  We characterize the single-electron energies and the wavefunction structure
of arrays with two, three, and four phosphorus atoms in silicon by implementing
atomistic tight-binding calculations and analyzing wavefunction overlaps to
identify the single-dopant states that hybridize to make the array states. The
energy spectrum and wavefunction overlap variation as a function of dopant
separation for these arrays shows that hybridization mostly occurs between
single-dopant states of the same type, with some cross-hybridization between
$A_1$ and $E$ states occurring at short separations. We also observe energy
crossings between hybrid states of different types as a function of impurity
separation. We then extract tunneling rates for electrons in different dopants
by mapping the state energies into hopping Hamiltonians in the site
representation. Significantly, we find that diagonal and nearest neighbor
tunneling rates are similar in magnitude in a square array. Our analysis also
accounts for the shift of the on-site energy at each phosphorus atom resulting
from the nuclear potential of the other dopants. This approach constitutes a
solid protocol to map the electron energies and wavefunction structure into
Fermi-Hubbard Hamiltonians needed to implement and validate analog quantum
simulations in these devices.
