International
Tables for Crystallography Volume B Reciprocal space Edited by U. Shmueli © International Union of Crystallography 2010 
International Tables for Crystallography (2010). Vol. B, ch. 1.2, pp. 1215
Section 1.2.7. Beyond the sphericalatom description: the atomcentred spherical harmonic expansion ^{a}Department of Chemistry, Natural Sciences & Mathematics Complex, State University of New York at Buffalo, Buffalo, New York 14260–3000, USA 
Even though the sphericalatom approximation is often adequate, atoms in a crystal are in a nonspherical environment; therefore, an accurate description of the atomic electron density requires nonspherical density functions. In general, such density functions can be written in terms of the three polar coordinates r, θ and ϕ. Under the assumption that the radial and angular parts can be separated, one obtains for the density function:
The angular functions Θ are based on the spherical harmonic functions defined by with , where are the associated Legendre polynomials (see Arfken, 1970).
The real spherical harmonic functions , , are obtained as a linear combination of : and The normalization constants are defined by the conditions which are appropriate for normalization of wavefunctions. An alternative definition is used for chargedensity basis functions: The functions and differ only in the normalization constants. For the spherically symmetric function , a population parameter equal to one corresponds to the function being populated by one electron. For the nonspherical functions with , a population parameter equal to one implies that one electron has shifted from the negative to the positive lobes of the function.
The functions and can be expressed in Cartesian coordinates, such that and where the are Cartesian functions. The relations between the various definitions of the real spherical harmonic functions are summarized by in which the direction of the arrows and the corresponding conversion factors define expressions of the type (1.2.7.4). The expressions for with are listed in Table 1.2.7.1, together with the normalization factors and .

The spherical harmonic functions are mutually orthogonal and form a complete set, which, if taken to sufficiently high order, can be used to describe any arbitrary angular function.
The spherical harmonic functions are often referred to as multipoles since each represents the components of the charge distribution , which gives nonzero contribution to the integral , where is an electrostatic multipole moment. Terms with increasing l are referred to as monopolar , dipolar , quadrupolar , octapolar , hexadecapolar , triacontadipolar and hexacontatetrapolar .
Sitesymmetry restrictions for the real spherical harmonics as given by Kara & KurkiSuonio (1981) are summarized in Table 1.2.7.2.

In cubic space groups, the spherical harmonic functions as defined by equations (1.2.7.2) are no longer linearly independent. The appropriate basis set for this symmetry consists of the `Kubic harmonics' of Von der Lage & Bethe (1947). Some loworder terms are listed in Table 1.2.7.3. Both wavefunction and densityfunction normalization factors are specified in Table 1.2.7.3.

A related basis set of angular functions has been proposed by Hirshfeld (1977). They are of the form , where is the angle with a specified set of polar axes. The Hirshfeld functions are identical to a sum of spherical harmonics with , , for , as shown elsewhere (Hirshfeld, 1977).
The radial functions can be selected in different manners. Several choices may be made, such aswhere the coefficient may be selected by examination of products of hydrogenic orbitals which give rise to a particular multipole (Hansen & Coppens, 1978). Values for the exponential coefficient may be taken from energyoptimized coefficients for isolated atoms available in the literature (Clementi & Raimondi, 1963). A standard set has been proposed by Hehre et al. (1969). In the bonded atom, such values are affected by changes in nuclear screening due to migrations of charge, as described in part by equation (1.2.6.1).
Other alternatives are:orwhere L is a Laguerre polynomial of order n and degree (2l + 2).
In summary, in the multipole formalism the atomic density is described byin which the leading terms are those of the kappa formalism [expressions (1.2.6.1), (1.2.6.2)]; the subscript p is either + or −.
The expansion in (1.2.7.6) is frequently truncated at the hexadecapolar (l = 4) level. For atoms at positions of high site symmetry the first allowed functions may occur at higher l values. For trigonally bonded atoms in organic molecules the l = 3 terms are often found to be the most significantly populated deformation functions.
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 bywhere 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.Arfken, G. (1970). Mathematical Models for Physicists, 2nd ed. New York, London: Academic Press.
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.
Clementi, E. & Raimondi, D. L. (1963). Atomic screening constants from SCF functions. J. Chem. Phys. 38, 2686–2689.
CohenTannoudji, C., Diu, B. & Laloe, F. (1977). Quantum Mechanics. New York: John Wiley and Paris: Hermann.
Hansen, N. K. & Coppens, P. (1978). Testing aspherical atom refinements on smallmolecule data sets. Acta Cryst. A34, 909–921.
Hehre, W. J., Stewart, R. F. & Pople, J. A. (1969). Selfconsistent molecular orbital methods. I. Use of Gaussian expansions of Slatertype atomic orbitals. J. Chem. Phys. 51, 2657–2664.
Hirshfeld, F. L. (1977). A deformation density refinement program. Isr. J. Chem. 16, 226–229.
Kara, M. & KurkiSuonio, K. (1981). Symmetrized multipole analysis of orientational distributions. Acta Cryst. A37, 201–210.
Stewart, R. F. (1980). Electron and Magnetization Densities in Molecules and Solids, edited by P. J. Becker, pp. 439–442. New York: Plenum.
Su, Z. & Coppens, P. (1990). Closedform expressions for Fourier–Bessel transforms of Slatertype functions. J. Appl. Cryst. 23, 71–73.
Von der Lage, F. C. & Bethe, H. A. (1947). A method for obtaining electronic functions and eigenvalues in solids with an application to sodium. Phys. Rev. 71, 612–622.
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.