Variational Optimization of the Second-Order Density Matrix Corresponding to a Seniority-Zero Configuration Interaction Wave Function

Abstract

We perform a direct variational determination of the second-order (two-particle) density matrix corresponding to a many-electron system, under a restricted set of the two-index N-representability P-, Q-, and G-conditions. In addition, we impose a set of necessary constraints that the two-particle density matrix must be derivable from a doubly occupied many-electron wave function, i.e., a singlet wave function for which the Slater determinant decomposition only contains determinants in which spatial orbitals are doubly occupied. We rederive the two-index N-representability conditions first found by Weinhold and Wilson and apply them to various benchmark systems (linear hydrogen chains, He, N2, and CN(-)). This work is motivated by the fact that a doubly occupied many-electron wave function captures in many cases the bulk of the static correlation. Compared to the general case, the structure of doubly occupied two-particle density matrices causes the associate semidefinite program to have a very favorable scaling as L(3), where L is the number of spatial orbitals. Since the doubly occupied Hilbert space depends on the choice of the orbitals, variational calculation steps of the two-particle density matrix are interspersed with orbital-optimization steps (based on Jacobi rotations in the space of the spatial orbitals). We also point to the importance of symmetry breaking of the orbitals when performing calculations in a doubly occupied framework.

Polynomial scaling approximations and dynamic correlation corrections to doubly occupied configuration interaction wave functions

“…12 shows the potential energy surface for N 2 in the cc-pVDZ basis. The methods compared are CCSD(T) in the MO basis and the CISDDT (Mmin CISD , CAS (10,18)). Density matrix renormalisation group (DMRG) 14,15 energies with FCI accuracy are added as a reference.…”

“…Reduced-cost DOCI solutions have previously been obtained either by a projected Schrödinger equation approach 7,8 or by using exactly solvable models 9 or a variant of the variational 2-RDM method projected on the seniority zero sector of the Hilbert space. 10 DOCI performs well at accounting for static correlation but fails in describing dynamic correlation 4 whereas CISD wave functions perform rather well for the latter. 2 A second aim of the paper is therefore to establish how methods based on the union of truncated DOCI and truncated one-electron excitations based CI spaces perform for both types of electron correlation at reduced computational cost.…”

Block product density matrix embedding theory for strongly correlated spin systems

“…We simply extend the original algorithm: the block product states and the unitary matrix are optimized successively until convergence is obtained. The unitary matrix is optimized by minimizing the variational energy λ through successive Jacobi-rotations [53].…”

Direct variational determination of the two-electron reduced density matrix for doubly occupied-configuration-interaction wave functions: The influence of three-index N-representability conditions

“…[18][19][20][21] The necessary and sufficient conditions for a 2-RDM to be N-representable are known, [22][23][24][25] but in practice, only a limited set of such constraints is used, commonly only the so-called P, Q, and G two-index N-representability conditions. Recently, we have reported 26,27 results of the v2RDM method under these P, Q, and G conditions approximating the 2-RDM arising from zero seniority number wave functions. We have performed that task formulating the optimization problem as a semidefinite program (SDP) [28][29][30][31] specifically adapted towards v2RDM 21,32,33 in which one maximizes a linear function on the intersection of a linear affine space and the convex cone of block-diagonal positive semidefinite matrices.…”