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 . 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.