  We present three distinct examples where phaseless auxiliary-field Quantum
Monte Carlo (ph-AFQMC) can be reliably performed with a single-determinant
trial wavefunction with essential symmetry breaking. We first utilized
essential time-reversal symmetry breaking with ph-AFQMC to compute the
triplet-singlet energy gap in the TS12 set. We found statistically better
performance of ph-AFQMC with complex-restricted orbitals than with
spin-unrestricted orbitals. We then showed the utilization of essential spin
symmetry breaking when computing the single-triplet gap of a known
biradicaloid, C$_{36}$. ph-AFQMC with spin-unrestricted Hartree-Fock
(ph-AFQMC+UHF) fails catastrophically even with spin-projection and predicts no
biradicaloid character. With approximate Brueckner orbitals obtained from
regularized orbital-optimized second-order M{\o}ller-Plesset perturbation
theory ($\kappa$-OOMP2), ph-AFQMC quantitatively captures strong biradicaloid
character of C$_{36}$. Lastly, we applied ph-AFQMC to the computation of the
quintet-triplet gap in a model iron porphyrin complex where brute-force methods
with a small active space fail to capture the triplet ground state. We show
unambiguously that neither triplet nor quintet is strongly correlated using
UHF, $\kappa$-OOMP2, and coupled-cluster with singles and doubles (CCSD)
performed on UHF and $\kappa$-OOMP2 orbitals. There is no essential symmetry
breaking in this problem. By virtue of this, we were able to perform
UHF+ph-AFQMC reliably with a cc-pVTZ basis set and predicted a triplet ground
state for this model geometry. The largest ph-AFQMC in this work correlated 186
electrons in 956 orbitals. Our work highlights the utility, scalability, and
accuracy of ph-AFQMC with a single determinant trial wavefunction with
essential symmetry breaking for systems mainly dominated by dynamical
correlation with little static correlation.
