State counting

To evaluate the state numbers and densities, a structure and set of frequencies have to be chosen for the transition complex. Provided the choice is made to match the experimentally derived Q lb Qrot /Qvib Qrot (obtained from the k , measured as a function of T), the computed k ( ) turn out to be insensitive to the details of the model that is selected. It means that, to a first approximation, the lifetimes of the excited molecules and the form of the low pressure fall-off are functions only of the entropies of the parent and its transition complex and that there are no adjustable parameters. This is advantageous to those whose aim is to calculate lifetimes, but evidently comparison of theory with experiment will not, in general, yield detailed information concerning the structures of the transition state. We return to these aspects later and presently consider the problem of evaluating the state densities, supposing that the structures and frequencies are known. 

The classical expression for the total states of s harmonic oscillators 

The cause of the failure of the classical equation is illustrated in Fig. 3 for a single oscillator. The classical W = C /hv is the full line through the origin, but the exact count is a step function, shown by the dotted line. The broken line, which bisects the steps at their mid-points, is an improved smoothed function with which to represent the number of states of the oscillator, as originally pointed out by Marcus and Rice [4] who named it the semi-classical approximation. 

The density of states for a single oscillator remains (hv) 1, and lVsemi ( ) may be recovered by suitable choice of integration limits. Thus 

Similarly, convoluting the individual densities of states, an equation for s oscillators is found, viz. 

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Figure A3.12.3. Hannonic RRKM imimolecular rate constants for C2Hj H+C2H4 dissociation classical state counting (solid curve), quantal state counting (dashed curve). Rate constant is in units of s and energy in kcal moK (Adapted from [17].)...

The chemical industry is one of the largest global industries with an annual sales value of 1,776 billion in 2005. The chemical industry in Europe counts for 2.5% within 24.5% overall industry contribution to the total GDP in the EU 15. Traditionally important markets for the chemical industry are Europe and the United States counting for 60% of global sales value in 2004 as shown in fig. 31 (CEFIC 2005). 

On the basis of these parameters we determined two possible transition states, 22 and 23. In transition state 22, the rhodium carbene is pointed away from the flip of the incipient cyclopentane ring (a chair-like transition state, counting the five carbons and the rhodium in the six-membered ring), whereas in 23 the rhodium carbene is pointed toward the flip of the incipient cyclopentane ring (a boat-hke transition state). As 10 (see Scheme 16.3) cyclizes to 12, in which the methyl and the phenyl are on the same face of the cyclopentane, we concluded that at the point of commitment to product formation, the transition state leading to cyclization must be chair-like 22 rather than boat-like 23. 

E-state indices, counts of atoms determined for E-state atom types, and fragment (SMF) descriptors. Individual structure-complexation property models obtained with nonlinear methods demonstrated a significantly better performance than the models built using MLR. However, the consensus models calculated by averaging several MLR models display a prediction performance as good as the most efficient nonlinear techniques. The use of SMF descriptors and E-state counts provided similar results, whereas E-state indices led to less significant models. For the best models, the RMSE of the log A- predictions is 1.3-1.6 for Ag+and 1.5-1.8 for Eu3+. 

The methods of state counting were further refined by Rabinovitch and his collaborators. By comparing variants of the semi-classical equation with computed, exact counts, it was discovered that accurate sums can be found using a single parameter expression. 

More recently, Beyer and Swinehart [9] have devised an algorithm for efficient computation of state counting, which is expected to find general application. It may also be used to include internal rotation [10]. 

Atom-type B-state counts were proposed as simple counts of the B-state atom types in a molecule [Butina, 2004]. 

Considering the assumptions in the phase-space model, this appears most appropriate for calculation of magnitudes of cross sections, and particularly of branching ratios. But it may be expected to fail with regard to product energy distributions, because these are fixed a priori by the process of state counting. 

In our PST calculation, the orbital angular momentum 1 of the dissociation must be included in the state count. The impact parameter b is a semiclas-sical function of the / quantum number in the following way ... 

Beyer-Swinehart density of vi brational states count... 

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Figure 9. Unimolecular decomposition rate constants ky for Mn(CO)x as a function of ion internal energy above threshold, Ef, calculated using RRKM theory employing Whitten-Rabinovitch state counting and bond energies listed in Table II, with log A = 15.

The equilibrium time t is useful in determining how long to wait before reading a new steady-state counting rate so that the indicated value has approached the true value within the error determined by counting statistics. The time taken for the ratemeter output to move from an initial rate Ii to within k standard deviations of the final rate I2 is [14]... 

As a simple preliminary test of the semiclassical adiabatic state counting algorithm, we consider the generalized harmonic system ... 

Figure 6.1 Cumulative density of states for a system of three coupled harmonic oscillators. The Hamiltonian is given by Equation (6.15) with (o —, oj = 4, (Mj = 5, c5i = 0.1, and di = O.IL The red line is the exact adiabatic state count using the levels given by Equation (6.19). The dotted green line is result of the semiclassical adiabatic method. Equation (6.12). The blue line is the traditional uncoupled harmonic description based on the potential minimum.

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