The most computationally intensive element of most electronic-structure calculations, be they Hartree-Fock, DFT, coupled-cluster, RDM, or any other method, is calculation of the four-center electron repulsion integrals (ERIs). As the number of four-center integrals for- mally scale as O(n^4), even a relatively small system will necessitate the calculations of numerous ERIs. Furthermore, memory and latency concerns make it inadvisable to store all calculated integrals. Instead, integrals are generally calculated as needed, erased, and if needed calculated again, further increasing the number of ERIs calculated. Thus rapid and efficient methods of evaluating ERIs are of great significance in quantum chemistry.,Over the years, numerous ingenious methods have been devised to speed the evaluation of Gaussian ERIs. One of the most successful and widespread of these is the Obara-Saika method, and its associated refinements due largely to Pople and Head-Gordon. Though the relationship between the Obara-Saika method and Schlegel’s derivative theorem have long been recognized, no paper to date has elucidated the exact nature of this relationship. Instead, most texts rely on rather obtuse proofs to the Obara-Saika method, essentially postulating the result and then proving it to be correct. In Ch. 2 we present an explicit derivation of the Obara-Saika method, beginning with the definition of Gaussian integrals and Schlegel’s derivative theorem, and proceeding to explicitly build the proof from that foundation.,We then shift tracks to discuss the use of density matrices in quantum chemistry. In Ch. 4 we explore the application of density matrices to excitonic transport. In Secs. 1.2.2 & 4.2.2 we introduce the use of a Lindblad master equation, and explain its application to open quantum systems. In particular, we discuss the role of dephasing noise in excitonic transport in one- and two-dimensional systems, and demonstrate how dephasing noise can be beneficial to said transport. We demonstrate that optimal dephasing rates exist, but that while in homogenous systems the optimal dephasing rate is determined by the coupling, whereas in inhomogeneous systems the optimal coupling rate is determined by the topologyof the system.,Recently, experimental verifications of the geometry of the benzene dication and the closely related hexamethylbenzene dications have been achieved. In both cases, the lowest energy geometry is a pentagonal-pyramid with a hexacoordinated carbon. In Ch. 3 we discuss the optimal geometries of the as of yet unstudied nitrobenzene and aniline dications. We demonstrate that the nitrobenzene dication structure is expected to closely resemble the parent benzene dication’s structure, while the aniline dication is expected to more closely resemble the neutral aniline structure. In the process, we demonstrate that the results obtained from reduced-density-matrix methods compare favorably with the results of the more expensive coupled-cluster calculations.