Computationally Affordable Methodstowards High-Dimensional MolecularSystems

The quest to understand the chemical processes that govern natural phenomena hasbeen a central theme in scientific research throughout the history of humanity. Fromthe large scales studied in astronomy and atmospheric sciences to the intrincate systemsin biochemistry, all such processes ultimately trace back to the dynamics of molecules atatomic and subatomic levels. Quantum mechanics marked a significant breakthroughin our ability to describe and predict molecular dynamics compared to the previousempirical or statistical approaches that previously dominated the field of chemistry.The central equations of molecular quantum mechanics at non-relativistic regimesare the Schrödinger equations. A range of theoretical and computational challengesarise when attempting to solve the Schrödinger equation. Among these challenges, themost prohibitive is the high-dimensionality inherent to the systems, which results fromthe necessity of characterizing the degrees of freedom for every constituent particle.This high-dimensionality leads to an extremely unfavorable scaling of the computationalcosts with the system size, a phenomenon commonly referred to as the curse of dimensionality.This thesis focuses on developing numerical methodologies that allow scalable computations.The content is divided in two different blocks. First, the time-independent vibrationalSchrödinger equation will be adressed. Then, we explore the dynamics of moleculesin the strong field physics limit. Concretely, we investigate the laser-induced electrondiffraction imaging technique.For the time-independent problem, we propose a methodology capable of creatingparametrizable families of orthonormal basis sets. The creation of such sets is done bythe introduction of a non-singular change of coordinates, modelled using normalizingflows (invertible neural networks). The normalizing-flows algorithm significantly enhancesthe approximation power of spectral methods by decoupling the different modes ofthe molecule. To obtain the optimal coordinate set, an optimization process of theapproximated energies to the variational limit is performed. The use of optimizationtechniques— that only require the physical domain and an underlying basis set— significanltyreduces the required expertise for performing vibrational calculations. Based on ananalysis of the optimal coordinates, we provide an explanation of how coordinate transformationsare affected by basis set truncation, number of target states and coordinate domain. Allthis intuition is used to propose two different transferable capabilites of normalizing-flow coordinates: transferability across basis set truncations and across molecules withsimilar structural motif. The transferability across basis set truncations allows for anefficient training algorithm, and for the production of some of the most accurate vibrationalresults for highly-excited target states in the literature. Transferability across differentmolecules opens the possibility to create a collection of preoptimized coordinates thatcan be applied in a broad range of quantum-chemistry calculations.In the second block, we aim to simulate the observables from the laser-inducedelectron diffraction imaging technique. In this case, the dynamics of the experimentare not solved by trying to find a direct solution of the time-dependent Schrödingerequation. Instead, we resolve the electron dynamics using the semiclassical model:an approximate model used in strong field physics that decomposes the continuumcomponent of the wavefunction of an ionized molecule into individual classical photoelectrontrajectories. The use of this approximation is motivated by the high-dimensionality ofthe systems, the complexity of the dynamics and long amplitude motions of photoelectrons.As part of the work, we implement all the necessary computational tools required toIconstruct the semiclassical model: a tunneling ionization theory, approximations of themolecular electric field, electron trajectory propagators and a detection algorithm. Asa single-active electron algorithm, the semiclassical model exhibits a favorable scalingwith respect to the size of the molecule, making it suitable for the study of complexsystems. We show that the algorithm is capable of replicating the experimentally measuredmomentum distributions, thereby validating its practical applicability.