# Lower Excited States of the Helium Isoelectronic Series

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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 [1]

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Contributed by: S. M. Blinder (January 2021)

Open content licensed under CC BY-NC-SA

## Snapshots

## Details

The contributions to the energy are given by

, ,

(Coulomb integral),

(exchange integral).

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 [2]. (The excited singlet energies for to 10 are estimated by extrapolation.)

References

[1] S. M. Blinder,* Introduction to Quantum Mechanics*, 2nd ed., Cambridge, MA: Elsevier, 2021 pp. 155–156.

[2] 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.

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