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This thesis is concerned with the development of a Projector Quantum Monte Carlo method for non-linear wavefunction ansatzes and its application to strongly correlated materials. This new approach is partially inspired by a prior application of the Full Configuration Interaction Quantum Monte Carlo (FCIQMC) method to the three-band (p-d) Hubbard model. Through repeated stochastic application of a projector FCIQMC projects out a stochastic description of the Full Configuration Interaction (FCI) ground state wavefunction, a linear combination of Slater determinants spanning the full Hilbert space. The study of the p-d Hubbard model demonstrates that the nature of this FCI expansion is profoundly affected by the choice of single-particle basis. In a counterintuitive manner, the effectiveness of a one-particle basis to produce a sparse, compact and rapidly converging FCI expansion is not necessarily paralleled by its ability to describe the physics of the system within a single determinant. The results suggest that with an appropriate basis, single-reference quantum chemical approaches may be able to describe many-body wavefunctions of strongly correlated materials. Furthermore, this thesis presents a reformulation of the projected imaginary time evolution of FCIQMC as a Lagrangian minimisation. This naturally allows for the optimisation of polynomial complex wavefunction ansatzes with a polynomial rather than exponential scaling with system size. The proposed approach blurs the line between traditional Variational and Projector Quantum Monte Carlo approaches whilst involving developments from the field of deep-learning neural networks which can be expressed as a modification of the projector. The ability of the developed approach to sample and optimise arbitrary non-linear wavefunctions is demonstrated with several classes of Tensor Network States all of which involve controlled approximations but still retain systematic improvability towards exactness. Thus, by applying the method to strongly-correlated Hubbard models, as well as ab-initio systems, including a fully periodic ab-initio graphene sheet, many-body wavefunctions and their one- and two-body static properties are obtained. The proposed approach can handle and simultaneously optimise large numbers of variational parameters, greatly exceeding those of alternative Variational Monte Carlo approaches.

Computational methods for predicting the binding affinity of small organic molecules tobiological macromolecules cover a vast range of theoretical and physical complexity. Generally, as the required accuracy increases so does the computational cost, thereby making the user choose a method that suits their needs within the parameters of the project. We present how WScore, a rigid-receptor docking program normally consigned to structure-based hit discovery in drug design projects, is systematically ameliorated to perform accurately enough for lead optimization with a set of ROCK1 complexes and congeneric ligands from a structure-activity relationship study. Initial WScore results from the Schrödinger 2019-3 release show poor correlation (R² ∼0.0), large errors in predicted binding affinity (RMSE = 2.30 kcal/mol), and bad native pose prediction (two RMSD > 4Å) for the six ROCK1 crystal structures and associated active congeneric ligands. Improvements to WScore’s treatment of desolvation, myriad code fixes, and a simple ensemble consensus scoring protocol improved the correlation (R² = 0.613), the predicted affinity accuracy (RMSE = 1.34 kcal/mol), and native pose prediction (one RMSD > 1.5Å). Then we evaluate a physically and thermodynamically rigorous free energy perturbation (FEP) method, FEP+, against CryoEM structures of the Machilis hrabei olfactory receptor, MhOR5, and associated dose-response assays of a panel of small molecules with the wild-type and mutants. Augmented with an induced-fit docking method, IFD-MD, FEP+ performs well for ligand mutating relative binding FEP (RBFEP) calculations which correlate with experimental log(EC50)with an R² = 0.551. Ligand absolute binding FEP (ABFEP) on a set of disparate ligands from the MhOR5 panel has poor correlation (R² = 0.106) for ligands with log(EC50) within the assay range. But qualitative predictions correctly identify the ligands with the lowest potency. Protein mutation calculations have no log(EC50) correlation and consistently fail to predict the loss of potency for a majority of MhOR5 single point mutations. Prediction of ligand efficacy (the magnitude of receptor response) is also an unsolved problem as the canonical active and inactive conformations of the receptor are absent in the FEP simulations. We believe that structural insights of the mutants for both bound and unbound (apo) states are required to better understand the shortcomings of the current FEP+ methods for protein mutation RBFEP. Finally, improvements to