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}
}
Free Publicly Available Full Text
Publisher's Version of Record
Other availability
Search WorldCat to find libraries that may hold this journal
Cited by: 6 works
Citation information provided by
Web of Science
Web of Science
Save to My Library
You must Sign In or Create an Account in order to save documents to your library.
Works referenced in this record:
Net atomic charges and molecular dipole moments from spherical-atom X-ray refinements, and the relation between atomic charge and shape
journal, January 1979
- Coppens, P.; Guru Row, T. N.; Leung, P.
- Acta Crystallographica Section A, Vol. 35, Issue 1
Testing aspherical atom refinements on small-molecule data sets
journal, November 1978
- Hansen, N. K.; Coppens, P.
- Acta Crystallographica Section A, Vol. 34, Issue 6
Difference densities by least-squares refinement. II. Tetracyanocyclobutane
journal, January 1975
- Harel, M.; Hirshfeld, F. L.
- Acta Crystallographica Section B Structural Crystallography and Crystal Chemistry, Vol. 31, Issue 1
Difference densities by least-squares refinement: fumaramic acid
journal, April 1971
- Hirshfeld, F. L.
- Acta Crystallographica Section B Structural Crystallography and Crystal Chemistry, Vol. 27, Issue 4
Appendix 2. A Deformation Density Refinement Program
journal, January 1977
- Hirshfeld, F. L.
- Israel Journal of Chemistry, Vol. 16, Issue 2-3
Some properties of the coupling coefficients of real spherical harmonics and their relation to Gaunt coefficients
journal, September 1996
- Homeier, Herbert H. H.; Steinborn, E. Otto
- Journal of Molecular Structure: THEOCHEM, Vol. 368
Introducing the Intel i860 64-bit microprocessor
journal, August 1989
- Kohn, L.; Margulis, N.
- IEEE Micro, Vol. 9, Issue 4
New Directions in Pseudoatom-Based X-Ray Charge Density Analysis
book, December 2010
- Koritsanszky, Tibor; Volkov, Anatoliy; Chodkiewicz, Michal
- Electron Density and Chemical Bonding II
Normalization factors for spherical harmonic density functions
journal, January 1988
- Paturle, A.; Coppens, P.
- Acta Crystallographica Section A Foundations of Crystallography, Vol. 44, Issue 1
Rotation matrices for real spherical harmonics: general rotations of atomic orbitals in space-fixed axes
journal, January 2007
- Pinchon, Didier; Hoggan, Philip E.
- Journal of Physics A: Mathematical and Theoretical, Vol. 40, Issue 7
Valence Structure from X‐Ray Diffraction Data: Physical Properties
journal, August 1972
- Stewart, Robert F.
- The Journal of Chemical Physics, Vol. 57, Issue 4
Electron population analysis with rigid pseudoatoms
journal, July 1976
- Stewart, R. F.
- Acta Crystallographica Section A, Vol. 32, Issue 4
On the calculation of the electrostatic potential, electric field and electric field gradient from the aspherical pseudoatom model
journal, August 2006
- Volkov, Anatoliy; King, Harry F.; Coppens, Philip
- Acta Crystallographica Section A Foundations of Crystallography, Vol. 62, Issue 5
On the basis-set dependence of local and integrated electron density properties: Application of a new computer program for quantum-chemical density analysis
journal, July 2009
- Volkov, Anatoliy; Koritsanszky, Tibor; Chodkiewicz, Michal
- Journal of Computational Chemistry, Vol. 30, Issue 9
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
journal, September 2018
- Nguyen, Daniel; Kisiel, Zbigniew; Volkov, Anatoliy
- Acta Crystallographica Section A Foundations and Advances, Vol. 74, Issue 5
Fast analytical evaluation of intermolecular electrostatic interaction energies using the pseudoatom representation of the electron density. II. The Fourier transform method
journal, April 2019
- Nguyen, Daniel; Volkov, Anatoliy
- Acta Crystallographica Section A Foundations and Advances, Vol. 75, Issue 3