Anharmonicity classical trajectories

Wolf R J and Hase W L 1980 Quasiperiodic trajectories for a multidimensional anharmonic classical Hamiltonian excited above the unimolecular threshold J. Chem. Phys. 73 3779-90... 

Such a method has recently been developed by Miller. et. al. (28). It uses short lengths of classical trajectory, calculated on an upside-down potential energy surface, to obtain a nonlocal correction to the classical (canonical) equilibrium probability density Peq(p, ) at each point then uses this corrected density to evaluate the rate constant via eq. 4. The method appears to handle the anharmonic tunneling in the reactions H+HH and D+HH fairly well (28), and can... 

The effect of sufficiently weak anharmonicities of the potential on this picture will be to distort the rectangle comprising the classical trajectories so that the motion occurs on a two-dimensional torus belonging to the three-dimensional constant energy subspace of the total four-dimensional phase space of the system [Arnold, 1978]. 

Figure 3.2 shows the same trajectory as in Figure 5.1 in a different representation together with the time-independent total dissociation wave-function StotiR,7)- The energy for the trajectory is the same as for the wavefunction. It comes as no surprise that, on the average, the wave-function closely follows the classical trajectory. Another example of this remarkable coincidence between quantum mechanical wavefunctions on one hand and classical trajectories on the other hand is shown in Figure 4.2. There, the time-dependent wavepacket (f) follows essentially the trajectory which starts at the equilibrium geometry of the ground electronic state. This is not unexpected in view of Equations (4.29) which state that the parameters of the center of the wavepacket obey the same equations of motion as the classical trajectory provided anharmonic effects are small. Figures 3.2 and 4.2 elucidate the correspondence between classical and quantum mechanics, at least for short times and fast evolving systems ...

Classical trajectory calculations on model ICN photodissociation. 464, 465 Effects of parent molecule bending and overall rotation Effects of product state anharmonicity in the theoretical descrip- 466 tion of the 234—266 nm photodissociation of ICN HgBr Rapid reaction of HgBr with Br shown to be responsible for 467... 

Figure 28. Anharmonic excited-state potential energy surface. The classical trajectory that originates from rest from the ground-state equilibrium geometry is shown superposed. 

If there are quanta in mode / and zero quanta in all the other modes, the state is called an overtone of the normal mode i. What does such a state correspond to in terms of a classical trajectory Consider the overtone of the antisymmetric stretch, again neglecting the bend. If all the energy in the overtone were in mode /, the trajectory would look like the anharmonic mode itself in figure Al.2.6. However, because of the unavoidable... 

The classical anharmonic RRKM rate constant for a fluxional molecule may be calculated from classical trajectories by following the initial decay of a microcanonical ensemble of states for the unimolecular reactant, as given by equation 1A3.12.41. Such a calculation has been performed for dissociation of the Alg and A1j3 clusters using a model analytic potential energy function written as a sum of Lennard-Jones and Axelrod-Teller potentials [30]. Stmctures of some of the Alg minima, for the potential function, are shown in figure A3.12.6. The deepest potential minimum has... 

The first classical trajectory study of unimolecular decomposition and intramolecular motion for realistic anharmonic molecular Hamiltonians was performed by Bunker [12,13]. Both intrinsic RRKM and non-RRKM dynamics was observed in these studies. Since this pioneering work, there have been numerous additional studies [9,17,30,M,65,66 and 62] from which two distinct types of intramolecular motion, chaotic and quasiperiodic [14], have been identified. Both are depicted in figure A3.12.7. Chaotic vibrational motion is not regular as predicted by the normal-mode model and, instead, there is energy transfer between the modes. If all the modes of the molecule participate in the chaotic motion and energy flow is sufficiently rapid, an initial microcanonical ensemble is maintained as the molecule dissociates and RRKM behaviour is observed [9]. For non-random excitation initial apparent non-RRKM behaviour is observed, but at longer times a microcanonical ensemble of states is formed and the probability of decomposition becomes that of RRKM theory. 

Hase and co-workers (136-140) have reported an extensive series of trajectory studies on overtone relaxation in benzene. Comparisons between our quantum studies for the five- and nine-mode benzene fragments of C,H and C3H, and the trajectory results were presented in Benzene I (103). Further comparisons for 16-mode and 21-mode benzene were presented in Benzene II and III (104,105). Clarke and Collins (141) also used classical trajectories to study overtone relaxation in benzene. Finally, Thompson et al. have also used trajectory methods to study energy flow from excited CH overtones (142,143) and from various excited CC stretch, CCH wag, and CCC bend normal modes. Several potential surfaces with varying degrees of anharmonicity were used. 

Anharmonic corrections have also been determined for unimolecular rate constants using classical mechanics. In a classical trajectory (Bunker, 1962, 1964) or a classical Monte Carlo simulation (Nyman et al., 1990 Schranz et al., 1991) of the unimolecular decomposition of a microcanonical ensemble of states for an energized molecule, the initial decomposition rate constant is that of RRKM theory, regardless of the molecule s intramolecular dynamics (Bunker, 1962 Bunker, 1964). This is because a... 

Chaotic behavior requires a nonhnearity in the equations of motion. For conservative mechanical systems, of which computing classical trajectories is, for us, the prime example. Section 5.2.2.1, the nonlinearity is due to the anharmonicity of the potential. In chemical kinetics" there are two sources of nonlinearity. One is when the concentrations are not uniform throughout the system so that diffusion must be taken into account. The other is if there is a feedback so that, for example, formation of products influences the reaction rate, see Problem H. As we shall see, this type of nonlinearity occurs naturally in many surface reactions and this is why we chose catalytic processes as an example. In both mechanical and chemical kinetics systems there is one more way to add nonlinear terms and this is by an external perturbation. For surface reactions this additional control can be implemented, for example, by modulating the gas-phase pressures of reactants and/or products."... 

Fig. 3. Classical trajectory histogram of P(t) x 10 s versus time for H-CEC-Cl with non-random excitation. The abscissa ranges from 0 to 4.5 x 10 s in each case. The atoms in parentheses refer to the part of the molecule initially excited. The dissociating bond is indicated in each plot. Dashed lines, are the anharmonic RRKM prediction. Total energy is 200 kcal/mol. (From Hase and Feng, reference 90. Reprinted with permission of the American Institute of Physics.)...

The phase space structures for two identical coupled anharmonic oscillators are relatively simple because the trajectories lie on the surface of a 2-dimensional manifold in a 4-dimensional phase space. The phase space of two identical 2-dimensional isotropic benders is 8-dimensional, the qualitative forms of the classifying trajectories are far more complicated, and there is a much wider range of possibilities for qualitative changes in the intramolecular dynamics. The classical mechanical polyad 7feff conveys unique insights into the dynamics encoded in the spectrum as represented by the Heff fit model. 

Does it make sense to associate a definite quantum number n. to each mode / in an anharmonic system In general, this is an extremely difficult question But remember that so far, we are speaking of the situation in some small vicinity of a minimum on the PES, where the Moser-Weinstein theorem guarantees the existence of the anharmonic normal modes. This essentially guarantees that quantum levels with low enoughvalues correspond to trajectories that lie on invariant tori. Since the levels are quantized, these must be special tori, each characterized by quantized values of the classical actions I. = n. + 5)/), which are constants of the motion on the invariant toms. As we shall see, the possibility of assigning a set of iV quantum numbers n- to a level, one for each mode, is a very special situation that holds only near the potential minimum, where the motion is described by the N anharmonic normal modes. However, let us continue for now with the region of the spectmm where this special situation applies. 

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... 

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