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. 1517
Section 1.2.8. Fourier transform of orbital products ^{a}Department of Chemistry, Natural Sciences & Mathematics Complex, State University of New York at Buffalo, Buffalo, New York 14260–3000, USA 
If the wavefunction is written as a sum over normalized Slater determinants, each representing an antisymmetrized combination of occupied molecular orbitals expressed as linear combinations of atomic orbitals , i.e. , the electron density is given by (Stewart, 1969a) with n_{i} = 1 or 2. The coefficients are the populations of the orbital product density functions and are given by
For a multiSlater determinant wavefunction the electron density is expressed in terms of the occupied natural spin orbitals, leading again to (1.2.8.2) but with noninteger values for the coefficients n_{i}.
The summation (1.2.8.1) consists of one and twocentre terms for which and are centred on the same or on different nuclei, respectively. The latter represent the overlap density, which is only significant if and have an appreciable value in the same region of space.
If the atomic basis consists of hydrogenic type s, p, d, f, … orbitals, the basis functions may be written asor which gives for corresponding values of the orbital productsand respectively, where it has been assumed that the radial function depends only on l.
Because the spherical harmonic functions form a complete set, their products can be expressed as a linear combination of spherical harmonics. The coefficients in this expansion are the Clebsch–Gordan coefficients (Condon & Shortley, 1957), defined by or the equivalent definition The vanish, unless is even, and .
The corresponding expression for is with and for , and and for and .
Values of C and for are given in Tables 1.2.8.1 and 1.2.8.2. They are valid for the functions and with normalization and .


By using (1.2.8.5a) or (1.2.8.5c), the onecentre orbital products are expressed as a sum of spherical harmonic functions. It follows that the onecentre orbital product density basis set is formally equivalent to the multipole description, both in real and in reciprocal space. To obtain the relation between orbital products and the chargedensity functions, the righthand side of (1.2.8.5c) has to be multiplied by the ratio of the normalization constants, as the wavefunctions and chargedensity functions are normalized in a different way as described by (1.2.7.3a) and (1.2.7.3b). Thus where . The normalization constants and are given in Table 1.2.7.1, while the coefficients in the expressions (1.2.8.6) are listed in Table 1.2.8.3.

Fourier transform of the electron density as described by (1.2.8.1) requires explicit expressions for the twocentre orbital product scattering. Such expressions are described in the literature for both Gaussian (Stewart, 1969b) and Slatertype (Bentley & Stewart, 1973; Avery & Ørmen, 1979) atomic orbitals. The expressions can also be used for Hartree–Fock atomic functions, as expansions in terms of Gaussian (Stewart, 1969b, 1970; Stewart & Hehre, 1970; Hehre et al., 1970) and Slatertype (Clementi & Roetti, 1974) functions are available for many atoms.
References
Avery, J. & Ørmen, P.J. (1979). Generalized scattering factors and generalized Fourier transforms. Acta Cryst. A35, 849–851.Bentley, J. & Stewart, R. F. (1973). Twocentre calculations for Xray scattering. J. Comput. Phys. 11, 127–145.
Clementi, E. & Roetti, C. (1974). Roothaan–Hartree–Fock atomic wavefunctions. At. Data Nucl. Data Tables, 14, 177–478.
Condon, E. V. & Shortley, G. H. (1957). The theory of atomic spectra. London, New York: Cambridge University Press.
Hehre, W. J., Ditchfield, R., Stewart, R. F. & Pople, J. A. (1970). Selfconsistent molecular orbital methods. IV. Use of Gaussian expansions of Slatertype orbitals. Extension to secondrow molecules. J. Chem. Phys. 52, 2769–2773.
Stewart, R. F. (1969a). Generalized Xray scattering factors. J. Chem. Phys. 51, 4569–4577.
Stewart, R. F. (1969b). Small Gaussian expansions of atomic orbitals. J. Chem. Phys. 50, 2485–2495.
Stewart, R. F. (1970). Small Gaussian expansions of Slatertype orbitals. J. Chem. Phys. 52, 431–438.
Stewart, R. F. & Hehre, W. J. (1970). Small Gaussian expansions of atomic orbitals: secondrow atoms. J. Chem. Phys. 52, 5243–5247.