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

International Tables for Crystallography (2006). Vol. C, ch. 8.7, p. 722

Table 8.7.3.3 

P. Coppens,a Z. Sub and P. J. Beckerc

a732 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

Table 8.7.3.3| top | pdf |
The matrix M−1 relating d-orbital occupancies Pij to multipole populations Plm (from Holladay, Leung & Coppens, 1983[link])

d-orbital populationsMultipole populations
P00P20P22+P40P42+P44+
[P_z^2] 0.200 1.039 0.00 1.396 0.00 0.00
Pxz 0.200 0.520 0.942 −0.931 1.108 0.00
Pyz 0.200 0.520 −0.942 −0.931 −1.108 0.00
[P_{x^2-y^2}] 0.200 −0.039 0.00 0.233 0.00 1.571
Pxy 0.200 −1.039 0.00 0.233 0.00 −1.571

Mixing terms.

 P21P21−P22+P22−P41+P41−P42+P42−P43+P43−P44−
[P_{z^{2}/xz}] 1.088 0.00 0.00 0.00 2.751 0.00 0.00 0.00 0.00 0.00 0.00
[P_{z^{2}/yz}] 0.00 1.088 0.00 0.00 0.00 2.751 0.00 0.00 0.00 0.00 0.00
[P_{z^2/x^2-y^2}] 0.00 0.00 −2.177 0.00 0.00 0.00 1.919 0.00 0.00 0.00 0.00
[P_{z^{2}/xy}] 0.00 0.00 0.00 −2.177 0.00 0.00 0.00 1.919 0.00 0.00 0.00
[P_{xz/yz}] 0.00 0.00 0.00 1.885 0.00 0.00 0.00 2.216 0.00 0.00 0.00
[P_{xz/x^{2}-y^{2}}] 1.885 0.00 0.00 0.00 −0.794 0.00 0.00 0.00 2.094 0.00 0.00
[P_{xz/xy}] 0.00 1.885 0.00 0.00 0.00 −0.794 0.00 0.00 0.00 2.094 0.00
[P_{yz/x^{2}-y^{2}}] 0.00 −1.885 0.00 0.00 0.00 0.794 0.00 0.00 0.00 2.094 0.00
[P_{yz/xy}] 1.885 0.00 0.00 0.00 −0.794 0.00 0.00 0.00 −2.094 0.00 0.00
[P_{x^{2}-y^{2}/xy}] 0.00 0.00 0.00 0.00 0.00 0.00 0.00 0.00 0.00 0.00 3.142