Moments of a charge distribution

Coppens, P.; Su, Z.; Becker, P. J.

doi:10.1107/97809553602060000615

International
Tables for
Crystallography
Volume C
Mathematical, physical and chemical tables
Edited by E. Prince

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International Tables for Crystallography (2006). Vol. C. ch. 8.7, pp. 716-718

Section 8.7.3.4.1. Moments of a charge distribution¹

P. Coppens,^a Z. Su^b and P. J. Becker^c

^a 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

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Use of the expectation value expression $[ \langle {O}\rangle =\textstyle\int \widehat {{O}}\rho({\bf r}) {\,{\rm d}}{\bf r},\eqno (8.7.3.15)]$ with the operator $[\widehat {{O}}=\widehat {\gamma }r_{\alpha _1}r_{\alpha _2}r_{\alpha _3}\ldots r_{\alpha _l}=r_{\alpha _1}r_{\alpha _2}r_{\alpha _3}\ldots r_{\alpha _l}]$ gives for the electrostatic moments of a charge distribution ρ(r) $[\mu _{\alpha _1\alpha _2\alpha _3\ldots \alpha _l}=\textstyle\int \rho({\bf r}) r_{\alpha _1}r_{\alpha _2}r_{\alpha _3}\ldots r_{\alpha _l}{\,{\rm d}}{\bf r},\eqno (8.7.3.16)]$ in which the $[r_\alpha ]$ are the three components of the vector r (α_i = 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 $[l\geq 2]$ . In the traceless definition, the quadrupole moment, $[\Theta _{\alpha \beta }]$ , is given by $[ \Theta_{\alpha \beta }= \textstyle{1\over2}\int \rho ({\bf r}) \left [3r_\alpha r_\beta -r^2\delta _{\alpha \beta }\right] {\,{\rm d}}r,\eqno (8.7.3.17)]$ where δ_αβ is the Kronecker delta function. The term $[\textstyle\int\rho({\bf r})r^2\,{\rm d}r]$ , 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. $[ { M}_{\alpha _1\alpha _2\ldots \alpha _l}^{\left (l\right) }= {\left (-1\right) ^l \over l!} \int \rho ({\bf r}) r^{2l+1} \displaystyle{\partial ^l \over \partial r_{\alpha _1}\partial r_{\alpha _2}\ldots \partial r_{\alpha _l}}\left (1\over r\right) {\,{\rm d}}{\bf r}.\eqno (8.7.3.18)]$

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) $[\eqalignno{ \Theta_{xx} &= \textstyle{3\over2}\mu _{xx}- {1\over 2} \left (\mu _{xx}+\mu _{yy}+\mu _{zz}\right) \cr &=\mu _{xx}- \textstyle{1\over2} \left (\mu _{yy}+\mu _{zz}\right), }]$ and $[ \Theta _{xy}= \textstyle{3\over2}\mu _{xy}.\eqno (8.7.3.19)]$ Expressions for the other elements are obtained by simple permutation of the indices.

For a site of point symmetry 1, the electrostatic moment $[\mu _{\alpha _1\alpha _2\alpha _3\ldots \alpha _l}]$ 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 $[{c}_{lmp}]$ (ITB, 2001) multiplied by [r^l] , which defines the spherical electrostatic moments $[ \Theta _{lmp}=\textstyle\int \rho({\bf r}) {c}_{lmp}r^l{\,{\rm d}}r. \eqno (8.7.3.20)]$

The expressions for $[{c}_{lmp}]$ are listed in Volume B of International Tables for Crystallography (ITB, 2001); for the l = 2 moment, the $[{c}_{lmp}]$ 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 $[\eqalignno{ \Theta _{zz} &=(1/2) \Theta _{20}, \cr \Theta _{xx} &=(1/2) [3\Theta _{22+}-(1/2) \Theta _{20}] , \cr \Theta _{yy} &=(1/2) [-3\Theta _{22+}-(1/2) \Theta _{20}] , \ \cr \Theta _{xz} &=(3/2) \Theta _{21+}, \cr \Theta _{yz} &=(3/2) \Theta _{21-}, \cr \Theta _{xy} &=(3/2) \Theta _{22-}. & (8.7.3.21)}]$

