qc-quad

Quantum-chemistry quadrature package.

This package makes easy to set up so-called Lebedev-Laikov and Gaussian-Legendre grids. These grids are suitable for numerical integration in quantum chemistry applications.

Lebedev-Laikov grids provide coordinates and weights for an optimal integration over a surface of unit sphere. These grids are Golden standard in quantum chemistry because of their accuracy and the minimal number of the integrand-function evaluations. The implementation of the grids is derived from a Matlab implementation by John Burkardt https://people.sc.fsu.edu/~jburkardt/m_src/sphere_lebedev_rule/sphere_lebedev_rule.html

Gauss-Legendre grids provide coordinates and weights for an optimal integration in one dimension, over a finite segment. The Gauss-Legendre grids are suitable for generic functions. This is in contrast to the behavior of wave-functions along radial distance. However, I provide these grids here in order to organize an integration of generic functions over spherical volumes. This integration maintains a nearly constant density of integration points over a ball volume. This integration was useful in computing atomic contributed volumes https://www.tandfonline.com/doi/abs/10.1080/00268979100100941 in my first-principle tools https://first-principle-tools.ew.r.appspot.com .

A motivation for composing this package instead of using several packages out there in the open-source community would be "doing things right", in a testable, malleable, maintainable, cross-platform and parsimonious way.

There are several open-source implementations of Lebedev-Laikov grids. The quantum-chemistry package Horton is using a Python binding to a C++ module implementing the grids https://github.com/theochem/horton. A sub-repository grid of theochem repositories offers the Lebedev-Laikov grids stored in .npy files. There is an exhaustive set of quadratures in repository https://github.com/sigma-py/quadpy . However, the code of quadpy is obfuscated. The repository https://github.com/Rufflewind/lebedev_laikov provides the grids via a Python binding to a source code in C language. Similarly, https://github.com/dftlibs/numgrid provides the grids through a binding to a Fortran source. There is a pure Python implementation of Lebedev-Laikov grids https://github.com/gabrielelanaro/pyquante/ . However, it provides the grids for the first 11 (6, 14, 26, 38, 50, 74, 86, 110, 146, 170, 194) out of 32 grids implemented by Dmitri Laikov. Finally, there is an issue in SciPy with a feature request of Lebedev grids https://github.com/scipy/scipy/issues/11929

Current package qc-quad is open source. It provides Lebedev-Laikov grids for 32 grids published by Dmitri Laikov. It is a pure Python package. This makes this implementation easily testable (pytest), extendable and usable.

References:

Axel Becke, "A multicenter numerical integration scheme for polyatomic molecules",
Journal of Chemical Physics,
Volume 88, Number 4, 15 February 1988, pages 2547-2553. 

Vyacheslav Lebedev, Dmitri 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.

Lawrence R. Dodd & Doros N. Theodorou, "Analytical treatment of the volume and surface area
of molecules formed by an arbitrary collection of unequal spheres intersected by planes",
Molecular Physics, 72:6, 1991, pages 1313-1345, DOI: 10.1080/00268979100100941

Installation

pip install qc-quad

Developer installation

Installation could be done with poetry = "^1.3.1"

poetry install

Usage

This is a part of some quantum-chemistry software. Some use cases can be seen in the tests.

Roadmap

Let's see what does the community thinks, but here are two points worth nothing.

Note, that to become truly quantum-chemical quadrature this package needs a smooth separation of space which depends on the geometry of the molecule. Please, see works by Axel Becke and onwards. Perhaps, a one-dimensional grid suitable for integrating exponentially decaying functions, together with a pruning scheme for Lebedev-Laikov grids would be suitable in this package. This should be easy to bolt in later for somebody who needs it.

The major difficulty in setting up of the Lebedev-Laikov grids is the absence of their definition for a given number of points. The grids are provided for minimum of 6 and maximum of 5810 points: 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 and 5810, 32 grids in total. Each of the 32 grids, is composed of a set of 6 kinds of grids composed of 6, 12, 8, 24, 24 and 48 points. To the best of my knowledge, no implementation out in the open-source offers any generalizations. For example, a construction of grids with more points by attaching the 48-point grids together or construction of a 36-point grid by using three of the 12-point grids is not implemented although this should be feasible.

Contributing

• PEP8 code formatting is mandatory.
• The bug fixes are welcome.
• Small improvements are welcome.
• Definition of the hard-coded coefficients is especially welcome.

Authors and acknowledgment

James Talman https://www.sciencedirect.com/science/article/abs/pii/0010465583901261 drew my attention to Lebedev-Laikov grids.

License

MIT license: no guarantee, free to use anywhere.

Project status

Initial release is done.