Unusual chemical bond and spectrum of beryllium dimer in ground $X^1\Sigma_g^+$ state

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.