Table 8.7.3.5

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, p. 723

Table 8.7.3.5

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

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

	P ₀₀	P ₂₋	P ₄₀	P ₄₃₊	P ₄₃₋
(a) In terms of d orbitals
P ₂₀	0.200	1.039	1.396	0.00	0.00
P ₂₁₊	0.200	0.520	−0.931	0.00	0.00
P ₂₁₋	0.200	0.520	−0.931	0.00	0.00
P ₂₂₊	0.200	−1.039	0.233	0.00	0.00
P ₂₂₋	0.200	−1.039	0.233	0.00	0.00
P _21+/22+	0.00	0.00	0.00	2.094	0.00
P _21+/22−	0.00	0.00	0.00	0.00	2.094
P _21−/22+	0.00	0.00	0.00	0.00	2.094
P _21−/22−	0.00	0.00	0.00	−2.094	0.00
(b) In terms of symmetry-adapted orbitals^†^‡
P₁(a_1g)	0.200	1.039	1.396	0.00
P₂(e_g)	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.