Second quantization as a tool for the calculation of effective hamiltonians

Abstract

Effective hamiltonians play an important role in the field of magnetism. Extensive use of irreducible tensor algebra is the common method to treat these hamiltonians. We present here a different method which is based on the second quantization formalism. The effective hamiltonian under consideration is transformed onto a second quantization representation. This form of the hamiltonian is directly used to calculate the corresponding hamiltonian matrix. Our procedure is easily applied to very different types of hamiltonians and basis sets. As an example, the present method is applied to the calculation of the exchange interaction between orbitally degenerate ions in Ti2 X 3 - 9, and is compared with other methods treating the same problem. We provide analytical expressions for the energy states in Ti2 X 3 - 9 for the ground manifold, 2 T 2 × 2 T 2 and for the excited manifolds, 2 T 2 × 2 E and 2 E × 2 E.