C. E. Schäffer

The angular overlap model, an attempt to revive the ligand field approaches

A general method for the calculation of orbital energy differences within a partially filled l-shell of a central ion in an inorganic chromophore is described. The angular overlap model covers σ, π and δ anti-bonding (as well as bonding) effects, and it is shown that the consequence of the model is identical with that of a singular contact term potential acting close to each ligand nucleus. Because of this the model can also treat chromophores containing different ligands, each new ligand contributing one σ and one π (and one δ) radial parameter. In this way fewer parameters are involved than in usual ligand field approaches. Selected symmetries of chromophores, cubic, pentagonal, tetragonal, and trigonal, are used to illustrate the applications of the model.

Some examples of different spectroscopic and magnetic properties connected with the chromium III — oxygen bond

Abstract The spectra are given of some chromium (III) complexes in while the chromium atom has oxygen as the six nearest neighbours. The ligand field parameter Δ is found to vary remarkably much and in a way different from what would be expected in view of the electrostatic ligand field model. Some unexpected spectroscopic and magnetic phenomena with polynuclear complexes are described.

Mixing s orbitals into p and d orbitals. An attempt at bridging the angular overlap model and the valence shell electron pair repulsion model. Critique of the cellular ligand-field model

Abstract The stereochemistries of main group molecules have been discussed by using the angular overlap model in its molecular orbital oriented form (MO-AOM). Either ligand-field stabilisation of the ground state s 2 p q −2 configuration, or s-p mixing, or both, provide a consistent bonding model for the stereochemistries. The transformation of the non-bonding orbitals into equivalent orbitals leads invariably to agreement with the lone-pair locations of the valence shell electron pair repulsion (VSEPR) model. The concepts of Hamiltonian-generated hybrids and pseudohemispherical molecular systems are found useful in this context. The MO-AOM formalism is also used for discussing s-d mixing in transition metal systems, and the energetic consequences within the ligand-field AOM (LF-AOM) are included. This is a second-order effect, which depends on squares and cross-products of radial parameters. It may still be quite large for tetragonal systems and for systems that deviate strongly from orthoaxiality. The usual ligand-additive property of the AOM is lost when the symmetry is lower than tetragonal and so is the energy separability into angular and radial factors. The cellular ligand-field model is found to be identical with the LF-AOM, except that its users consider it important not to acknowledge the formal hierarchy, MO-AOM⊃LF-AOM, as relevant. The unintelligible concept of an active coordination void is found to be unnecessary and insufficient.

Quantitative Comparison of the Symmetry Components of a Ligand Field: Illustrations of the Orthonormal Operators Formalism

A recently proposed formulation of ligand field models in terms of orthonormal operator sets allows a quantitative comparison of the various symmetry components of a ligand field operator in terms ...

The non-additive ligand field: An operator and parameter symmetry-analysis

The semi-empirical ligand field is a perturbation operator whose consequences are taken to first order using a basis set ofl functions. Since the basis spans an irreducible representation of the 3-dimensional rotation-inversion groupR3iit is useful to express the operator as a sum of components of irreducible tensor operators with respect to this group. IfR3iis reduced with respect to the molecular subgroup the electronic factor of each term in the sum must be totally symmetrical within this group. This choice of operator leads to thecrystal field parameterization without implying an electrostatic model. Alternatively a shift operator withinl space may be chosen as the essential part of the perturbation operator. This leads to theligand field parameterization. Between the two parameterizations there exists a one to one relationship, whose coefficients are proportional to 3l symbols. This relationship is given together with a brief discussion of the reasons for the proposed parameterizations.

The overlap concept of Clebsch-Gordan coefficients and Racah lemma constants

By virtue of the isomorphic relationship between simple tensor products |l1t1〉|l2t2〉, formed from othonormal bases of real functions, and the corresponding operators Sl1t1l2t2 = |l1t1〉〈l2t2| obtains analogous overlap expressions (scalar products) for functions and operators when using the trace definition of operator overlap. Defining R3-irreducible tensorial quantities [|l1{⊗}l2|]l3t3 (for functions) and [|l1{ ⊗ }l2|]l3t3 (for operators) by the same expansion formulae, one makes Clebsch-Gordan coefficients expressible as function overlaps of the type 〈l1t1|〉l2t2|[|l1{ ⊗ |l2{]l3t3 and as operator overlaps of the type 〈Sl1t1l2t2|[|l1{ ⊗ }l2|]l3t3〉. Similarly, Racah lemma constants (isoscalar factors) obtain overlap interpretations.

