I. INTRODUCTION 2 Atomic negative ions are representative of weakly-bound (typically having binding ener- gies one order of magnitude smaller than the neutral atoms or positive ions) fragile quantum systems. Due to the weak central field experienced by its outer valence electron, correla- tion is dominant and can no longer be treated as a small perturbation to the independent particle picture. Thus faithful estimation of binding energies, fine structure or excitation energies of negative ions pose considerable challenges to theoreticians and experimentalists. For example, very good precision on total energy is necessary to obtain modest values of the electron affinity. An impressive amount of theoretical as well as experimental works have been reported in the literature (see for example, [1, 2] for review) during the past three decades and interest in this area continues to grow further. Our focus is on the ground and excited states of negative ions of two most extensively studied systems (after possibly H − and He $^{−}$) from alkali and alkaline-earth groups, viz., Li, Be. Ever since the three long-lived bound states of Li − were reported through configuration- interaction (CI) calculations [3, 4], significant attention has been paid on their characteriza- tion and understanding. These three are respectively: the ground state [He]2s $^{2 1}$S e and two core-excited high-spin metastable bound states, 1s2s2p $^{2 5}$P $^{e}$, 1s2p $^{3 5}$S $^{o}$, each lying below the corresponding 1s2s2p $^{4}$P and 1s2p$^{2 4}$P parent states of Li. Thereafter a host of calculations including multiconfiguration Hartree-Fock (MCHF) [5], multiconfiguration interaction [6] as well as the recent saddle-point [7], variational Monte Carlo (VMC) [8], and other methods have been employed to determine the upper bound, relativistic and nonrelativistic energy, transition wavelength, oscillator strength, lifetime, fine structure, hyperfine parameter, etc., of these states. In general, these findings show good agreement with the experimental obser- vations [9, 10]. In contrast to Li − ground state, addition of a 2p electron to the ground state of Be 1s $^{2}$2s2 yields an unbound Be − [He]2s $^{2}$2p $^{2}$P o ground state; however, three metastable states [He]2s2p $^{2 4}$P $^{e}$, [He]2p $^{3 4}$S o and 1s2s2p $^{3 6}$S o were predicted, of which the first two have been observed experimentally [11–13]. Theoretical calculations include a variety of meth- ods, e.g., superposition of configurations [14], CI [15, 16], state-specific theory [17], MCHF [5, 18, 19], full core plus correlation [20], Rayleigh-Ritz variation method [21], VMC [8], etc., among others. In the last four decades, density functional theory (DFT) [22] has emerged as one of the