Density- and wavefunction-normalized Cartesian spherical harmonics forl≤ 20

Abstract

The widely used pseudoatom formalism [Stewart (1976).Acta Cryst. A32, 565–574; Hansen & Coppens (1978).Acta Cryst.A34, 909–921] 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 tol≤ 7 and the corresponding normalization coefficients were reported previously by Paturle & Coppens [Acta Cryst.(1988), A44, 6–7]. It was shown that the analytical form for normalization coefficients is available primarily forl≤ 4 [Hansen & Coppens, 1978; Paturle & Coppens, 1988; Coppens (1992).International Tables for Crystallography, Vol. B,Reciprocal space, 1st ed., edited by U. Shmueli, ch. 1.2. Dordrecht: Kluwer Academic Publishers; Coppens (1997).X-ray Charge Densities and Chemical Bonding. New York: Oxford University Press]. Only in very special cases it is possible to derive an analytical representation of the normalization coefficients for 4 <l≤ 7 (Paturle & Coppens, 1988). In most cases forl> 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 (Paturle & Coppens, 1988) method in the WolframMathematicasoftware to derive the Cartesian spherical harmonics forl≤ 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.