Phase space theory quantum

To determine the optimal value of quantum correction y, several criteria have been proposed, all of which are based on the idea that an appropriate classical theory should correctly reproduce long-time hmits of the electronic populations. (Since the populations are proportional to the mean energy of the corresponding electronic oscillator, this condition also conserves the ZPE of this oscillator.) Employing phase-space theory, it has been shown that this requirement leads to the condition that the state-specihc level densities... 

If redissociation into reactants is faster than stabilization, equations (3.15) and (3.16) simplify into a product of k,/k, and either kr or kcoll. Under these conditions, to obtain a theory for a total association rate coefficient, one must calculate both k,/k i and kr or kco . Three levels of theory have been proposed to calculate k, /k, . In the simplest theory, one assumes (Herbst 1980 a) that k, /k 3 is given by its thermal equilibrium value. In the next most complicated theory, the thermal equilibrium value is modified to incorporate some of the details of the collision. This approach, which has been called the modified thermal or quasi-thermal treatment, is primarily associated with Bates (1979, 1983 see also Herbst 1980 b). Finally, a theory which takes conservation of angular momentum rigorously into account and is capable of treating reactants in specific quantum states has been proposed. This approach, called the phase space theory, is associated mainly with Bowers and co-workers... 

The quantum yield of (3P0) production (IV-19) is less than 0.03 (180). Therefore, the most important process must be the formation of HgH. This conclusion is in disagreement with that of Yang et al. (1063), who have concluded from phase space theory that the production of vibrationaily excited H2 is most important. 

It must be emphasized that such phenomena are to be expected for a statistical system only in the regime of low level densities. Theories like RRKM and phase space theory (PST) (Pechukas and Light 1965) are applicable when such quantum fluctuations are absent for example, due to a large density of states and/or averaging over experimental parameter such as parent rotational levels in the case of incomplete expansion-cooling and/or the laser linewidth in ultrafast experiments. However, in the present case, it is unlikely that such phenomena can be invoked to explain why different rates are obtained when using ultrafast pump-probe methods that differ only in experimental detail. 

The statistical theories provide a relatively simple model of chemical reactions, as they bypass the complicated problem of detailed single-particle and quantum mechanical dynamics by introducing probabilistic assumptions. Their applicability is, however, connected with the collisional mechanism of the process in question, too. The statistical phase space theories, associated mostly with the work of Light (in Ref. 6) and Nikitin (see Ref. 17), contain the assumption of a long-lived complex formation and are thus best suited for the description of complex-mode processes. On the other hand, direct character of the process is an implicit dynamical assumption of the transition-state theory. 

The quantum mechanical phase space theory has also been developed [75—77] and applied to ion—molecule reactions, including some endothermic reactions [78, 79]. While results again seem to show that the theory is promising, the apparent future problem would be how to extend the treatment to systems involving more than three atoms. 

Figure 4.8 Vibrational, translational and rotational temperatures of carbon dimers emitted successively from thermally excited fullerene cations, as predicted by phase space theory with quantum densities of states.

While the results of trajectory calculations provide an accurate testing ground for more approximate theories, and, in the parameterised form developed by Su, Chesnavich and Bowers [25,26], a widely applied means of calculating capture rate coefficients for these more complex interactions, they provide less insight into reaction mechanisms and rate coefficient determinants than more analytic approaches. The simplest approach is provided by phase space theory (PST) which assumes an isotropic potential between the reactants [31]. The centrifugal term in the effective potential in (3.2) can be expressed in terms of the orbital angular momentum quantum number, , for the collision, so that the equation for Vejf (Rab) becomes ... 

Phase space theory quantitative formuiation We consider an atom-diatom collision where j is the rotational quantum number of the diatom and there are 2j -i-1 (degenerate, in the absence of a field) states of given j. Because J and M are conserved, detailed balance in the form of Eq. (6.37) can be applied to each term in die sum Eq. (6.41). By separating the terms that depend only on i or on/it follows that... 

Energy Constrained and Quantum Anharmonic Rice-Ramsperger-Kassel-Marcus and Phase Space Theory Rate Constants For AI3 Dissociation. 

For larger systems, various approximate schemes have been developed, called mixed methods as they treat parts of the system using different levels of theory. Of interest to us here are quantuin-seiniclassical methods, which use full quantum mechanics to treat the electrons, but use approximations based on trajectories in a classical phase space to describe the nuclear motion. The prefix quantum may be dropped, and we will talk of seiniclassical methods. There are a number of different approaches, but here we shall concentrate on the few that are suitable for direct dynamics molecular simulations. An overview of other methods is given in the introduction of [21]. 

Cao, J., Voth, G.A. The formulation of quantum statistical mechanics based on the Feynman path centroid density. I. Equilibrium properties. J. Chem. Phys. 100 (1994) 5093-5105 II Dynamical properties. J. Chem. Phys. 100 (1994) 5106-5117 III. Phase space formalism and nalysis of centroid molecular dynamics. J. Chem. Phys. 101 (1994) 6157-6167 IV. Algorithms for centroid molecular dynamics. J. Chem. Phys. 101 (1994) 6168-6183 V. Quantum instantaneous normal mode theory of liquids. J. Chem. Phys. 101 (1994) 6184 6192. 

In Sections IVA, VA, and VI the nonequilibrium probability distribution is given in phase space for steady-state thermodynamic flows, mechanical work, and quantum systems, respectively. (The second entropy derived in Section II gives the probability of fluctuations in macrostates, and as such it represents the nonequilibrium analogue of thermodynamic fluctuation theory.) The present phase space distribution differs from the Yamada-Kawasaki distribution in that... 

A theory for nonequilibrium quantum statistical mechanics can be developed using a time-dependent, Hermitian, Hamiltonian operator Hit). In the quantum case it is the wave functions [/ that are the microstates analogous to a point in phase space. The complex conjugate / plays the role of the conjugate point in phase space, since, according to Schrodinger, it has equal and opposite time derivative to v /. 

Newton, the limit h —> 0 is singular. The symmetries underlying quantum and classical dynamics - unitarity and symplecticity, respectively - are fundamentally incompatible with the opposing theory s notion of a physical state quantum-mechanically, a positive semi-definite density matrix classically, a positive phase-space distribution function. 

Since the birth of quantum theory, there has been considerable interest in the transition from quantum to classical mechanics. Because the two formulations are given in a different theoretical framework (nonlinear classical trajectories versus expectation values of linear operators), this transition is far more involved than the naive limit —> 0 suggests. By exploring the classical limit of quantum mechanics, new theoretical concepts have been developed, including path integrals [1], various phase-space representations of quantum mechanics [2], the semiclassical propagator and the trace formula [3], and the notion of quantum... 

In 1965, Joseph E. Mayer (Sidebar 13.5) and co-workers published a paper [M. Baur, J. R. Jordan, P. C. Jordan, and J. E. Mayer. Towards a Theory of Linear Nonequilibrium Statistical Mechanics. Ann. Phys. (NY) 65, 96-163 (1965)] in which the vectorial character of the thermodynamic formalism was suggested from a statistical mechanical origin. Although this paper attracted little attention at the time, its results suggest how thermodynamic geometry might be traced to the statistics of quantum mechanical phase-space distributions. 

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