TABLE I: Calculated total energies of ground and excited states of Li $^{−}$, Be− along with the literature data. Numbers in the parentheses denote absolute per cent deviations. See text for details. $^{a}$Ref. [32]. $^{b}$Ref. [14]. $^{c}$Ref. [8]. $^{d}$Ref. [6]. $^{e}$Ref. [7]. $^{f}$Ref. [3]. $^{g}$Ref. [15]. $^{h}$Ref. [20]. $^{i}$Ref. [21]. $^{j}$Ref. [16]. $^{k}$Ref. [5]. 5 the spiked harmonic oscillators, Hulthen and Yukawa potentials etc. [27]-[31]. In most of these cases, this has either outperformed the best results published so far for those systems or offered results of comparable accuracy to the best calculations. This is achieved through the following steps: (i) approximate a function by an Nth order polynomial (we use Leg- endre) f$_{N}$ ( x ) in such a way that the approximation is exact at collocation points x$_{j}$ , i.e., f$_{N}$ ( x$_{j}$ ) = f ( x$_{j}$ ) (ii) a transformation r = r ( x ) employed to map the semi-infinite domain r ∈ [0 , ∞ ] onto the finite domain x ∈ [ − 1 , 1] (iii) introducing an algebraic nonlinear mapping r = r ( x )= L (1 + x ) / (1 − x + α ), with α =2 L/r$_{max}$ , followed by a (iv) symmetrization proce- dure. Finally this leads to a symmetric eigenvalue problem which can be easily diagonalized efficiently by using standard available libraries (like NAG) to generate accurate eigenvalues and eigenfunctions, for both low as well as higher levels. Another notable feature is that this nonlinear optimal discretization maintains similar accuracy at both smaller and larger distances with significantly smaller number of points (all calculations done with 250 radial points), than those needed in usual finite-difference or finite element methods but at the same time promises faster convergence. A convergence criterion of 10 − 5 and 10 − 6 a.u. was imposed for the potential and energy to obtain all the results presented here. By performing a series of test calculations, a consistent set of GPS parameters were chosen ( α =25, N=250, r$_{max}$ =200 a.u.) which produced “stable” converged results. Ion State − E(a.u.) X-only XC This work Ref. This work Ref. Li − [He]2s $^{2 1}$S e 7.4278(0.005) 7.4282 a 7.4984(0.009) 7.4553 $^{b}$,7.5008$^{a}$, 7.4991c 1s2s2p $^{2 5}$P e 5.3640(0.006) 5.3643 c 5.3925(0.171) 5.3866 d , $^{e}$, 5.3833$^{c}$,5.3865$^{f}$, 5.3863k 1s2p $^{3 5}$S o 5.2223(0.004) 5.2225 c 5.2608(0.137) 5.2561 d , $^{e}$, 5.2536$^{c}$,5.2560$^{f}$, 5.2558k 1s2s2p3p $^{5}$P e 5.3289 5.3683(0.007) 5.3679 e Be − [He]2s2p $^{2 4}$P e 14.5078(0.008) 14.5090 g 14.5806(0.062) 14.5779 $^{h}$,14.5716$^{c}$, 14.5708$^{g}$,14.5769j [He]2p $^{3 4}$S o 14.3272(0.002) 14.3275 g 14.4081(0.049) 14.4063 $^{h}$,14.4010$^{c}$, 14.4002g 1s2s2p $^{3 6}$S o 10.4279(0.009) 10.4288 g 10.4758(0.092) 10.4662 $^{c}$, 10.4615$^{g}$, 10.4711i