Potential Energy Curves and Excited States of the C$_2$ Molecule by Auxiliary-Field Quantum Monte Carlo (AFQMC)*

The accurate determination of potential energy curves (PECs) and excited states represents two difficult problems in electronic structure calculations. We present AFQMC PECs of the challenging C$_2$ molecule, focusing on the ground state and two singlet low-lying excited states. AFQMC calculates a target many-body wave function (WF) by means of random walks in the space of Slater determinants. We employ truncated complete active space (CAS) trial WFs ($\Psi_T$) to guide the AFQMC projection to obtain the desired state. With the phase-free constraint,\footnote{S. Zhang and H. Krakauer, Phys. Rev. Lett. \textbf{90}, 136401 (2003)} the CAS $\Psi_T$ is effective in controlling the sign/phase problem, and filtering in the desired excited state. The AFQMC results are in very good agreement with exact results. Comparison with experimental spectroscopic constants will also be presented.