Reconstructive approaches to one- and two-electron density matrix theory

Novel computational methods for electronic structure theory are explored, in which the fundamental variable is either the one- or the two-electron reduced density matrix (1- or 2-RDM), rather than the electronic wavefunction. A unifying theme among these methods is density matrix reconstruction, that is, decoupling approximations that express higher-order density matrices as functionals of lower-order ones.; On the 2-RDM side, a connected (extensive) version of the Contracted Schrödinger Equation (CSE) is developed, in which the basic unknowns are the RDM cumulants through order four. Reconstruction functionals that neglect the 3- and 4-RDM cumulants are examined and revealed to be significantly less accurate than suggested by previous minimal-basis results. Exact 3-RDM cumulants for some four-electron systems are calculated and found to be comparable in importance to unconnected products of lower-order cumulants. Decoupling approximations for the 3- and 4-RDM cumulants are developed based upon a renormalized, diagrammatic perturbation theory for the three- and four-particle Green's functions, in which the effective, pairwise interaction is extracted from the two-particle cumulant. Diagram rules suitable for both the time-dependent and time-independent versions of this perturbation theory are derived.; Reconstructive approaches to natural orbital (1-RDM) functional theory are also examined, wherein the 2-RDM is parametrized in terms of the natural orbitals and their (generally fractional) occupancies. It is demonstrated, at the theorem level, that proposed “corrected Hartree” and “corrected Hartree-Fock” natural orbital functionals necessarily violate positivity of the 2-RDM, which is closely related to their failure to respect antisymmetry. Calculations demonstrate that negative eigenvalues of the 2-RDM are associated with a large, stabilizing (but ultimately spurious) contribution to the energy. Nevertheless, a partially self-interaction-corrected version of the corrected Hartree functional is shown to yield high-quality potential energy curves—comparable to multireference configuration interaction—for the strongly-correlated diatomic molecules HF and C₂. A new class of functionals that preserve both anti-symmetry and positivity are proposed and subjected to some preliminary tests.