  In light of the potential use of single-molecule magnets (SMMs) in emerging
quantum information science initiatives, we report first-principles
calculations of the magnetic exchange interactions in [$\mathrm{Mn}_{3}$]$_{2}$
dimers of $\mathrm{Mn}_3$ SMMs, connected by covalently-attached organic
linkers, that have been synthesized and studied experimentally by
magnetochemistry and EPR spectroscopy. Energy evaluations calibrated to
experimental results give the sign and order of magnitude of the exchange
coupling constant ($J_{12}$) between the two $\mathrm{Mn}_{3}$ units that match
with fits of magnetic susceptibility data and EPR spectra. Downfolding into the
$\mathrm{Mn}$ $d$-orbital basis, Wannier function analysis has shown that
magnetic interactions can be channeled by ligand groups that are bonded by van
der Waals interaction and/or by the linkers via covalent bonding of specific
systems, and effective tight-binding Hamiltonians are obtained. We call this
long-range coupling that involves a group of atoms a collective exchange.
Orbital projected spin density of states and alternative Wannier
transformations support this observation. To assess the sensitivity of $J_{12}$
to external pressure, stress-strain curves have been investigated for both
hydrostatic and uniaxial pressure, which have revealed a switch of $J_{12}$
from ferromagnetic to antiferromagnetic with increasing pressure.
