Lower Excited States of the Helium Isoelectronic Series
The lowest excited states of the helium isoelectronic series, He, , , … , , have the electron configurations and . The energies of these states can be computed quite accurately using the simple open-shell wavefunctions [more]
where the superscripts 1 and 3 correspond to the singlet and triplet states, respectively. The orbital functions can be approximated by the orthonormalized hydrogen-like functions, with variational parameters , and :
The parameter is replaced by when the function is used for the triplet state.
The energies of the singlet and triplet states are given by
where the lowercase notation , , indicates that the triplet-parametrized orbital is used. We seek to minimize these energies, in accordance with the variational principle, by optimization of the parameters and . We take as the parameter for the -orbital.
When values of the variational parameters and are selected, the graphic shows plots of the corresponding orbitals and the radial distribution functions (RDFs) for both singlet and triplet states. Also the computed energies are plotted as dashed blue and red lines on an energy-level diagram. The solid lines show the exact values of the energies, which are lower limits in the variational calculations. Checking "optimized values" gives the best computed results for the orbitals and energies. Atomic units are used throughout, with distances in bohrs and energies in hartrees.[less]
The contributions to the energy are given by
For , with the variational parameter replacing , the corresponding integrals are denoted , and . Since the Coulomb and exchange integrals have positive values, it would appear that the triplet energy is lower than the singlet, since the exchange integral occurs with a minus sign in the former. One must, however, also consider the comparative values of the other contributing integrals. Indeed, the is lower than the in the helium isoelectronic series, being the lowest excited state above the ground state. This is true for singlet-triplet energy ordering in other atoms and molecules in about 95% of cases, in conformity with Hund's rules.
The experimental atomic energies are calculated from data in the NIST Atomic Spectra Database . (The excited singlet energies for to 10 are estimated by extrapolation.)
 S. M. Blinder, Introduction to Quantum Mechanics, 2nd ed., Cambridge, MA: Elsevier, 2021 pp. 155–156.
 A. Kramida, Y. Ralchenko, J. Reader and NIST ASD Team, "NIST Atomic Spectra Database" (Version 5.8), Gaithersburg, MD: National Institute of Standards and Technology. (Dec 7, 2020) doi:10.18434/T4W30F.