The Renner vibronic-coupling problem in 4Π electronic states of linear molecules is analyzed with rigorous and systematic inclusion of spin-orbit (SO) coupling. The 8 × 8 Hamiltonian matrix of a 4Π state in the diabatic electronic representation has been constructed by a Taylor expansion in the bending normal mode up to second order. As previously found for 2Π states and 3Π states, SO-induced vibronic-coupling terms that are linear in the bending amplitude exist in addition to the quadratic electrostatic Renner coupling. The effects of the linear and quadratic Renner coupling on the four Kramers-degenerate potential energy surfaces of the 4Π state are discussed. The spectroscopic effects of the linear SO-vibronic-coupling mechanism have been analyzed by numerical calculations of vibronic spectra.

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