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. 723

Table 8.7.3.5 

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 8.7.3.5| top | pdf |
Orbital–multipole relations for trigonal complexes

 P00P2−P40P43+P43−
(a) In terms of d orbitals
P20 0.200 1.039 1.396 0.00 0.00
P21+ 0.200 0.520 −0.931 0.00 0.00
P21− 0.200 0.520 −0.931 0.00 0.00
P22+ 0.200 −1.039 0.233 0.00 0.00
P22− 0.200 −1.039 0.233 0.00 0.00
P21+/22+ 0.00 0.00 0.00 2.094 0.00
P21+/22− 0.00 0.00 0.00 0.00 2.094
P21−/22+ 0.00 0.00 0.00 0.00 2.094
P21−/22− 0.00 0.00 0.00 −2.094 0.00
(b) In terms of symmetry-adapted orbitals
P1(a1g) 0.200 1.039 1.396 0.00  
P2(eg) 0.400 −1.039 −0.310 −1.975  
[P_3(e_g^\prime)] 0.400 0.00 −1.087 1.975  
[P_4(e_{g^+}e'\hskip-3pt_g+e_{g^-}e'_{g^-})] 0.00 −2.942 2.193 1.397  
The electron density in terms of the symmetry-adapted orbitals is given by: [\rho_{3d}=P_1a^2_{1g}+\textstyle{1\over2}P_2(e^2_{g^+}-e^2_{g^-})+{1\over 2}P_4(e_{g^+}e'\hskip-3pt_g+e_{g^-}e'\hskip-3pt_{g^-}),]with: [a_{1g}=d_{z^2}]; [e_{g^+}=\sqrt(2/3)d_{x^2-y^2} - \sqrt(1/3)d_{xz}]; [e_{g^-}= \sqrt(2/3)d_{xy}+\sqrt(1/3)d_{yz}]; [e'\hskip-3pt_{g^+} = \sqrt(1/3)d_{x^2-y^2}+\sqrt(2/3)d_{xz}]; and [e'_{g^-}=\sqrt(1/3)d_{xy} -\sqrt(2/3)d_{yz}].
The signs given here imply a positive [e'\hskip-3pt_g] lobe in the positive xz quadrant. Care should be exercised in defining the coordinate system if this lobe is to point towards a ligand atom.