Splitting of Levels and Terms in a Chemical Environment

We may use the set of five d wave functions as a basis for a representation of the point group of a particular environment and thus determine the manner in which the set of d orbitals is split by this environment. Let us choose an octahedral environment for our first illustration. In order to determine the representation for which the set of d wave functions forms a basis, we must first find the elements of the matrices which express the effect upon the set of wave functions of each of the symmetry operations in the group the characters of these matrices will then be the characters of the representation we are seeking. 

Although the full symmetry of the octahedron is O, we can gain all required information about the d orbitals by using only the pure rotational subgroup O because Oh may be obtained from O by adding the inversion, i. However, we already know that d orbitals are even to inversion, so that it is only the pure rotational operations of O which bring us new information. 

We assume that the wave functions of a set of d orbitals are each of the general form specified by 9.2-1. We shall further assume that the spin function [jj% is entirely independent of the orbital functions and shall pay no further attention to it for the present. Since the radial function R(r) involves no directional variables, it is invariant to all operations in a point group and need concern us no further. The function 0(0) depends only upon the angle 0. Therefore, if all rotations are carried out about the axis from which 0 is measured (the z axis in Fig. 8.1), (0) will also be invariant. Thus, by always choosing the axes of rotation in this way (or, in other words, always quantizing the orbitals about the axis of rotation), only the function f ( ) will be altered by rotations. The explicit form of the 4 (0) function, aside from a normalizing constant, is 

If we take the function eim and rotate by an angle a, the function becomes ewiW+tt) Thus we can easily see that the set of 4 ( ) wave functions, I, becomes II on rotation by a  

This five-dimensional matrix is only a special case for a set of d functions, and clearly in the (21 + l)-fold set of functions (/ = 0 for an s level, 1 for a p level, 3 for an / level, and so on) we shall have 

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