Classical-trajectory method

Hase W L (ed) 1998 Comparisons of Classical and Quantum Dynamics (Adv. in Classical Trajectory Methods III) (Greenwich, CT JAI Press)... 

W L Hase (ed) 1992 Advances in Classical Trajectory Methods. 1. Intramolecular and Nonlinear Dynamics (London JAI)... 

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. 

A time-independent quantum mechanical study by Engel and coworkers (1985) gave qualitatively the same results as the time dependent analog and the quassi classical trajectory method. A quantum theory for probing transition-state absorption/emission has also been developed by Lee et al. (1989). 

One formalism which has been extensively used with classical trajectory methods to study gas-phase reactions has been the London-Eyring-Polanyi-Sato (LEPS) method . This is a semiempirical technique for generating potential energy surfaces which incorporates two-body interactions into a valence bond scheme. The combination of interactions for diatomic molecules in this formalism results in a many-body potential which displays correct asymptotic behavior, and which contains barriers for reaction. For the case of a diatomic molecule reacting with a surface, the surface is treated as one body of a three-body reaction, and so the two-body terms are composed of two atom-surface interactions and a gas-phase atom-atom potential. The LEPS formalism then introduces adjustable potential energy barriers into molecule-surface reactions. 

The problem of an unphysical flow of ZPE is not a specific feature of the mapping approach, but represents a general flaw of quasi-classical trajectory methods. Numerous approaches have been proposed to fix the ZPE problem [223]. They include a variety of active methods [i.e., the flow of ZPE is controlled and (if necessary) manipulated during the course of individual trajectories] and several passive methods that, for example, discard trajectories not satisfying predefined criteria. However, most of these techniques share the problem that they manipulate individual trajectories, whereas the conservation of ZPE should correspond to a virtue of the ensemble average of trajectories. 

W. L. Hase, Advances in Classical Trajectory Methods, Vol. 1, Jai Press, London, 1992. 

Recent work by Pritchard has concentrated on a state-to-state description of unimolecular reactions229 and an examination by classical trajectory methods of the effects of overall molecular rotation on the unimolecular rate. The latter calculations have revealed a most interesting aspect of computing in chaotic systems, namely, that the same algorithm gives different results on different machines for a trajectory with identical initial conditions, or even on the same machine with different releases of the same compiler. However, the ensemble average behavior, with an ensemble comprising 100 or more trajectories, is acceptably the same each time.230... 

Pattengill, M.D. (1979). Rotational excitation III Classical trajectory methods, in Atom-Molecule Collision Theory, ed. R.B. Bernstein (Plenum Press, New York). 

Porter, R.N. and Raff, L.M. (1976). Classical trajectory methods in molecular collisions, in Dynamics of Molecular Collisions, Part B, ed. W.H. Miller (Plenum Press, New York). 

Table 6.3 A comparison of different theoretical approaches to the evaluation of the thermal rate constant for the F + H2 —> HF + H reaction at T = 300 K. TST is transition-state theory (Example 6.2), QCT is the quasi-classical trajectory method [Chem. Phys. Lett. 254, 341 (1996)], and QM is (exact) quantum mechanics [J. Phys. Chem. 102, 341 (1998)].

A further advance occurred when Chesnavich et al. (1980) applied variational transition state theory (Chesnavich and Bowers 1982 Garrett and Truhlar 1979a,b,c,d Horiuti 1938 Keck 1967 Wigner 1937) to calculate the thermal rate coefficient for capture in a noncentral field. Under the assumptions that a classical mechanical treatment is valid and that the reactants are in equilibrium, this treatment provides an upper bound to the true rate coefficient. The upper bound was then compared to calculations by the classical trajectory method (Bunker 1971 Porter and Raff 1976 Raff and Thompson 1985 Truhlar and Muckerman 1979) of the true thermal rate coefficient for capture on the ion-dipole potential energy surface and to experimental data (Bohme 1979) on thermal ion-polar molecule rate coefficients. The results showed that the variational bound, the trajectory results, and the experimental upper bound were all in excellent agreement. Some time later, Su and Chesnavich (Su 1985 Su and Chesnavich 1982) parameterized the collision rate coefficient by using trajectory calculations. 

William L. Hase, Molecular Dynamics of Clusters, Surfaces, Liquids and Interfaces, Vol. 4, Adv. Classical Trajectory Methods, JAI, Stamford, CT, 1999. 

William L. Hase, Dynamics of Ion-Molecule Complexes, in Adv. Classical Trajectory Methods, Vol. 2, JAI Press, Greenwich, CT, 1994. 

A recent review of the dynamics of bimolecular reactions by Polanyi and Schreiber14 provides a detailed account of the classical trajectory method for treating the motion of a reactive system on a potential energy surface. Such classical motion studies on three-dimensional surfaces are now commonplace and have yielded a great deal of information regarding the microscopic dynamical behaviour... 

Extensive use has been made of classical trajectory methods to investigate various forms of the potential-energy surface for the reaction F + H2. Muckerman [518] has recently presented a very thorough review of potential-energy surfaces and classical trajectory studies for F + H2. The calculations all correctly predict vibrational population inversion, the value of and backward scattering of the products. Most calculations overestimate (FR) and those giving the lowest values of (Fr > use a potential-energy surface that unrealistically has wells in the entrance and exit valleys [519]. 

