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

International Tables for Crystallography (2006). Vol. B, ch. 1.2, pp. 14-15   | 1 | 2 |

## Section 1.2.7. Beyond the spherical-atom description: the atom-centred spherical harmonic expansion

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. Beyond the spherical-atom description: the atom-centred spherical harmonic expansion

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#### 1.2.7.1. Direct-space description of aspherical atoms

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Even though the spherical-atom approximation is often adequate, atoms in a crystal are in a non-spherical environment; therefore, an accurate description of the atomic electron density requires non-spherical 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 charge-density 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 non-spherical 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 .

 Table 1.2.7.1| top | pdf | Real spherical harmonic functions (x, y, z are direction cosines)
lSymbol CAngular function, Normalization for wavefunctions, §Normalization for density functions,
ExpressionNumerical valueExpressionNumerical value
0 00 1 1 0.28209 0.07958
1 0.48860 0.31831
2 20 0.31539 0.20675
1.09255 0.75
3 30 0.37318 0.24485
0.45705 0.32033
1.44531 1 1
0.59004 0.42441
4 40 0.10579 ‡‡ 0.06942
0.66905 0.47400
0.47309 0.33059
1.77013 1.25
0.62584 0.46875
5 50 0.11695 0.07674
0.45295 0.32298
2.39677 1.68750
0.48924 0.34515
2.07566 1.50000
0.65638 0.50930
6 60 0.06357 0.04171
0.58262 0.41721
0.46060 0.32611
0.92121 0.65132
0.50457 0.36104
10395 2.36662 1.75000
10395 0.68318 0.54687
7 70 0.06828 0.04480
0.09033 0.06488
0.22127 0.15732
0.15646 0.11092
1.03783 0.74044
0.51892 0.37723
135135 2.6460 2.00000
135135 0.70716 0.58205
Common factor such that
, , .
§As defined by where are Cartesian functions.
Paturle & Coppens (1988), as defined by where are Cartesian functions.
††ar = arctan (2).
‡‡ where .

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 non-zero 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 .

Site-symmetry restrictions for the real spherical harmonics as given by Kara & Kurki-Suonio (1981) are summarized in Table 1.2.7.2.

 Table 1.2.7.2| top | pdf | Index-picking rules of site-symmetric spherical harmonics (Kara & Kurki-Suonio, 1981)
 λ, μ and j are integers.
SymmetryChoice of coordinate axesIndices of allowed ,
1 Any
Any
2
m
222 ,
mm2
mmm
4
,
422 ,
4mm
2m ,
,
3
32
,

3m

6
622
6mm

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 low-order terms are listed in Table 1.2.7.3. Both wavefunction and density-function normalization factors are specified in Table 1.2.7.3.

 Table 1.2.7.3| top | pdf | Kubic Harmonic' functions
 (a) Coefficients in the expression with normalization (Kara & Kurki-Suonio, 1981).
Even l mp
l j0+2+4+6+8+10+
0 1 1
4 1
0.76376   0.64550
6 1
0.35355   −0.93541
6 2
0.82916   −0.55902
8 1
0.71807   0.38188   0.58184
10 1
0.41143   −0.58630   −0.69784
10 2
0.80202   0.15729   0.57622
l j   2− 4− 6− 8−
3 1   1
7 1
0.73598   0.41458
9 1
0.43301   −0.90139
9 2
0.84163   −0.54006
 (b) Coefficients and density normalization factors in the expression where (Su & Coppens, 1994).
Even l mp
l j   0+ 2+ 4+ 6+ 8+ 10+
0 1 1
4 1 0.43454 1
6 1 0.25220 1
6 2 0.020833   1
8 1 0.56292 1   1/5940
10 1 0.36490 1   1/5460
10 2 0.0095165 1
l j     2− 4− 6− 8−
3 1 0.066667   1
7 1 0.014612   1
9 1 0.0059569   1
9 2 0.00014800     1
 (c) Density-normalized Kubic harmonics as linear combinations of density-normalized spherical harmonic functions. Coefficients in the expression . Density-type normalization is defined as .
Even l mp
l j0+2+4+6+8+10+
0 1 1
4 1 0.78245   0.57939
6 1 0.37790   −0.91682
6 2   0.83848   −0.50000
l j 2− 4− 6− 8−
3 1 1
7 1 0.73145   0.63290
 (d) Index rules for cubic symmetries (Kurki-Suonio, 1977; Kara & Kurki-Suonio, 1981).
lj23 432
T O
0 1 × × × × ×
3 1 ×     ×
4 1 × × × × ×
6 1 × × × × ×
6 2 × ×
7 1 ×     ×
8 1 × × × × ×
9 1 ×     ×
9 2 ×   ×
10 1 × × × × ×
10 2 × ×

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 as where 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 energy-optimized 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: or where L is a Laguerre polynomial of order n and degree .

In summary, in the multipole formalism the atomic density is described by in 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 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 terms are often found to be the most significantly populated deformation functions.

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

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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.)
Arfken, G. (1970). Mathematical models for physicists, 2nd ed. New York, London: Academic Press.
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.
Clementi, E. & Raimondi, D. L. (1963). Atomic screening constants from SCF functions. J. Chem. Phys. 38, 2686–2689.
Cohen-Tannoudji, 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 small-molecule data sets. Acta Cryst. A34, 909–921.
Hehre, W. J., Stewart, R. F. & Pople, J. A. (1969). Self-consistent molecular orbital methods. I. Use of Gaussian expansions of Slater-type 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. & Kurki-Suonio, 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). Closed-form expressions for Fourier– Bessel transforms of Slater-type 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). 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.