Reaction coordinate parameter

Figure 2. Equipotential sections through the potential energy surface for an exchange reaction. The sections define ellipses if the surfaces are parabolic the top left set refer to the initial state (precursor complex) and the bottom right set refer to the final state (successor complex). The dashed line indicates the reaction coordinate. Parameters P and Pa reflect the state of polarization of the solvent, and coordinates d2 and da reflect the inner-shell configurations of the two reactants...

The results of the derivation (which is reproduced in Appendix A) are summarized in Figure 7. This figure applies to both reactive and resonance stabilized (such as benzene) systems. The compounds A and B are the reactant and product in a pericyclic reaction, or the two equivalent Kekule structures in an aromatic system. The parameter t, is the reaction coordinate in a pericyclic reaction or the coordinate interchanging two Kekule structures in aromatic (and antiaromatic) systems. The avoided crossing model [26-28] predicts that the two eigenfunctions of the two-state system may be fomred by in-phase and out-of-phase combinations of the noninteracting basic states A) and B). State A) differs from B) by the spin-pairing scheme. 

Use a pseudo reaction coordinate with one parameter constrained followed by a quasi-Newton optimization. 

Rate coefficients and kinetic parameters for iododeboronation were determined for the benzene- and thiophene-boronic acids, and the results are given in Table 256. The relative reactivities derived from this work correlated well with those obtained in a number of other electrophilic substitutions572, which is perhaps surprising in view of the large variation in the entropies of activation. These differences were explained by Brown et al.132 in terms of the transition state for the phenyl compound occurring earlier along the reaction coordinate than for the... 

In considering the equivalent of eq. (2.8) for multidimensional systems we will start by defining the relevant reaction coordinate, X, and the probability, P(X), that the system will be at different points, along X. The reaction coordinate can be taken rather arbitrarily as any well-defined parameter fe.g., X = (r23 - r12) in Fig. 1.7]. Once X is selected we can obtain P(X) by dividing the coordinate space into subsets according to the specific value of X and evaluating the one-dimensional function. 

There have been a number of investigations of the formulation of the problem of electron transfer accompanied by atom transfer particularly with regard to the simultaneous movement of the proton (which, in view of its small mass, may in fact be an atypical case). A possible model for such processes would assume a conservation of bond order along the reaction coordinates (Johnston, 1960). It is of interest that the results of such calculations are similar to those for electron transfer for weak coupling, although the interpretation of the process and parameters (such as a) are different. 

In qualitative terms, the reaction proceeds via an activated complex, the transition state, located at the top of the energy barrier between reactants and products. Reacting molecules are activated to the transition state by collisions with surrounding molecules. Crossing the barrier is only possible in the forward direction. The reaction event is described by a single parameter, called the reaction coordinate, which is usually a vibration. The reaction can thus be visualized as a journey over a potential energy surface (a mountain landscape) where the transition state lies at the saddle point (the col of a mountain pass). 

Fig. 2.5. Possible applications of a coupling parameter, A, in free energy calculations, (a) and (b) correspond, respectively, to simple and coupled modifications of torsional degrees of freedom, involved in the study of conformational equilibria (c) represents an intramolecular, end-to-end reaction coordinate that may be used, for instance, to model the folding of a short peptide (d) symbolizes the alteration of selected nonbonded interactions to estimate relative free energies, in the spirit of site-directed mutagenesis experiments (e) is a simple distance separating chemical species that can be employed in potential of mean force (PMF) calculations and (f) corresponds to the annihilation of selected nonbonded interactions for the estimation of e.g., free energies of solvation. In the examples (a), (b), and (e), the coupling parameter, A, is not independent of the Cartesian coordinates, x. Appropriate metric tensor correction should be considered through a relevant transformation into generalized coordinates...

Order Parameters, Reaction Coordinates, and Extended Ensembles... 

It is also possible to build the order parameter directly into the potential energy function. This would enable one to determine the free energy difference between a given potential and a reference potential. In a sense, this is complementary to the previous discussion of reaction coordinates rather than further subdividing configuration space by various order parameters, we are considering multiple configuration... 

The order parameter can be defined in two different ways. It can be either a function of atomic coordinates or just a parameter in the Hamiltonian. Examples of both types of order parameters are given in Sect. 2.8.1 in Chap. 2 and illustrated in Fig. 2.5. This distinction is theoretically important. In the first case, the order parameter is, in effect, a generalized coordinate, the evolution of which can be described by Newton s equations of motion. For example, in an association reaction between two molecules, we may choose as order parameter the distance between the two molecules. Ideally, we often would like to consider a reaction coordinate which measures the progress of a reaction. However, in many cases this coordinate is difficult to define, usually because it cannot be defined analytically and its numerical calculation is time consuming. This reaction coordinate is therefore often approximated by simpler order parameters. 

Free energy with several order parameters. The previous formulas can be extended to the case of multiple reaction coordinates i,..., p. For example, the derivative of the free energy becomes a gradient and... 

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