  Spin-lattice relaxation is a key open problem to understand the spin dynamics
of single-molecule magnets and molecular spin qubits. While modelling the
coupling between spin states and local vibrations allows to determine the more
relevant molecular vibrations for spin relaxation, this is not sufficient to
explain how energy is dissipated towards the thermal bath. Herein, we employ a
simple and efficient model to examine the coupling of local vibrational modes
with long-wavelength longitudinal and transverse phonons in the clock-like spin
qubit [Ho(W$_5$O$_{18}$)$_2$]$^{9-}$. We find that in crystals of this
polyoxometalate the vibrational mode previously found to be vibronically active
at low temperature does not couple significantly to lattice phonons. This means
that further intramolecular energy transfer via anharmonic vibrations is
necessary for spin relaxation in this system. Finally, we discuss implications
for the spin-phonon coupling of [Ho(W$_5$O$_{18}$)$_2$]$^{9-}$ deposited on a
MgO (001) substrate, offering a simple methodology that can be extrapolated to
estimate the effects on spin relaxation of different surfaces, including 2D
materials.

Atomic and Molecular Clusters