# Energies of Helium Isoelectronic Series Using Perimetric Coordinates

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The most accurate computations on the ground state of the helium atom and its isoelectronic series followed from the work of Pekeris [1]. For an S-state, the wavefunction depends on just three coordinates, say , and , which can be represented as the sides of a planar triangle. The *perimetric coordinates *, , have the advantage that they automatically satisfy the triangle inequalities and each independently varies from 0 to ∞. Pekeris's original computation made use of an expansion in perimetric coordinates containing 1058 terms, leading to the essentially exact nonrelativistic ground-state energy hartrees. In this Demonstration, we introduce the use of perimetric coordinates in computations on the two-electron isoelectronic series , , , …, , corresponding to in the Hamiltonian

Contributed by: S. M. Blinder (March 2019)

Open content licensed under CC BY-NC-SA

## Snapshots

## Details

Perimetric coordinates for two-electron systems:

, , ,

, ,

,

,

,

,

,

,

.

A more detailed introduction to the quantum theory of two-electron atoms is given in [2].

References

[1] C. L. Pekeris, "Ground State of Two-Electron Atoms," *Physical Review*, 112(5), 1958 pp. 1649–1658. doi:10.1103/PhysRev.112.1649.

[2] S. M. Blinder, *Introduction to Quantum Mechanics in Chemistry, Materials Science, and Biology*, Boston: Elsevier, 2004 Chapter 8.

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