Extension of ligand-field theory to encompass bridged structures. Emphasis on the angular overlap model

Abstract The concepts of weak and strong exchange fields are defined as a way of introducing ligand-field theory into problems associated with bridging. Then the molecular orbital angular overlap model (MO-AOM) is used to illuminate the concept of nephelauxetism and contribute to the understanding of the complementarity between charge transfer and electron transfer in bonding and spectroscopy. Charge transfer is associated with orthogonalization, electronic density and diffraction experiments; electron transfer with covalency, transfer of unpaired electron spins and population numbers of predominantly central-ion-localized orbitals. This discussion lends further support to the idea that the chemical concept of oxidation states in ligand-field complexes has an important physical meaning independent from the degree of charge transfer. This is illustrated by a number of chemical examples. It emerges that the MO-AOM has a mutual character in that not only can ligand orbitals be conceived as perturbers of central ion orbitals, but also vice versa. The perturbations are in pairs and have the same values angularly. The importance of the orthogonality of the AOM operators in this context is illustrated. This is also used to extend the MO-AOM to cover nonlinear ligation and bridging. The concept of angular overlap (AO) is given wider scope. The usual chemical distinction between the two limiting cases of bonding, the covalent bond and the heteropolar bond, is exhibited in the model description. The d-electron ligand-field-theory contribution to the problem of bridging emphasizes the usefulness of the concept of the parametrical d q model for this theory. For a bicentric system, for example, the electronic d q ⊗d q Hamiltonian of this model can be partitioned into AA and BB parts associated with the individual centers and a part, (AB+BA), associated with the (weak) coupling between the centers and its various symmetry/geometry-determined one-electron pathways, and this partition can be made at the orbital as well as at the d q -state level. The AA and BB parts can then be diagonalized and the (AB+BA) part rediagonalized, so as to follow the associated basis change. One is left with an almost diagonal description in cases of weak exchange coupling.

Phase-fixed double-group 3-Γ symbols. I. A novel exposition of the general theory of 3-Γ symbols and coupling coefficients

The present paper is the first in a series aiming at the establishment of a transparent and readily applicable Wigner-Racah algebra for all the noncommutative double groups.Starting from the Wigner-Eckart theorem in a very general setting, the theory of the fundamental quantities called here triple coefficients — and the closely related coupling coefficients — is developed and leads through a careful discussion of permutational symmetries to the concept of 3-Γ symbols. By basing the exposition on triple coefficients and by consistently using matrix representations, we obtain a notation and a terminology which enable a clear separation of permutational properties and problems concerning complex conjugation, and a more transparent discussion of tensor (Kronecker) product multiplicities.A particularly elegant formalism is obtained for a situation which generalizes that of the classical rotation-group Wigner-Racah algebra, viz., in which there is a fixed group element effecting (through the inner automorphism it defines) complex conjugation of all the standard irreducible matrix representations.

Phase-fixed double-group 3-Γ symbols. VI. Real 3-Γ symbols and coupling coefficients for the group hierarchy I*⊃ C5*

It is demonstrated that for the group-subgroup hierarchy I*⊃ C5*, one may choose standard irreducible matrix representations and corresponding all-real sets of 3-Γ symbols which obey a formalism just as elegant as the classical one for the 3-j symbols of the rotation double group. The 3-Γ symbols are phase-fixed by the specification of basis functions (or, equivalently, subduction coefficients) generating them and based on functions first given by McLellan.Other icosahedral double-group hierarchies are also briefly discussed.

Geometrical and algebraical invariances and general angular dependences of sets of s-p hybrid orbitals. Their angular overlap model relevance

Abstract An s-p hybrid orbital can be conceived as a linear combination of a scalar and a vector and can be written as h p ( θ , φ ) = h(sp p , θ , g 4) = h( χ , θ , ϕ )= s sin χ + p σ ( θ , φ ) cos χ where ρ = cot 2 χ determines the shape of the hybrid and p σ ( θ , φ ) = p x 〈p x |p σ ( θ , φ )〉 ang +p y 〈p y |p σ ( θ , φ ) 〉 ang +p z 〈p z |p σ ( θ , φ )〉 ang determines its direction. The coefficients to the unit vectors along the three Cartesian axes, written as p functions, are the angular overlaps between these unit vectors and the unit vector along the direction of the hybrid. The situation of four mutually orthogonal s-p hybrids has been analyzed by using these concepts and it has been found that these hybrids invariably lie pairwise in perpendicular planes. Moreover, if each hybrid h i with shape parameter χ i is replaced by a vector with the direction of the hybrid and the length sin2 χ i , then their vector sum is vanishing. The four-hybrid problem has three degrees of freedom determining shapes and relative directions. Formulas are given for a general analysis of the situation, and particularly for the analysis — on the basis of three experimentally determined angles — of the shape and direction of a single lone-pair, which is necessary in an angular overlap model (AOM) context. The formalism of the AOM is shown to function on the basis of hybrid orbitals, and this property of the AOM gives the clue to its handling of non-linearly ligating ligands.