Active-space two-electron reduced-density-matrix method: Complete active-space calculations without diagonalization of the $N$-electron Hamiltonian

Molecular systems in chemistry often have wave functions with substantial contributions from two-or-more electronic configurations. Because traditional complete-active-space self-consistent-field (CASSCF) methods scale exponentially with the number $N$ of active electrons, their applicability is limited to small active spaces. In this paper we develop an active-space variational two-electron reduced-density-matrix (2-RDM) method in which the expensive diagonalization is replaced by a variational 2-RDM calculation where the 2-RDM is constrained by approximate $N$-representability conditions. Optimization of the constrained 2-RDM is accomplished by large-scale semidefinite programming [Mazziotti, Phys. Rev. Lett. 93, 213001 (2004)]. Because the computational cost of the active-space 2-RDM method scales polynomially as $ra6$ where $ra$ is the number of active orbitals, the method can be applied to treat active spaces that are too large for conventional CASSCF. The active-space 2-RDM method performs two steps: (i) variational calculation of the 2-RDM in the active space and (ii) optimization of the active orbitals by Jacobi rotations. For large basis sets this two-step 2-RDM method is more efficient than the one-step, low-rank variational 2-RDM method [Gidofalvi and Mazziotti, J. Chem. Phys. 127, 244105 (2007)]. Applications are made to HF, $H2O$⁠, and $N2$ as well as $n$-acene chains for $n=2–8$⁠. When $n>4$⁠, the acenes cannot be treated by conventional CASSCF methods; for example, when $n=8$⁠, CASSCF requires optimization over approximately $1.47×1017$ configuration state functions. The natural occupation numbers of the $n$-acenes show the emergence of bi- and polyradical character with increasing chain length.