gau2grid

A collocation code for computing gaussians on a grid of the form:

out_Lp = x^l y^m z^n \sum_i coeff_i e^(exponent_i * (|center - p|)^2)

Where the returned matrix dimension are the angular momentum (L) by number of requested points (p).

import gau2grid
import numpy as np

# Build coordinates along the Z axis
>>> xyz = np.zeros((3, 5))
>>> xyz[2] = np.arange(5)

# Compute a 's' gaussian with a scaling and exponent of one at the origin
>>> ret = gau2grid.collocation(xyz, 0, [1], [1], [0, 0, 0])
>>> print(ret["PHI"])
[[  1.00000e+00   3.67879e-01   1.83156e-02   1.23409e-04   1.12535e-07]]

# Compute a 'p' gaussian with a scaling and exponent of one at the origin
>>> ret = gau2grid.collocation(xyz, 1, [1], [1], [0, 0, 0], spherical=False)
>>> print(ret["PHI"])
[[  0.00000e+00   0.00000e+00   0.00000e+00   0.00000e+00   0.00000e+00]
 [  0.00000e+00   0.00000e+00   0.00000e+00   0.00000e+00   0.00000e+00]
 [  0.00000e+00   3.67879e-01   3.66312e-02   3.70229e-04   4.50140e-07]]

# Note that the X and Y components are zero as they are orthogonal to our Z vector.

The returned matrix can be in either cartesian or regular solid harmonics. There are currently three algorithms in which to compute these collocation matrices:

• Optimize C: A autogenerated C library that optimizes for cache, vectorization, and sparsity. Fastest, requires compilation, found at gau2grid.collocation.
• Optimized/Generated NumPy: A exploratory tool to examine the sparsity in the gaussians. No compilation required, found at gau2grid.np_gen.collocation.
• NumPy Reference: A simple NumPy-based loop code. No compilation required, found at gau2grid.ref.collocation.

Output ordering

Ordering of the functions for each angular momentum is systematic based off various schemes detailed below. For cartesian ordering the only current option is the "row" or lexicographical order:

• L=0 (s): 0
• L=1 (p): x y z
• L=2 (d): xx xy xz yy yz zz

For spherical order there exists the "Gaussian" and "CCA" orderings:

• Gaussian: R_0, R^+_1, R^-_1, ..., R^+_l, R^-_l
• CCA: R^-_(l), R^-_(l-1), ..., R_0, ..., R^+_(l-1), R^+_l

The compiled ordering can be queried through the following two functions:

>>> gg.spherical_order()
'gaussian'
>>> gg.cartesian_order()
'cca'

Building Gau2Grid

The C library is built with CMake and has C no required dependancies other than the standard library. A CMake and build example can found below:

cmake -H. -Bobjdir
cd objdir; make -j2

Several common CMake options are as follow:

• -DPYTHON_EXECUTABLE - Path to the desired Python executable
• -DMAX_AM - Maximum angular momentum to compile to, default 6
• -DCMAKE_INSTALL_PREFIX - Installation directory
• -DCARTESIAN_ORDER - The cartesian ordering of the basis functions (row)
• -DSPHERICAL_ORDER - The spherical ordering of the basis functions (cca, gaussian)

Python installation

The gau2grid program (without the optimized C library) can be installed using the canonical setup.py script,

python setup.py install

Authors

This code was inspired by a number of folks and quite a few provided excellent advice.

• Daniel G. A. Smith - Code author
• Rob M. Parrish - Author of the Psi4 section which contains the original equations
• Lori A. Burns - CMake, building, and library linking
• Andy C. Simmonett - RSH coefficients
• Ben Pritchard - Generator and vectorization recommendations