Direct calculation of excited-state electronic energies and two-electron reduced density matrices from the anti-Hermitian contracted Schrödinger equation

Direct calculation of the ground-state two-electron reduced density matrix (2-RDM) and its energy has recently been achieved for many-electron atoms and molecules by solving the anti-Hermitian part of the contracted Schr\"odinger equation (ACSE) [D. A. Mazziotti, Phys. Rev. Lett. 97, 143002 (2006)]. In this paper the ACSE method is extended to computing the 2-RDMs and energies of excited states without the many-electron wave function. The contracted Schr\"odinger equation (CSE) is an important ingredient for excited-state 2-RDM methods because it is a stationary-state condition for both ground and excited states. We develop the theoretical framework for the ACSE as a stationary-state condition through its connections to the CSE and the Schr\"odinger equation. As in previous ground-state calculations, the indeterminacy of the ACSE is removed by reconstructing its 3-RDM as a functional of its 2-RDM through a cumulant theory for RDMs [D. A. Mazziotti, Chem. Phys. Lett. 289, 419 (1998)]. We calculate the initial 2-RDM from a multiconfiguration self-consistent-field calculation that includes multireference electron correlation, which can be especially important for excited states. The excited-state ACSE method is applied to computing absolute excited-state energies and vertical excitation energies of the molecules HF, H₂O, and N₂ as well as ground and excited potential-energy curves of HF. Comparisons are made to traditional multireference methods as well as full configuration interaction. Computed excited-state 2-RDMs nearly satisfy necessary N-representability conditions.