  ${}^3$He beam spin-echo experiments have been used to study surface
morphology, molecular and atomic surface diffusion, phonon dispersions, phason
dispersions and phase transitions of ionic liquids. However, the interactions
between ${}^3$He atoms and surfaces or their adsorbates are typically isotropic
and weak. To overcome these limitations, one can use molecules instead of
${}^3$He in surface spin-echo experiments. The molecular degrees of freedom,
such as rotation, may be exploited to provide additional insight into surfaces
and the behaviour of their adsorbates. Indeed, a recent experiment has shown
that $\textit{ortho}$-hydrogen can be used as a probe that is sensitive to the
orientation of a Cu(115) surface [Godsi et al., Nat. Comm. $\textbf{8}$, 15357
(2017)]. However, the additional degrees of freedom offered by molecules also
pose a theoretical challenge: a large manifold of molecular states and magnetic
field-induced couplings between internal states. Here, we present a fully
quantum mechanical approach to model molecular surface spin-echo experiments
and connect the experimental signal to the elements of the time-independent
molecule-surface scattering matrix. We present a one-dimensional transfer
matrix method that includes the molecular hyperfine degrees of freedom and
accounts for the spatial separation of the molecular wavepackets due to the
magnetic control fields. We apply the method to the case of
$\textit{ortho}$-hydrogen, show that the calculated experimental signal is
sensitive to the scattering matrix elements, and perform a preliminary
comparison to experiment. This work sets the stage for Bayesian optimization to
determine the scattering matrix elements from experimental measurements and for
a framework that describes molecular surface spin-echo experiments to study
dynamic surfaces.
