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 T-2(2g) + E-2(g) electronic multiplet is derived. The SO coupling is described by the microscopic Breit-Pauli operator. The 10 x 10 Hamiltonian matrices are explicitly given for all linear and quadratic electrostatic couplings and all linear SO-induced couplings. It is shown that the T-2(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 E-2(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 T-2(2g) - E-2(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. (C) 2012 American Institute of Physics. [http://dx.doi.org/10.1063/1.4751439]