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In this work, I study a variety of problems in electronic structure from a reduced density matrix perspective. In Chapter 2, I describe an extension of the current-constrained density matrix theory from its two-electron reduced density-matrix (2-RDM) formulation to a one-electron reduced density matrix (1-RDM) formulation. I demonstrate the current-constrained 1-RDM method through the computation of the theoretical, intrinsic resistance of acenes and phenacenes. In Chapter 3, I use reduced density matrix theory to study the electronic structure and conductivity of cyclo[18]carbon and its boron nitride analogue. I use the current-constrained matrix (1-RDM) theory to compute the molecular conductance in two cases: (1) conductance in the plane of the molecule and (2) conductance around the molecular ring as potentially driven by a magnetic field through the molecule's center. Off-diagonal long-range order (ODLRO) in the two-electron reduced density matrix (2-RDM) has long been recognized as a mathematical characteristic of conventional superconductors. The large eigenvalue of the 2-RDM has been shown to be a useful measure of this long-range order. In Chapter 4, I show that the cumulant 2-RDM also has a large eigenvalue in the limit of ODLRO. The largest eigenvalue of the cumulant 2-RDM is bounded from above by N. The large eigenvalue of the cumulant 2-RDM implies the large eigenvalue of the 2-RDM, and hence, is a natural measure of ODLRO that vanishes in the mean-field limit. In Chapter 5, I propose and implement a universal signature of the van der Waals interactions based on the cumulant part of the two-electron reduced density matrix (2-RDM). Due to the connected property of the cumulant, it can be used to detect the van der Waals interactions between two molecular moieties. In particular, I use the squared Frobenius norm of the cumulant of the 2-RDM, which has been previously shown to provide a size-extensive measure of the electron correlation. I study this signature of van der Waals forces in a collection of small molecules of varying geometries.




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