8.7.3.4.1.1. Moments as a function of the atomic multipole expansion

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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 ρ_total,i(r) = ρ_nuclear,i − ρ_e,i of atom i, $[ \mu _{\alpha _1\alpha _2\alpha _3\ldots \alpha _l}=\textstyle\int \rho _{{\rm total},{i}}({\bf r}) r_{\alpha _1}r_{\alpha_2}r_{\alpha _3}\ldots r_{\alpha _l}{\,{\rm d}}r,\eqno (8.7.3.22)]$ where the electronic part of the atomic charge distribution is defined by the multipole expansion $[\eqalignno{ \rho _{{e,i}}({\bf r}) &={P}_{{i,c}}\rho _{{\rm core}} (r) +{P}_{{i,v}}\kappa _{i}^3\rho _{ i,{\rm valence}}(\kappa _{i}r) \cr &\quad+\textstyle\sum \limits _{l=0}^{l_{\max }}\kappa _{i}^{\prime 3}{R}_{{i},l}(\kappa _{i}^{\prime }r) \textstyle\sum \limits _{m=0}^l\, \textstyle\sum \limits _p{P}_{{i},lmp}d_{lmp}(\theta, \varphi), & (8.7.3.23)}]$ where p = ± when m $[\gt \, 0]$ , and $[{R}_l(\kappa _{i}^{\prime }r)]$ is a radial function.

We get for the jth moment of the valence density $[\eqalignno{ \mu ^j &=\mu _{\alpha _1\alpha _2\alpha _3\ldots \alpha _j} \cr &=-\int \left [{P}_{{i,v}}\kappa _{i}^3\rho _{i,{\rm valence}}(\kappa _{i}r)+\textstyle\sum \limits _{l=0}^{l_{\max }}\kappa _{{i}}^{\prime 3}{R}_{{i},l}(\kappa _{i}'r) \right. \cr &\quad \times \left. \textstyle\sum \limits _{m=0}^l\textstyle\sum \limits _p{P}_{{i},lmp}{d}_{lmp}(\theta, \varphi ) \right] r_{\alpha _1}r_{\alpha _2}\ldots r_{\alpha _j}{\,{\rm d}}{\bf r}, & (8.7.3.24)}]$ in which the minus sign arises because of the negative charge of the electrons.

We will use the symbol $[\widehat {{O}}]$ for the moment operators. We get $[ \mu ^j=-\kappa _{i}^{\prime 3}\int \widehat {{O}}_j\sum \limits _{l=1}^{l_{\max }}\left [\textstyle\sum \limits _{m=0}^l\textstyle\sum \limits _p{P}_{lmp}d_{lmp}{R}_l\right] {\,{\rm d}}{\bf r},\eqno (8.7.3.25)]$ 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 $[ \Theta _{lmp}=-{P}_{lmp}\textstyle\int \widehat {{O}}_{lmp}\left [d_{lmp}{R}_l\right] {\,{\rm d}}{\bf r}.\eqno (8.7.3.26)]$

