Reaction mechanisms wave-function calculations

This is an introduction to the techniques used for the calculation of electronic excited states of molecules (sometimes called eximers). Specifically, these are methods for obtaining wave functions for the excited states of a molecule from which energies and other molecular properties can be calculated. These calculations are an important tool for the analysis of spectroscopy, reaction mechanisms, and other excited-state phenomena. 

A third matter to mention here is that the WKB approximation outlined above is limited in the realm in which it is valid. It is more applicable to protons than to electrons (Bockris and Sen, 1973). Other quantum mechanical methods of a quite different nature can be used13 (D. Miller, 1995) and have been applied to make numerical quantal calculations of the rate of redox reactions (Khan, Wright, and Bockris, 1977 Newton, 1986), but they depend on a knowledge of wave functions which, for electron levels in hydrated ions in solution, may still be too primitive for calculations of rate. 

Continuum solvation models consider the solvent as a homogeneous, isotropic, linear dielectric medium [104], The solute is considered to occupy a cavity in this medium. The ability of a bulk dielectric medium to be polarized and hence to exert an electric field back on the solute (this field is called the reaction field) is determined by the dielectric constant. The dielectric constant depends on the frequency of the applied field, and for equilibrium solvation we use the static dielectric constant that corresponds to a slowly changing field. In order to obtain accurate results, the solute charge distribution should be optimized in the presence of the field (the reaction field) exerted back on the solute by the dielectric medium. This is usually done by a quantum mechanical molecular orbital calculation called a self-consistent reaction field (SCRF) calculation, which is iterative since the reaction field depends on the distortion of the solute wave function and vice versa. While the assumption of linear homogeneous response is adequate for the solvent molecules at distant positions, it is a poor representation for the solute-solvent interaction in the first solvation shell. In this case, the solute sees the atomic-scale charge distribution of the solvent molecules and polarizes nonlinearly and system specifically on an atomic scale (see Figure 3.9). More generally, one could say that the breakdown of the linear response approximation is connected with the fact that the liquid medium is structured [105],... 

Once the gas phase Hamiltonian is parametrized as a function of the inner-sphere reaetion coordinate(s), the free energy is calculated as a function of the proton coordinate(s), the scalar solvent coordinates, and the inner-sphere reaction coordinate(s). Note that this approaeh assumes that the optimized geometries of the VB states are not significantly affected by the solvent. For proton transfer reactions, the proton donor-acceptor distance may be treated as an additional solute reaction coordinate that ean be incorporated into the molecular mechanical terms describing the diagonal matrix elements hf- and, in some cases, the off-diagonal matrix elements (/io)y. If the inner-sphere reaction coordinate represents a slow mode, it is treated in the same way as the solvent coordinates. As discussed throughout the literature, however, often the inner-sphere reaction coordinate must be treated quantum mechanically [27, 28]. In this case, the inner-sphere reaction coordinate is treated in the same way as the proton coordinate(s), and the vibrational wave functions depend explicitly on both the proton coordinate(s) and the inner-sphere reaction coordinate(s). 

Another advantage of the quantum calculations is that they provide a rigorous test of approximate methods for calculating dissociation rates, namely classical trajectories and statistical models. Two commonly used statistical theories are the Rice-Ramsperger-Kassel-Marcus (RRKM) theory and the statistical adiabatic channel model (SACM). The first one is thoroughly discussed in Chapter 2, while the second one is briefly reviewed in the Introduction. Moreover, the quantum mechanical approach is indispensable in analyzing the reaction mechanisms. A resonance state is characterized not only by its position, width and the distribution of product states, but also by an individual wave function. Analysis of the nodal structure of resonance wave functions gives direct access to the mechanisms of state- and mode-selectivity. 

Oran et al. [218,219] developed a global parameterized model which describes the chemical induction time as a function of temperature and pressure. Parameters of the induction time function were determined for stoichiometric hydrogen and methane in air mixtures. The parameters were fitted to numerical results obtained from the simulations based on detailed reaction mechanisms. This technique allowed a 22-times faster calculation of the induction time and reduced the simulation time in a onedimensional model by a factor of 7.5. The fitted model was used in two-dimensional shock-wave simulations. 

Electron-jump in reactions of alkali atoms is another example of non-adiabatic transitions. Several aspects of this mechanism have been explored in connection with experimental measurements (Herschbach, 1966 Kinsey, 1971). The role of vibrational motion in the electron-jump model has been investigated (Kendall and Grice, 1972) for alkali-dihalide reactions. It was assumed that the transition is sudden, and that reaction probabilities are proportional to the overlap (Franck-Condon) integral between vibrational wavefunctions of the dihalide X2 and vibrational or continuum wave-functions of the negative ion X2. Related calculations have been carried out by Grice and Herschbach (1973). Further developments on the electron-jump mechanism may be expected from analytical extensions of the Landau-Zener-Stueckelberg formula (Nikitin and Ovchinnikova, 1972 Delos and Thorson, 1972), and from computational studies with realistic atom-atom potentials (Evans and Lane, 1973 Redmon and Micha, 1974). 

The partition function and the sum or density of states are functions which are to statistical mechanics what the wave function is to quantum mechanics. Once they are known, all of the thermodynamic quantities of interest can be calculated. It is instructive to compare these two functions because they are closely related. Both provide a measure of the number of states in a system. The partition function is a quantity that is appropriate for thermal systems at a given temperature (canonical ensemble), whereas the sum and density of states are equivalent functions for systems at constant energy (microcanonical ensemble). In order to lay the groundwork for an understanding of these two functions as well as a number of other topics in the theory of unimolecular reactions, it is essential to review some basic ideas from classical and quantum statistical mechanics. 

In the usual quantum-mechanical implementation of the continuum solvation model, the electronic wave function and electronic probability density of the solute molecule M are allowed to change on going firom the gas phase to the solution phase, so as to achieve self-consistency between the M charge distribution and the solvent s reaction field. Any treatment in which such self-consistency is achieved is called a self-consistent reaction-field (SCRF) model. Many versions of SCRF models exist. These differ in how they choose the size and shape of the cavity that contains the solute molecule M and in how they calculate t nf... 

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