# Intrinsic Reaction Coordinate - What does 'Connectivity' really mean?

I am currently in the process of connecting transition state (TS) structures to other minima in order to determine a reaction pathway. Now I understand the basic concepts of this method. You start with your TS, run an IRC in both the Forward and Reverse directions, then you optimize the final structure in the successful IRC run.

My question is, what constitutes connectivity between a TS and a minimum energy structure?

Examples

1.) I run an IRC and I take the last resulting structure and optimize it. The optimization takes 4 steps and the geometry barely changed. Is this structure connected to the TS?

2.) I run an IRC and I take the last resulting structure and optimize it. The optimization takes 20 steps and the geometry changed fairly appreciably, however, connectivity of the atoms remains the same. Is this structure connected to the TS?

3.) I have a TS where a water is being formed next to some other molecule. I run the IRC and following the pathway shows you that the water forms and is being moved away from the molecule. The IRC successfully finishes so I take the last structure and optimize it. The optimization brings the water back towards the molecule to some particular spot where H-bonding occurs with the molecule. Is this structure connected to the TS?

Note

These examples were purposely laid out in order to determine at what point TS->Minima are or are not connected.

Various criteria may be applied in order to ascertain the connectivity of structures. A common solution is a path generated by requiring it to be a Newton path on the potential energy surface (E): $$(\nabla\nabla^\ast E(x))^{-1}\nabla E(x) = -\frac{dx}{dt}$$ Two points $x_0$ and $x_1$ are connected only if $\exists x:\mathbb{R}\to\Omega,\tau_0, \tau_1\in \mathbb{R}$ such that $x(t)$ is a Newton path and $x(\tau_i)=x_i$, i.e., starting and ending structures, where $\Omega$ is your space of structures (usually centered 3D coordinates). $x$ is then a parameterization of that path on $[\tau_0,\tau_1]\subset\mathbb{R}$.

Sometimes it is additionally required that there is at most 1 maximum on $E(x)$ between $\tau_0$ and $\tau_1$.

You may be interested in reading this page. In a literature search, you may want to read up on the work by Vanden-Eijnden, Bofill, Quapp, and Henkelman. This topic is too rich to discuss in this forum.

To address the specific examples, assuming the starting point is in the convex attraction basin of the nearest optimum and the used optimizer converges in such instances, the optimum is connected in the above sense. Most optimizers used in geometry optimizations are some variant of (quasi-) Newton-Raphson optimizer, so this criterion is given. There are several reasons why the IRC may not converge to the same optimum as the subsequently used geometry optimizer, including changes of basis sets, convergence tolerances, or in case of quasi-Newton methods, the Hessian guess.

The fact that the optimizer brings the water closer to the other molecule is no indication of lack of connectedness. Instead the IRC run did not end at an optimum. It is actually expected that the energy of two (neutral) molecules decreases as they approach each other from infinity even in the complete basis set limit (van der Waals forces). You just found the vdW/BSSE-bound pre/post-complex.

• Great answer, but your second link seems broken, could you fix that? – Felipe S. S. Schneider Jan 19 '17 at 11:53
• It seems the domain was sold. Sadly I don't remember whose publications I was thinking at the time (It's been almost 4 years :-). Probably Bofill, Quapp, van den Eijnden, Henkelman are good starting points in a literature search. – Deathbreath Jan 19 '17 at 15:38
• Thanks for taking the time to revise your answer. Unfortunately I still don't understand the first part, specifically your equation. Could you tell me what $dt$ refers to? Can I assume, that $\tau$ refers to a structure? What does it mean that the starting point is in the convex attraction basin? – Martin - マーチン Jan 23 '17 at 4:55
• @Martin-マーチン I've clarified above. Does this make sense to you now? Newton methods converge on convex functions, hence any convex region that includes an optimum is an attractive basin. – Deathbreath Jan 23 '17 at 15:53
• Yes, I think that is quite fine now. I'll let the bounty run for a few more days to give this a little more exposure, but I think it's yours. Thank you for your effort. – Martin - マーチン Jan 23 '17 at 16:26