2 Application of time - dependent density - functional theory to electron - ion coupling in ethylene

To examine the applicability of the time-dependent density-functional theory (TDDFT) for treating the electron-nucleus coupling in excited states, we calculate the strength distribution associated with the π-π * transition in ethylene. The observed optical transition strength at 7-8.5 eV region shows a complex structure arising from coupling to CC stretch motion, to torsional motion, and to Rydberg excitations. The mean energy of the observed peak is reproduced to about 0.2 eV accuracy by the TDDFT in the local density approximation (LDA). The reflection approximation is used to calculate the peak broadening. Roughly half of the broadening can be attributed to the fluctuation in the CC coordinate. The asymmetry in the line shape is also qualitatively reproduced by the CC coordinate fluctuation. We find, in agreement with other theoretical studies, that the torsional motion is responsible for the progression of weak transition strength extending from the peak down to about 6 eV. The LDA reproduces the strength in this region to about factor of 3. We conclude that the TDDFT is rather promising for calculating the electron nucleus coupling at short times.