GPU-accelerated linear algebra functions in an Auxiliary-Field Quantum Monte Carlo (AFQMC) program effect an average 50-fold reduction in GPU kernel compute time using optimized GPU library routines instead of custom made GPU kernels. Also MPI parallelization of the population control algorithm that destroys low-weight walkers has a bottleneck removed in large, multi-node AFQMC calculations.Computational methods for predicting the binding affinity of small organic molecules tobiological macromolecules cover a vast range of theoretical and physical complexity. Generally, as the required accuracy increases so does the computational cost, thereby making the user choose a method that suits their needs within the parameters of the project. We present how WScore, a rigid-receptor docking program normally consigned to structure-based hit discovery in drug design projects, is systematically ameliorated to perform accurately enough for lead optimization with a set of ROCK1 complexes and congeneric ligands from a structure-activity relationship study. Initial WScore results from the Schrödinger 2019-3 release show poor correlation (R² ∼0.0), large errors in predicted binding affinity (RMSE = 2.30 kcal/mol), and bad native pose prediction (two RMSD > 4Å) for the six ROCK1 crystal structures and associated active congeneric ligands. Improvements to WScore’s treatment of desolvation, myriad code fixes, and a simple ensemble consensus scoring protocol improved the correlation (R² = 0.613), the predicted affinity accuracy (RMSE = 1.34 kcal/mol), and native pose prediction (one RMSD > 1.5Å). Then we evaluate a physically and thermodynamically rigorous free energy perturbation (FEP) method, FEP+, against CryoEM structures of the Machilis hrabei olfactory receptor, MhOR5, and associated dose-response assays of a panel of small molecules with the wild-type and mutants. Augmented with an induced-fit docking method, IFD-MD, FEP+ performs well for ligand mutating relative binding FEP (RBFEP) calculations which correlate with experimental log(EC50)with an R² = 0.551. Ligand absolute binding FEP (ABFEP) on a set of disparate ligands from the MhOR5 panel has poor correlation (R² = 0.106) for ligands with log(EC50) within the assay range. But qualitative predictions correctly identify the ligands with the lowest potency. Protein mutation calculations have no log(EC50) correlation and consistently fail to predict the loss of potency for a majority of MhOR5 single point mutations. Prediction of ligand efficacy (the magnitude of receptor response) is also an unsolved problem as the canonical active and inactive conformations of the receptor are absent in the FEP simulations. We believe that structural insights of the mutants for both bound and unbound (apo) states are required to better understand the shortcomings of the current FEP+ methods for protein mutation RBFEP. Finally, improvements to GPU-accelerated linear algebra functions in an Auxiliary-Field Quantum Monte Carlo (AFQMC) program effect an average 50-fold reduction in GPU kernel compute time using optimized GPU library routines instead of custom made GPU kernels. Also MPI parallelization of the population control algorithm that destroys low-weight walkers has a bottleneck removed in large, multi-node AFQMC calculations.

Ce travail présente quelques résultats sur les systèmes de particules en interaction pour l'interprétation probabiliste des équations aux dérivées partielles, avec des applications à des questions de dynamique moléculaire et de chimie quantique. On présente notamment une méthode particulaire permettant d'analyser le processus de la force biaisante adaptative, utilisé en dynamique moléculaire pour le calcul de différences d'énergies libres. On étudie également la sensibilité de dynamiques stochastiques par rapport à un paramètre, en vue du calcul des forces dans l'approximation de Born-Oppenheimer pour rechercher l'état quantique fondamental de molécules. Enfin, on présente un schéma numérique basé sur un système de particules pour résoudre des lois de conservation scalaires, avec un terme de diffusion anormale se traduisant par une dynamique de sauts sur les particules / This work presents some results on stochastically interacting particle systems and probabilistic interpretations of partial differential equations with applications to molecular dynamics and quantum chemistry. We present a particle method allowing to analyze the adaptive biasing force process, used in molecular dynamics for the computation of free energy differences. We also study the sensitivity of stochastic dynamics with respect to some parameter, aiming at the computation of forces in the Born-Oppenheimer approximation for determining the fundamental quantum state of molecules. Finally, we present a numerical scheme based on a particle system for the resolution of scalar conservation laws with an anomalous diffusion term, corresponding to a jump dynamics on the particles