State-specific rate

If all the resonance states which fomi a microcanonical ensemble have random i, and are thus intrinsically unassignable, a situation arises which is caWtA. statistical state-specific behaviour [95]. Since the wavefunction coefficients of the i / are Gaussian random variables when projected onto (]). basis fiinctions for any zero-order representation [96], the distribution of the state-specific rate constants will be as statistical as possible. If these within the energy interval E E+ AE fomi a conthuious distribution, Levine [97] has argued that the probability of a particular k is given by the Porter-Thomas [98] distribution... 

If the state-specific rate constants are assumed continuous, equation (A3.12.65) can be written as [103]... 

In order to discuss predissociation dynamics, it is also important to derive a state-specific rate constant based on the measurements of absorption and dissociation cross sections. 

Conventional FTIR instruments, in which the interferometer mirror is translated at a constant velocity, are ideally suited to the analysis of steady state infrared emission. However, time resolution of the infrared emission is required in many applications, such as the measurement of absolute rate constants for the formation or subsequent relaxation of a vibrationally excited species. It is then necessary to follow the intensity of the emission (at a particular wavenumber if state-specific rate constants are required) as a function of time. For continuous-wave experiments, crude time resolution... 

At 2000 K and 1 atm, Hollander s state-specific rate constant becomes k. = 1.46 x 1010 exp(-AE/kT) s-1, where AE is the energy required for ionization. For each n-manifold state the fraction ionized by collisions is determined, as well as the fraction transferred to nearby n-manifold states in steps of An = 1. Then the fractions ionized from these nearby n-manifold states are calculated. In this way a total overall ionization rate is evaluated for each photo-excited d state. The total ionization rate always exceeds the state-specific rate, since some of the Na atoms transferred by collisions to the nearby n-manifold states are subsequently ionized. Table I summarizes the values used for the state-specific cross sections and the derived overall ionization and quenching rate constants for each n-manifold state. The required optical transition, ionization, and quenching rates can now be incorporated in the rate equation model. Figure 2 compares the results of the model calculation with the experimental values. 

Scanning the frequency of the dissociation laser and collecting the total OH fluorescence, while the state-selection and probe frequencies are kept fixed on specific transitions, produces a PHOFEX spectrum an example is displayed in the right-hand panel of Fig. 10. The lines correspond to specific resonance states with rotational quantum number J and projection quantum number if = 2 in vibrational state (6,0,0). If the individual lines are broader than the resolution of the laser system, one can determine the width from fitting the spectrum and thus determine the state-specific dissociation rate. If the true linewidth caused by dissociation is smaller than the resolution of the laser system, the rates can be extracted from time-resolved measurements. All three laser frequencies are fixed, and the OH probe laser used to detect a particular state of OH is delayed with respect to the dissociation laser. In this way one can monitor the appearance of the OH products as function of the delay time, in the same way as described above for N02- In contrast to NO2, however, the rate is a state-specific rate rather than an average rate, because of the high selectivity of the overtone... 

Figure 17 Calculated state-specific rates k — V/h for HCO and HNO vs. excess energy (small dots). The symbols indicate the fast (triangles), the slow (squares) and the average (diamonds) classical rates (see Sect. 8). The solid and dashed lines are the SACM and the classical RRKM rates, respectively. Fleprinted, with permission of the Bunsengesellschaft fiir Physikalische Chemie, from Ref. 39.

If the classical dynamics is ergodic and intrinsically RRKM, one might expect that the classical rate constant approximates the average rate of the quantum mechanical state-specific rates. That is indeed the case for the dissociation of HO2 (Fig. 12 of Ref. 60) the classical rate is only slightly smaller than the average quantum mechanical rate. The same holds also... 

In the quantum scattering approach the collision is modelled as a plane wave scattering off a force field which will in general not be isotropic. Incident and scattered waves interfere to give an overall steady state wavefunction from which bimolecular reaction cross-sections, cr, can be obtained. The characteristics of the incident wave are determined from the conditions of the collision and in general the reaction cross-section will be a function of the centre of mass collision velocity, u, and such internal quantum numbers that define the states of the colliding fragments, represented here as v and j. Once the reactive cross-sections are known the state specific rate coefficient, can be determined from. 

In what follows, we will discuss the IVR dynamics and VP event for a collinear nonrotating triatomic vdW complex A..BC. For this model, all the complications that arise from the partitioning of the total angular momentum of the complex into the relative angular momentum of the dissociation fragments and intrinsic angular momentum of the diatomic fragment do not appear, and the quantum (Q) state-specific rate constant of the VP event in Eq.(l) can be written in more details as... 

An interesting question is whether the large fluctuations in the quantum mechanical decay rates have an influence on the temperature and pressure dependent unimolecular rate constant P) defined within the strong collision model, in Eq. (2). In the state-specific quantum mechanical approach the integral over the smooth temperature dependent rate k E) is replaced by a sum over the state-specific rates fc,-. Applications have been done for HCO [93], HO2 [94-96], and HOCl [97]. The effect of a broad distribution of widths is to decrease the observed pressure dependent rate constant as compared to the delta function-like distribution, assumed by statistical theories [98,99]. The reason is that broad distributions favor small decay rates and the overall dissociation slows down. This trend, pronounced in the fall-of region, was clearly seen in a study of thermal rate constants in the unimolecular dissociation of HOCl [97]. The extremely... 

In this equation, Eo is the total enzyme active site concentration, and a is the steady-state specific rate of NADH formation, approached by a single exponential burst of amplitude with an apparent first-order rate constant A. is estimated by extrapolation of the steady-state portion of the progress curve to t = 0, and A is estimated by the usual logarithmic plot ... 

If this was the case one could generalize transition state theory to apply also to state specific rates. This is seen by application of detailed balance. If there is a transition state for each state of the products then, for the reversed reaction, each state of the reactants (i.e., the products of the forward reaction) will correlate to its own transition state. 

To conclude this section, for many reactant molecules it is expected that a micro-canonical ensemble of resonance states will contain states which exhibit mode-specific decay and can be identified by patterns (i.e., progressions) in the spectrum, as well as unassignable states with random i and, thus, state-specific rate constants with random fluctuations. In general, it is not expected that the ij , which form a microcanonical ensemble, will have identical k which equal the RRKM k(E). 

Figure 8.4 Porter-Thomas distribution of state specific rate constants, Eq. (8.15), for v 1, 2,4, 8, and In these plots x = k and (x) = k (Polik et al, 1990b).

The connection between the Porter-Thomas nonexponential N(r, E) distribution and RRKM theory is made through the parameters k and v. The average of the statistical state-specific rate constants k is expected to be similar to the RRKM rate constant k(E). This can be illustrated (Waite and Miller, 1980) by considering a separable (uncoupled) two-dimensional Hamilton H = + Hy whose decomposition path is... 

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