Variational two-electron reduced density matrix theory for many-electron atoms and molecules: Implementation of the spin- and symmetry-adapted $T_2$ condition through first-order semidefinite programming

The energy and properties of a many-electron atom or molecule may be directly computed from a variational optimization of a two-electron reduced density matrix (2RDM) that is constrained to represent many-electron quantum systems. In this paper we implement a variational 2RDM method with a representability constraint, known as the $T_2$ condition. The optimization of the 2RDM is performed with a first-order algorithm for semidefinite programming [D. A. Mazziotti, Phys. Rev. Lett. 93, 213001 (2004)] which, because of its lower computational cost in comparison to second-order methods, allows the treatment of larger basis sets. We also derive and implement a spin- and symmetry-adapted formulation of the $T_2$ condition that significantly decreases the size of the largest block in the $T_2$ matrix. The $T_2$ condition, originally derived by Erdahl [Int. J. Quantum Chem. 13, 697 (1978)], was recently applied via a second-order algorithm to atoms and molecules [Z. Zhao et al., J. Chem. Phys. 120, 2095 (2004)]. While these calculations were restricted to molecules at equilibrium geometries in minimal basis sets, we apply the 2RDM method with the $T_2$ condition to compute the electronic energies of molecules in both minimal and nonminimal basis sets at equilibrium as well as nonequilibrium geometries. Accurate potential energies curves are produced for BH, HF, and ${\mathrm{N}}_2$. Results are compared with the 2RDM method without the $T_2$ condition as well as several wave-function methods.

Figure 1

The 2-RDM method with the DQGT2 constraints yields a curve which is indistinguishable from the FCI curve for BH at all bond lengths.

Figure 2

The 2-RDM methods with DQG and DQGT2 as well as CCSD are compared in the bonding region and at intermediate stretches. Around equilibrium the DQGT2 method improves upon the energy errors of DQG and CCSD by factors of 5 and 2, respectively.

Figure 3

Potential-energy curves of HF are generated in a valence double-zeta basis set from the two variational 2RDM methods, one with the two-positivity conditions (DQG) and one with both the two-positivity and the $T_2$ conditions (DQGT2). Although the absolute magnitude of the error in the energy increases with bond distance for all approximate methods considered, it remains the smallest in the 2RDM method with DQGT2 constraints.

Figure 4

The 2RDM methods with DQG and DQGT2 as well as CCSD and CCSD(T) are compared for ${\mathrm{N}}_2$ at stretched geometries in a valence double-zeta basis set.