Symmetry Factoring of Secular Equations

Even with the simplifications that result from a drastic approximation such as the Hiickel approximation, the secular equation for the MOs of an n-atomic molecule will, in general, involve at least an unfactored nth-order determinant, as just illustrated in the case of naphthalene. It is clearly desirable to factor such determinants, and symmetry considerations provide a systematic and rigorous means of doing this. 

A secular equation such as 7.1-15 is derived from an array of the individual 

Suppose that, instead of writing the secular determinant from an n x n array of atomic orbitals, we use an n x n array of n orthonormal, linear combinations of the basis set orbitals. Suppose, furthermore—and this is the key—we require these linear combinations to be SALCs, that is, each one is required to be a function which forms a basis for an irreducible representation of the point group of the molecule. Then, as shown in Chapter 5, all integrals of the types 

The foregoing considerations lead to a three-step procedure for setting up a symmetry-factored secular equation  

Use the set of atomic orbitals as the basis for a representation of the group, and reduce this representation to its irreducible components. 

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