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, pp. 722-723

Section 8.7.3.6. Occupancies of transition-metal valence orbitals from multipole coefficients

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

8.7.3.6. Occupancies of transition-metal valence orbitals from multipole coefficients

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In general, the atom-centred density model functions describe both the valence and the two-centre overlap density. In the case of transition metals, the latter is often small, so that to a good approximation the atomic density can be expressed in terms of an atomic orbital basis set [d_{{i}} ], as well as in terms of the multipolar expansion. Thus, [\eqalignno{ \rho _d &=\textstyle\sum \limits _{{i}=1}^5\textstyle\sum \limits _{j\ge {i}}^5P_{{i}j} d_{i} d_j, \cr &=\textstyle\sum \limits _{l=0}^4\kappa ^{\prime 3}\left \{ R_l(\kappa ^{\prime }r) \textstyle\sum \limits _{m=0}^l\textstyle\sum \limits _pP_{lmp}d_{lmp}\right \}, & (8.7.3.76)}]in which [d_{lmp}] are the density functions.

The orbital products [d_{i}d_j] can be expressed as linear combinations of spherical harmonic functions, with coefficients listed in Volume B, Chapter 1.2[link] (ITB, 2001[link]), which leads to relations between the [P_{{i}j}] and [P_{lmp}]. In matrix notation, [{\bf P}_{lmp}={\bi M}{\bf P}_{{i}j},\eqno (8.7.3.77)]where [{\bf P}_{lmp}] is a vector containing the coefficients of the 15 spherical harmonic functions with l = 0, 2, or 4 that are generated by the products of d orbitals. The matrix M is also a function of the ratio of orbital and density-function normalization coefficients, given in Volume B, Chapter 1.2[link] (ITB, 2001[link]).

The d-orbital occupancies can be derived from the experimental multipole populations by the inverse expression, [ {\bf P}_{{i}j}={\bi M}^{-1}{\bf P}_{lmp}\eqno (8.7.3.78)](Holladay, Leung & Coppens, 1983[link]).

The matrix M−1 is given in Table 8.7.3.3[link]. Point-group-specific expressions can be derived by omission of symmetry-forbidden terms. Matrices for point group 4/mmm (square planar) and for trigonal point groups (3, [\bar{3}], 32, 3m, [\bar3m]) are listed in Tables 8.7.3.4[link] and 8.7.3.5[link], respectively. Point groups with and without vertical mirror planes are distinguished by the occurrence of both [d_{lm+}] and [d_{lm-}] functions in the latter case, and only [d_{lm+}] in the former, with m being restricted to n, the order of the rotation axis. The [d_{{ln}-}] functions can be eliminated by rotation of the coordinate system around a vertical axis through an angle [\psi_0] given by [\psi_0=(1/n)\arctan (P_{{ln}-}/P_{{ln}+})].

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

Table 8.7.3.4| top | pdf |
Orbital–multipole relations for square-planar complexes (point group D4h)

 P00P20P40P44+
P20 a1g 0.200 1.039 1.396 0.00
P21+ eg 0.200 0.520 −0.931 0.00
P21−   0.200 0.520 −0.931 0.00
P22+ b1g 0.200 −1.039 0.233 1.570
P22 b2g 0.200 −1.039 0.233 −1.570

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.

References

Holladay, A., Leung, P. C. & Coppens, P. (1983). Generalized relation between d-orbital occupancies of transition-metal atoms and electron-density multipole population parameters from X-ray diffraction data. Acta Cryst. A39, 377–387.Google Scholar
International Tables for Crystallography (2001). Vol. B, edited by U. Shmueli. Dordrecht: Kluwer Academic Publishers.Google Scholar








































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