Bohr orbit theory revisited. II. Energies for 1 S , 2 P , 3 D , and 4 F states of helium

A modified Bohr orbit procedure is used to calculate the energies for the 1S ground state and the 2 P , 3 D , and 4 F excited states of the helium atom. The energies are calculated from \documentclass{article}\pagestyle{empty}\begin{document}$ {{\int\limits_0^\pi {E\left(\Phi \right)P\left(\Phi \right)d\Phi } } \mathord{\left/ {\vphantom {{\int\limits_0^\pi {E\left(\Phi \right)P\left(\Phi \right)d\Phi }} {\int\limits_0^\pi {P\left(\Phi \right)d\Phi } }}} \right. \kern-\nulldelimiterspace} {\int\limits_0^\pi {P\left(\Phi \right)d\Phi } }} $\end{document} , in which E (ϕ) is the Bohr orbit energy for angle ϕ between the position vectors \documentclass{article}\pagestyle{empty}\begin{document}$ \vec r_1 $\end{document} r 1 and \documentclass{article}\pagestyle{empty}\begin{document}$ \vec r_2 $\end{document} r 2 , and P (ϕ) is a probability function for this angle. Numerical procedures are used to evaluate the integrals. Energies that range between −2.9082 and −2.9054 au are calculated for the 1S state (cf. −2.9037 au, exact). The Bohr energies for the excited states are −2.1318, −2.1240, −2.0562, −2.0555, −2.0314, and −2.0312 au, which are generally close to the exact energies of −2.1332, −2.1239, −2.0557, −2.0557, −2.0313, and −2.0313 au for the 2 3 P , 2 1 P , 3 3 D , 3 1 D, 4 3 F and 4 1 F states. Some relationships that exist between the Bohr theory and the Schrödinger local energies are discussed. Approximate Bohr orbit estimates for the energies of the 2 P states of He, Li + ,…,Ne 8+ are reported. The invariance of the two‐electron Bohr hamiltonian with respect to the interchange of the electron coordinates leads to two classical probability functions when the orbit quantum numbers for the two electrons differ.