  We present the optical and mechanical design of a mechanically compliant
quasi-two-dimensional photonic crystal cavity formed from thin-film silicon in
which a pair of linear nanoscale slots are used to create two coupled high-$Q$
optical resonances. The optical cavity supermodes, whose frequencies are
designed to lie in the $1500$~nm wavelength band, are shown to interact
strongly with mechanical resonances of the structure whose frequencies range
from a few MHz to a few GHz. Depending upon the symmetry of the mechanical
modes and the symmetry of the slot sizes, we show that the optomechanical
coupling between the optical supermodes can be either linear or quadratic in
the mechanical displacement amplitude. Tuning of the nanoscale slot size is
also shown to adjust the magnitude and sign of the cavity supermode splitting
$2J$, enabling near-resonant motional scattering between the two optical
supermodes and greatly enhancing the $x^2$-coupling strength. Specifically, for
the fundamental flexural mode of the central nanobeam of the structure at
$10$~MHz the per-phonon linear cross-mode coupling rate is calculated to be
$\tilde{g}_{+-}/2\pi = 1$~MHz, corresponding to a per-phonon $x^2$-coupling
rate of $\tilde{g}'/2\pi=1$~kHz for a mode splitting $2J/2\pi = 1$~GHz which is
greater than the radiation-limited supermode linewidths.
