Time-dependent Schrodinger equation, local

As long as the photodissociation reaction is fairly direct, the time-dependent formulation is fruitful and provides insight into both the process itself and the relationship of the final-state distributions to the absorption spectrum features. Moreover, solution of the time-dependent Schrodinger equation is feasible for these short-time evolutions, and total and partial cross sections may be calculated numerically.5 Finally, in those cases where the wavepacket remains well localized during the entire photodissociation process, a semi-classical gaussian wavepacket propagation will yield accurate results for the various physical quantities of interest.6... 

To study the electron-lattice coupling dynamics of any molecular system without any constrains concerning the relation between the localization energy, and the electronic coupling, /, it is necessary to simultaneously solve the time-dependent Schrodinger equation... 

We must, therefore, use the Schrodinger equation in its time-dependent form to describe the motion of the molecule, with the wave packet being initially localized on the PES, in space and time. If discrete travelling-wave solutions of the Schrodinger wave equation are combined, then they can be used to construct the required wave packet, which localizes it to a transient pulse. Assuming that a single-frequency wave solution of the time-dependent Schrodinger equation can be written as y(r, t) =A sin( r — rot), then the superposition wave-packet solution is Y (r, t) =... 

When a molecule is excited by an ultrashort laser pulse with an appropriate center frequency, a localized wave packet can be created in the excited electronic state because of the excitation of a coherent superposition of many vibrational-rotational states. It follows from fundamental laws that the d3mamics of molecular wave packets is governed by a time-dependent Schrodinger equation (eqn 2.29), where H is the relevant Hamiltonian of the given molecule. Because molecular potential-energy surfaces are anharmonic, this molecular wave packet tends to spread both in position (coordinates) and in momentum. However, in addition to expansion or defocusing, the wave packet also suffers delocalization at a certain instant of time. Coherent quantum... 

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