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

J Robert Michael1, Anatoliy Volkov1

  • 1Computational Science Program, Middle Tennessee State University, Murfreesboro, TN 37132, USA.

Acta Crystallographica. Section A, Foundations and Advances
|March 3, 2015
PubMed
Summary
This summary is machine-generated.

This study extends the use of spherical harmonics in X-ray charge density analysis by deriving accurate density normalization coefficients for higher angular momentum (l ≤ 20). This provides a more precise computational tool for understanding electron density distributions in crystals.

Keywords:
charge densitypseudoatom modelspherical harmonics

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Area of Science:

  • Crystallography and Materials Science
  • Quantum Chemistry and Computational Physics

Background:

  • The pseudoatom formalism in X-ray charge density studies relies on spherical harmonics to model electron density.
  • Existing analytical forms for density normalization coefficients are limited, primarily up to l ≤ 4, with numerical calculations used for higher l values.
  • Previous work provided analytical forms for density-normalized Cartesian spherical harmonics up to l ≤ 7, but analytical coefficients were scarce for l > 4.

Purpose of the Study:

  • To review and clarify notations for density-normalized spherical harmonics.
  • To derive analytical forms for Cartesian spherical harmonics up to l ≤ 20.
  • To determine density normalization coefficients to 35 significant figures and generate computational code.

Main Methods:

  • Literature review on density-normalized spherical harmonics and their notations.
  • Application of the Paturle-Coppens method using Wolfram Mathematica software.
  • Derivation of Cartesian spherical harmonics for l ≤ 20 and computation of normalization coefficients.
  • Generation of a Fortran90 code for practical application.

Main Results:

  • Analytical forms for Cartesian spherical harmonics derived for l ≤ 20.
  • Density normalization coefficients determined with high precision (35 significant figures).
  • A Fortran90 code is provided for researchers.

Conclusions:

  • The study provides a computationally robust and accurate method for handling higher-order spherical harmonics in charge density analysis.
  • The generated code and precise coefficients enhance the capabilities of experimental X-ray electron density studies.
  • This work benefits researchers in X-ray crystallography and those interested in the detailed analysis of electron density.