  We investigate "Posner molecules", calcium phosphate clusters with chemical
formula Ca$_9$(PO$_4$)$_6$. Originally identified in hydroxyapatite, Posner
molecules have also been observed as free-floating molecules $in$ $vitro$. The
formation and aggregation of Posner molecules have important implications for
bone growth, and may also play a role in other biological processes such as the
modulation of calcium and phosphate ion concentrations within the mitochondrial
matrix. In this work, we use a first-principles computational methodology to
study the structure of Posner molecules, their vibrational spectra, their
interactions with other cations, and the process of pairwise bonding.
Additionally, we show that the Posner molecule provides an ideal environment
for the six constituent $^{31}\text{P}$ nuclear spins to obtain very long spin
coherence times. $In$ $vitro$, the spins could provide a platform for
liquid-state nuclear magnetic resonance quantum computation. $In$ $vivo$, the
spins may have medical imaging applications. The spins have also been suggested
as "neural qubits" in a proposed mechanism for quantum processing in the brain.
