  We introduce a new quantum embedding method to explore spin-phonon
interactions in molecular magnets. This technique consolidates various
spin/phonon couplings into a limited number of collective degrees of freedom,
allowing for a fully quantum mechanical treatment. By precisely factorizing the
entire system into "system" and "bath" sub-ensembles, our approach simplifies a
previously intractable problem, making it solvable on modest-scale computers.
We demonstrate the effectiveness of this method by studying the spin relaxation
and dephasing times of the single-molecule qubit \ce{VOPc(OH)8}, which features
a lone unpaired electron on the central vanadium atom. By using this mode
projection method, we are able to perform numerical exact quantum dynamical
calculation on this system which allows us to follow the flow of quantum
information from the single spin qubit into the projected phonon degrees of
freedom. Our results demonstrate both the utility of the method and suggest how
one can engineer the environment as to further optimize the quantum properties
of a qubit system.
