Synopsis

IntegratorXX is a modern C++ library for the generation of atomic and molecular grids for quantum chemistry calculations. Among the most important applications of these grids is the evaluation of exchange--correlation (XC) related quantities (energies, potentials, etc) required for density functional theory calculations.

IntegratorXX provides a uniform interface for the generation of primitive, radial and solid angle quadratures, as well as there combination into spherical grids.

Design Goals

• Provide stable, reusable, and reproducible implementations of the various atomic and molecular grids commonly encountered in quantum chemistry calculations
• Develop a modern, modular, extensible C++ API to allow for the implementation and validation new atomic and molecular quadrature schemes.
• Provide complete C and Python interfaces to allow reusing the implementation in projects written in these languages

Dependencies

• CMake (3.17+)
• Modern C++ compiler (C++17 compliant)

Major Contributors

• David Williams-Young - LBNL (dbwy at lbl dot gov)
• Susi Lehtola - University of Helsinki

Implemented Quadratures

Here we list the quadratures currently implemented in IntegratorXX. Please refer to the source for appropriate references.

Primitive Quadratures

Primitive quadratures are those generated on a finite bound (e.g. Gauss quadrature rules). The general software design pattern of IntegratorXX is to build up higher-order quadrature rules (e.g. radial transformation, etc) from these primitive quadratures.

Quadrature Name	Domain	C++ Class
Gauss-Chebyshev (First Kind)	$(-1,1)$	GaussChebyshev1
Gauss-Chebyshev (Second Kind)	$[-1,1]$	GaussChebyshev2
Gauss-Chebyshev (Third Kind)	$(0,1)$	GaussChebyshev3
Gauss-Legendre	$[-1,1]$	GaussLegendre
Gauss-Lobatto	$[-1,1]$	GaussLobatto
Trapezoid Rule	$[0,1]$	UniformTrapezoid

Radial Quadratures

Radial quadratures are convolutions of primitive quadrature rules with a radial transformation scheme (mapping the natural domain of the primitive quadrature to positive semi-indefinite). The jacobian of the transformation is included in the radial weights while the radial component of the spherical volume element ($r^2$) is not.

Quadrature Name	Domain	C++ Class
Becke	$(0,\infty)$	Becke
Murray-Handy-Laming (MHL)	$(0,\infty)$	MurrayHandyLaming
Mura-Knowles (MK)	$(0,\infty)$	MuraKnowles
Treutler-Ahlrichs (TA, M3 + M4)	$(0,\infty)$	TreutlerAhlrichs

Angular Quadratures

Angular quadratures integrate over $S^2$ (solid angle). These have typically been manually constructed to integrate spherical harmonics up to a specific order, and are thus only compatible witch specific grid orders (see note below).

Quadrature Name	Domain	C++ Class
Ahrens-Beylkin	$S^2$	AhrensBeylkin
Delley	$S^2$	Delley
Lebedev-Laikov	$S^2$	LebedevLaikov
Womersley	$S^2$	Womersley

A Note on Angular Quadratures

All of the currently implemented angular quadrature schemes are only compatible with specific grid orders corresponding to specific algebraic orders of spherical harmonics they integrate exactly. The construction of the angular grids takes the number of points as argument, and will fail if the grid order is incompatible. As these magic numbers are different for each of the quadratures, we provide a set of look-up functions which can safely produce compatible grid orders:

using angular_type = LebedevLaikov<double>; // FP64 LL grid, similar for other implementations
using traits = IntegratorXX::quadrature_traits<angular_type>;

auto npts  = traits::npts_by_algebraic_order(order); // Return the grid order associated with a particular algebraic order
auto order = traits::algebraic_order_by_npts(npts);  // Return the algebratic order associated with a grid order
auto next_order = traits::next_algebraic_order(order); // Return the next largest (inclusive) algebratic order compatible with `angular_type` 

Radial Pruning Schemes

For the generation of spherical quadratures, IntegratorXX additionally supports the following radial pruning schemes:

Name	Description	C++ Specifier
Unpruned	Do not perform pruning	PruningScheme::Unpruned
Robust	The Psi4 "robust" pruning scheme	PruningScheme::Robust
Treutler	The Treutler-Ahlrichs pruning scheme	PruntinScheme::Treutler

Example Usage

Many example usages for 1-d quadratures (i.e. primitive and radial) can be found in test/1d_quadratures.cxx and test/spherical_generator.cxx. Below is a simple invocation example for the generation of an atomic sphere via the runtime generator:

using namespace IntegratorXX;                         // Import namespace
auto rad_spec = radial_from_string("MuraKnowles");    // MK Radial scheme
auto ang_spec = angular_from_string("AhrensBeylkin"); // AH Angular scheme
size_t nrad   = 99;
size_t nang   = 372;
double rscal  = 2.0;

// Generate Grid Specification
UnprunedSphericalGridSpecification unp{
  rad_spec, nrad, rscal, ang_spec, nang
};
auto pruning_spec = create_pruned_spec(PruningScheme::Robust, unp); 

// Generate Quadrature
auto sph_quad = SphericalGridFactory::generate_grid(pruning_spec);

size_t npts = sph_quad->npts();
const auto& points  = sph_quad->points();  // std::vector<std::array<double,3>>
const auto& weights = sph_quad->weights(); // std::vector<double>

Header-only builds

With the exception of the runtime grid generator, the entirety of the grid specification in IntegratorXX is header-only and can operate without pre-compiled components. By default, the runtime generator is pre-compiled to improve the efficiency of the compilation process and to avoid excessive build times in complex projects with aggressive compiler optimization. N.B. it is highly recommend that users maintain this default behavior to avoid excessive compilation sizes and build times.

IntegratorXX also allows for header-only use of the runtime generator by setting INTEGRATORXX_HEADER_ONLY=ON. This feature also allows for circumvention of the CMake build system by simply including the requisite implementation header.

To use the runtime generator header-only, one needs to include <integratorxx/generators/impl/impl.hpp> exactly once per project, otherwise duplicate / incompatible symbols will occur.

Contributing and Bug Reports

We welcome any and all contributions and encourage bug reports. Please use the Issue and Pull Request features as appropriate.

License

IntegratorXX is made freely available under the terms of the 3-Clause BSD license. See LICENSE.txt for details.