Reaction dynamics nuclear wave function

Equation (28) is still exact. To introduce the classical-path approximation, we assume that the nuclear dynamics of the system can be described by classical trajectories that is, the position operator x is approximated by its mean value, namely, the trajectory x t). As a consequence, the quantum-mechanical operators of the nuclear dynamics (e.g., Eh (x)) become classical functions that depend parametrically on x t). In the same way, the nuclear wave functions dk x,t) become complex-valued coefficients dk x t),t). As the electronic dynamics is evaluated along the classical path of the nuclei, the approximation thus accounts for the reaction of the quantum DoE to the dynamics of the classical DoF. 

The descriptions of molecular dynamics and the theory of chemical reactions in gas and condensed phases are based on the concept of potential energy function (hypersurface) [1,2] rooted in the Bom-Oppenheimer (BO) approach [3]. The parametric dependance of the electronic wave function with respect to nuclear coordinates is the basic idea on which the BO framework rest. In this paper, a different approach is taken. The electronic state functions are taken to be independent from the instantaneous nuclear positions. As a first step, we consider molecular systems which are characterized by stationary nuclear configurations belonging to particular symmetry groups. The corresponding electronic stationary states must always transform according to given irreduci-... 

From a theoretical point of view, in the study of atom-atom or atom-molecule collisions one needs to solve the Schrodinger equation, both for nuclear and electronic motions. When the nuclei move at much lower velocities than those of the electrons inside the atoms or molecules, both motions (nuclear and electronic) can be separated via the Born-Oppenheimer approximation. This approach leads to a wave function for each electronic state, which describes the nuclear motion and enables us to calculate the electronic energy as a function of the intemuclear distance, i.e. the potential energy V r). Therefore, V r) can be obtained by solving the electronic Schrodinger equation for each inter-nuclear distance. As a result, the nuclear motion, which we shall see is the way chemical reactions take place, is a dynamical problem that can be solved by using either quantum or classical mechanics. 

The standard theoretical treatment of chemical reaction dynamics is based on the separation of the total molecular motion into fast and slow parts. The fast motion corresponds to the motion of the electrons and the slow motion corresponds to the motion of the nuclei. The theoretical foundation for the separation of the electronic and nuclear motion was first developed by Born and Oppenheimer. In this approach, the total molecular wave function is expanded in terms of a set of electronic eigenfunctions which depend parametrically on the nuclear coordinates. The expansion coefficients are the... 

When considering reaction paths on the PE surfaces of excited states, as required for the rationalization of photochemistry [4], two major additional complications arise. First, reliable ab initio energy calculations for excited states are typically much more involved than ground-state calculations. Secondly, multi-dimensional surface crossings are the rule rather than the exception for excited electronic states. The concept of an isolated Born-Oppenheimer(BO) surface, which is usually assumed from the outset in reaction-path theory, is thus not appropriate for excited-state dynamics. At surface crossings (so-called conical intersections [5-7]) the adiabatic PE surfaces exhibit non-differentiable cusps, which preclude the application of the established methods of mathematical reaction-path theory [T3]. As an alternative to non-differentiable adiabatic PE surfaces, so-called diabatic surfaces [8] may be introduced, which are smooth functions of the nuclear coordinates. However, the definition of these diabatic surfaces and associated wave functions is not unique and involves some subtleties [9-11]. 

Once a potential surface has been obtained, a variety of dynamics methods can be used to determine reaction cross sections and rate constants. The "exact approach would involve solving the nuclear-motion Schrodinger equation for the appropriate scattering wave function. Because of the complexity associated with doing this, such exact solutions have been confined to the H4-H2 reaction. Much progress in the past year has been reported in the determination of approximate quantum solutions in three dimensions, so some of these applications are described in section II.B. Many more atom-diatom reactions have been treated exactly using one-dimensional models. 

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