For our study, we adopted a wave-function based electron-correlated approach employing the semi- empirical valence-electron CNDO/2 model Hamiltonian developed by Pople and coworkers[31–33]. The method- ology adopted in this work is discussed in detail in our earlier papers[28–30], therefore, we present only a brief description of it here. As compared to our earlier works on boron-based clusters[29, 30] where we had used the INDO model, the choice of CNDO/2 model here has to do with the fact that INDO parameters are not available for the third row atoms like aluminium in the original Pople approach[34]. Our calculations are initiated at the Hartree-Fock (HF) level, within the CNDO/2 model, using a computer program de- veloped recently by us[28]. The CNDO-HF molecular orbitals (MOs) thus obtained, are used to transform the Hamiltonian from the original atomic-orbital (AO) to the MO representation, which is subsequently used in the post-HF correlated calculations. The transformed CNDO/2 Hamiltonian matrix elements in the MO repre- sentation are supplied to the computer program package MELD[35], which is used to perform the correlated calculations using the multi-reference singles-doubles configuration-interaction (MRSDCI) approach. Using the ground- and excited-state wave functions obtained from the MRSDCI calculations, electric dipole matrix elements are computed and subsequently utilized to compute the linear absorption spectrum, under the electric-dipole approximation, assuming a Lorentzian line shape. By analyzing the excited states contributing to the peaks of the computed spectrum obtained from a given calculation, bigger MRSDCI calculations are performed with a larger number of reference states. This procedure is repeated until the computed spectrum converges within an acceptable tolerance. In the past, we have used such an iterative MRSDCI approach on a number of conjugated polymers to perform large-scale correlated calculations of their linear and nonlinear optical spectra[36]. 

In this section we present and discuss our results on the electronic structure and optical properties computed using our CNDO-MRSDCI approach. However, before that, to benchmark our methodology we present and dis- cuss our CNDO-CI results on the simplest hydride of aluminium, namely AlH. 

First we performed the geometry optimization of the AlH molecule at the CNDO-HF level, by calculating the total energy of the system for different bond lengths, and 

## III. RESULTS AND DISCUSSION 

## II. THEORY 

## A. AlH Molecule 

then locating the minimum. Our optimized bond length was found to be 1.75 Å which is 0.1 Å larger than the reported experimental value of 1.65 Å[14, 15]. However, our aim here is to compute the optically-active excited states of the molecule, and as we will show later that this much of difference in the geometry leads to insignificant differences in their excitation energies. 

In the CNDO/2 model Al has nine Slater-type basis functions (1s, 3p, 5d), while H has only one basis func- tion, leading to ten basis functions in all. Being a va- lence electron approach, the total number of electrons being explicitly considered for AlH within the CNDO/2 model is four, with Al contributing three electrons, and H one. Therefore, with four electrons, and ten basis functions full-CI (FCI) calculations for the system are feasible, because in the singlet subspace the total num- ber of all possible configuration state functions (CSFs) is only 825. Thus, the FCI results presented here are exact within the chosen model (CNDO/2), and any disagree- ments with the experiments will indicate the deficiency of the model rather than that of the CI expansion. 

The CNDO-HF calculations predict the Mulliken pop- ulation of Al(H) atoms to be +0.24(-0.24), implying a partial ionic character to this system, with Al be- ing the electron donor, fully consistent with previous works[23]. The absorption spectrum of AlH is well-known and has been studied extensively over the years both experimentally[16] and theoretically[15, 17, 18]. The ground state of the system is classified as X $^{1}$Σ+ while the dipole connected excited states are A $^{1}$Π , C $^{1}$Σ+ , D $^{1}$Σ+ , and E $^{1}$Π . The first excited state A $^{1}$Π corresponds to HOMO( H )$_{→}$LUMO( L ) transition which is 5 σ → 2 π in nature. The reported experimental value of this transi- tion is 2.91 eV, with which our calculations are in good agreement with the computed values of 3.11 eV, and 3.12 eV at the bond lengths of 1.75 Å (our optimized bond length) and 1.65 Å (experimental bond length) respectively. This also shows that small differences in geometry have negligible impact on the calculated exci- tation energy of the lowest excited state. The C and D Σ -bands correspond to Rydberg-type excitations and, therefore, cannot be computed using CNDO approach, because they require the use of extremely diffused basis functions[15]. The E $^{1}$Π band corresponds to | H − 1 → L 〉 ( 4 σ → 2 π ) excitation, and has been measured close to 6.9 eV[16], while our calculation predicts its value to be 7.1 eV, thus overestimating it by 0.2 eV. We note that quantitatively speaking, the disagreement between our results, and the experimental ones, both for the A $^{1}$Π , and E $^{1}$Π bands is 0.2 eV, which is quite satisfying, given the semiempirical nature of this approach. Having bench- marked the CNDO-CI approach for the case of the AlH molecule, next we present the results of our calculations for Al 4 H 6 . 

