  We explore the extended Koopmans' theorem (EKT) within the phaseless
auxiliary-field quantum Monte Carlo (AFQMC) method. The EKT allows for the
direct calculation of electron addition and removal spectral functions using
reduced density matrices of the $N$-particle system, and avoids the need for
analytic continuation. The lowest level of EKT with AFQMC, called EKT1-AFQMC,
is benchmarked using small molecules, 14-electron and 54-electron uniform
electron gas supercells, and diamond at the $\Gamma$-point. Via comparison with
numerically exact results (when possible) and coupled-cluster methods, we find
that EKT1-AFQMC can reproduce the qualitative features of spectral functions
for Koopmans-like charge excitations with errors in peak locations of less than
0.25 eV in a finite basis. We also note the numerical difficulties that arise
in the EKT1-AFQMC eigenvalue problem, especially when back-propagated
quantities are very noisy. We show how a systematic higher order EKT approach
can correct errors in EKT1-based theories with respect to the satellite region
of the spectral function. Our work will be of use for the study of low-energy
charge excitations and spectral functions in correlated molecules and solids
where AFQMC can be reliably performed.
