Accurately computing excited states and lattice dependent tight-binding models from first principles simulations

The core problem of condensed matter physics is the understanding of how diverse macroscopic quantum phenomenon can emerge from a basic set of constituent particles and their interactions. Fortunately, quantum mechanics provides a universal bridge between the microscopic electrons and macroscopic quantum phenomenon via the ab initio Hamiltonian and Hilbert space. The eigenstates and eigenenergies of the ab initio Hamiltonian exactly correspond to the excited states and energies of a given material, and can be used to compute physical observables like conductivity, optical gaps, and magnetization. However, computing the eigenstates and eigenspectrum of the ab initio Hamiltonian through brute force methods is generally computationally intractable, except in special cases of small molecules like H2. Instead, approximate methods have been developed and used to great success in accurately computing the eigenspectrum and eigenstates of the ab initio Hamiltonian, falling under the categories of first principles methods and effective model Hamiltonians. While both first principles methods and effective model Hamiltonians have been used to accurately compute properties of real materials, significant avenues of research still remain. The biggest avenue for discovery is first principles excited states methods, with accurate ground state techniques like quantum Monte Carlo lacking a mature excited state counterpart. Accompanying this large avenue for change is the constant need for adaptation and development of methods to keep up with the rapid rate of novel materials discoveries. The construction of interacting effective Hamiltonians from ab initio calculations is the predominant challenge in the effective model approach. To this end, my thesis has been oriented around advancing the state of the art in first principles excited state computation and effective models with lattice effects. First, I present my work on investigating a new trial wave function for use in quantum Monte Carlo (QMC). The new non-orthogonal determinant wave function expands the possibilities of accurate QMC calculations, as the quality of the trial wave function is a primarily limiting factor of accuracy in QMC calculations. Next, I present my work on developing a stable statistical estimate for gradients used in QMC wave function optimization. This efficient method is simple to integrate into existing QMC codes, and improves the efficiency of QMC wave function optimization, an integral component of QMC calculations. Following that, I present my work on creating a novel method for computing excited states in QMC. The novel method addresses short comings of state-of-the-art QMC methods by allowing for state specific optimization with high accuracy and computational efficiency. I conclude by demonstrating my work building a tight-binding model for twisted bilayer graphene with lattice interactions from density functional theory (DFT). The work provided a pipeline for developing accurate DFT models with electron-lattice interaction, and demonstrated the importance of these interactions in the quantitative and qualitative description of the flat bands in magic angle TBLG. My work demonstrates the power of ab initio techniques in accurately computing excitations and eigenstates of complex quantum systems, and provides a concrete stepping stone towards the ad- vancement of accurate and efficient first principles methods and effective model Hamiltonians