Occupancies of transition-metal valence orbitals from multipole coefficients

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. 722-723

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

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

8.7.3.6. 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 (ITB, 2001), 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 (ITB, 2001).

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).

The matrix M⁻¹ is given in Table 8.7.3.3. 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 and 8.7.3.5, 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 P_ij to multipole populations P_lm (from Holladay, Leung & Coppens, 1983)

d-orbital populations	Multipole populations
d-orbital populations	P₀₀	P₂₀	P₂₂₊	P₄₀	P₄₂₊	P₄₄₊
	0.200	1.039	0.00	1.396	0.00	0.00
P_xz	0.200	0.520	0.942	−0.931	1.108	0.00
P_yz	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
P_xy	0.200	−1.039	0.00	0.233	0.00	−1.571

Mixing terms.

	P₂₁	P₂₁₋	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 D_4h)

		P₀₀	P₂₀	P₄₀	P₄₄₊
P₂₀	a_1g	0.200	1.039	1.396	0.00
P₂₁₊	e_g	0.200	0.520	−0.931	0.00
P₂₁₋		0.200	0.520	−0.931	0.00
P₂₂₊	b_1g	0.200	−1.039	0.233	1.570
P₂₂	b_2g	0.200	−1.039	0.233	−1.570

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

	P₀₀	P₂₋	P₄₀	P₄₃₊	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
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.

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

International Tables for Crystallography (2006). Vol. C. ch. 8.7, pp. 722-723