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Numerical integration grid for molecules.

Project description

Who are the people behind this code?


  • Radovan Bast


  • Roberto Di Remigio (OS X testing, streamlined Travis testing, better C++, error handling)

For a list of all the contributions see


  • Simon Neville (reporting issues)
  • Jaime Axel Rosal Sandberg (reporting issues)

This tool uses SPHERE_LEBEDEV_RULE, a C library written by John Burkardt which computes a Lebedev quadrature rule over the surface of the unit sphere in 3D, see also:

This library uses and acknowledges the MolSSI BSE (, which is a rewrite of the Basis Set Exchange ( and is a collaboration between the Molecular Sciences Software Institute ( and the Environmental Molecular Sciences Laboratory (


If you use this tool in a program or publication, please acknowledge its author(s). The metadata necessary for citing this tool can be found in the CITATION.cff file. For more information CITATION.cff files, see

Would you like to contribute?

Yes please! Please follow this excellent guide. We do not require any formal copyright assignment or contributor license agreement. Any contributions intentionally sent upstream are presumed to be offered under terms of the Mozilla Public License Version 2.0.


  • CMake
  • C and C++ compiler
  • Fortran compiler (to build the optional Fortran interface)
  • CFFI (to access the optional Python interface)
  • pytest (to test the optional Python interface)


Installing via pip

pip install numgrid

Building and testing from sources

Fetch the code:

git clone

Install Python dependencies (optional):

pipenv install

Build the code:

cd numgrid
./setup --fc=gfortran --cc=gcc --cxx=g++
cd build
make test

The Python interface is automatically tested by Travis CI:


The library provides a context-aware C interface. In addition it also provides a Fortran and Python interfaces as thin layers on top of the C interface:

Python: numgrid/
   \     Fortran: numgrid/numgrid.f90
    \   /
  C interface: numgrid/numgrid.h


Coordinates are in bohr.


Grid computation is done per atom/basis type and proceeds in five steps:

  • Create atom
  • Get number of points (depends on basis set range)
  • Allocate memory to hold the grid
  • Compute grid on this atom in a molecular environment
  • Free atom and its memory

The Python interface takes care of the allocation and deallocation part but the essential point is that memory management is happening on the client side.

If you have many atom centers that have the same atom type and same basis set, it will make sense to create only one atom object and then reuse this object to compute the grid on all atoms with the same basis type.

It is no problem to create several atom objects at the same time.

Python example

The Python interface is generated using CFFI.

As an example let us generate a grid for the water molecule:

import numgrid

radial_precision = 1.0e-12
min_num_angular_points = 86
max_num_angular_points = 302

num_centers = 3
proton_charges = [8, 1, 1]

x_coordinates_bohr = [0.0, 1.43, -1.43]
y_coordinates_bohr = [0.0, 0.0, 0.0]
z_coordinates_bohr = [0.0, 1.1, 1.1]

# cc-pVDZ basis
alpha_max = [11720.0, 13.01, 13.01]  # O, H, H
max_l_quantum_numbers = [2, 1, 1]  # O, H, H
alpha_min = [[0.3023, 0.2753, 1.185],  # O
             [0.122, 0.727],  # H
             [0.122, 0.727]]  # H

for center_index in range(num_centers):
    context = numgrid.new_atom_grid(radial_precision,

    num_points = numgrid.get_num_grid_points(context)

    # generate an atomic grid in the molecular environment
    x, y, z, w = numgrid.get_grid(context,

    num_radial_points = numgrid.get_num_radial_grid_points(context)

    # generate an isolated radial grid
    r, w = numgrid.get_radial_grid(context)


# generate an isolated angular grid
x, y, z, w = numgrid.get_angular_grid(num_angular_grid_points=14)

Avoiding explicit exponent ranges

Using the Python interface you can choose to not provide explicit exponent ranges and instead specify the basis set which is then fetched directly from using the wonderful MolSSI BSE:

context = numgrid.new_atom_grid_bse(radial_precision=1.0e-12,

Saving grid in Numpy format

The current API makes is relatively easy to export the computed grid in Numpy format.

In this example we save the radial grid positions and weights to two separate files in Numpy format:

import numgrid
import numpy as np

# we assume the context is created
# ...
r, w = numgrid.get_radial_grid(context)'radial_grid_r.npy', r)'radial_grid_w.npy', w)


To see a real example, have a look at the C++ test case.

Creating a new atom grid

context_t *numgrid_new_atom_grid(const double radial_precision,
                                 const int min_num_angular_points,
                                 const int max_num_angular_points,
                                 const int proton_charge,
                                 const double alpha_max,
                                 const int max_l_quantum_number,
                                 const double alpha_min[]);

The smaller the radial_precision, the better grid.

