## 1 Introduction  
Theoretical study of the absorption processes [1], photoionization [2], elastic [3], and inelastic [4] scattering of an X-ray photon by a molecular system is a complex problem that requires an exact description of the excited states with emerging vacancies in deep and subvalent electronic shells. Existing methods for calculating wave functions can be classified according to the form of their representation as multicenter (MC) [5] and single-center (SC) [6]. In the MC representation, molecular orbitals (MO) in different regions of space are described with respect to the centers associated with the nuclei of the molecular system. In this approach, the main limitation is related to the choice of the basic set of wave functions affecting the accuracy and amount of calculation [7, 8]. In the calculations, the construction of basic functions in the form of a superposition of Slater [9] or Gaussian [10] orbitals are widely used. The most well-known realization of the MC approach is the method of linear combination of atomic orbitals (MO LCAO) [11, 12]. Its equations were first written by [PERSON] [5]. In this case, the problem of minimizing the energy functional is reduced to finding the coefficients for the expansion of MO in basic atomic orbitals. A fairly complete overview on the use of the MO LCAO presented in [13], shows that the main drawback of the method is the principal difficulties in obtaining excited states of molecular systems caused by the problem of choosing the optimal basis set. Another approach is to extend the methods for calculating the wave functions of atomic electrons to molecular systems that have been developed in the SC method of MO calculation [6; 14; 15]. The SC approach is based on the MO representation in the form of an expansion in the basis of functions centered relative on one point of space. The application of the SC method has a number of advantages, especially when studying electron-photon interaction processes in many-electron systems [15; 16; 17]. From the computational point of view, the simplicity of using SC in comparison with MC is due to the developed theory and methods of atomic calculations. The transition to molecular systems is determined by introducing summation over the orbital angular momenta, since one-electron atomic wave function is formally the SC component of the MO decomposition. This approach avoids the problems associated with the redundancy of the basis, or at least controls them, in contrast to the MC approach. In addition, the absence of multicenter integrals makes it possible to substantially simplify the procedure for performing calculations. Historically, the first way to implement the SC method was to represent the partial harmonics of the SC expansion in the form of a linear combination of Slater-type orbitals and directly minimize the expressions for the total energy [6; 11]. In a different approach, described in detail in [15; 16], another variant of the SC method is proposed, which reduces to solving the secular equation. In this case, both the discrete function and the continuous spectrum function are used as basis functions, which are a solution of the Hartree-Fock equations. This realization of the SC approach is not free from the problems of choosing optimal basis sets, as well as from the slow convergence of the SC expansion series of the MO. Moreover, it has a strong influence on the energy of the state of the partial SC harmonics of the MO decomposition with large orbital moment values under low contribution of these harmonics to the function norm. The last two drawbacks can also be attributed to the SC method in general. The application of the variational principle for obtaining the wave functions of electrons in the case of an atom leads to the well-known system of integro-differential Hartree-Fock equations. In molecular SC systems, the representation allows to obtain a system of integro-differential equations connecting different partial harmonics of MO.