Substitution with R_l = {(κ′ζ)^n(l)+3/[n(l) +2] !}r^n(l)exp(−ζr) and $[\widehat {{O}}_{lmp}={c}_{lmp}r^l]$ and subsequent integration over r gives $[ \Theta _{lmp}=-{P}_{lmp} {1\over(\kappa ^{\prime }\zeta ) ^l}\, {[n(l)+l+2] ! \over [n(l) +2] !}\, {1\over{D}_{lm}{M}_{lm}}\int {y}_{lmp}^2\sin \theta \,{\,{\rm d}}\theta {\,{\rm d}}\varphi, \eqno (8.7.3.27)]$ where the definitions $[ d_{lmp}={L}_{lm}{c}_{lmp}=\left ({{L}_{lm} \over {M}_{lm}}\right) {y}_{lmp}\quad {\rm and}\quad {c}_{lmp}=\left ({1\over {M}_{lm}}\right) {y}_{lmp}\eqno (8.7.3.28)]$ have been used (ITB, 2001). Since the $[{y}_{lmp}]$ functions are wavefunction normalized, we obtain $[ \Theta _{lmp}=-{P}_{lmp} {1\over(\kappa ^{\prime }\zeta ) ^l}\, { [n(l) +l+2] ! \over [n(l) +2] !}\, {{L}_{lm} \over \left ({M}_{lm}\right) ^2}.\eqno (8.7.3.29)]$ Application to dipolar terms with n(l) = 2, L_lm = 1/π and M_lm = (3/4π)^1/2 gives the x component of the atomic dipole moment as $[ \mu _x=-\int {P}_{11+}{\,d}_{11+}{R}_1x{\,{\rm d}}{\bf r}=- \displaystyle{20 \over 3\kappa ^{\prime }\zeta }{P}_{11+}.\eqno (8.7.3.30)]$ For the atomic quadrupole moments in the spherical definition, we obtain directly, using n(l) = 2, l = 2 in (8.7.3.29), $[ \Theta _{20}=- {30 \over \left (\kappa ^{\prime }\zeta \right) ^2}\, {{L}_{20}\over \left ({M}_{20}\right) ^2}\,{P}_{20}=- {36\sqrt {3} \over \left (\kappa ^{\prime }\zeta \right) ^2}{P}_{20},\eqno (8.7.3.31)]$ and, for the other elements, $[\Theta _{2mp}=-{30\over \left (\kappa ^{\prime }\zeta \right) ^2}\, {{L}_{2m}\over \left ({M}_{2m}\right) ^2}{P}_{2mp}=- {6\pi \over\left (\kappa ^{\prime }\zeta \right) ^2}{P}_{2mp}. \eqno (8.7.3.32)]$

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 $[\eqalignno{ \Theta _{zz} &=- {18\sqrt {3}\over \left (\kappa ^{\prime }\zeta \right) ^2}{P}_{20}, \cr \Theta _{yy} &=+{9\over\left (\kappa ^{\prime }\zeta \right) ^2}\left (\sqrt {3}{P}_{20}+\pi {P}_{22+}\right), \cr \Theta _{xx} &= {9\over \left (\kappa ^{\prime }\zeta \right) ^2}\left (\sqrt {3}{P}_{20}-\pi {P}_{22+}\right),}]$ and $[ \Theta _{xz}=-{9\pi\over{\left (\kappa ^{\prime }\zeta \right) ^2}}{P}_{21+}, \eqno (8.7.3.33)]$ and analogously for the other off-diagonal elements.

