Energies of Helium Isoelectronic Series Using Perimetric Coordinates

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
.
The wavefunctions considered by Pekeris were expansions in the form
.
We consider a much more modest version with
,
where , , can be chosen such as to minimize the variational integral
.
The optimized results can be obtained directly by checking "show optimized values."
The corresponding ionization energies of each atom is also shown, given by , where is the ground-state energy of the single-electron ion.
The atomic energies and ionization energies are represented on bar graphs, with the exact nonrelativistic values written in for reference. The variationally determined energy must, of necessity, be higher (less negative) than the exact value.

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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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