| Description |
for Lebedev quadratures on the surface of the unit sphere at double precision.
**********Relative error is generally expected to be ~2.0E-14 [1]********
Lebedev quadratures are superbly accurate and efficient quadrature rules for approximating integrals of the form $v = \iint_{4\pi} f(\Omega) \ \ud \Omega$, where $\Omega$ is the solid angle on the surface of the unit sphere. Lebedev quadratures integrate all spherical harmonics up to $l = order$, where $degree \approx order(order+1)/3$. These grids may be easily combined with radial quadratures to provide robust cubature formulae. For example, see 'A. Becke, 1988c, J. Chem. Phys., 88(4), pp. 2547' (The first paper on tractable molecular Density Functional Theory methods, of which Lebedev grids and numerical cubature are an intrinsic part).
@param degree - positive integer specifying number of points in the requested quadrature. Allowed values are (degree -> order):
degree: { 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 };
order: {3,5,7,9,11,13,15,17,19,21,23,25,27,29,31,35,41,47,53,59,65,71,77, 83,89,95,101,107,113,119,125,131};
@return leb_tmp - struct containing fields:
x - x values of quadrature, constrained to unit sphere
y - y values of quadrature, constrained to unit sphere
z - z values of quadrature, constrained to unit sphere
w - quadrature weights, normalized to $4\pi$.
@example: $\int_S x^2+y^2-z^2 \ud \Omega = 4.188790204786399$
f = @(x,y,z) x.^2+y.^2-z.^2;
leb = getLebedevSphere(590);
v = f(leb.x,leb.y,leb.z);
int = sum(v.*leb.w);
@citation - Translated from a Fortran code kindly provided by Christoph van Wuellen (Ruhr-Universitaet, Bochum, Germany), which in turn came from the original C routines coded by Dmitri Laikov (Moscow State University, Moscow, Russia). The MATLAB implementation of this code is designed for benchmarking of new DFT integration techniques to be implemented in the open source Psi4 ab initio quantum chemistry program.
As per Professor Wuellen's request, any papers published using this code or its derivatives are requested to include the following citation:
[1] V.I. Lebedev, and D.N. Laikov
"A quadrature formula for the sphere of the 131st
algebraic order of accuracy"
Doklady Mathematics, Vol. 59, No. 3, 1999, pp. 477-481. |
