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We explore the extended Koopmans' theorem (EKT) within the phaselessauxiliary-field quantum Monte Carlo (AFQMC) method. The EKT allows for thedirect calculation of electron addition and removal spectral functions usingreduced density matrices of the $N$-particle system, and avoids the need foranalytic continuation. The lowest level of EKT with AFQMC, called EKT1-AFQMC,is benchmarked using small molecules, 14-electron and 54-electron uniformelectron gas supercells, and diamond at the $\Gamma$-point. Via comparison withnumerically exact results (when possible) and coupled-cluster methods, we findthat EKT1-AFQMC can reproduce the qualitative features of spectral functionsfor Koopmans-like charge excitations with errors in peak locations of less than0.25 eV in a finite basis. We also note the numerical difficulties that arisein the EKT1-AFQMC eigenvalue problem, especially when back-propagatedquantities are very noisy. We show how a systematic higher order EKT approachcan correct errors in EKT1-based theories with respect to the satellite regionof the spectral function. Our work will be of use for the study of low-energycharge excitations and spectral functions in correlated molecules and solidswhere AFQMC can be reliably performed.

Spectral Functions from Auxiliary-Field Quantum Monte Carlo without Analytic Continuation: The Extended Koopmans' Theorem Approach