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

The vibronic Hamiltonians describing linear and quadratic Jahn-Teller coupling as well as spin-orbit coupling in ME states of trigonal systems are derived for the spin multiplicities M=3,4,5, starting from the microscopic Breit-Pauli spin-orbit operator. The 2M×2M 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 ME × 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 2M 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.