International
Tables for Crystallography Volume C Mathematical, physical and chemical tables Edited by E. Prince © International Union of Crystallography 2006 |
International Tables for Crystallography (2006). Vol. C. ch. 8.7, pp. 716-718
Section 8.7.3.4.1. Moments of a charge distribution1a 732 NSM Building, Department of Chemistry, State University of New York at Buffalo, Buffalo, NY 14260-3000, USA,bDigital Equipment Co., 129 Parker Street, PKO1/C22, Maynard, MA 01754-2122, USA, and cEcole Centrale Paris, Centre de Recherche, Grand Voie des Vignes, F-92295 Châtenay Malabry CEDEX, France |
8.7.3.4.1. Moments of a charge distribution1
Use of the expectation value expression with the operator gives for the electrostatic moments of a charge distribution ρ(r) in which the are the three components of the vector r (αi = 1, 2, 3), and the integral is over the complete volume of the distribution.
For l = 0, (8.7.3.16) represents the integral over the charge distribution, which is the total charge, a scalar function described as the monopole. The higher moments are, in ascending order of l, the dipole, a vector, the quadrupole, a second-rank tensor, and the octupole, a third-rank tensor. Successively higher moments are named the hexadecapole (l = 4), the tricontadipole (l = 5), and the hexacontatetrapole (l = 6). An alternative, traceless, definition is often used for moments with . In the traceless definition, the quadrupole moment, , is given by where δαβ is the Kronecker delta function. The term , which is subtracted from the diagonal elements of the tensor, corresponds to the spherically averaged second moment of the distribution.
Expression (8.7.3.17) is a special case of the following general expression for the lth-rank traceless tensor elements.
Though the traceless moments can be derived from the unabridged moments, the converse is not the case because the information on the spherically averaged moments is no longer present in the traceless moments. The general relations between the traceless moments and the unabridged moments follow from (8.7.3.18). For the quadrupole moments, we obtain with (8.7.3.17) and Expressions for the other elements are obtained by simple permutation of the indices.
For a site of point symmetry 1, the electrostatic moment of order l has (l + 1)(l + 2)/2 unique elements. In the traceless definition, not all elements are independent. Because the trace of the tensor has been set to zero, only 2l + 1 independent components remain. For the quadrupole there are 5 independent components of the form (8.7.3.19).
In a different form, the traceless moment operators can be written as the Cartesian spherical harmonics (ITB, 2001) multiplied by , which defines the spherical electrostatic moments
The expressions for are listed in Volume B of International Tables for Crystallography (ITB, 2001); for the l = 2 moment, the have the well known form 3z2 − 1, xz, yz, (x2 − y2)/2, and xy, where x, y and z are the components of a unit vector from the origin to the point being described. The spherical electrostatic moments have (2l + 1) components, which equals the number of independent components in the traceless definition (8.7.3.18), as it should. The linear relationships are
In the multipole model [expression (8.7.3.7)], the charge density is a sum of atom-centred density functions, and the moments of a whole distribution can be written as a sum over the atomic moments plus a contribution due to the shift to a common origin. An atomic moment is obtained by integration over the charge distributions ρtotal,i(r) = ρnuclear,i − ρe,i of atom i, where the electronic part of the atomic charge distribution is defined by the multipole expansion where p = ± when m , and is a radial function.
We get for the jth moment of the valence density in which the minus sign arises because of the negative charge of the electrons.
We will use the symbol for the moment operators. We get where, as before, p = ±. The requirement that the integrand be totally symmetric means that only the dipolar terms in the multipole expansion contribute to the dipole moment. If we use the traceless definition of the higher moments, or the equivalent definition of the moments in terms of the spherical harmonic functions, only the quadrupolar terms of the multipole expansion will contribute to the quadrupole moment; more generally, in the traceless definition the lth-order multipoles are the sole contributors to the lth moments. In terms of the spherical moments, we get
Substitution with Rl = {(κ′ζ)n(l)+3/[n(l) +2] !}rn(l)exp(−ζr) and and subsequent integration over r gives where the definitions have been used (ITB, 2001). Since the functions are wavefunction normalized, we obtain Application to dipolar terms with n(l) = 2, Llm = 1/π and Mlm = (3/4π)1/2 gives the x component of the atomic dipole moment as For the atomic quadrupole moments in the spherical definition, we obtain directly, using n(l) = 2, l = 2 in (8.7.3.29), and, for the other elements,
As the traceless quadrupole moments are linear combinations of the spherical quadrupole moments, the corresponding expressions follow directly from (8.7.3.31), (8.7.3.32) and (8.7.3.21). We obtain with n(2) = 2 and and analogously for the other off-diagonal elements.
