Application The Jahn-Teller Effect

In 1937 Jahn and Teller made the claim that degenerate states of molecules are intrinsically unstable [6, 7], 

Theorem 13 Non-linear molecules in a spatially-degenerate electronic state are subject to spontaneous symmetry-breaking forces that distort the molecule to a geometry of lower symmetry, where the degeneracy is removed. 

The theorem is based on a perturbation of the Hamiltonian by smaU displacements of the nuclei. A high-symmetry geometry is chosen as the origin, and the nuclear displacements are described by normal modes which transform as irreps of the point group. The nuclear positions are parameters in the electronic Hamiltonian. One has, to second-order  

The partial derivatives with respect to the normal modes will affect only the electrostatic Vtie term in the Hamiltonian. These operators are thus electrostatic one-electron operators. At the coordinate origin, the electronic state is degenerate, and is described by a set of wavefunctions, FaYa), where TL is a degenerate irrep. The energies as functions of the coordinates are obtained by diagonalizing the Hamiltonian matrix, H, with elements  

The matrix in H is also called the Jahn-Teller (JT) matrix. The linear terms in this matrix are of type  

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