Moments as a function of the atomic multipole expansion

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. 716-717

Section 8.7.3.4.1.1. Moments as a function of the atomic multipole expansion

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

8.7.3.4.1.1. 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.

References

International Tables for Crystallography (2001). Vol. B, edited by U. Shmueli. Dordrecht: Kluwer Academic Publishers.Google Scholar

International Tables for Crystallography (2006). Vol. C. ch. 8.7, pp. 716-717