Spin and symmetry adaptation of the variational two-electron reduced-density-matrix method

The variational two-electron reduced-density-matrix (2-RDM) method computes the ground-state energy and 2-RDM of an atom or molecule without calculation of the many-electron wave function. Recently, the computational efficiency of the 2-RDM method has been significantly enhanced through the use of a first-order algorithm for semidefinite programming [Mazziotti, Phys. Rev. Lett. 93, 213001 (2004)]. In this paper we develop a spin- and symmetry-adapted formulation of the method that further improves its efficiency by incorporating both the spin and spatial symmetries of many-electron atoms and molecules. While previous work on density-matrix symmetry focused on only one form of the 2-RDM, the variational method employs three different forms of the 2-RDM, known as the D, Q, and G matrices, to restrict the 2-RDM to be approximately N-representable, that is representable by an N-electron wave function. We apply spin symmetries to the three forms of the 2-RDM, each of which breaks into four diagonal spin-blocks, namely one singlet and three triplet blocks. If the molecules have point-group symmetry, each of the 2-RDMs may be further subdivided into smaller diagonal blocks according to the spatial symmetry of the basis functions. The subdivision of the 2-RDMs into diagonal blocks generates significant computational savings in both floating-point operations and memory storage. Calculations illustrate the computational savings. Spin adaptation also enforces the correct expectation value of the S2 operator, which in earlier work is applied as a separate constraint.