skip to main content

DOE PAGESDOE PAGES

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

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:
 [1] ;  [2]
  1. Middle Tennessee State University, Murphreesboro, TN (United States). Computational Science Program
  2. Middle Tennessee State University, Murphreesboro, TN (United States). Computational Science Program and Dept. of Chemistry
Publication Date:
OSTI Identifier:
1172304
Grant/Contract Number:
SC0005094
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
Research Org:
Middle Tennessee State University, Murphreesboro, TN (United States)
Sponsoring Org:
USDOE Office of Science (SC), Biological and Environmental Research (BER) (SC-23)
Country of Publication:
United States
Language:
English
Subject:
74 ATOMIC AND MOLECULAR PHYSICS X-ray diffraction, charge density, spherical harmonics