The AOM as a Parameterisation Scheme

The AOM is concerned purely with the representation of the matrix elements of the ligand-field potential part of the ligand-field Hamiltonian in a basis of d-orbitals. The ligand-field potential which we discussed in Sect. 5, is divided into a superposition of nonoverlapping contributions by dividing the molecular space into a set of nonoverlapping cells. 

If we define transformed d-orbitals (d/ as a linear combination of the original d-orbitals using the matrices, 

The question now arises given some hermitian matrix V and the unitary structure matrices for each cell in the molecule, one can ask is the set of parameters / unique, or are there many distinct sets of AOM parameters (e/) that generate the same matrix V If this latter situation obtains, and in the absence of a physical interpretation of (6-4) it normally does, one may then enquire as to how a computed set of AOM parameters can acquire physico-chemical significance. Although AOM parameters are routinely computed by the best-fit parameter fitting procedure sketched above, the mathematical structure of Eq. (6-4), which to be sure is quite simple, has only recently been discussed  

Equation (6-4) is to be interpreted as a set of equations, one for each (i, /), that give matrix elements Vij as a linear transformation of the set of AOM parameters. Although V is of dimension n and contains matrix elements, it is of course hermitian and there are (at most) A n(n + 1) independent matrix elements. Thus Eq. (6-4) is a set of Vi n n H- 1) linear equations on the other hand there are altogether Nn AOM parameters ef. A simple change of notation is helpful here we regard the Vi n(n + 1) matrix elements V,j and the Nn AOM parameters as sets of scalars r/a, Mp respectively, and rewrite (6-4) in the familiar form of a system of linear equations, 

N = J4(n + 1) since for rf-orbitals, n - 5, this requires N = 3, whereas as noted above we almost always have N A, and often N 6 as a result, many different sets of AOM parameters e( can generate the same ligand-field matrix Vij - rja. This is simply a reflection of the fact that as the AOM equation stands, we usually have more unknowns than independent equations if we are given V and search for the set e. In order to guarantee a unique set AOM parameters Eq. (6-4) must be supplemented by a set of equations of constraint. 

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