How to obtain reactant lifetime using transition state theory?

I've read this paper and the authors calculate life times of an intermediate using Quasi-Classical-Trajectories (QCT), RRKM and transition state theory (TST). Table 3 on page 4755 show the results.

If I understand correctly for the QCT calculations they take all (several hundred in that case) trajectories and just do statistical analysis of the lifetime of that intermediate. That perfectly makes sense and they come up with a value +/- error.

For TST they also calculate a lifetime of that intermediate, assuming it is in equilibrium with it's surrounding. This makes me think that they used the reaction barrier from intermediate to product for their calculation of the lifetime. How to do this? Up to now I've only used TST in combination with Eyring equation for calculation of rate constants.

As far as I understand, there are two things that I should answer for you.

Calculating rate constants with any trajectory methods is different than using TST. In fact, the whole point of using QCT is so we can account for nonstatistical effects. These nonstatistical effects are usually extremely important, as it is also mentioned in the article

... This dynamic matching leads to pronounced nonstatistical effects on the lifetimes...

Any time you are using TST, you are assuming there are no nonstatistical effects whatsoever. This approximation is probably never true, but it leads to surprisingly good numerical results. The way you calculate reaction rates in QCT is by integrating the reaction probability using a Monte Carlo integration formula, cleverly sampling the initial conditions of the reactants. I usually recommend a good book of W. Miller. The book itself is a bit outdated, but the derivation by Porter and Raff in the first chapter is beautiful.

Now you asked about the determination of the lifetime. The article, as I understand it, talks mostly about the lifetime of the activated complex. This can be done because in QCT we know the coordinates of atoms in every single timestep, so we can compare them to some given values to decide whether we are in the activated complex region or not. From the article:

Transition zones were defined for the C–C forming (3‡) and the C–N breaking (5‡) steps (see Supporting Information for details). [...] The lifetime of intermediate 4b was defined as the elapsed time between departure from the transition zone 3b‡ and entrance in transition zones 5b‡

So, long story short: QCT determines lifetimes differently from TST, this is because TST is inherently statistical, while QCT also considers nonstatistical effects. Instead, QCT calculations rely on defining geometries for different states, and simply calculating how many integration steps the system spends there.

• Thank you for the answer. I understand and/or knew the things you explained here and it doesn't really answers my question. I know how to calculate a rate constant using TST (eyring equation) and I know how to determine the lifetime of this intermediate using QCT. What I don't know is how to calculate the lifetime using TST (and how to calculate rate constants using a trajectory - if that's even possible). How do I get to the lifetime of this intermediate using TST? – DSVA May 21 '17 at 14:10
• As for calculating the rate constant by QCT, I'd still refer you to the Raff-Porter chapter I linked earlier, it is well explained there. As I mentioned in my comment, you do a Monte Carlo sampling of the initial condition, and if you choose the right distributions, then the reaction rate calculation reduces to counting the ratio of reactive collisions to nonreactive collisions. Note that this is true only for a well-derived Monte Carlo sampling, that is what is found in the book I mentioned earlier. Another reference can be found at Gaussan help page: gaussian.com/glossary/hase96 – Ezze May 21 '17 at 14:15
• I am not quite sure about determining the lifetime of the activated complex by TST calculations, as I have no real experience in TST methods. I have a hard time imagining an activated complex with a non-zero lifetime tho. It is a local energy maximum, thus a metastable configuration. I'd make a wild guess that it can have no lifetime whatsoever. Someone who is more experienced than me should correct me tho. – Ezze May 21 '17 at 14:17
• thanks, this sounds very interesting I will look into that. Let's see if someone else knows how to do this. I'm able to calculate the lifetime of the activated complex but not the reactant. – DSVA May 21 '17 at 14:24
• If you can calculate the reaction rate, isn't the reactant lifetime just the reciprocal of the rate coefficient? – Ezze May 21 '17 at 14:34