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

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10.1063/1.4751439

### Abstract

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 ^{2} *T* _{2g } + ^{2} *E* _{ 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 ^{2} *T* _{2g }manifold exhibits, in addition to the well-known electrostaticJTeffects, 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 ^{2} *E* _{ 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 ^{2} *T* _{2g } − ^{2} *E* _{ g } pseudo-JT coupling via the *t* _{2g } mode. The relevance of these analytic results for the static and dynamic JTeffects in octahedral complexes containing heavy elements is discussed.

© 2012 American Institute of Physics

Received 16 May 2012
Accepted 24 August 2012
Published online 17 September 2012

Article outline:

I. INTRODUCTION

II. ELECTRONIC HAMILTONIAN, ELECTRONIC BASIS FUNCTIONS, VIBRATIONAL NORMAL MODES AND JT SELECTION RULES

III. (^{2} *T* _{2g } + ^{2} *E* _{ g }) × (*t* _{2g } + *e* _{ g }) JT AND PJT HAMILTONIAN

A. Taylor expansion of the electronic Hamiltonian and calculation of matrix elements

B. ^{2} *T* _{2g } × (*t* _{2g } + *e* _{ g }) JT Hamiltonian

C. ^{2} *E* _{ g } × (*t* _{2g } + *e* _{ g }) JT Hamiltonian

D. (^{2} *T* _{2g } + ^{2} *E* _{ g }) × (*t* _{2g } + *e* _{ g }) PJT coupling

IV. CONCLUSIONS

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/content/aip/journal/jcp/137/11/10.1063/1.4751439

2012-09-17

2014-04-21

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