  A procedure for defining virtual spaces, and the periodic one-electron and
two-electron integrals, for plane-wave second quantized Hamiltonians has been
developed and demonstrated using full configuration interaction (FCI)
simulations and variational quantum eigensolver (VQE) circuits on Quantinuum's
ion trap quantum computers accessed through Microsoft's Azure Quantum service.
This work is an extension to periodic systems of a new class of algorithms in
which the virtual spaces were generated by optimizing orbitals from small
pairwise CI Hamiltonians, which we term as correlation optimized virtual
orbitals with the abbreviation COVOs. In this extension, the integration of the
first Brillouin zone is automatically incorporated into the two-electron
integrals. With these procedures we have been able to derive virtual spaces,
containing only a few orbitals, that were able to capture a significant amount
of correlation. The focus in this manuscript is on comparing the simulations of
small molecules calculated with plane-wave basis sets with large periodic unit
cells at the $\Gamma$-point, including images, to results for plane-wave basis
sets with aperiodic unit cells. The results for this approach were promising as
we were able to obtain good agreement between periodic and aperiodic results
for an LiH molecule. Simulations performed on the Quantinuum H1-1 quantum
computer were able to produce surprisingly good energies, reproducing the FCI
values for the 1 COVO Hamiltonian to within 11 milliHartree (6.9 kcal/mol),
when corrected for noise.
