Approaches to solve the schrodinger equation for multi-electron systems in momentum space

In this study, when hydrogen or a hydrogen-like atom is placed in a uniform electric field, shifts in energy levels are calculated in momentum space by using the Fourier transform of hydrogen type orbitals. By choosing the electric field in the z-direction, the z average value to be calculated to form the Hamiltonian matrix elements is written in the momentum space. The average value obtained is found as the product of the angular and radial integrals. To calculate the angular integrals, recurrence relations of the spherical harmonics especially in dipole transitions and derivative relations of spherical harmonics with respect to angles are used. The radial integrals are calculated analytically using the recurrence relations and series expressions of the Gegenbauer polynomials. The analytical expression obtained for the average value is quite convenient for numerical calculations. As a result, Hamiltonian matrix elements are obtained using the Mathematica programming language and the eigenvalues and eigenvectors of the matrix are calculated by using the Jacobi method. The results found are in complete agreement with the results in the configurations space. Molecular integrals must be calculated to determine the physical and chemical properties of atoms and molecules. All molecular integrals are written in terms of overlap integrals. Overlap integrals are easily written in terms of angular and radial integrals and as known, angular integrals are easily calculated while radial integrals pose problems. In this study, overlap integrals in momentum space are stated by using Fourier transformation for Slater type orbitals. Recurrence relations are obtained for radial integrals in the created expression and the algorithm written using the initial values of these relations is calculated by running the program coded in Mathematica programming language. The obtained values are in agreement with the literature.