Variational Calculations on the Helium Isoelectronic Series
The Schrödinger equation for a helium-like (two-electron) atom or ion is given by , expressed in atomic units , assuming infinite nuclear mass and neglecting relativistic corrections. Atomic numbers from 1 () and 2 () through 10 () are considered. Unlike the one-electron Schrödinger equation, this problem cannot be solved analytically. E. A. Hylleraas, around 1930, carried out variational calculations giving ground state energies and ionization potentials in essential agreement with experimental results. This, at least to physicists and chemists, could be considered a "proof" of the general validity of the Schrödinger equation. (By contrast, the Bohr theory gave correct energies for the hydrogen atom but failed miserably for helium and heavier atoms.) For spherically symmetrical states, the helium problem reduces to just three independent variables. Hylleraas introduced the variables , , and considered linear variational functions of the form , with a common choice for the exponential parameter. For basis functions, the energies are obtained by solution of an secular equation. This Demonstration can handle a maximum of 10 basis functions, actually going beyond Hylleraas' original capability. You can vary the exponential parameter and number of basis functions for given to compute the ground state energy in Hartree atomic units and the first ionization potential (IP) in electron volts. These are compared with the most accurate "exact" nonrelativistic results, shown in parentheses. The logo showing two electrons in Bohr orbits is for decorative purposes only.
Snapshot 1: the simplest one-parameter variational approximation for the helium atom
Snapshot 2: to demonstrate the stability of the hydride ion (energy less than -.5 hartree) requires at least three basis functions
Snapshot 3: for higher atomic numbers, such as , simple functions give relatively accurate energies
Reference: H. A. Bethe and E. E. Salpeter, Quantum Mechanics of One- and Two-Electron Atoms, Berlin: Springer-Verlag, 1957 pp. 146–154.