Cartesian-Hyperspherical Hartree-Fock method for numerical observation of stabilized Langmuir states

Based on the fact that hyperspherical radial dynamics of the Helium atom can be slow comparing to other coordinates we formulate adiabatic Hartree-Fock equations for the Hydrogen atom in both circularly polarized and the magnetic fields for the Langmuir states both in the circularly polarized and magnetic fields. The time-dependent Hartree-Fock equations are solved as functions of the effective parameter $z$ (approximately the hyperspherical radius of the suspected configuration) and then the time-averaged energy is minimized to obtained the electron equilibrium in third spatial direction. The transverse part of the wave function is then found by solving two-dimensional Schr{\"o}dinger equation with the effective potential found from the minimalization.