Topological Representations of Jahn-Teller Distortions

The underlying group theory for the study of JT distortions uses a direct product [85,86] T X r of a suitable irreducible representation (irrep) T of the symmetry group G of the undistorted nuclear configuration. The characters of r X r are the squares of the characters of the corresponding operations of G in the irrep E according to the following equation where g is any operation in G  

If the dimension of T is d, then the dimension of E x E is d. Furthermore, the direct product representation E x E can be reduced into a sum of two lowerdimensional representations, namely the symmetric part [E x E]s of dimension /2d(d + 1) and the antisymmetric part E x E a of dimension l/2d(d — 1). The characters of the symmetric and antisymmetric parts of the direct product E x E can be determined by the following formulas  

According to the rule of the JT instability [87], the JT distortion coordinates of a degenerate electronic state corresponding to an irreducible representation (irrep) E must span representations A belonging to the non-totally symmetric part of the symmetrized direct product of E, i.e. 

Note that for a /-dimensional irrep E the total dimension of the corresponding representation A is [d(d + l)/2] - 1. 

The reduction of the representation A to a sum of irreps depends on the symmetry point group G of the undistorted nuclear configuration. A system of JT distortions corresponding to a given E in a point group G is conventionally described [24] as a JT problem of the type E (gi A or E 0 k where E is the representation of G containing the degenerate electronic state and the X s are the irreps of G contained in the representation A defined by Equation (1.20). The irreps for the JT problems of interest are listed in Table 1.2. 

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