8.7.3.4.1.2. Molecular moments based on the deformation density

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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 $[\eqalignno{ \mu _{xx}\left (\rho _{{\rm total}}\right) &=\textstyle\int \rho x^2{\,{\rm d}}{\bf r} \cr &=\textstyle\int \rho _{{\rm promolecule}}x^2{\,{\rm d}}{\bf r}+\int \Delta\rho x^2{\,{\rm d}}{\bf r.} & (8.7.3.34)}]$ The promolecule is the sum over spherical atom densities, or $[\eqalignno{ \textstyle\int \rho _{{\rm promolecule}}x^2{\,{\rm d}}{\bf r} &= \textstyle\int \textstyle\sum \limits _{{i}}\rho _{{\rm spherical\ atom,}i}x^2{\,{\rm d}}{\bf r} \cr &= \textstyle\sum \limits _{i} \textstyle\int \rho _{{\rm spherical\ atom,}i}x^2{\,{\rm d}}{\bf r.} & (8.7.3.35)}]$ If R_i = (X_i, Y_i, Z_i) is the position vector for atom i, each single-atom contribution can be rewritten as $[\eqalignno{ \mu _{{i},\,xx,\,{\rm spherical\ atom}} &= \textstyle\int \rho _{i{\rm,\, spherical\ atom}}x^2{\,{\rm d}}{\bf r} \cr &= \textstyle\int \rho _{i{\rm,\, spherical\ atom}} (x-X_{i}) ^2{\,{\rm d}}{\bf r} \cr &\quad+X_{i} \textstyle\int \rho _{i{\rm,\, spherical\ atom}}2(x-X_{{i}}) {\,{\rm d}}{\bf r} \cr &\quad+X_{i}^2 \textstyle\int \rho _{i{\rm,\, spherical\ atom}}{\,{\rm d}}{\bf r.}\ & (8.7.3.36)}]$ 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 $[\langle(x-X) {_{i}^2}\rangle ={1\over3}\langle r_{{i}}^2\rangle]$ , and $[\langle r_{i}^2\rangle =\int \rho _{i}(r) r^2{\,{\rm d}}{\bf r},]$ $[ \textstyle\int \rho _{{\rm promolecule}}x^2{\,{\rm d}}{\bf r}= \textstyle{1\over3}\textstyle\sum \limits _{{\rm atoms}}\langle r^2\rangle _{{\rm spherical\ atom}},\eqno (8.7.3.37)]$ and, by substitution in (8.7.3.34), $[ \mu _{xx}(\rho _{{\rm tot}}) =\mu _{xx} (\Delta\rho) + \textstyle{1\over3}\textstyle\sum \limits _{{\rm atoms}}\langle r^2\rangle _{{\rm spherical\ atom}},\ \eqno (8.7.3.38a) ]$ with $[ \mu _{xx}(\Delta\rho) =\textstyle\sum \limits _{i}\left(\textstyle\int \Delta\rho _{i}x^2{\,{\rm d}}{\bf r}+2X_{i}\mu _{i}+X_{{i}}^2q_{i}\right), \eqno (8.7.3.38b) ]$ in which $[\mu _{i}]$ and $[q_{i}]$ 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 μ_αβ and the traceless moments $[\Theta]$ _αβ of the deformation density can be illustrated as follows. From (8.7.3.17), we may write $[ \Theta _{\alpha \beta }\left (\Delta\rho \right) = \textstyle{3\over2}\mu _{\alpha \beta }\left (\Delta\rho \right) - {1\over2}\delta _{\alpha \beta }\int \Delta\rho \,r^2{\,{\rm d}}{\bf r}.\eqno (8.7.3.39)]$ 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 $[ (\Delta\rho) _{{\rm spherical}} = \kappa ^3{P}_{{\rm valence}}\rho _{{\rm valence}}(\kappa r) -{P}_{{\rm valence}}^0\rho _{{\rm valence}}(r) \eqno (8.7.3.40)]$ and $[\eqalignno{ \textstyle\int \Delta\rho \,r^2{\,{\rm d}}{\bf r} &=\textstyle\int \textstyle\sum \limits _{i}\left [\kappa _{i}^3{P}_{{\rm valence,}i}\rho _{{\rm valence,}i}\left (\kappa _{i}r\right) \right.\cr &\left. \quad -{P}_{{\rm valence,}i}^0\rho _{{\rm valence,}i}\left (r\right) \right] r^2{\,{\rm d}}{\bf r} \cr &=\textstyle\sum \limits _{i}\left ({P}_{{\rm valence,}i}/\kappa _{{i}}^2-{P}_{{\rm valence}}^0\right) \left \langle r_{i}^2\right \rangle _{{\rm spherical\ valence\ shell}} \cr &\quad+{R}_{i}^2\left ({P}_{{\rm valence,}i}-{P}_{{\rm valence,}i}^0\right), & (8.7.3.41)}]$ which, on substitution in (8.7.3.39), gives the required relation.

8.7.3.4.1.3. The effect of an origin shift on the outer moments

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In general, the multipole moments depend on the choice of origin. This can be seen as follows. Substitution of $[{\bf r}_\alpha ^{\prime }={\bf r}_{\alpha} - {\bf R}_{\alpha}]$ in (8.7.3.16) corresponds to a shift of origin by R_α, or X, Y, Z in the original coordinate system. In three dimensions, we get, for the first moment, the charge q, $[ q^{\prime }=q,\eqno (8.7.3.42)]$ and for the transformed first and second moments $[\eqalignno{ \mu _x^{\prime } &=\mu _x-q{\bf X}\semi \ \mu _y^{\prime }=\mu _y-q{\bf Y}\semi \ \mu _z^{\prime }=\mu _{z}-q{\bf Z}\semi \cr \mu _{\alpha \alpha }^{\prime } &=\mu _{\alpha \alpha }-2\mu _\alpha {R}_\alpha +q{R}_\alpha ^2\semi \cr \mu _{\alpha \beta }^{\prime } &=\mu _{\alpha \beta }-\mu _\alpha {R}_\beta -\mu _\beta {R}_\alpha +q{R}_\alpha {R}_\beta. & (8.7.3.43)}]$

