Nuclear dynamics vibrational wave function

For bound state systems, eigenfunctions of the nuclear Hamiltonian can be found by diagonalization of the Hamiltonian matiix in Eq. (11). These functions are the possible nuclear states of the system, that is, the vibrational states. If these states are used as a basis set, the wave function after excitation is a superposition of these vibrational states, with expansion coefficients given by the Frank-Condon overlaps. In this picture, the dynamics in Figure 4 can be described by the time evolution of these expansion coefficients, a simple phase factor. The periodic motion in coordinate space is thus related to a discrete spectrum in energy space. 

Transient vibrational dynamics. Perturbation theory yields an intuitive picture of adsorbate relaxation the loss of a vibrational quantum and associated nodal structure in the nuclear wave function is coupled to an irreversible transfer of momentum to the metallic electrons (see Fig. 2). To obtain time-resolved information about the dynamical processes at work, it is nonetheless necessary to go beyond this simple model. In the past decades, classical molecular dynamics has been hugely successful at shedding light on the transient vibrational evolution in a variety of adsorbate-surface systems (see, e.g., ref. 54-56). The methods of choice for including non-adiabatic effects on the dynamics can be divided in two main families friction-lype... 

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