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Transfer matrix theory of surface spin echo experiments with molecules

${}^3$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.