Stability and metastability of clusters in a reactive atmosphere: theoretical evidence for unexpected stoichiometries of Mg$_M$O$_x$

In heterogeneous catalysis, materials function at finite temperature and in an atmosphere of reactive molecules at finite pressure. As a first step towards understanding the catalytic behavior of metal-oxide clusters, we study the $(T,p)$ dependence of the composition, structure, and stability of the various isomers for each size $M$ of Mg$_M$O$_x$ clusters in an oxygen atmosphere. The calculations are performed via a massively parallel genetic algorithm in a cascade approach. With the term ``cascade'', we identify a multistep procedure in which successive steps employ higher levels of theory, with each next level using information obtained at the lower level. We find that small clusters ($M < 5$) are in thermodynamic equilibrium when $x>M$. The non-stoichiometric clusters exhibit peculiar magnetic behavior, suggesting the possibility of tuning magnetic properties by changing environmental pressure and temperature conditions. Furthermore, we show that density-functional theory (DFT) with a hybrid exchange-correlation (xc) functional is needed for predicting accurate phase diagrams of metal-oxide clusters. Neither a (sophisticated) force field nor DFT with (semi)local xc functionals are sufficient for even a qualitative prediction.