Nuclear dynamics the Schrodinger equation

On the theoretical side, advances have also been made both in methodology and in concepts. For example, new and powerful techniques for the solution of the time-dependent Schrodinger equation (see Section 1.1) have been developed. New concepts for laser control of chemical reactions have been introduced where, for example, one laser pulse can create a non-stationary nuclear state that can be intercepted or redirected with a second laser pulse at a precisely timed delay. 

The theoretical foundation for reaction dynamics is quantum mechanics and statistical mechanics. In addition, in the description of nuclear motion, concepts from classical mechanics play an important role. A few results of molecular quantum mechanics and statistical mechanics are summarized in the next two sections. In the second part of the book, we will return to concepts and results of particular relevance to condensed-phase dynamics. 

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. 

We consider a system of K electrons and N nuclei, interacting through Coulomb forces. The basic equation of motion in quantum mechanics, the time-dependent Schrodinger equation, can be written in the form 

The translational motion of the particles as a whole (i.e., the center-of-mass motion) can be separated out. This is done by a change of variables from riab, R ab to i CM and r, R, where Rcm gives the position of the center of mass and r, R are internal coordinates that describe the relative position of the electrons with respect to the nuclei and the relative position of the nuclei, respectively. This coordinate transformation implies 

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