Different forms of effective Hamiltonians

The eigenvalues of atomic or molecular Hamiltonians often occur in clusters, corresponding to various kinds of fine structure. A convenient description of these clusters is obtained by transforming the original Hamiltonian to a block-diagonal effective Hamiltonian, where the eigenvalues of each diagonal block correspond to the set of levels within a cluster. At each step in the perturbation expansion, the off-diagonal blocks of the transformation operator are determined by the perturbation equation, but the diagonal blocks are not determined. Thus various solutions are possible. In particular, the Bloch solution gives an effective Hamiltonian that is non-Hermitian and is unsatisfactory for use in fitting observed energy levels. Unitary transformations give Hermitian effective Hamiltonians that are still not unique. This non-uniqueness can lead to indeterminacies in fitting the energy levels.