  This review outlines the main results which show the dual nature of the
chemical bond in diatomic beryllium molecule in the ground $X^1\Sigma_g^+$
state. It has been shown that the beryllium atoms are covalently bound at
low-lying vibrational energy levels ({\nu}=0-4), while at higher ones
({\nu}=5-11) they are bound by van der Waals forces near the right turning
points. High precision ab initio quantum calculations of Be$_2$ resulted in the
development of the modified expanded Morse oscillator potential function which
contains all twelve vibrational energy levels [A.V. Mitin, Chem. Phys. Lett.
682, 30 (2017)]. The dual nature of chemical bond in Be$_2$ is evidenced as a
sharp corner on the attractive branch of the ground state potential curve.
Moreover, it has been found that the Douglas-Kroll-Hess relativistic
corrections also show a sharp corner when presented in dependence on the
internuclear separation. The difference in energy between the extrapolated and
calculated multi-reference configuration interaction energies in dependence on
the internuclear separation also exhibits singular point in the same region.
The other problems of ab initio quantum calculations of the beryllium dimer are
also discussed. Calculated spectrum of vibrational-rotational bound states and
new metastable states of the beryllium dimer in the ground state important for
laser spectroscopy are presented. The vibration problem was solved for the
modified expanded Morse oscillator potential function and for the potential
function obtained with Slater-type orbitals [M. Lesiuk et al, Chem. Theory
Comput. 15, 2470 (2019)]. The theoretical upper and lower estimates of the
spectrum of vibrational-rotational bound states and the spectrum of
rotational-vibrational metastable states with complex-valued energy eigenvalues
and the scattering length in the beryllium dimer are presented.
