The 3E × E, 4E × E and 5E × E Jahn–Teller Hamiltonians of trigonal systems

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The vibronic Hamiltonians describing linear and quadratic Jahn–Teller coupling as well as spin–orbit coupling in M E states of trigonal systems are derived for the spin multiplicities M = 3 , 4 , 5 , starting from the microscopic Breit–Pauli spin–orbit operator. The 2 M × 2 M Hamiltonian matrices in a diabatic spin-electronic basis are obtained by the expansion of the Hamiltonian in powers of the Jahn–Teller active normal mode. It is shown that the M E × E Jahn–Teller Hamiltonians can be made block-diagonal by the transformation to an alternative diabatic basis which mixes electronic orbital and spin projections. The adiabatic potential-energy surfaces and the adiabatic electronic wave functions are obtained in analytical form. In systems with an odd number of electrons (and thus even spin multiplicity) the Jahn–Teller effect is quenched by strong spin–orbit coupling and the adiabatic potential-energy surfaces are strictly two-fold degenerate (Kramer’s degeneracy). In systems with an even number of electrons (and thus odd spin multiplicity), two of the 2 M adiabatic potentials exhibit an E × E Jahn–Teller effect which is unaffected by the spin–orbit interaction. The geometric phases of the adiabatic electronic wave functions are those of the 2E × E Jahn–Teller effect with and without spin–orbit coupling, respectively.