Nuclear reaction coordinate

Both these processes can be considered to occur in several distinct stages as follows (i) formation of precursor state where the reacting centers are geometrically positioned for electron transfer, (ii) activation of nuclear reaction coordinates to form the transition state, (iii) electron tunneling, (iv) nuclear deactivation to form a successor state, and (v) dissociation of successor state to form the eventual products. At least for weak-overlap reactions, step (iii) will occur sufficiently rapidly (< 10 16s) so that the nuclear coordinates remain essentially fixed. The "elementary electron-transfer step associated with the unimolecular rate constant kel [eqn. (10)] comprises stages (ii)—(iv). 

The energies E [Pn] in Eq. [35] depend on the nuclear solvent polarization that serves as a three-dimensional (3D) nuclear reaction coordinate driving electronic transitions. The two-state model actually sets up two directions the vector of the differential field AS b and the off-diagonal field Sab-Therefore, only two projections of Vn need to be considered the longitudinal field parallel to ASab and the transverse field perpendicular to ASab- In the case when the directions of the differential and off-diagonal fields coincide, one needs to consider only the longitudinal field, and the theory can be formulated in terms of the scalar reaction coordinate... 

Fig. 1. Electron transfer in the normal Marcus region. Potential energies of the reactant and product as a function of the nuclear (reaction) coordinate the zero-order (left) and first-order (right) representations.

Figure 1. Free-energy profile for simple electrochemical reaction + e - Y plotted against the nuclear-reaction coordinate. Fig. lA shows the overall electrochemical free-energy profile I, bulk reactant P, precursor state S, successor state 11, bulk product. Figs. IB and C show the components of the free-energy profile arising from the solution species (Y, Y ), and transferring electron, respectively.

Figure 2. Comparison between component potential-energy surfaces for elementary electrochemical exchange reaction for which the reaction entropy AS is positive (A) and resultant free-energy profile (B), plotted against the nuclear-reaction coordinate.

We refer to this as the QTST estimate for the quantum electron transfer rate. Dynamical simulations allow us to test whether this formula applies to ET reactions in general. Here we have employed the electronic coordinate o-(t) to define the reaction coordinate and the centroid. An approach employing a nuclear reaction coordinate and centroid has been put forward in Ref. 37, leading to conclusions similar to those for the case of an electronic centroid. 

Electi ocyclic reactions are examples of cases where ic-electiDn bonds transform to sigma ones [32,49,55]. A prototype is the cyclization of butadiene to cyclobutene (Fig. 8, lower panel). In this four electron system, phase inversion occurs if no new nodes are fomred along the reaction coordinate. Therefore, when the ring closure is disrotatory, the system is Hiickel type, and the reaction a phase-inverting one. If, however, the motion is conrotatory, a new node is formed along the reaction coordinate just as in the HCl + H system. The reaction is now Mdbius type, and phase preserving. This result, which is in line with the Woodward-Hoffmann rules and with Zimmerman s Mdbius-Huckel model [20], was obtained without consideration of nuclear symmetry. This conclusion was previously reached by Goddard [22,39]. 

The reaction coordinate is one specific path along the complete potential energy surface associated with the nuclear positions. It is possible to do a series of calculations representing a grid of points on the potential energy surface. The saddle point can then be found by inspection or more accurately by using mathematical techniques to interpolate between the grid points. 

Quantum tunnelling in chemical reactions can be visualised in terms of a reaction coordinate diagram (Figure 2.4). As we have seen, classical transitions are achieved by thermal activation - nuclear (i.e. atomic position) displacement along the R curve distorts the geometry so that the... 

The calculation of the potential of mean force, AF(z), along the reaction coordinate z, requires statistical sampling by Monte Carlo or molecular dynamics simulations that incorporate nuclear quantum effects employing an adequate potential energy function. In our approach, we use combined QM/MM methods to describe the potential energy function and Feynman path integral approaches to model nuclear quantum effects. 

Intraanchor reactions, conical intersection, two-state systems, 437-438 Intramolecular electron transfer, electron nuclear dynamics (END), 349-351 Intrinsic reaction coordinate (IRC), direct molecular dynamics, theoretical background, 358-361... 

There are two major approaches to including nonequilibrium effects in reaction rate calculations. The first approach treats the inability of the solvent to maintain its equilibrium solvation as the system moves along the reaction coordinate as a frictional drag on the reacting solute system.97, 100 The second approach adds one or more collective solvent coordinate to the nuclear coordinates of the solute.101 107 When these solvent coordinates are... 

Next consider the solute. We will again call the relevant solute nuclear coordinate s and the characteristic time now xs. For a thermally activated bamer crossing, where s is the reaction coordinate for passage over an effective potential VeffH ) at temperature T, a reasonable expression is... 

Figure 1. An example of ground state nuclear tunneling along the reaction coordinate (R.C.). The reactant well (R) is on the left side and the product well (P) is on the right. The blue and red lines describe a light and a heavy isotope probability function, respectively.

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