Variational Two-electron Reduced Density Matrix Theory for Many-electron Atoms and Molecules: Implementation of the Spin- and Symmetry-adapted T2 Condition through First-order Semidefinite Programming

Abstract 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 (2-RDM) that is constrained to represent many-electron quantum systems. In this paper we implement a variational 2-RDM method with a representability constraint, known as the T2 condition. The optimization of the 2-RDM is performed with a first-order algorithm for semidefinite programming [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 spinand symmetry-adapted formulation of the T2 condition that significantly decreases the size of the largest block in the T2 matrix. The T2 condition, originally derived by Erdahl [Int. J. Quantum Chem. 13, 697 (1978)], was recently applied via a second-order algorithm to atoms and molecules [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 2-RDM method with the T2 condition to compute the electronic energies of molecules in both minimal and non-minimal basis sets at equilibrium as well as non-equilibrium geometries. Accurate potential energies curves are produced for BH, HF, and N2. Results are compared with the 2-RDM method without the T2 condition as well as several wavefunction methods.