The effect of an origin shift on the outer moments

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. 717-718

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

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

8.7.3.4.1.3. 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.

References

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

International Tables for Crystallography (2006). Vol. C. ch. 8.7, pp. 717-718