We have performed very accurate quantum mechanical calculations of the five lowest S states of the beryllium atom. In the nonrelativistic part of the calculations we used the variational method and we explicitly included the nuclear motion in the Schrödinger equation. The nonrelativistic wave functions of the five states were expanded in terms of explicitly correlated Gaussian functions. These wave functions were used to calculate the leading ␣ 2 relativistic correction ͑␣ is the fine structure constant͒ and the ␣ 3 quantum electrodynamics ͑QED͒ correction. We also estimated the ␣ 4 QED correction by calculating its dominant component. A comparison of the experimental transition frequencies with the frequencies obtained based on the energies calculated in this work shows an excellent agreement.

Variational calculations employing explicitly correlated Gaussian functions and explicitly including the nuclear motion [i.e., without assuming the Born-Oppenheimer (BO) approximation] have been performed to determine the lowest singlet transition energy in the 9Be atom. The non-BO wave functions were used to calculate the alpha2 relativistic corrections (alpha=1/137.035,999,679). With those corrections and with the alpha3 and alpha4 QED corrections determined previously by others, we obtained 54,677.35 cm(-1) for the 3(1)S-->2(1)S transition energy. This result falls within the error bracket for the experimental transition of 54,677.26(10) cm(-1). This is the first time an electronic transition of Be has been calculated from first principles with the experimental accuracy.

Very accurate quantum mechanical calculations of the pure vibrational spectrum of the molecular ion are reported and compared with newly obtained pure vibrational transitions extracted from the available experimental data. The calculations are performed without assuming the Born-Oppenheimer approximation regarding separability of the nuclear and electronic motions and include the first order relativistic mass-velocity and Darwin corrections. For the two lowest transitions, whose experimental energies are established with the highest precision, the calculated and the experimental results show very good agreement.