Studying electronic behavior, whether it's electronic structure or dynamics, is crucial to better understanding many physical and chemical phenomena. While traditional wavefunction methods have been successful for a variety of systems, the computational cost can be prohibitive. A different approach is to use the reduced density matrix (RDM) perspective, which allows for a decrease in computational cost without sacrificing accuracy. Here, the RDM perspective is used to consider two different problems: reducing the computational cost of the variational 2-electron reduced density matrix method and generalizing the treatment of open quantum systems to treat systems of multiple fermions. To approach the first problem, I will outline several approximations to the Schoedinger equation, with a particular focus on the variational 2-RDM method constrained to the pair space. I will then present results using this method with orbital localization to recover size extensivity in molecular and polymer chains. Finally, I will use this method in conjunction with the complete active space self-consistent field method to analyze strong correlation in systems up to 80 electrons in 80 orbitals. For the second problem, I derive constraints on the Lindbladian matrices to generalize the Lindblad treatment of Markovian open quantum systems to treat systems of multiple fermions. I then generalize the Lindblad theory to treat non-Markovian open quantum systems, and introduce further constraints to generalize this method to treat systems of multiple fermions. Both of these projects take steps towards more efficient and more accurate use of reduced density matrix mechanics to treat problems in electronic structure and dynamics.




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