Density- and wavefunction-normalized Cartesian spherical harmonics for l ≤ 20
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 forl ≤ 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 Mathematicasoftware 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.
- Authors:
-
- Middle Tennessee State University, Murphreesboro, TN (United States). Computational Science Program
- Middle Tennessee State University, Murphreesboro, TN (United States). Computational Science Program and Dept. of Chemistry
- Publication Date:
- Research Org.:
- Middle Tennessee State Univ., Murfreesboro, TN (United States)
- Sponsoring Org.:
- USDOE Office of Science (SC), Biological and Environmental Research (BER)
- OSTI Identifier:
- 1172304
- Grant/Contract Number:
- SC0005094
- Resource Type:
- Accepted Manuscript
- Journal Name:
- Acta Crystallographica. Section A, Foundations and Advances (Online)
- Additional Journal Information:
- Journal Name: Acta Crystallographica. Section A, Foundations and Advances (Online); Journal Volume: 71; Journal Issue: 2; Journal ID: ISSN 2053-2733
- Publisher:
- International Union of Crystallography
- Country of Publication:
- United States
- Language:
- English
- Subject:
- 74 ATOMIC AND MOLECULAR PHYSICS; X-ray diffraction, charge density, spherical harmonics
Citation Formats
Michael, J. Robert, and Volkov, Anatoliy. Density- and wavefunction-normalized Cartesian spherical harmonics for l ≤ 20. United States: N. p., 2015.
Web. doi:10.1107/S2053273314024838.
Michael, J. Robert, & Volkov, Anatoliy. Density- and wavefunction-normalized Cartesian spherical harmonics for l ≤ 20. United States. https://doi.org/10.1107/S2053273314024838
Michael, J. Robert, and Volkov, Anatoliy. Sun .
"Density- and wavefunction-normalized Cartesian spherical harmonics for l ≤ 20". United States. https://doi.org/10.1107/S2053273314024838. https://www.osti.gov/servlets/purl/1172304.
@article{osti_1172304,
title = {Density- and wavefunction-normalized Cartesian spherical harmonics for l ≤ 20},
author = {Michael, J. Robert and Volkov, Anatoliy},
abstractNote = {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 forl ≤ 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 Mathematicasoftware 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.},
doi = {10.1107/S2053273314024838},
journal = {Acta Crystallographica. Section A, Foundations and Advances (Online)},
number = 2,
volume = 71,
place = {United States},
year = {Sun Mar 01 00:00:00 EST 2015},
month = {Sun Mar 01 00:00:00 EST 2015}
}
Web of Science
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Works referencing / citing this record:
Fast analytical evaluation of intermolecular electrostatic interaction energies using the pseudoatom representation of the electron density. I. The Löwdin α-function method
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