Variational two-electron reduced-density-matrix theory: Partial 3-positivity conditions for N-representability

Variationally calculating the ground state of a many-electron quantum system using only the two-electron reduced-density-matrix (2-RDM) requires N-representability conditions that constrain the 2-RDM to correspond to an N-electron wave function. A systematic hierarchy of N-representability conditions, known as p-positivity conditions, has been developed [ D. A. Mazziotti and R. M. Erdahl Phys. Rev. A 63 042113 (2001)], and many-electron atoms and molecules in nonminimal basis sets have been solved with useful accuracy by a variational 2-RDM method with 2-positivity conditions [ D. A. Mazziotti Phys. Rev. Lett. 93 213001 (2004)]. This paper considers two forms of partial 3-positivity conditions, the lifting conditions and the T₁∕T₂ conditions, to further enhance the accuracy of the 2-RDM methods without the computational cost of full 3-positivity conditions. Variational 2-RDM methods with different N-representability constraints including 2-positivity conditions, the two types of partial 3-positivity conditions, as well as the complete 3-positivity conditions are applied to compute the ground state of the Lipkin spin model. The energies and 2-RDMs are compared to the results from full and truncated configuration interaction, many-body perturbation theory, and couple cluster theory with single and double excitations. Implications of using partial 3-positivity for variational 2-RDM calculations of many-electron atoms and molecules will be discussed.