The moments derived from the total density ρ(r) and from the deformation density Δρ(r) are not identical. To illustrate the relation for the diagonal elements of the second-moment tensor, we rewrite the xx element as The promolecule is the sum over spherical atom densities, or If Ri = (Xi, Yi, Zi) is the position vector for atom i, each single-atom contribution can be rewritten as Since the last two integrals are proportional to the atomic dipole moment and its net charge, respectively, they will be zero for neutral spherical atoms. Substitution in (8.7.3.35) gives, with , and and, by substitution in (8.7.3.34), with in which and are the atomic dipole moment and the charge on atom i, respectively.
The last term in (8.7.3.38a) can be derived rapidly from analytical expressions for the atomic wavefunctions. Results for Hartree–Fock wavefunctions have been tabulated by Boyd (1977). Since the off-diagonal elements of the second-moment tensor vanish for the spherical atom, the second term in (8.7.3.38a) disappears, and the off-diagonal elements are identical for the total and deformation densities.
The relation between the second moments μαβ and the traceless moments αβ of the deformation density can be illustrated as follows. From (8.7.3.17), we may write Only the spherical density terms contribute to the integral on the right. Assuming for the moment that the spherical deformation is represented by the valence-shell distortion (i.e. neglect of the second monopole in the aspherical atom expansion), we have, with density functions ρ normalized to 1, for each atom and which, on substitution in (8.7.3.39), gives the required relation.
In general, the multipole moments depend on the choice of origin. This can be seen as follows. Substitution of in (8.7.3.16) corresponds to a shift of origin by Rα, or X, Y, Z in the original coordinate system. In three dimensions, we get, for the first moment, the charge q, and for the transformed first and second moments
For the traceless quadrupole moments, the corresponding equations are obtained by substitution of and r′ = r − R into (8.7.3.17), which gives Similar expressions for the higher moments are reported in the literature (Buckingham, 1970).
We note that the first non-vanishing moment is origin-independent. Thus, the dipole moment of a neutral molecule, but not that of an ion, is independent of origin; the quadrupole moment of a molecule without charge and dipole moment is not dependent on the choice of origin and so on. The molecular electric moments are commonly reported with respect to the centre of mass.
The moments of a molecule or of a molecular fragment are obtained from the sum over the atomic moments, plus a contribution due to the shift to a common origin for all but the monopoles. If individual atomic coordinate systems are used, as is common if chemical constraints are applied in the least-squares refinement, they must be rotated to have a common orientation. Expressions for coordinate system rotations have been given by Cromer, Larson & Stewart (1976) and by Su & Coppens (1994a).
The transformation to a common coordinate origin requires use of the origin-shift expressions (8.7.3.42)–(8.7.3.44), with, for an atom at , . The first three moments summed over the atoms i located at become and with α, β = x, y, z; and expressions equivalent to (8.7.3.44) for the traceless components αβ.
Expression (8.7.3.16) for the outer moment of a distribution within a volume element may be written as with , and integration over the volume .
Replacement of ρ(r) by the Fourier summation over the structure factors gives where is the product of l coordinates according to (8.7.3.16), and μl represents the moment of the static distribution if the F(h) are the structure factors on an absolute scale after deconvolution of thermal motion. Otherwise, the moment of the thermally averaged density is obtained.
The integral is defined as the shape transform S of the volume For regularly shaped volumes, the integral can be evaluated analytically. A volume of complex shape may be subdivided into integrable subvolumes such as parallelepipeds. By choosing the subvolumes sufficiently small, a desired boundary surface can be closely approximated.
If the origin of each subvolume is located at , relative to a coordinate system origin at P, the total electronic moment relative to this origin is given by
Expressions for for and a subvolume parallelepipedal shape are given in Table 8.7.3.2. Since the spherical order Bessel functions that appear in the expressions generally decrease with increasing x, the moments are strongly dependent on the low-order reflections in a data set. An example is the shape transform for the dipole moment. Relative to an origin O, A shift of origin by leads to in agreement with (8.7.3.46).
|
References
Boyd, R. J. (1977). The radial density function for the neutral atoms from helium to xenon. Can. J. Phys. 55, 452–455.Google ScholarBuckingham, A. D. (1970). Physical chemistry. An advanced treatise, Vol. 4. Molecular properties, edited by D. Henderson, pp. 349–386. New York: Academic Press.Google Scholar
Cromer, D. T., Larson, A. C. & Stewart, R. F. (1976). Crystal structure refinements with generalized scattering factors. J. Chem. Phys. 65, 336–349.Google Scholar
International Tables for Crystallography (2001). Vol. B, edited by U. Shmueli. Dordrecht: Kluwer Academic Publishers.Google Scholar
Moss, G. & Coppens, P. (1981). Pseudomolecular electrostatic potentials from X-ray diffraction data. In Molecular electrostatic potentials in chemistry and biochemistry, edited by P. Politzer & D. Truhlar. New York: Plenum.Google Scholar
Su, Z. & Coppens, P. (1994a). Rotation of real spherical harmonics. Acta Cryst. A50, 636–643.Google Scholar