Adiabatic theory of reactions

The adiabatic approximation for reaction dynamics assumes that motion along the reaction coordinate is slow compared to the other modes of the system and the latter adjust rapidly to changes in the potential from motion along the reaction coordinate. This approximation is the same as the Born-Oppenheimer electronically adiabatic separation of electronic and nuclear motion, except that here we 

A convenient choice of the reaction path is the MEP in isoinertial coordinates, because by construction the gradient of the potential V(s,u) is tangent to s and there is no coupling between s and u through second order. Therefore the potential can be conveniently approximated by 

As a first approximation we will assume that the reaction-path curvature can be neglected, but we will eliminate this approximation after Eq. (27.22) because the curvature of the reaction path is very important for tunneling. 

Treating bound modes quantum mechanically, the adiabatic separation between s and u is equivalent to assuming that quantum states in bound modes orthogonal to s do not change throughout the reaction (as s progresses from reactants to products). The reaction dynamics is then described by motion on a one-mathematical-dimensional vibrationally and rotationally adiabatic potential 

If the reaction coordinate is treated classically, the probability for reaction on a state n,A) at a total energy E is zero if the energy is below the maximum in the adiabatic potential for that state, and 1 otherwise  

---

One expects to observe a barrier resonance associated with each vibra-tionally adiabatic barrier for a given chemical reaction. Since the adiabatic theory of reactions is closely related to the rate of reaction, it is perhaps not surprising that Truhlar and coworkers [44, 55] have demonstrated that the cumulative reaction probability, NR(E), shows the influence barrier resonances. Specifically, dNR/dE shows peaks at each resonance energy and Nr(E) itself shows a staircase structure with a unit step at each QBS energy. It is a more unexpected result that the properties of the QBS seem to also imprint on other reaction observables such as the state-to-state cross sections [1,56] and even can even influence the helicity states of the products [57-59]. This more general influence of the QBS on scattering observables makes possible the direct verification of the existence of barrier-states based on molecular beam experiments. 

Like Eq. (27.2), Eqs. (27.11) and (27.12) are also hybrid quantized expressions in which the bound modes are treated quantum mechanically but the reaction coordinate motion is treated classically. Whereas it is difficult to see how quantum mechanical effects on reaction coordinate motion can be included in VTST, the path forward is straightforward in the adiabatic theory, since the one-dimensional scattering problem can be treated quantum mechanically. Since Eq. (27.12) is equivalent to the expression for the rate constant obtained from microcanonical variational theory [7, 15], the quantum correction factor obtained for the adiabatic theory of reactions can also be used in VTST. 

R.T. Skodje, The adiabatic theory of heavy-light-heavy chemical reactions, Annu. Rev. Phys. Chem. 44 (1993) 145. 

Statistical methods represent a background for, e.g., the theory of the activated complex (239), the RRKM theory of unimolecular decay (240), the quasi-equilibrium theory of mass spectra (241), and the phase space theory of reaction kinetics (242). These theories yield results in terms of the total reaction cross-sections or detailed macroscopic rate constants. The RRKM and the phase space theory can be obtained as special cases of the single adiabatic channel model (SACM) developed by Quack and Troe (243). The SACM of unimolecular decay provides information on the distribution of the relative kinetic energy of the products released as well as on their angular distributions. 

Electron-transfer kinetics in solutions have often been analyzed and interpreted in the framework of the general adiabatic theory of Marcus (43). Although electron-transfer dynamics are not always characterized by a classical rate constant (44), a general formulation of the chemical reaction concerns the rate constant k, which can be expressed as ... 

Adiabatic Statistical Theory of Reaction Rates 5.1. Exact Formulations of the Adiabatic Rate Theory... 

The expressions (106.Ill) and (124.Ill) represent two equivalent formulations of an exact adiabatic statistical theory of reaction rates based on two different definitions of the activated complex . Therefore, there exists the relation... 

The two equivalent adiabatic expressions (106.HI) and (124. HI) represent alternatives of the acciarate formulation of the statistical theory of reaction rates, which rest on two other definitions of the activated complex as a virtual state. In general, they do not involve the Arrhenius exponential factor which includes the classical activation energy. 

In this situation the rate constant of inner-sphere redox reactions can be evaluated by applying the adiabatic formulation of reaction rate theory and by using, for instance, the basic equation (103.III) which yields (since g = 1)... 

The oscillator model for proton transfer was first developed by DOGONADZE and KUSNETSO /147/. The above treatment proposed by CHRISTOV /37e/ is based on the general "theory of reaction rates applied to the two-frequency oscillator model. It reproduces the essential results of earlier work concerning electronically and protonically non-adiabatic reactions and yields, moreover, simple, explicit expressions for adiabatic reactions never derived before. This shows the utility of certain new methods in calculating reaction probabilities, developed in Chapter II, which allow an application of the most suitable formulations of the rate theory. 

ABSTRACT. A review is given of recent applications of the distorted wave (DW) method to the theory of chemical reactions. A brief account of the following topics is included the formal DW theory of reactions, static and adiabatic methods for choosing the distortion potentials, and the removal of the 3 Euler angles from the 6 dimensional DW integral. Applications of various DW theories to the H+F2 0( P)+H2. 0( P) -C(CH3) 4. 

J. M. Bowman, A. Kuppermann, J. T. Adams, and D. G. Truhlar, A direct test of the vibrationally adiabatic theory of chemical reactions, Chem. Phys. Lett. 20 229 (1973). 

D. G. Truhlar, Adiabatic theory of chemical reactions, J. Chem. Phys. 53 2041 (1970). 

Baer M 1985 The theory of electronic non-adiabatic transitions in chemical reactions Theory of Chemical Reaction Dynamics vol II, ed M Baer (Boca Raton, FL CRC Press) p 281... 

One way to overcome this problem is to start by setting up the ensemble of trajectories (or wavepacket) at the transition state. If these bajectories are then run back in time into the reactants region, they can be used to set up the distribution of initial conditions that reach the barrier. These can then be run forward to completion, that is, into the products, and by using transition state theory a reaction rate obtained [145]. These ideas have also been recently extended to non-adiabatic systems [146]. 

Rather than using transition state theory or trajectory calculations, it is possible to use a statistical description of reactions to compute the rate constant. There are a number of techniques that can be considered variants of the statistical adiabatic channel model (SACM). This is, in essence, the examination of many possible reaction paths, none of which would necessarily be seen in a trajectory calculation. By examining paths that are easier to determine than the trajectory path and giving them statistical weights, the whole potential energy surface is accounted for and the rate constant can be computed. 

---