Product energy distribution canonical

Figure 9.1 The calculated microcanonical translational product energy distributions for a system of s classical oscillators at the indicated excess energies. The equivalent canonical RED at a system energy k jT = 400 cm is a simple exponential function, exp(—EJkaf ).

Note the large vibrational heat bath which insures that the translational and rotational product energy distributions are well described by a canonical distribution at a temperature T which is the product system temperature. 

Finally, the total available energy E is related to the temperature of the products by Eq. (9.18). This equation is not exactly correct because the average translational energy can vary between k T and 2k T (Klots, 1994c). However, this difference has a negligible effect on the calculated T for all but very small molecules. In the latter case, the whole concept of expressing the translational or rotational product energy distributions as a canonical one breaks down in any case. 

Experimental product rotational and translational energy distributions derived from energy-selected dissociation reactions can frequently be characterized by a temperature which implies that the distribution is a canonical one. This is found even when rotationally cold reactants are prepared in a state-selective manner. How can this be We illustrate the origin of these canonical distributions by calculating the rotational and vibrational distributions for a system of classical harmonic oscillators. 

The microcanonical [Eq. (9.14)] and the canonical [Eq. (9.15)] translational energy distributions are compared graphically in figure 9.1 for the case of three molecules with 3,5, and 25 classical harmonic oscillators at a constant product temperature, k T = 400 cm (ca. 580 K). Because of the different product heat capacities, the excess energy varies for the three different reactions. It is evident that the largest molecule with 25 product oscillators results in microcanonical and canonical distributions that are nearly indistinguishable. As the size of the molecule decreases, the discrepancy between the two distributions increases. The same trends would be evident had we used the more correct quantum vibrational density of states for the microcanonical translational PEDs. 

In summary, the PED of any subset of energy levels will be given by a corresponding canonical distribution if the subset is small compared to the total energy sink. Furthermore, even when the canonical and microcanonical distributions differ, the average energies will be the same. This justifies the use of Eq. (9.18) to calculate the product temperature. 

The canonical probability distribution of potential energy Pnvt( o T) is then given by the product of the density of states n(E) and the Boltzmann weight factor Wb(E T) ... 

If now we select an ideal gas (no intcrmolecular forces) which is placed in some external force field, the potential energy is simply a sum of the individual potential energies of each molecule and the canonical distribution can be expressed as a product ... 

Once the final multicanonical weight factor has been derived it provides the distribution for the production simulation in which high-energy configurations will be sampled adequately and high-energy barriers can be crossed with ease. Moreover, from this single simulation it is possible to derive the canonical distribution f canon(7 ,E) at any temperature (hence the name multicanonical ) ... 

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