  The transition temperature between the low-temperature alpha phase of tin to
beta tin is close to the room temperature (Tab =13C), and the difference in
cohesive energy of the two phases at 0 K of about dEcoh=0.02 eV/atom is at the
limit of the accuracy of DFT (density functional theory) with available
exchange-correlation functionals. It is however critically important to model
the relative phase energies correctly for any reasonable description of
phenomena and technologies involving these phases, for example, the performance
of tin electrodes in electrochemical batteries. Here, we show that several
commonly used and converged DFT setups using the most practical and widely used
PBE functional result in dEcoh of about 0.04 eV/atom, with different types of
basis sets and with different models of core electrons (all-electron or
pseudopotentials of different types), which leads to a significant
overestimation of Tab. We show that this is due to the errors in relative
positions of s and p -like bands, which, combined with different populations of
these bands in alpha and beta Sn, leads to overstabilization of alpha tin. We
show that this error can be effectively corrected by applying a Hubbard +U
correction to s -like states, whereby correct cohesive energies of both alpha
and beta Sn can be obtained with the same computational scheme. We quantify for
the first time the effects of anharmonicity on dEcoh and find that it is
negligible.
