Determination of the q-Centroid

Since 4 ) = 0 if q, is a promoting mode, the part of (51)23 involving 1 (t) and kr vanishes. The remaining expressions (51)24, and that that corre- 

Our discussion up to this point has been restricted to determination of the decay rates of nonradiative transitions up to third order of perturbation theory. We have established how the transition probabilities may be reduced by the knowledge of the average of the density of states weighted vibrational overlap factor and its symmetry properties. Assuming that the promoting modes are nontotally symmetric (ns) and separated from the totally symmetric accepting modes (ts), we have found that 

After these preparations, the q entroid may now be determined by those qo and qopi appearing in (5.56)-(5.59), which likewise cause the sum in (5.55) to vanish. First, we considerthenontotally symmetric (ns) vibrations. Since Equations 5.56 and 5.59 contain directly the nontotally symmetric mode qo, and Equations 5.58 are nonzero if q is ns and p is totally symmetric (ts), the q entroid condition for the ns modes is given by 

For totally symmetric vibrations, we need only Equations 5.57. The g-centroid is then determined by setting ( 1)22 + ( )(SI)22 = 0. This gives 

Let us now return to the basic relations (5.27) through (5.32), which play the role of generalized displacement parameters. To gain some insight into the significant feature of these quantities, we consider the zero-temperature limit We assume that only the vibrationless level of the electronic exdted state is occupied with an appreciable probability in thermal equiUhrium, and therefore u , = 1 (ji = 1,2). Furthermore, we make the special assumption by setting = raj,. Under these circumstances, Equation 5.27 reduces to the simpler form 

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It is not clear what the most useful applications of such procedure will turn out to be. We have already mentioned the determination of the q-centroid for evaluating... 

To the design points obtained are added center points (centroids) of two-, three-,. .., and (q-l)-dimensional faces of the polyhedron and its center point. Coordinates of a central point are determined by taking average coordinates of previously chosen vertices ... 

The R- or w-dependence of second-order constants (o,p,q, 7, etc.) is more important than that of the off-diagonal perturbation parameters themselves (as determined from a direct fit to perturbed levels) because, in their definitions as sums over the interactions with energetically remote levels [Eqs. (5.5.1a) - (5.5.3a)], the off-diagonal matrix elements are taken between nondegenerate vibrational levels. Thus the ii-centroid, R vi, will vary with v and v and will not be equal to Rc- Since a+ and b are not independent of RuV and RvVi is not independent of v at fixed v, it is not permissible to remove the electronic factor from the summation over remote perturbers. Furthermore, the denominator of each term in the perturbation summation depends on an energy difference. This can cause a strong v, J-dependence of the second-order constants, especially when the vibrational levels of the state under consideration are near the energy... 

By virtue of a number of complementary perspectives [42-44,49], it turns out that the reduced centroid density at the dividing surface along some reaction coordinate q is the central quantity in determining the... 

The probability PXqr q ) fo move the reaction coordinate centroid variable from the reactant configuration to the transition state is readily calculated [108] by PIMC or PIMD techniques [17-19] combined with umbrella sampling [77,108,123] of the reaction coordinate centroid variable. In the latter computational technique, a number of windows are set up which confine the path centroid variable of the reaction coordinate to different regions. These windows connect in a piecewise fashion the possible centroid positions in going from the reactant state to the transition state. A series of Monte Carlo calculations are then performed, one for each window, and the centroid probability distribution in each window is determined. These individual window distributions are then smoothly joined to calculate the overall probability function in Eq. (4.11). An equivalent approach is to calculate the centroid mean force and integrate it from the reactant well to barrier top (i.e., a reversible work approach for the calculation of the quantum activation free energy [109,124]). 

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