Quantum graphs

At first I was a bit sceptical about the idea of using quantum graphs in solving the vibrational Schredinger equation. A fresh paper by Csaba Fabri and Attila Csaszar "Vibrational quantum graphs and their application to the quantum dynamics of CH5+" in PCCP is a novelty, there is no doubt about it. Nice creative work.

But then I decided to understand the topic a little bit deeper and despite an unexpectedly impressive results given in the paper, I'm probably too short sighted to see potential for further extending this approach. This is a zero-order reduced-dimensionality model. A natural extension to its present version would be to add a non-uniform 1D potential to every edge in the graph. This potential is well approximated as flat, due to extremely low flip and twist barriers, but still, I would add this 1D slices of the PES along isomerisation coordinates. In this way we are no longer getting sin(x) type solutions from every edge in the graph.

Another issue is how to account for couplings between modes. If there is a way of making this graph to include all couplings in the full-dimensional PES then there is a question of computational complexity of such an approach. The whole idea smells like quantum path integral formulation of Quantum mechanics or instanton theory by Althrope and J. Richardson. I think these quantum graphs are a subspace of this theory.

It is still early days and I'm hesitating if to pursue this idea. It may be big, it may be just a curiosity though.