Numerical approximation of quantum canonical statistical observables with mean-field molecular dynamics and machine learning

Molecular electronic structure calculations are fundamental to modern quantum chemistry and materials science, offering detailed quantum-mechanical descriptions of electron-molecule interactions. Central to these calculations is solving the electronic Schrödinger equation, under the renowned Born-Oppenheimer approximation, where the eigenvalues represent system energy levels and the eigenfunctions describe electron wave functions. Given the complexity of electron-electron interactions, exact solutions are often limited to simple systems. Therefore, reliable numerical approximations of electronic eigenstates are crucial for bridging theoretical predictions with experimental observations, enabling accurate simulations of molecular properties, chemical reactivity, and material behaviour. Numerical analysis plays a pivotal role in this context, providing essential insights for refining computational methods and enhancing the accuracy of electronic structure calculations. To accurately model electron-nuclei systems at high temperatures, it is important to account for contributions from electronic excited states. Particularly, we address this challenge by employing a mean-field Hamiltonian dynamics method, which incorporates the contributions of each electronic eigenstate into the effective potential energy surface, weighted by their respective canonical equilibrium probabilities under the Gibbs distribution. This thesis presents four papers that delve into the mean-field molecular dynamics framework. In Paper A, we examine the canonical mean-field molecular dynamics approximation of correlation functions between quantum observables. Based on the Weyl quantization from semiclassical analysis, we provide an error estimate along with numerical validations for this classical mean-field approximation scheme. In Paper B, we investigate the neural network approximation of target potential functions in the molecular dynamics of Hamiltonian systems, using a data set sampled from the corresponding equilibrium Gibbs ...