Variable reaction coordinate

This reaction was investigated by Klippenstein and Harding [57] using multireference configuration interaction quantum chemistry (CAS + 1 + 2) to define the PES, variable reaction coordinate TST to determine microcanonical rate coefficients, and a one-dimensional (ID) master equation to evaluate the temperature and pressure dependence of the reaction kinetics. There are no experimental investigations of pathway branching in this reaction. 

Potential energy surfaces in variable reaction coordinate RRKM theory... 

Figure 3.5. Schematic fast variable reaction coordinate potentials. (A) Short time frozen solvent [14,23] regime. At the shortest times typical reactions conform to Eq. (3.36), and thus are driven by the instantaneous potential V(S x) = gas phase potential t/(x) + cage potential v S-,x). The double well form of F(S .v) reflects the frozen solvent s capacity to transiently confine the solute. (B) Partially relaxed regime. At longer times, the short time picture of Eq. (3.36) breaks down due to a solvent relaxation [11,16] aimed at restoring thermodynamic equilibrium. This relaxation converts the instantaneous potential into the less confining total fast variable potential that can further drive the solute toward products.

In the context of association reactions, an algorithm in which the reaction coordinate definition is optimized along with the dividing surface along a one-parameter sequence of paths is called variable reaction coordinate (VRC) variational transition state theory... 

Notably, their discussions about non-adiabaticities have been for a reaction coordinate assumed to be the separation between the centers-of-mass of the two reactants. The use of a more general reaction coordinate, as in some VTST methods, may account for some of the non-adiabaticities. Indeed, reaction coordinate optimizations in the variable reaction coordinate (VRC)-TST approach have often found a reduction by a factor of two or more relative to that obtained for the separation between the centers-of-mass. However, these reductions are generally for short range separations where the long-range potential expansions are inappropriate. 

For radical-radical reactions, the full mode coupling and anharmonicity effects for the relative and overall rotational motions must be explicitly accounted for. We have derived a direct variable reaction coordinate transition state theory approach that appears to 3deld accurate rate coefficients for a number of alkyl radical reactions.This approach is analogous to that embodied in Eq. (4.10) for the long-range transition state, but includes variational optimizations of the form of the reaction coordinate and does not make the large orbital moment of inertia assumption. A detailed description of this approach was provided in some of our recent articles. 

S. Robertson, A. F. Wagner, and D. M. Wardlaw, /. Phys. Chem. A, 106, 2598 (2002). Flexible Transition State Theory for a Variable Reaction Coordinate Analytical Expressions and an Application. 

Y. Georgievskii and S. J. Klippenstein, /. Chem. Phys., 118, 5442 (2003). Variable Reaction Coordinate Transition State Theory Analytic Results and Application to the C2H3 -F H —> C2H4 Reaction. 

Kramers solution of the barrier crossing problem [45] is discussed at length in chapter A3.8 dealing with condensed-phase reaction dynamics. As the starting point to derive its simplest version one may use the Langevin equation, a stochastic differential equation for the time evolution of a slow variable, the reaction coordinate r, subject to a rapidly statistically fluctuating force F caused by microscopic solute-solvent interactions under the influence of an external force field generated by the PES F for the reaction... 

The general criterion of chemical reaction equiUbria is the same as that for phase equiUbria, namely that the total Gibbs energy of a closed system be a minimum at constant, uniform T and P (eq. 212). If the T and P of a siagle-phase, chemically reactive system are constant, then the quantities capable of change are the mole numbers, n. The iadependentiy variable quantities are just the r reaction coordinates, and thus the equiUbrium state is characterized by the rnecessary derivative conditions (and subject to the material balance constraints of equation 235) where j = 1,11,.. ., r ... 

Let X be the normalized progress variable in a system subject to the Marcus equation (Eq. 5-69), so jc = -t- AG°/8AGo, where has the significance of a in Eq. (5-67). Then deduce this equation, which describes the energy change, relative to the reactant, over the reaction coordinate ... 

The entries in the table are arranged in order of increasing reaction coordinate or distance along the reaction path (the reaction coordinate is a composite variable spanning all of the degrees of freedom of the potential energy surface). The energy and optimized variable values are listed for each point (in this case, as Cartesian coordinates). The first and last entries correspond to the final points on each side of the reaction path. 

The second use of Equations (2.36) is to eliminate some of the composition variables from rate expressions. For example, 0i-A(a,b) can be converted to i A a) if Equation (2.36) can be applied to each and every point in the reactor. Reactors for which this is possible are said to preserve local stoichiometry. This does not apply to real reactors if there are internal mixing or separation processes, such as molecular diffusion, that distinguish between types of molecules. Neither does it apply to multiple reactions, although this restriction can be relaxed through use of the reaction coordinate method described in the next section. 

Rates of addition to carbonyls (or expulsion to regenerate a carbonyl) can be estimated by appropriate forms of Marcus Theory. " These reactions are often subject to general acid/base catalysis, so that it is commonly necessary to use Multidimensional Marcus Theory (MMT) - to allow for the variable importance of different proton transfer modes. This approach treats a concerted reaction as the result of several orthogonal processes, each of which has its own reaction coordinate and its own intrinsic barrier independent of the other coordinates. If an intrinsic barrier for the simple addition process is available then this is a satisfactory procedure. Intrinsic barriers are generally insensitive to the reactivity of the species, although for very reactive carbonyl compounds one finds that the intrinsic barrier becomes variable. ... 

Where A F(z) is the free energy at z relative to that at the reactant state minimum zr, and the ensemble average < > is obtained by a quantum mechanical effective potential [15]. Note that the inherent nature of quantum mechanics is at odds with a potential of mean force as a function of a finite reaction coordinate. Nevertheless, the reaction coordinate function z[r] can be evaluated from the path centroids r, first recognized by Feynman and Flibbs as the most classical-like variable in quantum statistical mechanics and later explored by many researchers [14, 15]. 

Tx and Tx are the kinetic energies of the atomic coordinates and X variables, respectively. The As are treated as volumeless particles with mass mx. Since the X variables are associated with the chemical reaction coordinates , the A-dynamics method can utilize the power of specific biasing potentials in the umbrella sampling method to overcome sampling problems that require conventional FEP calculations to be performed in multiple steps. 

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