  Rhenium diboride is a recently recognized ultra-incompressible superhard
material. Here we report the electronic (e), phonon (p), e-p coupling and
thermal properties of ReB$_2$ from first-principles density-functional theory
(DFT) calculations and neutron scattering measurements. Our calculated elastic
constants ($c_{11}$ = 641 GPa, $c_{12}$ = 159 GPa, $c_{13}$ = 128 GPa, $c_{33}$
= 1037 GPa, and $c_{44}$ = 271 GPa), bulk modulus ($B$ $\approx$ 350 GPa) and
hardness ($H$ $\approx$ 46 GPa) are in good agreement with the reported
experimental data. The calculated phonon density of states (DOS) agrees very
well with our neutron vibrational spectroscopy result. Electronic and phonon
analysis indicates that the strong covalent B-B and Re-B bonding is the main
reason for the super incompressibility and hardness of ReB$_2$. The thermal
expansion coefficients, calculated within the quasi-harmonic approximation and
measured by neutron powder diffraction, are found to be nearly isotropic in $a$
and $c$ directions and only slightly larger than that of diamond in terms of
magnitude. The excellent agreement found between calculations and experimental
measurements indicate that first-principles calculations capture the main
interactions in this class of superhard materials, and thus can be used to
search, predict, and design new materials with desired properties.
