Variational reduced-density-matrix calculations on radicals: An alternative approach to open-shell ab initio quantum chemistry

An alternative approach to open-shell molecular calculations using the variational two-electron reduced-density-matrix (2-RDM) theory [Mazziotti, Phys. Rev. Lett. 93, 213001 (2004)] is presented. The energy and 2-RDM of the open-shell molecule (or radical) are computed from the limit of dissociating one or more hydrogen atoms from a molecule in a singlet state. Because the ground-state energy of an “infinitely” separated hydrogen atom in a given finite basis is known, we can determine the energy of the radical by subtracting the energy of one or more hydrogen atoms from the energy of the total dissociated system. The 2-RDM is constrained to have singlet symmetry in all calculations. Two sets of $N$-representability conditions are employed: (i) two-positivity conditions, and (ii) two-positivity conditions plus the $T_2$ condition, which is a subset of the three-positivity conditions. Optimization of the energy with respect to the 2-RDM is performed with a first-order algorithm for solving the semidefinite program within the variational 2-RDM method. We present calculations of several radicals near equilibrium as well as the dissociation curves of the diatomic radicals CH and OH.

Figure 1

The shapes of the potential energy curves of the CH radical from the 2-RDM methods with DQG and DQGT2 conditions as well as the approximate wave-function methods UMP2 and UCCSD are compared to the shape of the FCI curve. The potential-energy curves of the approximate methods are shifted by a constant to make them agree with the FCI curve at equilibrium or $1.20\mathrm{\AA }$. The 2-RDM method with the DQGT2 conditions yields a potential curve that, within the graph, is indistinguishable in its contour from the FCI curve. Absolute errors are reported in Table .

Figure 2

The shapes of the potential energy curves of the OH radical from the 2-RDM methods with DQG and DQGT2 conditions as well as the approximate wave-function methods UMP2 and UCCSD are compared to the shape of the FCI curve. The potential-energy curves of the approximate methods are shifted by a constant to make them agree with the FCI curve at equilibrium or $1.00\mathrm{\AA }$. The 2-RDM method with the DQGT2 conditions yields a potential curve that, within the graph, is indistinguishable in its contour from the FCI curve. Absolute errors are reported in Table .

Figure 3

The expectation values $⟨{\widehat{S}}^2⟩$ for both CH and OH from the 2-RDM/exact and UCCSD methods are shown as functions of internuclear distance. Spin contamination becomes significant in UCCSD at $1.30\mathrm{\AA }$ for OH and $1.50\mathrm{\AA }$ for CH. Unlike unrestricted methods, the 2-RDM method preserves the correct spin symmetry of the 2-RDM at all internuclear separations.