B. Al$_{4}$H$_{6}$ We first carried out the geometry optimization of Al 4 H 6 within a density functional theory (DFT) based approach, using Gaussian03 computational package[37] employing B3LYP functional, and 6-311+g(d) basis set. The optimized geometry is presented in Fig.1, and it is virtually identical to that reported by Li et al .[6] As noted by Li et al .[6] also, this structure is very similar to the computed structure of the borane, B 4 H 6 [38] which still has not been synthesized. We used this DFT optimized geometry to perform the calculations of the photoabsorp- tion spectrum within our CNDO-CI approach, as well as the ab initio TDDFT method. . Before discussing our calculated spectra, we present some of the molecular orbitals (MOs) of the system close to the Fermi level. In Fig. 2 MOs obtained from the CNDO-HF calculations are presented. Although, here we have not presented the ab initio B3LYP MOs which were used to perform the TDDFT calculations, we note that among the occupied orbitals, H − 3 , H − 1 , and H obtained from the CNDO-HF and the DFT calcula- tions were qualitatively very similar. Mulliken charges of various atoms indicate Al to be in slightly cationic state, while the H atoms carry small negative charges. Therefore, we conclude that the bonding in the system is largely covalent, with some polar character. The covalent nature of the bonding is also obvious from the MO plots, which show that there is significant amount of charge density in the region between the atoms. Next, we present and discuss the linear optical ab- sorption spectrum of the system computed using the CNDO-CI and the ab initio TDDFT methods. The cal- culations were performed using the geometry shown in Fig. 1, which as mentioned above, was obtained at the Figure 1: Optimized geometry of Al 4 H 6 obtained from the ab initio DFT calculations (see text for details). Each bond length (in Å) has also been indicated. B3LYP/6-311+g(d) level of theory. For the CNDO-CI calculations the total number of valence electrons was 18 and the total number of basis functions was 42, thus rul- ing out the FCI calculations. Therefore, as mentioned in section II, the MRSDCI method was adopted to com- pute correlated wave functions and energies of the ground and the optically active excited states. The calculations were performed without the use of point-group symme- try, therefore, the CI expansion became fairly large scale, with the largest CI calculation consisting of 1.3 millions CSFs. The CI matrix was iteratively diagonalized[35] to obtain 60 lowest roots which required several hours of CPU time. The dipole moments connecting these excited states to the ground state were used to compute the pho- toabsorption spectrum presented here. Before discussing our final spectrum, in Fig. 3 we present the convergence of our optical absorption results with respect to the size of the CI matrix, represented in this case by the num- ber of reference configurations (N ref ) from which singly- and doubly- excited CSFs were generated to obtain the MRSDCI expansion. For example, in our largest MRS- DCI calculation we used N ref =22 which generated 1.3 million total CSFs. The convergence of our calculations with respect to N ref is obvious from Fig. 3, in which the spectra corresponding to N ref =14 and N ref =22 are virtually indistinguishable. In Fig. 4, we present our final linear optical absorp- tion spectrum of Al 4 H 6 computed using the CNDO-CI method. Before discussing our spectrum in detail, we Figure 2: (Color online) Molecular orbitals (iso plots) of Al 4 H 6 from HOMO-3 to LUMO+3, obtained from the CNDO-HF calculations. 3 (a) HOMO-3 (b) HOMO-2 (c) HOMO-1 (d) HOMO (e) LUMO (f) LUMO+1 (g) LUMO+2 (h) LUMO+3