Semiclassical modeling of the states and spectra of para helium singlets with an algebraic formula for excited states (Q1098604)

This paper presents a semiclassical model for the spectra of Para Helium. For the ground state two electrons are considered as paired and the position at which their orbital motions are equidistant from the nucleus at some time T, r, is such that the centripetal and electrostatic forces balance. In addition at this time T it is assumed that the speed of each electron satisfies the Bohr relationship \(mvr=nh/2\pi\) \(n=1,2,3... \). The pairing of the electrons is assumed to give rise to a generalized Coulombic force \(F_{12}=e^ 2/(2r)^{2+\alpha}.\) These assumptions lead to two nonlinear simultaneous equations for \(\alpha\) and r which are solved using Newton's method. This model provides an estimate for energy. This is improved by considering r to be the average distance of an electron from the nucleus and for a distance of 2r assuming that the velocity of any electron is zero. This is the bounding constraint. A calculation of the energy for \(v=0\), a radius of 2r for one electron and a new value of \(\alpha\) from a least squares relation between r and \(\alpha\), is given. This turns out to be a very accurate estimate. Additional reasonable semiclassical assumptions give excited state values with surprising accuracy and algebraic formulae for excited state estimates.