Title: Algorithms for discrete and continuous quantum systems
Description: This thesis is divided into three chapters. In the first chapter we outline a simple and numerically inexpensive approach to describe the spectral features of the single-impurity Anderson model. The method combines aspects of the density matrix embedding theory (DMET) approach with a spectral broadening approach inspired by those used in numerical renormalization group (NRG) methods. At zero temperature for a wide range of U , the spectral function produced by this approach is found to be in good agreement with general expectations as well as more advanced and complex numerical methods such as DMRG-based schemes. The theory developed here is simply transferable to more complex impurity problems The second chapter outlines the density matrix embedding methodology in the context of electronic structure applications. We formulate analytical gradients for energies obtained from DMET focusing on two scenarios: RHF-in-RHF embedding and FCI-in-RHF embedding. The former involves solving the small embedded system at Restricted Hartree-Fock (RHF) level of theory.

This serves to check the validity of the formulas by reproducing the RHF results on the full system for energies and gradients. The latter scenario employs full configuration interaction (FCI) as a high level solver for the small embedded system. Our results show that only Hellmann-Feynman terms, which involve derivatives of one- and two-electron terms in the atomic orbital basis, are required to calculate energy gradients in both cases. We applied our methodology to the problem of H 10 ring dissociation where the analytical gradients matched those obtained numerically. The gradient formulation is applicable to geometry optimization of strongly correlated molecules and solids. It can also be used in ab initio molecular dynamics where forces on nuclei are obtained from DMET energy gradients. In the final chapter we focus on the study of finite-temperature equilibrium properties of quantum systems in continuous space. We formulate an expansion of the partition function in continuous-time and use Monte Carlo to sample terms in the resulting infinite series.

Such a strategy has been highly successful in quantum lattice models but has found scant application in off-lattice systems. The Monte Carlo estimate of the average energy of quantum particles in continuous space subject to simple model potentials is found to converge with low statistical error to the exact solutions even when very high perturbation order is required. We outline two ways in which the algorithm can be applied to more complex problems. First, by drawing an analogy between the formulation in continuous-time with the discretized, Trotter factorized version of standard path integral Monte Carlo (PIMC). This allows one to use the suite of standard PIMC moves to carry out the position sampling required to obtain the weight of each time configuration. Finally, we propose an alternate route by fitting the many-particle potential with multidimensional Gaussians which provides an analytical form for the position integrals.