International
Tables for
Crystallography
Volume B
Reciprocal space
Edited by U. Shmueli

International Tables for Crystallography (2006). Vol. B. ch. 1.2, p. 15   | 1 | 2 |

## Section 1.2.7.2. Reciprocal-space description of aspherical atoms

P. Coppensa*

aDepartment of Chemistry, Natural Sciences & Mathematics Complex, State University of New York at Buffalo, Buffalo, New York 14260-3000, USA
Correspondence e-mail: coppens@acsu.buffalo.edu

#### 1.2.7.2. Reciprocal-space description of aspherical atoms

| top | pdf |

The aspherical-atom 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; Cohen-Tannoudji 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 lth-order 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 scattering-factor tables (IT IV, 1974). Expressions for the evaluation of using the radial function (1.2.7.5ac) have been given by Stewart (1980) and in closed form for (1.2.7.5a) by Avery & Watson (1977) and Su & Coppens (1990). The closed-form expressions are listed in Table 1.2.7.4.

 Table 1.2.7.4| top | pdf | Closed-form expressions for Fourier transform of Slater-type functions (Avery & Watson, 1977; Su & Coppens, 1990)
N
k 1 2 3 4 5 6 7 8
0
1
2
3
4
5
6
7

Expressions (1.2.7.8) show that the Fourier transform of a direct-space spherical harmonic function is a reciprocal-space spherical harmonic function with the same l, m, or, in other words, the spherical harmonic functions are Fourier-transform invariant.

The scattering factors of the aspherical density functions in the multipole expansion (1.2.7.6) are thus given by

The reciprocal-space 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 X-ray Crystallography (1974). Vol. IV. Birmingham: Kynoch Press. (Present distributor Kluwer Academic Publishers, Dordrecht.)Google Scholar
Arfken, G. (1970). Mathematical models for physicists, 2nd ed. New York, London: Academic Press.Google Scholar
Avery, J. & Watson, K. J. (1977). Generalized X-ray scattering factors. Simple closed-form expressions for the one-centre case with Slater-type orbitals. Acta Cryst. A33, 679–680.Google Scholar
Cohen-Tannoudji, 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). Closed-form expressions for Fourier– Bessel transforms of Slater-type functions. J. Appl. Cryst. 23, 71–73.Google Scholar
Weiss, R. J. & Freeman, A. J. (1959). X-ray 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