Reaction dynamics classical trajectory approach

Raff, L.M. and Thompson, D.L. (1985). The classical trajectory approach to reactive scattering, in Theory of Chemical Reaction Dynamics, Vol. Ill, ed. M. Baer (CRC Press, Bow Ruton). 

Some useful reviews of quasiclassical and semiclassical dynamics include D. G. Truhlar and J. T. Muckerman, in Atom-Molecule Collision Theory. R. B. Bernstein, Ed., Plenum Press, New York, 1979, pp. 505-566. Reactive Scattering Cross Sections. 111. Quasiclassical and Semiclassical Methods. L. M. Raff and D. L. Thompson, in Theory of (%emical Reaction Dynamics, M. Baer, Ed., CRC Press, Boca Raton, FL, 1986, Vol. Ill, pp. 1—121. The Classical Trajectory Approach to Reactive Scattering. M. S. Child, in Theory of Chemical Reaction Dynamics, M. Baer, Ed., CRC Press, Boca Raton, FL, 1986, Vol. Ill, pp. 247-279. Semiclassical Reactive Scattering. 

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. 

At first sight, the easiest approach is to fit a set of points near the saddle point to some analytical expression. Derivatives of the fitted function can then be used to locate the saddle point. This method has been well used for small molecules (see Sana, 1981). An accurate fit to a large portion of the potential energy surface is also needed for the study of reaction dynamics by classical or semi-classical trajectory methods. 

Fig. 3. Vibrational population distributions of N2 formed in associative desorption of N-atoms from ruthenium, (a) Predictions of a classical trajectory based theory adhering to the Born-Oppenheimer approximation, (b) Predictions of a molecular dynamics with electron friction theory taking into account interactions of the reacting molecule with the electron bath, (c) Born—Oppenheimer potential energy surface, (d) Experimentally-observed distribution. The qualitative failure of the electronically adiabatic approach provides some of the best available evidence that chemical reactions at metal surfaces are subject to strong electronically nonadiabatic influences. (See Refs. 44 and 45.)...

Considerable use continues to be made of classical trajectory calculations in relating the experimentally determined attributes of electronically adiabatic reactions to the features in the potential energy surface that determine these properties. However, over the past 3 or 4 years, considerable progress has been made with semiclassical and quantum mechanical calculations with the result that it is now possible to predict with some degree of confidence the situations in which a purely classical approach to the collision dynamics will give acceptable results. Application of the semiclassical method, which utilises classical dynamics plus the superposition of probability amplitudes [456], has been pioneered by Marcus [457-466] and by Miller [456, 467-476],... 

A difficulty with the nonequilibrium approach is that one must estimate the time constant or time constants for solvent equilibration with the solvent. This may be estimated from solvent viscosities, from diffusion constants, or from classical trajectory calculations with explicit solvent. Estimating the time constant for solvation dynamics presents new issues because there is more than one relevant time scale [69, 80]. Fortunately, though, the solvation relaxation time seems to depend mostly on the solvent, not the solute. Thus it is very reasonable to assume it is a constant along the reaction path. 

S. Chapman, The classical trajectory-surface-hopping approach to charge-transfer processes, State-Selected and State-to-State, Ion-Molecule Reaction Dynamics. Part 2 Theory, Advances in Chemical Physics LXXXII (M. Baer and C. Y. Ng, eds.), Wiley, New York, 1992, p. 423. 

Our theoretical understanding of unimolecular reactions has kept pace with the advances in experimental techniques. As a result of powerful computers, the geometry of the reacting molecule as well as its vibrational frequencies can now be calculated as a function of the reaction coordinate from the beginning to the end of the reaction. The results of these calculations provide sufficient information to treat the reaction with much greater sophistication than was previously possible. In addition, accurate potential energy surfaces have permitted reactions to be investigated by classical trajectory and quantum dynamics methods, approaches that are not inherently limited by the assumptions of the statistical theory. 

Reaction path methods have great promise for future progress because they can be made direct or automated. Direct ab initio dynamics is taken to mean the ab initio evaluation of the molecular energy whenever it is required in a dynamical calculation. At present this is prohibitively expensive. A more traditional approach uses electronic structure methods to provide information about a PES which is then fitted to a functional form. The dynamics employed thereafter is restricted only by the limitations of current dynamical theories. However, the process of fitting functional forms to a molecular PES is difficult, unsystematic, and extremely time consuming. The reaction path approaches that use interpolation of the reaction path data are affordable methods aimed toward direct dynamics, as they avoid the process of fitting a functional form for the PES. Such methods have been automated (programmed) and are therefore readily usable. Thus reaction path methods have been applied with various levels of ab initio theory to statistical theories of the reaction rate, to approximate quantum dynamics, and to classical trajectory studies of reactions. 

An alternative way to obtain the spectral density is by numerical simulation. It is possible, at least in principle, to include the intramolecular modes in this case, although it is rarely done [198]. A standard approach [33-36,41] utilizes molecular dynamics (MD) trajectories to compute the classical real time correlation function of the reaction coordinate from which the spectral density is calculated by the cosine transformation [classical limit of Eq. (9.3)]. The correspondence between the quantum and the classical densities of states via J(co) is a key for the evaluation of the quantum rate constant, that is, one can use the quantum expression for /Cj2 with the classically computed J(co). This is true only for a purely harmonic system [199]. Real solvent modes are anharmonic, although the response may well be linear. The spectral density of the harmonic system is temperature independent. For real nonlinear systems, J co) can strongly depend on temperature [200]. Thus, in a classical simulation one cannot assess equilibrium quantum populations correctly, which may result in serious errors in the computed high-frequency part of the spectrum. Song and Marcus [37] compared the results of several simulations for water available at that time in the literature [34,201] with experimental data [190]. The comparison was not in favor of those simulations. In particular, they failed to predict... 

Wigner s dynamical perspective on TST also led naturally to the development of efficient methods to calculate the transmission coefficient (or Wigner s recrossing factor y). This approach is outlined by Keck [31, 32] for recombination reactions and by Anderson and coworkers [33, 34] for bimolecular reactions. This work helped elucidate the connection between classical TST and classical trajectory calculations. 

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