It should be emphasized that classical trajectories methods at present can be considered as fairly standard techniques for studying the dynamical behaviour of small molecular systems (either triatomic or tetraatomic). As a consequence many technical points have already been discussed in great detail in the literature " and they will not be discussed here. Such technical questions are, for instance ... 

In light of previous experimental and theoretical work on the F f H2 reaction, it can be seen why an experisient of this complexity is necessary in order to observe dynamic resonances in this reaction. The energetics for this reaction and its isotopic variants are displayed in Figure 1. Chemical laser (11) and infrared chemiluminescence (12) studies have shown that the HF product vibrational distribution is hi ly inverted, with most of the population in v=2 and v°°3. A previous crossed molecular beam study of the F + D2 reaction showed predominantly back-scattered DF product (13). These observations were combined with the temperature dependence of the rate constants from an early kinetics experiment (14) in the derivation of the semiempirical Muckerman 5 (M5) potential energy surface (15) using classical trajectory methods. Although an ab initio surface has been calculated (16), H5 has been the most widely used surface for the F H2 reaction over the last several years. 

According to the classical trajectory method, one assumes that the input stochastic amplitudes are of the form... 

For better comparison of theoretical predictions for different-order processes, we have plotted the quantum Fano factors for both interacting modes in the no-energy-transfer regime with N = 2 — 5 and r = 5 in Fig. 7. One can see that all curves start from F w(0) = 1 for the input coherent fields and become quasistationary after some relaxations. The quantum and semiclassical Fano factors coincide for high-intensity fields and longer times, specifically for t > 50/(Og), where il will be defined later by Eq. (54). In Fig. 17, we observe that all fundamental modes remain super-Poissonian [F (t) >1], whereas the iVth harmonics become sub-Poissonian (F (t) < 1). The most suppressed noise is observed for the third harmonic with the Fano factor 0.81. In Fig. 7, we have included the predictions of the classical trajectory method (plotted by dotted lines) to show that they properly fit the exact quantum results (full curves) for the evolution times t > 50/(Og). The small residual differences result from the fact that the amplitude r was chosen to be relatively small (r = 5). This value does not precisely fulfill the condition r> 1. We have taken r = 5 as a compromise between the asymptotic value r oo and computational complexity to manipulate the matrices of dimensions 1000 x 1000. Unfortunately, we cannot increase amplitude r arbitrary due to computational limitations. 

Now, on applying the classical trajectory method, one should perform averaging over all solutions (49) and (52) to calculate the required statistical moments. Here, we calculate the first and second-order field intensity moments necessary for determination of the Fano factors. The mean intensities of the fundamental and harmonic modes are simply given by h = N2r2 and % = r2, respectively. The second-order moments of field intensity are found to be... 

Stine and Marcus.63 These results are in excellent agreement (as few %) with the accurate quantum mechanical values obtained by Secrest and Johnson24 even for extremely weak transitions with a probability as small as 10" n. It is thus encouraging that the semiclassical model is able to describe such quantum-like phenomena for which ordinary (i.e. real-valued) classical trajectory methods would clearly be inapplicable. 

In many cases, too, the semiclassical model provides a quantitative description of the quantum effects in molecular systems, although there will surely be situations for which it fails quantitatively or is at best awkward to apply. From the numerical examples which have been carried out thus far— and more are needed before a definitive conclusion can be reached—it appears that the most practically useful contribution of classical S-matrix theory is the ability to describe classically forbidden processes i.e. although completely classical (e.g. Monte Carlo) methods seem to be adequate for treating classically allowed processes, they are not meaningful for classically forbidden ones. (Purely classical treatments will not of course describe quantum interference effects which are present in classically allowed processes, but under most practical conditions these are quenched.) The semiclassical approach thus widens the class of phenomena to which classical trajectory methods can be applied. 

The simplest three atom ion-molecule reactions are those of H+ with molecular hydrogen and its various isotopic variants. This is an interesting reaction not only because of its simplicity but the reaction involves a deep potential well represented by the stable H3 ion and hence one has the interesting possibility that a long-lived collision complex may be involved. Some preliminary work on such reactions as D+(HD, H)D2, D+(HD, D)HD+ and D+(HD, D2)H+ has been reported.25 However, it is only recently that the complete details of the combined theoretical and experimental study of these reactions has become available.26,27 The associated theoretical treatment of these reactions is the trajectory surface hopping TSH technique extended from the classical trajectory method.28... 

The trajectory surface hopping method is an additional extension of the classical trajectory method. Potential energy surfaces are constructed for each electronic state involved in the collision. In addition, a function has to be obtained that defines the locus of points at which hops between surfaces can occur. Still another function is necessary which gives the probability of such jumps as a function of nuclear positions and velocities.28 Diatomics-in-molecules surfaces approximated the two lowest singlet potential surfaces of H3. The surfaces have been shown30 to be in good agreement with accurate ab initio calculations by Conroy.31... 

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