Spin-orbit coupling patterns induced by twist and pyramidalization modes in C2H4:: A quantitative study and a qualitative analysis

A study of the spin-orbit coupling (SOC) mechanisms which couple the triplet pi pi* state (T-1) to the singlet ground state (S-0) in ethylene is carried out at a variety of computational levels and basis sets, using the full Breit-Pauli (BP) SOC Hamiltonian, the one-electron mean-field (MF) operator, and the approximate one-electron operator based on an effective nuclear charge, Z*. The basis set and wave functions requirements needed for good quality SOC calculations are elucidated by studying the SOC interaction using single- and multireference CI as well as MCSCF wavefunctions, with basis sets ranging from the minimal STO-3G all the way to an extended one with quadruple zeta and polarization quality. Two archetype distortion modes of ethylene were considered: a twist mode which changes the symmetry from D-2h to D-2 and then to D-2d and pyramidalization modes which change the ethylene symmetry to C-2v (Syn-pyramidalization) or C-2h (anti-pyramidalization), as well as C-s (i.e., a mono-pyramidalization distortion). It is found that both the twist and syn-pyramidalization distortions-of ethylene promote a nonzero SOC interaction, which involves an interplay between one-center and two-center SOC terms. In the twist distortion, the interplay is strong because the one-center terms arise from a residual incomplete cancelation of the two;on-site interactions. In contrast, in the syn-pyramidalization distortion the interplay is weak, because the one-center terms add up. Consequently, the syn-pyramidalization promotes SOC matrix elements which exceed 6 cm(-1), while the twist mode has a weaker SOC on the order of 2 cm(-1). Zero SOC is obtained for distortion which involve either a 90 degrees twist, or an anti-pyramidalization. The monopyramidalization distortion leads to SOC which is ca. 50% of that which is generated by the syn-pyramidalization. A qualitative analysis based on symmetry and electronic structure enables to understand these trends. A simple physical model, which enables us to carry out the Vectorial summation of SOC in a pictorial manner, is constructed and used to explain the trends in the twist and syn-pyramidalization modes.