Eigenstates and Commutation Relations

Our goal is to find a set of states that are simultaneous eigenstates for the molecular Hamiltonian and for the set of symmetry operators. Symmetry operators must commute with the Hamiltonian but not necessarily among themselves. A symmetry operator must be unitary, since, obviously, it cannot change the length of a state vector. Suppose we have an eigenstate of the Hamiltonian  

For a single symmetry operator we can immediately find the possible eigenvalues. Any symmetry operator A is cyclic, that is there exists a number n, such that A = E, the identity operation. Suppose X) is an eigenstate of A with eigenvalue X  

An operator of order n thus has precisely n distinct eigenvalues. We are in this section assuming that we are dealing with integer spin systems so that a rotation through 2 n equals the identity. Halfinteger spin will be treated in the following section. 

The problem is different when the generators do not commute. As a specific example we can take the case of the point group D4. The two generators are C4 and C2. They do not commute but obey the relation  

This relation follows straightforwardly from geometric inspection. 

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