Computation classical

One of the simplest chemical reactions involving a barrier, H2 + H —> [H—H—H] —> II + H2, has been investigated in some detail in a number of publications. The theoretical description of this hydrogen abstraction sequence turns out to be quite involved for post-Hartree-Fock methods and is anything but a trivial task for density functional theory approaches. Table 13-7 shows results reported by Johnson et al., 1994, and Csonka and Johnson, 1998, for computed classical barrier heights (without consideration of zero-point vibrational corrections or tunneling effects) obtained with various methods. The CCSD(T) result of 9.9 kcal/mol is probably very accurate and serves as a reference (the experimental barrier, which of course includes zero-point energy contributions, amounts to 9.7 kcal/mol). 

Table 13-7. Computed classical barrier heights AE [kcal/mol] for the reaction H2 + H — [ II II 111 —> H2 + H (6-311++G(,3pd) basis set) data compiled from Johnson et al., 1994, and Csonka and Johnson, 1998.

Second, AEQM xgj ) is calculated as the difference between the QM subsystem energy computed form the ab initio calculation (Eqm(QM)), and the QM/MM electrostatic energy computed classically (Eeiectrostatics(QM/MM)) [13]. 

Besides the experimental studies, the theoretical approaches have also been developed to study the details of state-to-state chemistry to probing the transition-state. Polanyi and coworkers computed classical trajectories to follow the dynamics of... 

Fig. 2.3. Computed classical deflection function, x(b), of He-Ar pairs at a low (/), two intermediate (i) and a high (h) translational speed.

The expressions, Eqs. 5.9 through 5.16, are quite general, wave mechanical formulae of the spectral moments, which for small order n are suitable for numerical computations classical approximations may be derived readily from these. 

Figure 4 Percentage of dissociation occurring at each site on a Cu(l 0 0) surface as a function of initial molecular energy [45], The results, computed classically, show that dissociation in the vibrational ground-state, v = 0, occurs preferentially at the bridge site, while that in the vibrational excited state, v = 1, occurs preferentially at the atop site.

As discussed above, the conventional approach to VER in liquids involves a classical molecular dynamics simulation of the solute (with one or more vibrational modes constrained to be rigid) in the solvent. The required time-correlation functions are computed classically and then... 

An algorithm to compute classical trajectories using boundary value formulation is presented and discussed. It is based on an optimization of a functional of the complete trajectory. This functional can be the usual classical action, and is approximated by discrete and sequential sets of coordinates. In contrast to initial value formulation, the pre-specified end points of the trajectories are useful for computing rare trajectories. Each of the boundary-value trajectories ends at desired products. A difficulty in applying boundary value formulation is the high computational cost of optimizing the whole trajectory in contrast to the calculation of one temporal frame at a time in initial value formulation. 

The above discussion suggests that we cannot minimize the action directly to compute classical trajectories. We focus instead on the residuals (13). One boundary value formulation that we frequently use minimizes the squares of the residual vectors, i.e., we define a target function T that we wish to minimize as a function of all the intermediate coordinates Xj,j = 2,..., N — 1. 

The expression we derive below leads to an action and to a stationary (minimum) condition on the classical path. The optimal path is a discrete approximation to a classical trajectory. Interestingly, in the integral limit (an infinitesimal time step), the action below was used already by Gauss ( ) to compute classical trajectories [14]. At variance with Gauss we keep a finite At. 

If we had a theory for this factor, we could calculate quantum relaxation rates using computed classical correlation functions. For the bilinear model (13.11) we get, using (13.29) and (13.30)... 

The last three years have seen considerable interest in the development of semiclassical methods for treating complex molecular collisions, i.e. those which involve inelastic or reactive processes. One of the reasons for this activity is that the recent work, primarily that of Miller and that of Marcus,2 has shown how numerically computed classical trajectories can be used as input to the semiclassical theory, so that it is not necessary to make any dynamical approximations when applying these semiclassical approaches... 

Table 3.1. Shown are the average energy transfers for the col linear hard-sphere-atom/harmonic-oscillator model computed classically (CM) and quantum mechanically (QM). The energies are in terms of multiples of the oscillator spacing. The atom/ oscillator mass ratio was 0.02 in this problem...

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."... 

The previous simple analysis example follows a pre-computing classical approach where a simple linearized lumped parameter model of the system was developed. In the pre-computing or classical approach this simple model was solved by the application of analytical methods such as Laplace transforms and frequency-response analysis. For completeness, these methods will be briefly introduced here. The interested reader should refer to the texts that take this pre-computing classical approach, such as... 

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