For the traceless quadrupole moments, the corresponding equations are obtained by substitution of $[{\bf r}_\alpha ^{\prime }={\bf r}_{\alpha}-{\bf R}_{\alpha}]$ and r′ = r − R into (8.7.3.17), which gives $[\eqalignno{ \Theta _{\alpha \beta }^{\prime } &=\Theta _{\alpha \beta }+ \textstyle{1\over2}\left (3{\bf R}_\alpha {\bf R}_\beta -{\bf R}^2\delta _{\alpha \beta }\right) q \cr &\quad - \textstyle{3\over2}\left ({\bf R}_\beta \mu _\alpha +{\bf R}_\alpha \mu _\beta \right) +\sum \limits _\gamma \left ({\bf R}_\gamma \mu _\gamma \right) \delta _{\alpha \beta }. & (8.7.3.44)}]$ 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.

8.7.3.4.1.4. Total moments as a sum over the pseudoatom moments

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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 $[{\bf r}_{{i}}]$ , $[{\bf R}=-{\bf r}_i]$ . The first three moments summed over the atoms i located at $[{\bf r}_{i}]$ become $[\eqalignno{ q_{{\rm total}} &=\textstyle\sum q_{i}, &(8.7.3.45)\cr \mu _{{\rm total}} &=\textstyle\sum \limits _{i}\mu _{i}+\textstyle\sum {\bf r}_{{i}}q_{i}, &(8.7.3.46)}%fd8.7.3.46]$ and $[ \mu _{\alpha \beta, {\rm total}}=\textstyle\sum \limits _{i}\big (\mu _{\alpha \beta {i}}+r_{\beta {i}}^{\prime }\mu _\alpha +r_{\alpha {i}}^{\prime }\mu _\beta +r_{\alpha {i}}^{\prime }r_{\beta {i}}^{\prime }q_{i}\big) \eqno (8.7.3.47)]$ with α, β = x, y, z; and expressions equivalent to (8.7.3.44) for the traceless components $[\Theta]$ _αβ.

8.7.3.4.1.5. Electrostatic moments of a subvolume of space by Fourier summation

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Expression (8.7.3.16) for the outer moment of a distribution within a volume element [V_T] may be written as $[ \mu _{\alpha _1\alpha _2\ldots \alpha _l}=\textstyle\int\limits _{V_T}\rho({\bf r}) \widehat {\gamma }_{\alpha _1\alpha _2\alpha _3\ldots \alpha _l}{\,{\rm d}}{\bf r}, \eqno (8.7.3.16)]$ with $[\widehat {\gamma }_{\alpha _1\alpha _2\alpha _3\ldots \alpha _l}=r_{\alpha _1}r_{\alpha _2}r_{\alpha _3}\ldots r_{\alpha _l}]$ , and integration over the volume [V_T] .

Replacement of ρ(r) by the Fourier summation over the structure factors gives $[\eqalignno{ \mu _{(V_T) }^l &= {1\over V}\int\limits _{V_T}\widehat {\gamma}_l\sum F({\bf h}) \exp (-2\pi i{\bf h\cdot r}) \,{\,{\rm d}}{\bf r} \cr &={1\over V}\sum F({\bf h}) \int\limits _{V_T}\widehat {\gamma }_l\exp(-2\pi i{\bf h\cdot r}) \,{\,{\rm d}}{\bf r},& (8.7.3.48)}]$ where $[\widehat\gamma_l]$ is the product of l coordinates according to (8.7.3.16), and μ^l 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 $[\int _{V_T}\widehat {\gamma }_l]$ $[\exp (-2\pi i{\bf h}\cdot {\bf r}){\,{\rm d}}{\bf r}]$ is defined as the shape transform S of the volume [V_T.] $[ \mu ^l(V_T) = {1\over V}\sum F({\bf h}) S_{V_T}(\widehat {\gamma }_l,{\bf h}). ]$ 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 $[{\bf r}_{i}]$ , relative to a coordinate system origin at P, the total electronic moment relative to this origin is given by $[ \mu ^l\left (\sum \limits _{i}V_{T,{i}}\right) = {1\over V}\sum_{{\bf h}}F({\bf h}) S_{V_T}(\widehat {\gamma }_l,{\bf h}) \sum _{i}\exp (-2\pi i{\bf h}\cdot {\bf r}_{i}) . \eqno (8.7.3.49)]$

