Quantum-classical mixed mode equations

In order to obtain a more compact formulation of the mixed quantum-classical equations we use a Hamilton-Jacobi-like formalism for the propagation of the quantum degree of freedom as in earlier studies [23], A similar approach has been introduced by Nettesheim, Schiitte and coworkers [54, 55, 56], TTie formalism presented here is based on recent investigations of the present authors [23], This formalism can be summarized as follows. Starting from the Hamiltonian Eqn. (2.2) and averaging over the x- and y-mode, respectively, gives... 

Let us suppose it is possible to treat the nuclear subsystem in a molecule classically and electronic subsystem quantum mechanically. This type of theoretical framework is called a mixed quantum-classical representation. Such a mixed representation can find many applications in science. For instance, a fast mode such as the proton dynamics in a protein should be considered as a quantum subsystem, while the rest of the skeletal structure can be treated as a classical subsystem [3, 484, 485]. It is quite important in this context to establish the correct equations of motion for each of the subsystems and to ask what are their rigorous solutions and how the quantum effects penetrate into the classical subsystems. By studying the quantum-electron and classical-nucleus nonadiabatic dynamics as deeply as possible, we will see how such rigorous solutions, if any, look like qualitatively and quantitatively. This is one of the main aims of this book. 

Because the mapping approach treats electronic and nuclear dynamics on the same dynamical footing, its classical limit can be employed to study the phase-space properties of a nonadiabatic system. With this end in mind, we adopt a onemode two-state spin-boson system (Model IVa), which is mapped on a classical system with two degrees of freedom (DoF). Studying various Poincare surfaces of section, a detailed phase-space analysis of the problem is given, showing that the model exhibits mixed classical dynamics [123]. Furthermore, a number of periodic orbits (i.e., solutions of the classical equation of motion that return to their initial conditions) of the nonadiabatic system are identified and discussed [125]. It is shown that these vibronic periodic orbits can be used to analyze the nonadiabatic quantum dynamics [126]. Finally, a three-mode model of nonadiabatic photoisomerization (Model III) is employed to demonstrate the applicability of the concept of vibronic periodic orbits to multidimensional dynamics [127]. 

The TDSCF approximation is a good starting point for a mixed quantum mechanical/classical treatment. Let us assume that R is the classical and r the quantum mechanical mode. Then, the wavefunction r(r t) describing the vibration of the fragment molecule is a solution of the time-dependent Schrodinger equation... 

Classically, we may set the energy of an oscillator at any value. The frequency for a normal mode is not affected by the energy. Quantum mechanically, only certain energies are possible according to Equation 4.67. However, the energy differences between different states are equal. For this reason, the wave packet moves with the classical frequency. As we will see in Chapter 7, any energy can be given to the wave packet, jnst as in the classical case. For example, the first and second states can be mixed in any proportions, but if the coefficient for the second state is very small, the wave packet will hardly move. Just a small disturbance is introduced. 

So far aU we have written about are classical problems, but there are definite quantum aspects to the proton transfer reaction and, in addition, to the interaction of systems with light and dissipation mechanisms. Vibrations cannot always be treated classically either. At the outset, it should be stated that, unlike the classical description, which allows a relatively simple mechanism for dissipation by means of the Fokker-Planck equation, quantum mechanics does not allow such a possibihty. In addition, aU treatments based on a mixed description of a classical and a quantum system are fundamentally flawed [29]. In the remainder of this chapter, we nevertheless introduce a possible description that looks formaUy the same in classical and quantum mechanics and has some features that make it possible to introduce an elementary mechanism of decay to the equilibrium state. And, equaUy important, it gives a mechanism for averaging over the strongly coupled vibrational modes, as weU as a unified description of the interaction with light. 

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