Potential energy surface normalization factor

Figure 13 Potential energy surfaces for electron transfer reactions. Hamionic oscillator potential energy functions for reactants and product are shown, including the nuclear wave functions, which are shaded. The dark shaded region indicates the magnitude of overlap of the nuclear wave functions, which is the Franck-Condon factor, (a) is the normal region, (b) is the activationless region and (c) is the inverted region as defined in the text. (Ref. 72. Reproduced by permission of Namre Publishing Group, www.nature.com)...

For simple fluids composed of small molecules, the current equilibrium simulation techniques are highly accurate and the computational tools exist for carrying out long enough simulations of a sufficiently large system to obtain experimental accuracy. Thus, for these systems the accuracy of the potential energy surface is normally the factor that will restrict the accuracy that can be achieved for the determination of their properties. 

When porphyrins with much higher triplet energies such as palladium octaethylporphyrin (17 Et = 44.8 kcal mol" ) were used as sensitizers, even the cis trans isomerization of stilbene took place as a quantum chain process = 1-6) [95]. The high quantum efficiencies were explained by a quantum chain process in which the metalloporphyrin serves as both an energy donor and an acceptor. Since the quantum yield of cis trans isomerization of 1,2-diphenylpropene = 0-37) remained as a normal value under the same experimental conditions as those of stilbene, the potential energy surface of the triplet state is an important factor for occurrence of the quantum chain cis-trans isomerization. That is, in 1,2-diphenylpropene the triplet state exists exclusively as a perpendicular conformation, where the triplet state and the ground state lay very close in energy and the deactivation can only take place thermally. 

Q is the usual partition function of the activated complex referred to the minimum in the potential of the normal molecule as the zero of energy, Q is the partition function qf the three rotations and three translations of the normal molecule, Ea IS the activation energy of the reaction as measured from the minimum of the normal molecule potential energy surface to the minimum of the activated complex, 0 is the zero-point energy of the activated complex, and the v( s are the vibrational frequencies, of the normal molecule. Moreover, A the rate of deactivation of active molecules to normal molecules, has been set equal to the collision number Z times an efficiency factor y, assumed to be isotope independent. 

This approach allows for a fully quantum mechanical treatment of the dynamics, avoiding the nse of quantum correction factors used to denote classical dynamical approaches, with the concession that the potential energy surface must be expanded, ignoring higher order nonlinearity in the mode coupling. The potential energy surface is expanded with respect to the normal coordinates of the system, q, and bath, 01, and their freqnencies up to third and fourth order nonlinear conpling ... 

Figure 3 shows the absolute values of the autocorrelation functions for three different offsets AQ, defining three different initial positions for 4> on the final state potential energy surface in Figure 2. The slope of the potential surface at the initial position determines the decrease of the autocorrelation function from its initial value of 1, and it depends on the offset AQ between the minima of the potentials along the normal coordinate in Figures 1 and 2. For an offset AQ of zero, the center of the wavepacket (f> encounters a flat potential surface. No decrease of the absolute value of the autocorrelation is expected with time, as the overlap remains 1 at all times. The slow decrease seen for the solid line in Figure 3 is therefore caused by the damping factor F and the calculated spectrum is narrow. For an offset AQ of 1, the decrease at short times is faster, due to the nonzero slope of the potential surface at Q = l, and the calculated spectrum shows a large overall bandwidth. This trend is even more pronounced for the larger offset Ag = 3, the...

Comparisons of TST rate constants with those computed using accurate quantum scattering methods with the same potential energy surface have shown that TST is normally quite accurate (usually to within a factor of two of the accurate rate constant) providing the most sophisticated versions of the method are used, A major approximation of TST is that recrossing of trajectories back from products to reactants is not important. There is evidence from trajectory calculations on systems such as... 

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