Rotation degeneracy

In Eq. (44), gei(T ) is the ratio of transition state and reactant electronic partition functions [31] and the rotational degeneracy factor = (2ji + l)(2/2 + 1) for heteronuclear diatomics, and will also include nuclear spin considerations in the case of homonuclear diatomics. 

Figure 3.23. State-resolved associative desorption of D2 from Cu(l 11). (a) average desorbing kinetic energy (Ey) as a function of v,J quantum state, (b) state-resolved desorbing flux Df(v, J, Ts = 925 K) normalized by the rotational degeneracy and plotted in a manner such that a Boltzmann distribution is linear. The straight lines correspond to a rotational temperature T3 = Ts for each v state. From Ref. [33].

The molecular symmetry number (a) is a measure of the rotational degeneracy of the molecule. It is defined as the number of indistinguishable positions that can be obtained by rigidly rotating the molecule about its center mass. Symmetry numbers for spherical, conical, and cylindrical molecules, shapes with infinite axes of rotation, have o values of approximately 200, 20, and 20, respectively. Chemicals with no axes of symmetry have o = l. 

But remember that H+H is the simplest of all reactions. Moving more in the direction of true chemistry, consider next a reaction for which only two nuclei are hydrogens (instead of three) — the F+H reaction. This reaction is over 1 eV exothermic in going from the reactant valley, over a small (1 kcal) barrier, to the product valley. The exothermicity of reaction means that there are several energetically accessible (open) vibrational channels for this system even at the threshold for reaction. If we include all the rotational levels with each vibration, and the proper (2j+l) rotational degeneracies, we have an unthinkably large number of coupled equations to solve — over 1200 channels. (See Fig. 10.) To solve this problem, we must... 

The rotational degeneracy g/ is 2J+ 1, and the nuclear-spin degeneracy gj varies with rotational level only when the molecule contains symmetrically equivalent nuclei. 

The total rate is obtained by summing over all the allowed values of , given by n( ). The rotational density of states (or the rotational degeneracy) for a spherical top product is (2j +1)2. The value of n( ) can be deduced from figure 7.23. For j < J, the number of allowed values increases as 2y + 1. However, beyond j = J, the number of allowed values is constant at 2J + 1. The PST dissociation rate summed over all the allowed values is thus given by... 

Rotational Degeneracy for Molecule Plus Atom Products... 

However, the cross section for association of the fragments with orbital angular momentum, f, is proportional to (2i -I- 1). In addition, the product rotor has a rotational degeneracy of 2j + 1 if it is a linear product or (2j -f 1) in case of a spherical top. Thus the cross section for y -I- — 7 is proportional to... 

The restricted domain of j and is shown in the vs. 7 graph in figure 9.7. It is evident that the allowed values of depend upon the relative values of j and J. Hence when the products consist of a linear molecule plus an atom the cross section, or the rotational degeneracy, is given by... 

We note now the difference between the prior and the PST rotational degeneracies. For the case of the NO2 dissociation, the prior probability in Eq. (9.24) for production of Jno was (27 0 " O- This is identical to the PST result when j J which is the high 7 limit case [Eq. (9.36a)]. On the other hand, for cold samples in which 0, the PST result predicts a probability that is independent of 7 0 [Eq. (9.36b)]. A similar reduced dimensionality is apparent in the case of the spherical top plus an atom. 

Rotational Degeneracies for a Sphere Plus a Diatom or Sphere... 

Finally, when both products are spherical tops, the rotational degeneracy is... 

Now that we have expressions for the rotational degeneracies, the PST PEDs are obtained in the same manner as those for the prior distributions. That is the PED for products having angular momenta 7, and 72 is given simply by... 

A major difference between the prior and PST product energy distributions is that in PST the rotational degeneracies of the 0 atom and NO are intimately intertwined. On the other hand, in the prior distribution, the two terms are independent of each other so that the rotational degeneracies of the O atom and NO are simply multiplied together as in Eq. (9.24). 

Equation (9.83) is not strictly valid in the limit as 7- 0 because we approximated the correct rotational degeneracies of 2y + 1 (for the sphere) with 2y. When these degeneracies are treated correctly, the sums of states in the limit of 7 = 0 are the same as given above. 

Figure 2 is a schematic diagram illustrating how one includes transition operators associated with the rotational, degeneracy and dynamical group of the three-dimensional harmonic oscillator. 

Thus the overall rotational degeneracy is (2/+ 1) [i.e., (2/+ 1) fold-degenerate in both k and rrij quantum numbers]. 

Fig.3.5 Rotational distribution of a given vibrational manifold a) pure Boltzmann distribution, b) Rotational degeneracy factor, c) Total population distribution.

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