The correlated insulating and superconducting phases of magic angle twisted bilayer graphene (TBG) have received intense research attention in the past few years. Since each moir e unit cell of magic angle TBG contains around ten thousand carbon atoms, to take into account electron correlations among different moir e unit cells, a faithful atomistic model of TBG would involve hundreds of thousands of carbon atoms. This is extremely challenging for numerical studies of TBG even at the level of tight-binding models. As a result, the Bistritzer-MacDonald (BM) model, a continuum tight-binding model, has become a widely adopted starting point for further numerical studies. The BM model reveals that the flat bands of interest are energetically separated from the other bands and protected by $C_{2z}\mc T$ symmetry. Therefore, a reasonable starting point involves projecting the interacting models onto the flat bands and studying the approximate symmetries. This gives rise to the ``interacting Bistritzer-MacDonald'' (IBM) model, which takes the form of an extended Hubbard model with pairwise long-range interactions. Although the IBM model is not uniquely defined, and a unified physical description of the correlated phases has yet to emerge, such a downfolding procedure has been used by a number of recent works for studying phase diagrams of TBG beyond the tight-binding approximation.

This dissertation aims to present a comprehensive review and detailed proof of the magic angle twisted bilayer graphene, its Hamiltonian, and corresponding symmetries. The unique symmetries of TBG enable the use of various numerical approximations, including the Hartree-Fock method. The geometric and electronic symmetries of TBG restrict the ground state space. Consequently, Hartree-Fock provides a remarkably accurate approximation in the chiral limit where a $U(4)\times U(4)$ symmetry exists. This accuracy is attributed to the existence of a specific $U(4)\times U(4)$ symmetric ground state within the space of Slater determinant states. Understanding the symmetries of the BM Hamiltonian is crucial for selecting numerical states to approximate the ground states of TBG. A detailed proof of these symmetries enhances our comprehension of the electronic properties of twisted bilayer graphene, validates the accuracy of the HF method, and facilitates the computation of correlated states via exact diagonalization. Furthermore, the discussion about symmetries justifies the accuracy of quantum chemistry approaches beyond the HF method and exact diagonalization. Following the theoretical discussions, this dissertation delves into post-Hartree-Fock calculations. These calculations may become significant when long-range Coulomb interactions are introduced to the IBM model, and the interaction energy scale exceeds the energy dispersion. The techniques to handle such long-range interactions are well studied in the ab initio quantum chemistry community. Employing mature quantum chemistry software packages, this dissertation conducts both HF and post-HF calculations equally for the ground state and excited state properties of the IBM model at the correlated electron level. This approach offers two significant advantages: 1) In cases involving large systems where exact diagonalization proves impractical, HF and post-HF methods provide a viable alternative on a similar level. 2) For both integer and non-integer fillings, the electronic structures of tBLG are investigated using coupled-cluster-based methods such as CCSD and CCSD(T), along with the quantum chemistry density matrix renormalization group (QC-DMRG) method.