Spatial Symmetry in VB Theory

The point symmetry group of the molecule is denoted by 9 (Dnu or Cnv in the present case), and it is necessary to produce from the functions (35) wavefunctions which form bases for irreducible representations A of rd. We note first of all that since all the orbitals are localized on one or other of the atoms forming the molecule, the application of a spatial symmetry operation 52 of rS is equivalent to a permutation of the orbitals on the equivalent atoms amongst themselves, possibly multiplied by a rotation of the orbitals on the central atom. Hence with every operation 52 we may associate a certain permutation of the orbitals, Pr, in which the bar emphasizes that one permutes the orbitals themselves and not the electron co-ordinates. Thus, 

We may disregard the closed-shell cores of the atoms since these play no role in the construction of symmetry-adapted wavefunctions, and concentrate attention upon the valence electrons. In the simplest case, with one valence electron per atom, we have a configuration 0102 / n of N singly-occupied, non-degenerate valence orbitals which is then said to form a covalent structure for the molecule. Then under any spatial symmetry operation (%, a VB function Vsu-.k transforms as 

In this expression, g is the order of the group 2 , and making use of equation (41), we obtain 

The set of matrices Us Pr) in equation (41) form a reducible representation of the group which is reduced into its irreducible components A by the coefficients a above. If we denote the irreducible representations of iAn and 2 by 17 Is- iVl and Z)(/1 , respectively, then this reduction can be written symbolically as 

The coefficients determine the number of times a particular molecular multiplet 2s+ y1 occurs in this decomposition. This is just the dimension of the secular equation (23) which has to be solved in the symmetry-adapted basis. The cSA can be determined in the usual manner from character tables for the groups SAn and St.  

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