QUESTION: How can the 2-D hindered rotations in loose transition states be handled using DenSum?
7. QUESTION: How can the 2-D hindered rotations in loose transition states be handled using DenSum?
ANSWER: There are at least three different ways 2D-hindered rotors for loose transition states can be handled by DenSum. The first is by using the restricted Gorin prescription of Smith
43 and Golden 62 (see Benson's discussion of the Hindered Gorin Model), which is based on a modified version of the Gorin Model. 80 The second is to replace the restricted rotor formulation
with one based on particle-in-a-box. The third is to use two 1-D hindered rotors (implemented in DenSum) instead of one 2-D hindered rotor (2-D hindered rotors are not implemented in DenSum).
Smith and Golden 43 use a "hindrance parameter" to modify the moment of inertia of a 2-
D free rotor: the Hindered Gorin Model. They find the value of the hindrance empirically by varying it until they achieve a fit with experimental rate constant data. In my opinion, a potential drawback of the Smith and Golden approach is that the rotor model has no zero point energy, and restricting the range of rotational motion results in a model that is more like a particle-in-a-box, which has a finite zero point energy. It is for this reason that I added the particle-in-a-box degree of freedom type to DenSum. The zero point energy is important because of the role it plays in isotopic reactions. To use particle-in-a-box energy levels (instead of free rotor levels), one can empirically vary the "frequency" parameter until agreement with experiment is achieved, just as done by Smith and Golden.
The third alternative is to use two 1-D hindered rotors for each 2-D internal rotation. Here, the moment of inertia can be used directly in DenSum and the hindrance potential is then varied until agreement with the rate constant data is achieved. I've done a few brief tests of this approach and it seems to work well. The use of an actual hindered rotor is attractive to me, because it seems more physically realistic (one of these days we'll investigate this using quantum chemical calculations). The drawback is that state degeneracies may not be calculated correctly.
I don't know if there is a significant difference among rate constants calculated using the methods described above. Each of the methods is probably quite adequate in fitting almost any experimental data (a detailed comparison of the methods would be quite interesting). However, making predictions for temperatures where no experimental data are available requires formulation of an ad hoc model for hindrance as a function of temperature. Jordan, Smith, and
Gilbert 81 have formulated such a model (based on free rotors) and have tested it for a few cases; it is possible that their method could be used with Densum. The Marcus and Wardlaw
approach 82,83 is more predictive, but only if an accurate multi-dimensional potential energy surface is available; their approach is difficult to implement (and cannot be done in the present
84 version of DenSum). Miller and Klippenstein 85 have used VariFlex software, which implements an extended version of the Marcus-Wardlaw approach.
Appendix. Theoretical Basis
Much of this Appendix is based on an expanded and corrected version of the paper originally published in the International Journal of Chemical Kinetics: J.R. Barker, Multiple- well, multiple-reaction-path unimolecular reaction systems. I. MultiWell computer program suite. Int. J. Chem. Kinetics, 33, 232-45 (2001).