Quantum trajectory approach

We have seen that the quantum trajectory approach (as does the single configuration self-consistent field approach) leads to equations of motion where one degree of freedom feels the average interaction through an Ehrenfest type average (see, e.g., equations 29, 33, 34, and 51). Thus the interaction or the correlation between the two modes is described approximately. In the limit of narrow wavepackets a term such as the one appearing in equation (51) would approach the classical expression. Assume for instance an exponential interaction potential such that V(r, R) = C exp(—a(R - Xr)), then... 

The big advantage of the Gaussian wavepacket method over the swarm of trajectory approach is that a wave function is being used, which can be easily manipulated to obtain quantum mechanical information such as the spechum, or reaction cross-sections. The initial Gaussian wave packet is chosen so that it... 

Reality suggests that a quantum dynamics rather than classical dynamics computation on the surface would be desirable, but much of chemistry is expected to be explainable with classical mechanics only, having derived a potential energy surface with quantum mechanics. This is because we are now only interested in the motion of atoms rather than electrons. Since atoms are much heavier than electrons it is possible to treat their motion classically. Quantum scattering approaches for small systems are available now, but most chemical phenomena is still treated by a classical approach. A chemical reaction or interaction is a classical trajectory on a potential surface. Such treatments leave out phenomena such as tunneling but are still the state of the art in much of computational chemistry. 

In addition to experiments, a range of theoretical techniques are available to calculate thermochemical information and reaction rates for homogeneous gas-phase reactions. These techniques include ab initio electronic structure calculations and semi-empirical approximations, transition state theory, RRKM theory, quantum mechanical reactive scattering, and the classical trajectory approach. Although still computationally intensive, such techniques have proved themselves useful in calculating gas-phase reaction energies, pathways, and rates. Some of the same approaches have been applied to surface kinetics and thermochemistry but with necessarily much less rigor. 

The classical theory is a valuable complement of the quantum mechanical approaches. It is best suited for fast and direct photodissociation. Quantum mechanical effects, however, such as resonances or interferences inherently cannot be described by classical mechanics. The obvious extension is a semiclassical theory (Miller 1974, 1975) which incorporates the quantum mechanical superposition principle without the complexity of full quantum mechanical calculations. All ingredients are derived solely from classical trajectories. For an application in photodissociation see Gray and Child (1984). 

For practical reasons, a quasi-classical approximation to the quantum dynamics described by Eq. (1.10) is often sought. In the quasi-classical trajectory approach (discussed in Section 4.1) only one aspect of the quantum nature of the process is incorporated in the calculation the initial conditions for the trajectories are sampled in accord with the quantized vibrational and rotational energy levels of the reactants. 

Obviously, purely quantum mechanical effects cannot be described when one replaces the time evolution by classical mechanics. Thus, the quasi-classical trajectory approach exhibits, e.g., the following deficiencies (i) zero-point energies are not conserved properly (they can, e.g., be converted to translational energy), (ii) quantum mechanical tunneling cannot be described. 

Fig. 4.1.2 Harmonic oscillator with the energy E = p2/(2m) + (1/2)kq2 (which is the equation for an ellipse in the (q,p)-space). In the quasi-classical trajectory approach, E is chosen as one of the quantum energies, and all points on the ellipse may be chosen as initial conditions in a calculation, i.e., corresponding to all phases a [0, 27r].

Let us make a final comment, concerning the violation of the Green-Kubo relation. There is a close connection between the breakdown of this fundamental prescription of nonequilibrium statistical physics and the breakdown of the agreement between the density and trajectory approach. We have seen that the CTRW theory, which rests on trajectories undergoing abrupt and unpredictable jumps, establishes the pdf time evolution on the basis of v /(f), whereas the density approach to GME, resting on the Liouville equation, either classical or quantum, and on the convenient contraction over the irrelevant degrees of freedom, eventually establishes the pdf time evolution on the basis of a correlation function, the correlation function in the dynamical case... 

Another advantage of the quantum calculations is that they provide a rigorous test of approximate methods for calculating dissociation rates, namely classical trajectories and statistical models. Two commonly used statistical theories are the Rice-Ramsperger-Kassel-Marcus (RRKM) theory and the statistical adiabatic channel model (SACM). The first one is thoroughly discussed in Chapter 2, while the second one is briefly reviewed in the Introduction. Moreover, the quantum mechanical approach is indispensable in analyzing the reaction mechanisms. A resonance state is characterized not only by its position, width and the distribution of product states, but also by an individual wave function. Analysis of the nodal structure of resonance wave functions gives direct access to the mechanisms of state- and mode-selectivity. 

In this chapter we elucidate the state-specific perspective of unimolec-ular decomposition of real polyatomic molecules. We will emphasize the quantum mechanical approach and the interpretation of the results of state-of-the-art experiments and calculations in terms of the quantum dynamics of the dissociating molecule. The basis of our discussion is the resonance formulation of unimolecular decay (Sect. 2). Summaries of experimental and numerical methods appropriate for investigating resonances and their decay are the subjects of Sects. 3 and 4, respectively. Sections 5 and 6 are the main parts of the chapter here, the dissociation rates for several prototype systems are contrasted. In Sect. 5 we shall discuss the mode-specific dissociation of HCO and HOCl, while Sect. 6 concentrates on statistical state-specific dissociation represented by D2CO and NO2. Vibrational and rotational product state distributions and the information they carry about the fragmentation step will be discussed in Sect. 7. Our description would be incomplete without alluding to the dissociation dynamics of larger molecules. For them, the only available dynamical method is the use of classical trajectories (Sect. 8). The conclusions and outlook are summarized in Sect. 9. 

It can be applied in the analysis of almost every nonlinear quantum process. Even external pumping and energy losses can be easily described. In the classical trajectory approach to SHG [27], deterministic solutions of the classical SHG are used, while quantum noise of initial fields is artificially simulated by Gaussian distribution. One can study the time evolution of the bunch of... 

Recently [8-11] an alternative treatment to mix quantum mechanics with classical mechanics, based on Bohmian quantum trajectories was proposed. Briefly, the quantum subsystem is described by a time-dependent Schrodinger equation that depends parametrically on classical variables. This is similar to other approaches discussed above. The difference comes from the way the classical trajectories are calculated. In our approach, which was called mixed quantum-classical Bohmian (MQCB) trajectories, the wave packet is used to define de Broglie-Bohm quantum trajectories [12] which in turn are used to calculate the force acting on the classical variables. 

Three novel approaches to the simulation of NA dynamics of large chemical systems have been presented [20-22]. The approaches extend the standard quantum-classical NA MD to incorporate quantum effects of the solvent subsystem that have been traditionally treated by classical mechanics. These effects include quantum trajectory branching (wave packet splitting), loss of quantum coherence directly related to the Franck-Condon overlap contribution to the NA transition probability, and ZPE of nuclear motion that contributes to the NA coupling and must be preserved during the equilibration of the energy released by the NA transition. 

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