Quantum Reaction Rate Theory

An important avenue of future research in quantum reactive scattering theory is the development of reduced dimensionality theories [21], which are able to estimate the reactivity of relatively complex systems by fixing or averaging over spectator modes. The difficult aspect of this work is in determining coordinate systems for which the spectator modes are indeed globally inert. 

The use of quantum reactive scattering theory to predict rate constants for chemical reactions is presently relegated to the 3 or 4 atom world. It is crucial to extend reaction rate theory to larger systems. One possibility for this is to use classical 

---

The phase space structure of classical molecular dynamics is extensively used in developing classical reaction rate theory. If the quanmm reaction dynamics can also be viewed from a phase-space perspective, then a quantum reaction rate theory can use a significant amount of the classical language and the quantum-classical correspondence in reaction rate theory can be closely examined. This is indeed possible by use of, for example, the Wigner function approach. For simplicity let us consider a Hamiltonian system with only one DOF. Generalization to many-dimensional systems is straightforward. The Wigner function associated with a density operator /)( / is defined by... 

To gain more insight into quantum reaction rate theory, we below make a detailed comparison between the rigorous quantum rate theory and the quantized ARRKM theory. It is significant that the rigorous quantum results [Eqs. (359) and (361)] are very similar to the results from the quantized ARRKM theory [Eqs. (346) and (351)]. In particular, with the three assumptions... 

Eq. (351) can be transformed to Eq. (359). Further identifying Ns with 2ti p( ), Eq. (346) becomes identical with Eq. (361). Hence, under certain circumstances the quantized ARRKM theory is equivalent to the rigorous quantum reaction rate theory. A number of remarks are in order. First, assumption (a) is automatically satisfied by definition. Second, assumption (b) implies that Fw in the quantized ARRKM theory be the direct analog of the quantum flux operator in the flux-flux autocorrelation formalism. Third, assumption (c) requires that the action of the operator 0jy(V5 v) at any particular time, say at time zero, is equivalent to the action of the projector P i) at time infinity. Regarding 0vi (V5 v) as the analog... 

It is thus evident that the experimental results considered in sect. 4 above are fully consistent with the interpretation based on absolute reaction rate theory. Alternatively, consistency is equally well established with the quantum mechanical treatment of Buhks et al. [117] which will be considered in Sect. 6. This treatment considers the spin-state conversion in terms of a radiationless non-adiabatic multiphonon process. Both approaches imply that the predominant geometric changes associated with the spin-state conversion involve a radial compression of the metal-ligand bonds (for the HS -> LS transformation). 

Hie possibility that a particle with energy Jess than the barrier height can penetrate is a quantum-mechanical phenomenon known as the tunnel effect. A number of examples are known in physics and chemistry. The problem illustrated here with a rectangular barrier was used by Eyring to estimate the rates of chemical reactions. ft forms the basis of what is known as the absolute reaction-rate theory. Another, more recent example is the inversion of the ammonia molecule, which was exploited in the ammonia maser - the fbiemnner of the laser (see Section 9.4,1). 

As a result of the development of quantum mechanics, another theoretical approach to chemical reaction rates has been developed which gives a deeper understanding of the reaction process. It is known as the Absolute Reaction Rate Theory orthe Transition State Theory or, more commonly, as the Activated Complex Theory (ACT), developed by H. Eyring and M. Polanyi in 1935. According to ACT, the bimolecular reaction between two molecules A2 and B2 passes through the formation of the so-called activated complex which then decomposes to yield the product AB, as illustrated below ... 

CLASSICAL, SEMICLASSICAL, AND QUANTUM MECHANICAL UNIMOLECULAR REACTION RATE THEORY... 

Recent years have also witnessed exciting developments in the active control of unimolecular reactions [30,31]. Reactants can be prepared and their evolution interfered with on very short time scales, and coherent hght sources can be used to imprint information on molecular systems so as to produce more or less of specified products. Because a well-controlled unimolecular reaction is highly nonstatistical and presents an excellent example in which any statistical theory of the reaction dynamics would terribly fail, it is instmctive to comment on how to view the vast control possibihties, on the one hand, and various statistical theories of reaction rate, on the other hand. Note first that a controlled unimolecular reaction, most often subject to one or more external fields and manipulated within a very short time scale, undergoes nonequilibrium processes and is therefore not expected to be describable by any unimolecular reaction rate theory that assumes the existence of an equilibrium distribution of the internal energy of the molecule. Second, strong deviations Ifom statistical behavior in an uncontrolled unimolecular reaction can imply the existence of order in chaos and thus more possibilities for inexpensive active control of product formation. Third, most control scenarios rely on quantum interference effects that are neglected in classical reaction rate theory. Clearly, then, studies of controlled reaction dynamics and studies of statistical reaction rate theory complement each other. 

Equations (359) and (361) indicate that the reason that the flux-flux autocorrelation formalism gives exact quantum reaction rate constants is simply that all the dynamical information from time zero to time inflnity has been included. Indeed, as shown by Eqs. (360) and (362), in the classical limit the flux-flux autocorrelation formalism requires us to follow all classical trajectories until t = +00 so as to rigorously tell which trajectory is reactive and which trajectory is nonreactive. Evidently, then, the flux-flux autocorrelation formalism is not a statistical reaction rate theory insofar as no approximation to the reaction dynamics is made. 

As shown above, classical unimolecular reaction rate theory is based upon our knowledge of the qualitative nature of the classical dynamics. For example, it is essential to examine the rate of energy transport between different DOFs compared with the rate of crossing the intermolecular separatrix. This is also the case if one attempts to develop a quantum statistical theory of unimolecular reaction rate to replace exact quantum dynamics calculations that are usually too demanding, such as the quantum wave packet dynamics approach, the flux-flux autocorrelation formalism, and others. As such, understanding quantum dynamics in classically chaotic systems in general and quantization effects on chaotic transport in particular is extremely important. 

In this chapter we have reviewed the development of unimolecular reaction rate theory for systems that exhibit deterministic chaos. Our attention is focused on a number of classical statistical theories developed in our group. These theories, applicable to two- or three-dimensional systems, have predicted reaction rate constants that are in good agreement with experimental data. We have also introduced some quantum and semiclassical approaches to unimolecular reaction rate theory and presented some interesting results on the quantum-classical difference in energy transport in classically chaotic systems. There exist numerous other studies that are not considered in this chapter but are of general interest to unimolecular reaction rate theory. 

Special Interests Radioactivity, application of quantum mechanics to chemistry theory of reaction rates theory of liquids, rheology molecular biology, theory of flames, optical rotation... 

Another indication that the reactive state is moderately long lived is that the photoaquation of trans-[Co(en)2Cl2y shows an apparent activation energy of 5.2 kcal/mole (20), Since, as was noted earlier, the quantum eflBciency depends inversely on knr, which should also be activated, the implication is that Ecr is significantly greater than 5.2 kcal/mole. A species with such an activation energy should, by reaction rate theory, have a lifetime that is relatively long compared with vibrational times. 

Absolute reaction rate theory is a theory that aims to provide explanations for both the activation energy and the pre-exponential factor A (the frequency factor ) in the rate equation from first principles. The underlying theories that it uses are quantum mechanics and statistical mechanics. The rate formula of the absolute theory of reaction rates is given in terms of the partition functions Z of the reactants and the transition state by... 

---