For min_num_angular_points and max_num_angular_points, see “Angular grid” below.

alpha_max is the steepest basis set exponent.

alpha_min is an array of the size max_l_quantum_number + 1 and holds the smallest exponents for each angular momentum. If an angular momentum set is missing “in the middle”, provide 0.0. In other words, imagine that you have a basis set which only contains s and d functions and no p functions and let us assume that the most diffuse s function has the exponent 0.1 and the most diffuse d function has the exponent 0.2, then alpha_min would be an array of three numbers holding {0.1, 0.0, 0.2}.

Get number of grid points on current atom

The following two functions are probably self-explaining. We need to provide the context which refers to a specific atom object.

int numgrid_get_num_grid_points(const context_t *context);

int numgrid_get_num_radial_grid_points(const context_t *context);

Get grid on current atom, scaled by Becke partitioning

We assume that grid_x_bohr, grid_y_bohr, grid_z_bohr, and grid_w are allocated by the caller and have the length that equals the number of grid points.

x_coordinates_bohr, y_coordinates_bohr, z_coordinates_bohr, and proton_charges refer to the molecular environment and have the size num_centers.

Using center_index we tell the code which of the atom centers is the one we have computed the grid for.

void numgrid_get_grid(const context_t *context,
                      const int num_centers,
                      const int center_index,
                      const double x_coordinates_bohr[],
                      const double y_coordinates_bohr[],
                      const double z_coordinates_bohr[],
                      const int proton_charges[],
                      double grid_x_bohr[],
                      double grid_y_bohr[],
                      double grid_z_bohr[],
                      double grid_w[]);

Get radial grid on current atom

We assume that radial_grid_r_bohr and radial_grid_w are allocated by the caller and have both the length that equals the number of radial grid points.

void numgrid_get_radial_grid(const context_t *context,
                             double radial_grid_r_bohr[],
                             double radial_grid_w[]);

Get angular grid

This does not refer to any specific atom and does not require any context.

num_angular_grid_points has to be one of the many supported Lebedev grids (see table on the bottom of this page) and the code will assume that the grid arrays are allocated by the caller and have at least the size num_angular_grid_points.

void numgrid_get_angular_grid(const int num_angular_grid_points,
                              double angular_grid_x_bohr[],
                              double angular_grid_y_bohr[],
                              double angular_grid_z_bohr[],
                              double angular_grid_w[]);

Destroy the atom and deallocate all data

void numgrid_free_atom_grid(context_t *context);

Fortran API

Closely follows the C API. To see a real example, have a look at the Fortran test case.


The design decision was to not parallelize the library but rather parallelize over the atom/basis types by the caller. This simplifies modularity and code reuse.

Space partitioning

The molecular integration grid is generated from atom-centered grids by scaling the grid weights according to the Becke partitioning scheme, JCP 88, 2547 (1988). The default Becke hardness is 3.

Radial grid

The radial grid is generated according to Lindh, Malmqvist, and Gagliardi, TCA 106, 178 (2001).

The motivation for this choice is the nice feature of the above scheme that the range of the radial grid is basis set dependent. The precision can be tuned with one single radial precision parameter. The smaller the radial precision, the better quality grid you obtain.

The basis set (more precisely the Gaussian primitives/exponents) are used to generate the atomic radial grid range. This means that a more diffuse basis set generates a more diffuse radial grid.

If you need a grid but you do not have a basis set or choose not to use a specific one, then you can feed the library with a fantasy basis set consisting of just two primitives. You can then adjust the range by making the exponents more steep or more diffuse.

Angular grid

The angular grid is generated according to Lebedev and Laikov [A quadrature formula for the sphere of the 131st algebraic order of accuracy, Russian Academy of Sciences Doklady Mathematics, Volume 59, Number 3, 1999, pages 477-481].

The angular grid is pruned. The pruning is a primitive linear interpolation between the minimum number and the maximum number of angular points per radial shell. The maximum number is reached at 0.2 times the Bragg radius of the center.

The higher the values for minimum and maximum number of angular points, the better.

For the minimum and maximum number of angular points the code will use the following table and select the closest number with at least the desired precision:

{6,    14,   26,   38,   50,   74,   86,   110,  146,
 170,  194,  230,  266,  302,  350,  434,  590,  770,
 974,  1202, 1454, 1730, 2030, 2354, 2702, 3074, 3470,
 3890, 4334, 4802, 5294, 5810}

Taking the same number for the minimum and maximum number of angular points switches off pruning.

How to include Numgrid in a CMake project

There are multiple ways to achieve this. Here is how to include the library using FetchContent:

cmake_minimum_required(VERSION 3.11 FATAL_ERROR)

project(example LANGUAGES CXX)


  GIT_TAG        e14bf969d68e7847f5e40f36816f61f245211a9b


if(NOT numgrid_POPULATED)

add_executable(example "")



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