Expressions for $[S_{V_T}]$ for $[l\leq 2]$ and a subvolume parallelepipedal shape are given in Table 8.7.3.2. Since the spherical order Bessel functions [j_n(x)] 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, $[ S(\widehat {\gamma }_1,{\bf h}) =\textstyle\int\limits_{V_t}{\bf r}_0\exp (-2\pi i{\bf h}\cdot {\bf r}_0){\,{\rm d}}{\bf r}.]$ A shift of origin by $[-{\bf r}_{i}]$ leads to $[\eqalignno{ \mu ^{1^{\prime }}(V_T) &= {1\over V}\sum F({\bf h}) \left[S_{V_T}(\widehat {\gamma}_1,{\bf h}) +{\bf r}_1\int \exp (-2\pi i{{\bf h}\cdot r}_0) {\,{\rm d}}{\bf r}\right] \cr &\quad\times \exp (-2\pi i{\bf h}\cdot {\bf r}_1) \cr &=\mu ^1(V_T) +{\bf r}_1q, }]$ in agreement with (8.7.3.46).

Table 8.7.3.2| top | pdf |
Expressions for the shape factors S for a parallelepiped with edges δ_x, δ_y, and δ_z (from Moss & Coppens, 1981)

j₀ and j₁ are the zero- and first-order spherical Bessel functions: j₀(x) = sin x/x, j₁(x) = sin x/x² − cos x/x; V_T is volume of integration.

$[\hat y]$	Property	$[S[\hat y({\bf r}), {\bf h}]]$
1	Charge	$[V_T\,j_0(2\pi h_x\delta_x)\, j_0(2\pi h_y\delta_y)\, j_0(2\pi h_z\delta_z)]$
r_α	Dipole μ_α	$[\eqalign { -i&V_T\delta_\alpha\,j_1(2\pi h_\alpha\delta_\alpha) \cr & \times\, j_0(2\pi h_\beta\delta_\beta)\,j_0(2\pi h_y\delta_y)}]$
r_αr_β	Second moment μ_αβ off-diagonal	$[\eqalign { -V_T&\delta_\alpha\delta_\beta\,j_1 (2\pi h_\alpha\delta_\alpha) \cr \times\,& j_1(2\pi h_\beta\delta_\beta)\,j_0(2\pi h_\gamma \delta_\gamma) }]$
r_αr_α	Second moment μ_αα diagonal	$[\eqalign { -V_T&\delta^2_\alpha\bigg\{\displaystyle{j_1(2\pi h_\alpha\delta_\alpha) \over \pi h_\alpha \delta_\alpha} - j_0(2\pi h_\alpha\delta_\alpha)\bigg\} \cr \quad \times\, &j_0(2\pi h_\beta\delta_\beta)\,j_0(2\pi h_\gamma\delta_\gamma) }]$

References

Boyd, R. J. (1977). The radial density function for the neutral atoms from helium to xenon. Can. J. Phys. 55, 452–455.Google Scholar

Buckingham, 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

International Tables for Crystallography (2006). Vol. C. ch. 8.7, pp. 716-718

Section 8.7.3.4.1. Moments of a charge distribution1

8.7.3.4.1. Moments of a charge distribution1

8.7.3.4.1.1. Moments as a function of the atomic multipole expansion

8.7.3.4.1.2. Molecular moments based on the deformation density

8.7.3.4.1.3. The effect of an origin shift on the outer moments

8.7.3.4.1.4. Total moments as a sum over the pseudoatom moments

8.7.3.4.1.5. Electrostatic moments of a subvolume of space by Fourier summation

References

Section 8.7.3.4.1. Moments of a charge distribution¹

8.7.3.4.1. Moments of a charge distribution¹