Toward an automatic assignment of vibrational eigenstates in protonated water clusters

Water exhibits quantum effects that are not completely understood. One

striking example is the pH value of 7, which would be 8.5 if water were

classical. The Zundel ion (H₅O₂⁺) is a prototypical building block

of acidic water that is useful for studying these quantum effects. The

strong anharmonic and fluxional nature of this 15-dimensional system is

difficult to simulate, but recently we were able to use novel methods

based on Tensor Tree Network States (TTNSs) to obtain more than one

thousand vibrational eigenstates of the Zundel ion to high accuracy. By

assigning these eigenstates in terms of approximate quantum numbers, we

can gain deep insight into the structure of the states and vibrational

spectrum. However, assignment of these wavefunctions is cumbersome, as

it involves manual inspection of wavefunction cuts and analysis of

excitation patterns. In addition, assigning excitations in

near-degenerate coordinates requires suitable coordinates that expose

the underlying symmetry of the wavefunction. Here, we outline three ways

to improve this assignment: (1) Natural orbital decompositions of

subsets of the wavefunctions. (2) Transformations to different

curvilinear coordinates that decouple the degenerate vibrational modes.

(3) Unsupervised machine learning by clustering large sets of

eigenstates. With these improvements, we are able to more accurately and

faster assign the eigenstates of the Zundel ion, including those with

strong coupling. This will lead to a systematic understanding of the

vibrational quantum effects of protonated water clusters. Our proposed

techniques are general and can be applied to any vibrational system.