## 4 Results and Discussions  
The program code realizing the above iterative scheme, as well as the results of calculating the ground and excited states are available at the electronic address [33]. The hydrogen fluoride (HF) molecule was chosen for testing the numerical procedure due to the following circumstances:  
1. A relatively small number of molecule shells allows the exploration of the possibility of the developed iterative scheme and the procedure for taking into account the higher harmonics in the SC expansion of MO when they are implemented on the ECM with limited CPU time;
2. The spectrum of free states and the structure of the wave functions have been thoroughly investigated [34]. In the calculations, the reference system associated with the F atom is chosen. To systemize the one-electron states, the quantum numbers characteristic for molecules of the point symmetry group \(C_{\text{cov}}\) are used. The MO calculations for the HF molecule are performed for the equilibrium distance between the nuclei of hydrogen atoms and fluorine \(R_{0}=1.733\) a.u. [35]. Comparative calculations of the energy characteristics (one-electron MO energies and total energy) of the ground state \(1\sigma^{2}2\sigma^{2}3\sigma^{2}1\tau^{4}\) of the HF molecule are given in Table 1.  
In section (a) of Table 1, the calculation was performed via the MC method of MO LCAO using the General Atomic and Molecular Electronic Structure System [12]. The calculation results in section (b) of Table 1 were obtained in this work using the MO expanded into a series (1) and limited \(l\leq 8\). They show sufficient accuracy which correlate with the calculation methods. The use of higher harmonics of the ligand wave functions for an accurate description of MO in the molecular nuclei region (see Section 2.2) gives an additional contribution to the total energy and regularly changes the energy level of the \(1\sigma\)-state of the HF molecule. The changes in the energy quantities observed in section (c) of Table 1 are primarily due to the contribution of the \(1\)s state of the H atom, which is a ligand in this case. Construction of MO is determined by the selection of \(l_{0}\) which includes the ligand functions in the MO structure. Figures 1 and 2 illustrate the procedure of determining \(l_{0}\) with respect to the calculation (section (c), Table 1). Figure 1 displays SC expansion of MO \(2\sigma\) of the ground state of HF molecule (calculation is performed for the SC series (1) with \(l\leq 60\)). Figure 2 displays the \begin{table}
\begin{tabular}{c c c c c c} \hline \hline
**Type** & \(\mathbf{1}\sigma\) & \(\mathbf{2}\sigma\) & \(\mathbf{3}\sigma\) & \(\mathbf{1}\pi\) & **Total Energy** \\ \hline a & \(-26.2923\) & \(-1.5990\) & \(-0.7674\) & \(-0.6464\) & \(-100.0577\) \\ b & \(-26.2905\) & \(-1.5995\) & \(-0.7656\) & \(-0.6492\) & \(-100.0594\) \\ c & \(-26.2915\) & \(-1.5995\) & \(-0.7656\) & \(-0.6492\) & \(-100.0616\) \\ \hline \hline \end{tabular}
\end{table} 
Table 1: One-electron MO energy and the total energy of the HF molecule (in atomic units). SC expansion of the wave function of H atom 1s state (the solution of the Hartree-Fock equations), decomposed in the molecular coordinate system (see Section 2.3). The example of MO 2\(\sigma\) shows (see Figure 1b) that the form of radial parts of the partial harmonics of SC expansion of MO starting with \(l=8\) (\(l\) is the partial harmonic denoted in the figure) is in good  
Figure 1: Radial parts of partial harmonics of SC expansion of 2\(\sigma\)–MO of the ground HF molecule state. Calculation of the ground state is performed for the equilibrium distance (\(R_{0}=1.733\) a.u. [35]) between the atom nuclei of H and F: (**a**) SC expansion of MO into a series (1) c \(l\leq 60\); and (**b**) SC expansion of MO into a series (1) c \(l\geq 8\).