Coupled electron-nuclear quantum dynamics through and around a conical intersection

In solving the time-dependent Schrödinger equation for a coupled electron-nuclear system, we study the motion of wave packets in a model which exhibits a conical intersection (CoIn) of adiabatic potential energy surfaces. Three different situations are studied. In the first case, an efficient non-adiabatic transition takes place while the wave packet passes the region of the CoIn. It is demonstrated that during these times, the nuclear probability density retains its Gaussian shape and the electronic density remains approximately constant. Second, dynamics are regarded where non-adiabatic transitions do not take place, and the nuclear dynamics follows a circle around the location of the CoIn. During this motion, the electronic density is shown to rotate. The comparison with the Born-Oppenheimer nuclear dynamics reveals the geometrical phase being associated with the circular motion. This phase is clearly revealed by an analysis of time-dependent autocorrelation functions and spectra obtained from the numerically exact and the Born-Oppenheimer calculation. The intermediate situation with a small non-adiabatic transition probability is characterized by wave-packet splitting into several fractions.