  In periodic systems, the Hartree-Fock (HF) exchange energy exhibits the
slowest convergence of all HF energy components as the system size approaches
the thermodynamic limit. We demonstrate that the recently proposed staggered
mesh method for Fock exchange energy [Xing, Li, and Lin, Math. Comp., 2024],
which is specifically designed to sidestep certain singularities in exchange
energy evaluation, can expedite the finite-size convergence rate for the exact
exchange energy across a range of insulators and semiconductors when compared
to the regular and truncated Coulomb methods. This remains true even for two
computationally cheaper versions of this new method, which we call Non-SCF and
Split-SCF staggered mesh. Additionally, a sequence of numerical tests on simple
solids showcases the staggered mesh method's ability to improve convergence
towards the thermodynamic limit for band gaps, bulk moduli, equilibrium lattice
dimensions, energies, and phonon force constants.
