  Vibrational dynamics of adsorbates near surfaces plays both an important role
for applied surface science and as model lab for studying fundamental problems
of open quantum systems. We employ a previously developed model for the
relaxation of a D-Si-Si bending mode at a D:Si(100)-(2$\times$1) surface,
induced by a "bath" of more than $2000$ phonon modes [U. Lorenz, P. Saalfrank,
Chem. Phys. {\bf 482}, 69 (2017)], to extend previous work along various
directions. First, we use a Hierarchical Effective Mode (HEM) model [E.W.
Fischer, F. Bouakline, M. Werther, P. Saalfrank, J. Chem. Phys. {\bf 153},
064704 (2020)] to study relaxation of higher excited vibrational states than
hitherto done, by solving a high-dimensional system-bath time-dependent
Schr\"odinger equation (TDSE). In the HEM approach, (many) real bath modes are
replaced by (much less) effective bath modes. Accordingly, we are able to
examine scaling laws for vibrational relaxation lifetimes for a realistic
surface science problem. Second, we compare the performance of the multilayer
multiconfigurational time-dependent Hartree (ML-MCTDH) approach with the
recently developed coherent-state based multi-Davydov D2 {\it ansatz} [N. Zhou,
Z. Huang, J. Zhu, V. Chernyak, Y. Zhao, {J. Chem. Phys.} {\bf 143}, 014113
(2015)]. Both approaches work well, with some computational advantages for the
latter in the presented context. Third, we apply open-system density matrix
theory in comparison with basically "exact" solutions of the multi-mode TDSEs.
Specifically, we use an open-system Liouville-von Neumann (LvN) equation
treating vibration-phonon coupling as Markovian dissipation in Lindblad form to
quantify effects beyond the Born-Markov approximation.
