| An N-particle system can be described without approximation using a two-particle reduced density matrix (2-RDM) if the particles are indistinguishable and interact pairwise. If a 2-RDM is determined without using an N-particle wavefunction, the computational advantage of using a two-particle representation is realized. Directly minimizing the energy as a functional of the 2-RDM, however, does not yield a 2-RDM that describes an N -particle system (N-representable). In this thesis we discuss two approaches for determining approximately N-representable 2-RDMS without using wavefunctions. In the first approach, known as the variational 2-RDM method, we minimize the energy as a function of the 2-RDM while explicitly enforcing a subset of N-representability conditions on the 2-RDM to obtain a lower bound to the exact energy. We apply this approach to the strongly correlated quantum phase transition in the 1D transverse Ising model and demonstrate that compared to the exact solution, the variational 2-RDM method provides an accurate description of the Ising lattice and is an alternative tool for locating a quantum phase transition using only the ground electronic state. The second approach used in this work is the parametric 2-RDM method. The 2-RDM is parameterized using a reference 2-RDM and a subset of N-representability conditions. We apply the method to several isomerization reactions including those of oxywater, ammonia oxide, carbonic acid and diazene. Results obtained using the parametric 2-RDM method reproduce experimental results in predicting the equilibrium ratio of cis- to trans-carbonic acid isomers and describe the multireference transition state in the isomerization of diazene as well as multireference wavefunction methods. We extend the 2-RDM parameterization to describe electronic systems in arbitrary spin states and demonstrate similar accuracy at equilibrium and nonequilibrium geometries as was seen in describing singlet electronic states. Lastly, we reparameterize the 2-RDM to obtain a new parameterization that scales with the fourth power of the size of the basis set and improves on the accuracy of second-order Moller-Plesset perturbation theory, particularly for electronic systems that are strongly correlated. |
