Posner molecules: from atomic structure to nuclear spins

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.