Jahn-Teller, pseudo-Jahn-Teller, and spin-orbit coupling Hamiltonian of a d electron in an octahedral environment

Starting from the model of a single d-electron in an octahedral crystal environment, the Hamiltonian for linear and quadratic Jahn-Teller (JT) coupling and zeroth order as well as linear spin-orbit (SO) coupling in the ²T _2g + ²E _g electronic multiplet is derived. The SO coupling is described by the microscopic Breit-Pauli operator. The 10 × 10 Hamiltonian matrices are explicitly given for all linear and quadratic electrostatic couplings and all linear SO-induced couplings. It is shown that the ²T _2g manifold exhibits, in addition to the well-known electrostatic JT effects, linear JT couplings which are of relativistic origin, that is, they arise from the SO operator. While only the e _g mode is JT-active in the ²E _g state in the nonrelativistic approximation, the t _2g mode becomes JT-active through the SO operator. Both electrostatic as well as relativistic forces contribute to the ²T _2g - ²E _g pseudo-JT coupling via the t _2g mode. The relevance of these analytic results for the static and dynamic JT effects in octahedral complexes containing heavy elements is discussed.