High-dimensional quantum dynamics enhanced by machine learning tools

• Applicant: Dr. David Mendive-Tapia, Ph.D.
• Subject Area: Theoretical Chemistry: Electronic Structure, Dynamics, Simulation
• Term: since 2025
• Project identifier: Deutsche Forschungsgemeinschaft (DFG) - Project number 567014199

The multi-configuration time-dependent Hartree (MCTDH) method has emerged as the reference approach for very accurate or even numerically exact results in high dimensional quantum dynamics. However, nowadays the main bottleneck in the development of MCTDH is not about reaching a limit of what can be done with the available computational resources, rather the significant attached dedicated work - in terms of manpower - as compared to more straightforward approaches. This is a especially cumbersome problem when using internal coordinates (i.e. chemically meaningful coordinates such as bond lengths, bond angles and dihedral angles) rather than normal modes or Cartesian coordinates. The key attribute to address here is the need to minimize the degree of coupling between the individual coordinates. Importantly, this cannot be inferred from one single geometry/point, rather from the global information of the entire space spanned by the nuclear probability distribution. In general, this space will span 3N-6 dimensions (where N is the number of atoms) and it can be fairly complex. One such case would be floppy anharmonic molecules with many local minima and low energy barriers. Interestingly, even though the analysis of probability distributions forms a substantial core of many machine learning tools, addressing the coordinate problem in the field of quantum dynamics from this point of view is largely unexplored, which opens up several interesting possibilities. Therefore, the objectives of this project can then be summarized as: 1) Characterize and understand from a data analysis perspective the probability distribution described by the nuclear wave-function - i.e. via sampling distributions and correlation matrices - when employing tensor-network decompositions (multi-layer trees) such as MCTDH. 2) Expand on the previous work and investigate how more sophisticated machine learning tools - such as dimensionality reduction (DR), variations auto-encoders (VAEs) and graph neural networks (GNNs) - can be applied to find compact multi-layered representations of the wave-function and optimal sets of nuclear coordinates in the many possible representations of interest for a chemist. 3) Develop a general, yet automatic protocol, for the use of internal coordinates in quantum dynamics simulations comparable to the ones employed in electronic structure theory. Last, in order to proof the utility of the approach to the general scientific community, two problems object of a detailed study have been carefully handpicked. These seek a balance between expecting a significant achievement based on previous experience tacking similar case studies and showcasing the usefulness of having an optimal taylor-made coordinate system: 1) Characterizing the IR spectrum of a floppy anharmonic system with an unusual high density of Fermi resonant states. 2) Study the proton transfer of a molecule involving large amplitude motions on several excited electronic states.

• DFG Programme: Research Grants