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version (39.2 KB) by Robert Parrish
Produces Lebedev Grids of up to the 131st order for integration on the unit sphere.


Updated 26 Mar 2010

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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.

Cite As

Robert Parrish (2020). getLebedevSphere (, MATLAB Central File Exchange. Retrieved .

Comments and Ratings (3)

Roman Balabin

Just great! More than helpful.

Hannes Helmholz

Derek O'Connor

Prof V I Lebedev died last week.
From NA Digest, V. 10, # 13
Vyacheslav Ivanovich Lebedev, January 27, 1930 - March 22, 2010

Prof. Vyacheslav Ivanovich Lebedev died suddenly on his way to work in Moscow, Russia on March 22, 2010.

Prof. V.I Lebedev was a Ph.D. student of S.L. Sobolev. He worked at the
Kurchatov Institute and Soviet/Russian Academy of Sciences (RAS), and
taught students at the Moscow State University and Moscow Institute of
Physics and Technology. He authored over hundred papers and several
books, most noticeably, "Numerical methods in the theory of neutron
transport" jointly with G.I. Marchuk and "Functional Analysis in
Computational Mathematics," based on his lectures. He graduated over 15

Derek O'Connor

MATLAB Release Compatibility
Created with R2007a
Compatible with any release
Platform Compatibility
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Inspired: Geometric light field model

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