Internal rotation in Jahn-Teller coupled systems: The ethene and allene cations

Graphical abstract: A vibronic Hamiltonian has been derived to treat the Jahn-Teller effect in molecules, such as the allene cation, in which one of the active modes is the torsion or an internal rotation. The potential energy surfaces are expressed using Fourier series instead of Taylor series. Research highlights: → The Ex(b1+b2) Jahn-Teller effect in the 2E ground state of the singly-charged cations of ethene and allene is investigated. → A theoretical model is presented that allows a joint treatment of the Jahn-Teller effect and of the large-amplitude torsional motion. → The vibronic model is based on the expression of the potential energy surfaces using Fourier series instead of Taylor series. - Abstract: The electronic ground state of singly-charged cations of the cumulene family (H2CCnCH2, n = 0, 1, ...) is doubly degenerate at the D2d geometry and is subject to an E x b or E x (b1 + b2) Jahn-Teller effect. One of the Jahn-Teller active modes is the torsion, i.e., the disrotatory motion of the two terminal CH2 entities. The 2π periodicity of the torsional motion and the presence of maxima in the potential energy path along the torsional coordinate at planar geometries poses problems in the usual treatment of the Jahn-Teller effect of these cations, which relies on a Taylor expansion of the potential energy surfaces at the point of electronic degeneracy (D2d). A vibronic Hamiltonian has been derived to treat the E x b and E x (b1 + b2) Jahn-Teller effects in molecules in which one of the Jahn-Teller active modes is the torsion or an internal rotation. The potential energy surfaces are expressed using Fourier series instead of Taylor series which enables a joint treatment of the Jahn-Teller effect and of hindered internal rotations. The resulting Hamiltonian has been used to predict the vibronic energy level structure of C2H4+ and C3H4+ and intensity distributions in the photoelectron spectra of C2H4 and C3H4. In particular, splittings arise from tunneling through the potential energy barrier at the D2h geometry. The predictions are compared with available spectroscopic data.