatable :: grid_points_radial (n_points_radial_grid)
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double precision :: dr_radial_integral
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File: :file:`grid_becke.irp.f`
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points in [0,1] to map the radial integral [0,\infty]
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.. c:var:: index_final_points
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.. code:: text
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double precision, allocatable :: final_grid_points (3,n_points_final_grid)
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double precision, allocatable :: final_weight_functions_at_final_grid_points (n_points_final_grid)
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integer, allocatable :: index_final_points (3,n_points_final_grid)
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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_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:: index_final_points_reverse
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.. code:: text
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double precision, allocatable :: final_grid_points (3,n_points_final_grid)
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double precision, allocatable :: final_weight_functions_at_final_grid_points (n_points_final_grid)
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integer, allocatable :: index_final_points (3,n_points_final_grid)
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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_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:: m_knowles
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.. code:: text
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integer :: m_knowles
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File: :file:`grid_becke.irp.f`
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value of the "m" parameter in the equation (7) of the paper of Knowles (JCP, 104, 1996)
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.. c:var:: n_points_final_grid
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.. code:: text
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integer :: n_points_final_grid
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File: :file:`grid_becke_vector.irp.f`
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Number of points which are non zero
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.. c:var:: n_points_grid_per_atom
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.. code:: text
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integer :: n_points_grid_per_atom
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File: :file:`grid_becke.irp.f`
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Number of grid points per atom
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.. c:var:: weight_functions_at_grid_points
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.. code:: text
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double precision, allocatable :: weight_functions_at_grid_points (n_points_integration_angular,n_points_radial_grid,nucl_num)
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File: :file:`grid_becke.irp.f`
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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.
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.. c:var:: weights_angular_points
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.. code:: text
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double precision, allocatable :: angular_quadrature_points (n_points_integration_angular,3)
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double precision, allocatable :: weights_angular_points (n_points_integration_angular)
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File: :file:`grid_becke.irp.f`
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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
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Subroutines / functions
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-----------------------
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.. c:function:: cell_function_becke
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.. code:: text
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double precision function cell_function_becke(r,atom_number)
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File: :file:`step_function_becke.irp.f`
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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
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.. c:function:: derivative_knowles_function
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.. code:: text
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double precision function derivative_knowles_function(alpha,m,x)
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File: :file:`integration_radial.irp.f`
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Derivative of the function proposed by Knowles (JCP, 104, 1996) for distributing the radial points
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.. c:function:: example_becke_numerical_grid
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.. code:: text
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subroutine example_becke_numerical_grid
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File: :file:`example.irp.f`
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subroutine that illustrates the main features available in becke_numerical_grid
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.. c:function:: f_function_becke
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.. code:: text
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double precision function f_function_becke(x)
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File: :file:`step_function_becke.irp.f`
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.. c:function:: knowles_function
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.. code:: text
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double precision function knowles_function(alpha,m,x)
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File: :file:`integration_radial.irp.f`
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Function proposed by Knowles (JCP, 104, 1996) for distributing the radial points : the Log "m" function ( equation (7) in the paper )
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.. c:function:: step_function_becke
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.. code:: text
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double precision function step_function_becke(x)
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File: :file:`step_function_becke.irp.f`
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Step function of the Becke paper (1988, JCP,88(4))
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