Electrostatic multipole contributions to the binding energy of electrons

The interaction of an electron with a local static charge distribution (e.g., an atom or molecule) is dominated by the radial 1/r Coulomb potential at large distances. The second-order effect is generated by the non-central electric dipole contribution cosθ/r². The third-order effect is attributed to the electric quadrupole potential (3cos²θ-1)/2r³. In this study, we used the tridiagonal representation approach to produce a reasonably accurate account for the combined effects of every contribution to the binding energy of the electron with an effective quadrupole interaction. As a result, we obtained the bound states of a valence electron in an atom with both electric dipole and quadrupole moments.