Born-Oppenheimer approximation transition state theory

It is important to point out here, in an early chapter, that the Born-Oppenheimer approximation leads to several of the major applications of isotope effect theory. For example the measurement of isotope effects on vapor pressures of isotopomers leads to an understanding of the differences in the isotope independent force fields of liquids (or solids) and the corresponding vapor molecules with which they are in equilibrium through use of statistical mechanical theories which involve vibrational motions on isotope independent potential functions. Similarly, when one goes on to the consideration of isotope effects on rate constants, one can obtain information about the isotope independent force constants which characterize the transition state, and how they compare with those of the reactants. 

The well-known Born-Oppenheimer approximation (BOA) assumes all couplings Kpa between the PES are identically zero. In this case, the dynamics is described simply as nuclear motion on a single adiabatic PES and is the fundamental basis for most traditional descriptions of chemistry, e.g., transition state theory (TST). Because the nuclear system remains on a single adiabatic PES, this is also often referred to as the adiabatic approximation. 

Let us assume the availability of a useful body of quantitative data for rates of decay of excited states to give new species. How do we generalize this information in terms of chemical structure so as to gain some predictive insight For reasons explained earlier, I prefer to look to the theory of radiationless transitions, rather than to the theory of thermal rate processes, for inspiration. Radiationless decay has been discussed recently by a number of authors.16-22 In this volume, Jortner, Rice, and Hochstrasser 23 have presented a detailed theoretical analysis of the problem, with special attention to the consequences of the failure of the Born-Oppenheimer approximation. They arrive at a number of conclusions with which I concur. Perhaps the most important is, "... the theory of photochemical processes outlined is at a preliminary stage of development. Extension of that theory should be of both conceptual and practical value. The term electronic relaxation has been applied to the process of radiationless decay. 

Studies of the Pgl electron spectra have shown that, in spite of all the mentioned complications, many Pgl systems with molecular targets can still be well described within the theory of simple Pgl, if only the possibility of vibrational transitions is incorporated into the function r(fl). Within the Born-Oppenheimer approximation for both the projectile-target motion and the intramolecular motion, this is done in the following way. We denote by r,(rt) the width belonging to a certain final electronic state, defined as in (11.85). Then r,( ) can, at any distance R, be decomposed as... 

In the framework of the Born-Oppenheimer approximation, radiationless transitions from one surface to another are impossible. (See, e.g., Michl and BonaCit -Koutecky, 1990.) It is therefore necessary to go beyond the Born-Oppenheimer approximation and to include the interaction between different electronic molecular states through the nuclear motion in order to be able to describe such transitions. Using the time-dependent perturbation theory for the rate constant of a transition between a pair of states one arrives at... 

The ability to predict accurate potential surfaces means that we are in a position to investigate the nature of the kinetic barriers, including the activated complexes, of key surface processes. In fact, Born-Oppenheimer potential surfaces can be used not only with the transition-state approach to kinetics but also with the much more general and exact collision theory (e.g., scattering S-matrix ilieory). While methods based on collision and scattering theory have pointed out deficiencies in the traditional transition-state theory (TST), they have also served to uphold many of TST s simple claims. In turn, new generalized transition state theories have been born. For complex systems, the transition-dale approach, while admittedly approximate, has been well established. 

The reaction rates were calculated using variational transition state theory, which includes a multidimensional tunneling approximation. The Born-Oppenheimer potential on the MEP is called EMEP(s), where s is the reaction... 

The theory of multi-oscillator electron transitions developed in the works [1, 2, 5-7] is based on the Born-Oppenheimer s adiabatic approach where the electron and nuclear variables are divided. Therefore, the matrix element describing the transition is a product of the electron and oscillator matrix elements. The oscillator matrix element depends only on overlapping of the initial and final vibration wave functions and does not depend on the electron transition type. The basic assumptions of the adiabatic approach and the approximate oscillator terms of the nuclear subsystem are considered in the following section. Then, in the subsequent sections, it will be shown that many vibrations take part in the transition due to relative change of the vibration system in the initial and final states. This change is defined by the following factors the displacement of the equilibrium positions in the... 

In Chapters 4 and 5 we made use of the theory of radiationless transitions developed by Robinson and Frosch." In this theory the transition is considered to be due to a time-dependent intramolecular perturbation on non-stationary Born-Oppenheimer states. Henry and Kasha > and Jortner and co-workers< > have pointed out that the Born-Oppenheimer (BO) approximation is only valid if the energy difference between the BO states is large relative to the vibronic matrix element connecting these states. When there are near-degenerate or degenerate zeroth-order vibronic states belonging to different configurations the BO approximation fails. 

Deutsche and Moscowitz, a number erf semiempirical models have been proposed to explain features in experimental spectra. Although none has enjoyed any but the most modest successes, experience is gradually accumulating on the limits of applicability. Some of these are reviewed below. It is readily shown that within the Born-Oppenheimer (BO) approximation, the electronic contribution to the magnetic dipole transition moment associated with a vibrational transition of a molecule in its ground electronic state vanishes. A non-BO theory of VCD intensities was independently developed by several groups.2° 2" ... 

The application of PMO theory to photo-pericyclic reactions has been discussed by Dougherty 1971), and he draws attention to the consequences of the breakdown in the Born-Oppenheimer (BO) approximation in certain excited state processes. The BO approximation assumes that nuclear motions are very slow compared with the speed of electronic transitions, so that effectively the nuclei behave as if they were fixed in space (cf. Franck-Condon principle). Since electronic motions are thus independent of nuclear motions, then the potential energy versus reaction co-ordinate curves for the ground state and first-excited state processes should roughly parallel one another. However, the relative energies of the bonding wells (i,e, the hollows in the P.E. surface corresponding to reactant and product) are reversed on the excited state surface so that the transition state on this surface moves from one side of the reaction co-ordinate to the other. 

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