# 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.

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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): $$\nabla\nabla^\ast E(x)\nabla E(x) = -\frac{dx}{dt}$$ 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 $x$ is a Newton path and $x(\tau_i)=x_i$, where $\Omega$ is your space of structures (usually centered 3D coordinates)
You may additionally require that there is at most 1 maximum on $E(x)$ between $\tau_0$ and $\tau_1$.