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 $T_1∕T_2$ 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.