Schrodinger equation quantum-classical framework

Once this equation is solved for all relevant regions of the nuclear configuration space, in the BO framework, the nuclear motion can be treated either via a classical mechanical analysis with the help of computer simulations [6], or it can be treated quantum mechanically for simple models [54], In the latter scheme, the nuclear Schrodinger equation must be solved ... 

In the present section we are concerned with genuine internal noise. We consider a closed, isolated many-body system, whose evolution is given by a Schrodinger equation. Remember that in the classical case in III.2 we gave a macroscopic description in terms of a reduced set of macroscopic variables, which obey an autonomous set of differential equations. These equations are approximate and deviations appear in the form of fluctuations, which are a vestige of the large number of eliminated microscopic variables. Our task is to carry out this program in the framework of quantum mechanics. 

Conventional methods based on quantum mechanical models use matrix diagonalization to find a self-consistent solution of the time-independent Schrodinger equation. Unfortunately, the cost of matrix diagonalization grows extremely rapidly with the number of atoms in the system. Consequently, methods based on quantum mechanical models tend to be computationally expensive. As a result, the zeolite framework is often treated as a cluster instead of as a periodical system. To overcome this obstacle, hybrid models have been put forward in which the problem is circumvented the reaction center is described in a quantum mechanical way, whereas the surroundings are described in a classical way. ... 

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