Nuclear dynamics time-dependent Schrodinger equation

Obviously, the BO or the adiabatic states only serve as a basis, albeit a useful basis if they are determined accurately, for such evolving states, and one may ask whether another, less costly, basis could be just as useful. The electron nuclear dynamics (END) theory [1-4] treats the simultaneous dynamics of electrons and nuclei and may be characterized as a time-dependent, fully nonadiabatic approach to direct dynamics. The END equations that approximate the time-dependent Schrodinger equation are derived by employing the time-dependent variational principle (TDVP). 

A mixed quantum classical description of EET does not represent a unique approach. On the one hand side, as already indicated, one may solve the time-dependent Schrodinger equation responsible for the electronic states of the system and couple it to the classical nuclear dynamics. Alternatively, one may also start from the full quantum theory and derive rate equations where, in a second step, the transfer rates are transformed in a mixed description (this is the standard procedure when considering linear or nonlinear optical response functions). Such alternative ways have been already studied in discussing the linear absorbance of a CC in [9] and the computation of the Forster-rate in [10]. 

The reader is assumed to be familiar with some of the basic concepts of quantum mechanics. At this point we will therefore just briefly consider a few central concepts, including the time-dependent Schrodinger equation for nuclear dynamics. This equation allows us to focus on the nuclear motion associated with a chemical reaction. 

The dynamics of nuclear spins can be treated by a time-dependent Schrodinger equation, where the Hamiltonian contains terms for each constituting relaxation mechanism ... 

For a complete treatment of a laser-driven molecule, one must solve the many-body, multidimensional time-dependent Schrodinger equation (TDSE). This represents a tremendous task and direct wavepacket simulations of nuclear and electronic motions under an intense laser pulse is presently restricted to a few bodies (at most three or four) and/or to a model of low dimensionality [27]. For a more general treatment, an approximate separation of variables between electrons (fast subsystem) and nuclei (slow subsystem) is customarily made, in the spirit of the BO approximation. To lay out the ideas underlying this approximation as adapted to field-driven molecular dynamics, we will consider from now on a molecule consisting of Nn nuclei (labeled a, p,...) and Ne electrons (labeled /, j,...), with position vectors Ro, and r respectively, defined in the center of mass (rotating) body-fixed coordinate system, in a classical field E(f) of the form Eof t) cos cot). The full semiclassical length gauge Hamiltonian is written, for a system of electrons and nuclei, as [4]... 

The main reason for introducing quantum TSTs was that simpler descriptions of the rate constant had treated the nuclei as classical particles. While this is often an appropriate point of view, it fails completely in some cases, and almost always when hydrogen is involved in bond breaking or formation. Indeed, strong nuclear quantum effects have been observed in many reactions and recently even in several enzymatic reactions [19-21]. In such situations, one should treat some or all nuclei quantum mechanically. The rate constant is a dynamic quantity and hence very difficult to compute quantum mechanically because solving the time-dependent Schrodinger equation exactly is possible only for a handful of atoms [22]. Fortunately, the longtime quantum dynamic effects on the thermal rate constant of most chemical reactions... 

In this chapter we present the time-dependent quantum wave packet approaches that can be used to compute rate constants for both nonadiabatic and adiabatic chemical reactions. The emphasis is placed on our recently developed time-dependent quantum wave packet methods for dealing with nonadiabatic processes in tri-atomic and tetra-atomic reaction systems. Quantum wave packet studies and rate constants computations of nonadiabatic reaction processes have been dynamically achieved by implementing nuclear wave packet propagation on multiple electronic states, in combination with the coupled diabatic PESs constructed from ab initio calculations. To this end, newly developed propagators are incorporated into the solution of the time-dependent Schrodinger equation in matrix formulism. Applications of the nonadiabatic time-dependent wave packet approaches and the adiabatic ones to the rate constant computations of the nonadiabatic tri-atomic F (P3/2, P1/2) + D2 (v = 0,... 

Worth GA, Robb MA, Lasorne B (2008) Solving the time-dependent Schrodinger equation for nuclear motion in one step direct dynamics of non-adiabatic systems. Mol Phys 106 2077... 

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