Applications of reduced density matrix mechanics to strongly correlated systems and quantum tomography

The work in this dissertation approaches quantum chemical systems from two complementary perspectives: (1) calculation of the quantum state from first principles and (2) reconstruction of quantum states from experimental data. Reduced Density Matrix (RDM) mechanics provides a unifying framework for these two approaches.;The 2-particle RDM (2-RDM) contains complete information to characterize the electronic structure of molecular systems. Methods for computing the 2-RDM from solving the anti-Hermitian part of the contracted Schrodinger equation (ACSE) are extended to allow treatment of chemical systems in arbitrary spin and electronic states. The resulting ACSE technology is used to study sigmatropic shifts of propene and acetone enolate, electrocyclic rearrangement of allyl radical, and a conical intersection on the potential energy surface of triplet methylene. An efficient method for geometry optimization is developed using the Hellmann-Feynman theorem. The possibility of exploiting locality of the 1- and 2-RDM to develop linear scaling theories is discussed.;Experimental measurements can provide rich information about a quantum system and direct access to the RDM without solving the Schrodinger equation or ACSE. Inversion of experimental data can produce an estimate of the RDM, but this estimate may be unphysical or incomplete. A methodology based on semidefinite programming (SDP) is developed for reconstructing the RDM from experimental data which leads naturally to non-negative solutions - a critical attribute of physically realistic solutions. In addition, this methodology enforces special additional constraints, known as N-representability conditions, which ensure that the RDM represents an N-electron system. his methodology is used reconstruct the time-dependent 1-RDM describing the exciton-transfer dynamics in a photosynthetic light harvesting complex.