International
Tables for Crystallography Volume B Reciprocal space Edited by U. Shmueli © International Union of Crystallography 2006 
International Tables for Crystallography (2006). Vol. B. ch. 1.2, p. 15
Section 1.2.7.2. Reciprocalspace description of aspherical atoms ^{a}Department of Chemistry, Natural Sciences & Mathematics Complex, State University of New York at Buffalo, Buffalo, New York 142603000, USA 
The asphericalatom form factor is obtained by substitution of (1.2.7.6) in expression (1.2.4.3a): In order to evaluate the integral, the scattering operator must be written as an expansion of products of spherical harmonic functions. In terms of the complex spherical harmonic functions, the appropriate expression is (Weiss & Freeman, 1959; CohenTannoudji et al., 1977)
The Fourier transform of the product of a complex spherical harmonic function with normalization and an arbitrary radial function follows from the orthonormality properties of the spherical harmonic functions, and is given by where is the lthorder spherical Bessel function (Arfken, 1970), and θ and ϕ, β and γ are the angular coordinates of r and S, respectively.
For the Fourier transform of the real spherical harmonic functions, the scattering operator is expressed in terms of the real spherical harmonics: which leads to Since occurs on both sides, the expression is independent of the normalization selected. Therefore, for the Fourier transform of the density functions
In (1.2.7.8b) and (1.2.7.8c), , the Fourier–Bessel transform, is the radial integral defined as of which in expression (1.2.4.3) is a special case. The functions for Hartree–Fock valence shells of the atoms are tabulated in scatteringfactor tables (IT IV, 1974). Expressions for the evaluation of using the radial function (1.2.7.5a–c) have been given by Stewart (1980) and in closed form for (1.2.7.5a) by Avery & Watson (1977) and Su & Coppens (1990). The closedform expressions are listed in Table 1.2.7.4.

Expressions (1.2.7.8) show that the Fourier transform of a directspace spherical harmonic function is a reciprocalspace spherical harmonic function with the same l, m, or, in other words, the spherical harmonic functions are Fouriertransform invariant.
The scattering factors of the aspherical density functions in the multipole expansion (1.2.7.6) are thus given by
The reciprocalspace spherical harmonic functions in this expression are identical to the functions given in Table 1.2.7.1, except for the replacement of the direction cosines x, y and z by the direction cosines of the scattering vector S.
References
International Tables for Xray Crystallography (1974). Vol. IV. Birmingham: Kynoch Press. (Present distributor Kluwer Academic Publishers, Dordrecht.)Google ScholarArfken, G. (1970). Mathematical models for physicists, 2nd ed. New York, London: Academic Press.Google Scholar
Avery, J. & Watson, K. J. (1977). Generalized Xray scattering factors. Simple closedform expressions for the onecentre case with Slatertype orbitals. Acta Cryst. A33, 679–680.Google Scholar
CohenTannoudji, C., Diu, B. & Laloe, F. (1977). Quantum mechanics. New York: John Wiley and Paris: Hermann.Google Scholar
Stewart, R. F. (1980). Electron and magnetization densities in molecules and solids, edited by P. J. Becker, pp. 439–442. New York: Plenum.Google Scholar
Su, Z. & Coppens, P. (1990). Closedform expressions for Fourier– Bessel transforms of Slatertype functions. J. Appl. Cryst. 23, 71–73.Google Scholar
Weiss, R. J. & Freeman, A. J. (1959). Xray and neutron scattering for electrons in a crystalline field and the determination of outer electron configurations in iron and nickel. J. Phys. Chem. Solids, 10, 147–161.Google Scholar