Theory and applications of generalized operator transforms for diagonalization of spin Hamiltonians

Abstract

A generalized transform formalism for vector operators is devised for diagonalization of a rather wide class of spin hamiltonians. The operator technique leads to equations for transformation matrices, for which analytical solutions are given. These allow analytical formulation of the transformed electron Zeeman term, the sum of the magnetic hyperfine and nuclear Zeeman term, the electric quadrupole term and the electronic and nuclear Zeeman coupling terms. The angular dependence of energy eigenvalues, frequencies and line strengths of ESR and ENDOR transitions to first order will be expressed as compact bilinear and quadratic forms of the columns of the matrix relating the molecular coordinate system to the laboratory system. Thereby the explicit calculation of rotation matrices may be completely avoided, though the latter formally express the operator transforms. The generalized operator transform is also carried out for the off-diagonal blocks originating from hyperfine interaction terms. This allows the second order energy terms to be expressed explicitly as compact hermitean forms of a simple structure, in particular the explicit structure of mixing terms between hyperfine interactions of different (sets of) nuclei is obtained. The relation to the conventional Bleaney transform is discussed and the analogy to the generalized operator transform is worked out. Furthermore, a set of practical formulae is collected for experiments with angularly dependent ENDOR (ESR) crystal spectra. A comparison of exact (numerical) diagonalization with the numerical accuracy of the generalized transform method is given. The latter is shown to approximate exact transition frequencies and line strength for any situation with high precision. Finally two examples are discussed, which illustrate the generalized transform treatment of the second order terms.