Density- and wavefunction-normalized Cartesian spherical harmonics for l ≤ 20

Michael, J. Robert; Volkov, Anatoliy

doi:10.1107/s2053273314024838

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Density- and wavefunction-normalized Cartesian spherical harmonics for l ≤ 20

Michael, J. Robert·Volkov, Anatoliy

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Abstract

The widely used pseudoatom formalism in experimental X-ray charge-density studies makes use of real spherical harmonics when describing the angular component of aspherical deformations of the atomic electron density in molecules and crystals. The analytical form of the density-normalized Cartesian spherical harmonic functions for up to l ≤ 7 and the corresponding normalization coefficients were reported previously by Paturle & Coppens. It was shown that the analytical form for normalization coefficients is available primarily for l ≤ 4. Only in very special cases it is possible to derive an analytical representation of the normalization coefficients for 4 < l ≤ 7. In most cases for l > 4 the density normalization coefficients were calculated numerically to within seven significant figures. In this study we review the literature on the density-normalized spherical harmonics, clarify the existing notations, use the Paturle–Coppens method in the Wolfram Mathematica software to derive the Cartesian spherical harmonics for l ≤ 20 and determine the density normalization coefficients to 35 significant figures, and computer-generate a Fortran90 code. The article primarily targets researchers who work in the field of experimental X-ray electron density, but may be of some use to all who are interested in Cartesian spherical harmonics.

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74 ATOMIC AND MOLECULAR PHYSICS X-ray diffraction, charge density, spherical harmonics

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Michael, J. Robert, Volkov, Anatoliy. 2015-03-01. Density- and wavefunction-normalized Cartesian spherical harmonics for l ≤ 20. https://doi.org/10.1107/s2053273314024838

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