One possible extension of the present work would be to use complex GHF determinants as trial wavefunctions in ph-AFQMC. Chemical systems with documented real to complex GHF instabilities include the asymmetric CO 2 stretch, 49 fullerene molecules, and tetrahedra such as H 4 and Fe 4 S 4 . 100 Another possible extension of the present work would be to improve the trial wavefunction by using a linear combination of GHF determinants that may be obtained using non-orthogonal Configuration Interaction 101,102 or symmetry-projection techniques. More work is also needed to obtain a better understanding of the GHF-solution space (akin to what has previously been done for UHF solutions ) as the results of subsequent perturbation theory, Coupled Cluster, and ph-AFQMC depend directly on using a "suitable" GHF solution as the trial (see the Supplemental Information for a presentation of multiple competing GHF states in the case of stretched O 2 ).