## 1 Introduction
This approach is most accurate due to the elimination of the problem of selecting basic sets for describing the MO. In works [17; 18; 19; 20; 21; 22], this method was realized when calculating the wave function of an excited photoelectron of a discrete and continuous spectrum on the basis of a system of coupled integro-differential equations. Herewith, despite the application of the widely known method of solving the differential equations of Numerov [23], the developed approach has a number of drawbacks related to the use of the approximation of the frozen core, and also the restriction of the SC expansion of the MO to the harmonics. It makes an essential contribution to the expansion of the wave function, due to its slow convergence. In particular, the accuracy of describing vacancies in the excited and ionized states restricts the application of the approximations of \"Z + 1\" and \"frozen core\". In calculating the weakly excited states of molecular systems, the delta-SCF [24] method based on the DFT theory has been developed, requiring the search for a suitable exchange-correlation potential for an accurate description of the electron-electron interaction. At the same time, the application of the method is limited when solving a wide class of problems of elastic and inelastic X-ray photon scattering, as well as photoexcitation processes, which is due to the need to obtain highly-excited electronic states. At the same time, the slow convergence of the MO in the SC representation has a significant effect on the accuracy of the calculated values. According to the estimate made in [18], in order to describe the one-electron energies of MO molecules with ligands of the second period of the Mendeleyev periodic table with an accuracy of \(\sim\)3%, one must take into account the partial harmonics with orbital angular momentum \(1\geq 50\) in the SC expansion of MO. The increase in the number of terms of the SC expansion leads to an increase in the system of differential equations, which causes more computational capacity giving an insignificant contribution of the obtained higher harmonics of the expansion to the observed quantities, but important for the energy characteristics of the systems [22]. In [17; 25; 26] the problem of slow convergence of the SC of MO expansion was solved by \"effective\" accounting of the influence of partial harmonics with large values of the orbital angular momentum, by increasing the participation coefficient of the last from the accounted partial harmonics in the limited SC expansion when calculating the photoelectron MO. At the same time, such an \"effective\" technique in calculating individual spectroscopic quantities does not allow to correctly describe the electron-electron Coulomb interaction, including the photoabsorption cross-section. The paper proposes an iterative scheme for calculating the excited and ionized states of molecular systems obtained outside the framework of the widely used approximations of \"Z + 1\" and \"frozen core\". The numerical procedure is based on solving a system of coupled integro-differential equations for all MOs included in the electronic configuration of the state. The higher harmonics of MO are represented as a composition of the wave functions of the electron shells of atoms expanded in a series of spherical harmonics. This ensures the required description accuracy of electron density near the nuclei of the molecular system. The proposed computational procedure is based on a combination of three-point difference Numerov scheme and Thomas algorithm. The theoretical basis and details of the iteration scheme are described. The efficiency of higher harmonics consideration in the MO decomposition is illustrated by the example of calculating the energy characteristics of the main electronic configuration of a hydrogen fluoride molecule with an increase in the ligand charge. The space expansion of electron density is also presented. The energy characteristics of the ionized states of the \(\sigma\)-symmetry molecule are estimated.