| We study the reduced density matrix method, a variational approach for electronic structure calculations based on the two-body reduced density matrix. This method minimizes the ground state energy with respect to the two-body reduced density matrix subject to some conditions which it must satisfy, known as N-representability conditions. The resulting optimization problem is a semidefinite program, a convex optimization problem for which computational methods have greatly advanced during the past decade. Two significant advances are reported in this thesis. First, we formulate the reduced density matrix method using the dual formulation of semidefinite programming instead of the previously-used primal one; this results in substantial computational savings and makes it possible to study larger systems than was done previously. Second, in addition to the previously-used P, Q and G conditions we investigate a pair of positive semidefinite conditions that has a three-index form; we call them the T1 and T2 conditions. We find that the inclusion of the T1 and T2 conditions gives a significant improvement over results previously obtained using only the P, Q and G conditions; and provides in all cases we have studied (47 molecules) more accurate results than other more familiar methods: Hartree-Fork; 2nd order Moller-Plesset method (MP2), singly and doubly substituted configuration interaction (SDCI), quadratic configuration interaction including single and double substitutions (QCISD), Brueckner doubles (with triples) (BD(T)) and coupled cluster singles and doubles with perturbational treatment of triples (CCSD(T)). |
