Reaction mechanisms Born-Oppenheimer approximation

Marcus uses the Born-Oppenheimer approximation to separate electronic and nuclear motions, the only exception being at S in the case of nonadiabatic reactions. Classical equilibrium statistical mechanics is used to calculate the probability of arriving at the activated complex only vibrational quantum effects are treated approximately. The result is... 

While studies of specific acid catalysis of redox cofactors shed light on the intricacies of the electron transfer process [54], the conditions required for preprotonation of the cofactor are highly acidic (pH < 0), and would not generally be found in biological systems. There are, however, systems such as Qb reduction in the Rhodobacter sphaeroides reaction center [55], where kinetic data indicate proton transfer prior to or simultaneous with electron transfer. This would seem to indicate that a general acid process is operative. At first glance, this sort of mechanism would seem to be contrary to the Born-Oppenheimer approximation. This apparent paradox can be avoided, however, if quantum chemical (nonadiabatic) processes are considered. 

The modem theory of chemical reaction is based on the concept of the potential energy surface, which assumes that the Born-Oppenheimer adiabatic approximation [16] is obeyed. However, in systems subjected to the Jahn-Teller effect, adiabatic potentials have the physical meaning of the potential energy of nuclei only under the condition that non-adiabatic corrections are small [28]. In the vicinity of the locally symmetric intermediate, these corrections will be very large. The complete description of nuclear motion, i.e. of the mechanism of the chemical reaction, can be obtained only from Schroedinger s equation without applying the Born-Oppenheimer approximation in the vicinity of the locally... 

Electro-Nuclear Quantum Mechanics Beyond the Born-Oppenheimer Approximation. Towards a Quantum Electronic Theory of Chemical Reaction Mechanisms... 

Electro-nuclear quantum mechanics beyond the Born-Oppenheimer approximation. Towards a quantum electronic theory of chemical reaction mechanisms... 

In this review, almost all of the simulations we have described use only classical mechanics to describe the nuclear motion of the reaction system. However, a more accurate analysis of many reactions, including some of the ones that have already been simulated via purely classical mechanics, will ultimately require some infusion of quantum mechanical methods. This infusion has already taken place in several different types of reaction dynamics electron transfer in solution, > i> 2 HI photodissociation in rare gas clusters and solids,i i 22 >2 ° I2 photodissociation in Ar fluid,and the dynamics of electron solvation.22-24 Since calculation of the quantum dynamics of a full solvent is at present too time-consuming, all of these calculations involve a quantum solute in a classical solvent. (For a system where the solvent is treated quantum mechanically, see the quantum Monte Carlo treatment of an electron transfer reaction in water by Bader et al. O) As more complex reaaions are investigated, the techniques used in these studies will need to be extended to take into account effects involving electron dynamics such as curve crossing, the interaction of multiple electronic surfaces and other breakdowns of the Born-Oppenheimer approximation, the effect of solvent and solute polarization, and ultimately the actual detailed dynamics of the time evolution of the electronic degrees of freedom. 

The use of classical simulations on the basis of the Born-Oppenheimer approximation is problematic when dealing with metals, even in the absence of electrochemical reactions involving the transfer of electrons, which can certainly not be approximated by classical mechanics. 

The potential energy surface, i.e., the variation of the energy of a system as a function of the positions of all its constituent atoms, is fundamental to the quantitative description of chemical structures and reaction processes. The quantum mechanical evaluation of potential surfaces is based on the use of the Born-Oppenheimer approximation (e g., see Hehre et al. 1986 Lasaga and Gibbs 1990). In the Bom-Oppenheimer approximation, the positions of the nuclei in the system, R, are fixed and the wave equation is solved for the wavefunction of the electrons. The energy, E, will then be a function of the atomic positions E R), i.e., the solutions will produce a potential surface. If we know E R) accurately, we can predict the detailed atomic forces and the chemical behavior of the entire system. 

From a theoretical point of view, in the study of atom-atom or atom-molecule collisions one needs to solve the Schrodinger equation, both for nuclear and electronic motions. When the nuclei move at much lower velocities than those of the electrons inside the atoms or molecules, both motions (nuclear and electronic) can be separated via the Born-Oppenheimer approximation. This approach leads to a wave function for each electronic state, which describes the nuclear motion and enables us to calculate the electronic energy as a function of the intemuclear distance, i.e. the potential energy V r). Therefore, V r) can be obtained by solving the electronic Schrodinger equation for each inter-nuclear distance. As a result, the nuclear motion, which we shall see is the way chemical reactions take place, is a dynamical problem that can be solved by using either quantum or classical mechanics. 

Moreover, we refer to these kinds of concepts as force field calculations (molecular mechanics) which approximate the potential field (Born-Oppenheimer approximation) by "classical energy relations and adjustable parameters. These methods have successfully accompanied and completed the ab initio calculations until now. For the literature covering these methods and their results, we refer to other surveys. Because of the use of analytical potentials, the procedures are not as time-consuming as ab initio methods. However, their importance is placed behind the conceptually stronger ab initio methods, and they are not suited to localize structures between the minimizers on the PES as it is of primary importance for the kinetic characteristic of a chemical reaction. 

The study of molecular systems using quantum mechanics is based on the Born-Oppenheimer approximation. This approximation relies on the fact that the electrons, because of their smaller mass, move much faster than the heavier nuclei, so they follow the motion of the nuclei adiabatically, whereas the latter move on the average potential of the former. The Born-Oppenheimer approximation is sufficient to describe most chemical processes. In fact, our notion of molecular structure is based on the Born-Oppenheimer approximation, because the molecular structure is formed by nuclei being placed in fixed positions. There are, however, essential nonadiabatic processes in nature that cannot be described within this approximation. Nonadiabatic processes are ubiquitous in photophysics and photochemistry, and they govern such important phenomena as photosynthesis, vision, and charge-transfer reactions. 

In this review we consider reactions for which auxiliary assumption (1), the Born-Oppenheimer approximation, is met or is assumed to be met. Furthermore, we assume that energy transfer processes are occurring fast enough to replenish the populations of depleted reactant states, so g — 1 for all gas-phase reactions considered here. Therefore, the true quantum mechanical rate constant is given by... 

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