Two-Parameter Variational Functions for the Helium Isoelectronic Series
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The simple variational treatment of the helium atom and, more generally, the related series of isoelectronic ions, is a standard topic in quantum chemistry texts. The Hamiltonian for helium-like systems 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 here. Unlike the one-electron Schrödinger equation, this two-electron 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 (see the Demonstration "Variational Calculations on the Helium Isoelectronic Series"). This, at least to physicists and chemists, could be considered a "proof" of the general validity of the Schrödinger equation. (In contrast, the Bohr theory gave correct energies for the hydrogen atom but failed miserably for helium and heavier atoms.)[more]
We focus in this Demonstration on much simpler variational approximations to the helium wavefunction [1–3]. The approximate energy is then given by , where the denominator enables us to use non-normalized wavefunctions. The classic first approximation is a product of scaled hydrogen-like functions (): . Optimizing the parameter so that the energy is a minimum, such that , it is found that (1.6875 for helium), giving an energy ( hartrees for helium). This is compared with the exact nonrelativistic value for the helium ground-state energy, hartrees. The first ionization potential is the energy difference with the hydrogen-like ion, which has an energy of exactly . Thus, expressed in electron volts, eV, compared to the exact value 24.592 eV.
The function first proposed by Eckert (1930) represents an "open shell" modification, in which the "inner" and "outer" orbitals are allowed to have different shielding constants. The last two functions include explicit dependence on the coordinate (Hylleraas, 1929; Baber and Hassé, 1937), which attempts to account for some of the correlation energy—the difference between the instantaneous and averaged interactions between the electrons.
The virial theorem for a Coulombic system requires that the average potential and kinetic energies, given by and , respectively, satisfy the ratio . This is true for the exact solution and serves as an additional criterion for a variational function [3, 4].
You can choose the parameters and or and that optimize the computed energy and . The results are displayed on two bar graphs, which show how close you come to the exact values.[less]
Snapshot 1: optimized one-parameter function for helium
Snapshot 2: optimized open-shell function
Snapshot 3: best approximate helium function in this Demonstration
Snapshot 4: simple result for neon; higher values of make electron repulsion less significant
Snapshot 5: simplest function for hydride ion ; implies dissociation to H +
Snapshot 6: open-shell function; shows existence of weakly-bound
 W.-K. Li, G.-D. Zhou, and T. Mak, Advanced Structural Inorganic Chemistry, Oxford: University Press, 2008, pp. 46–48.
 W.-K. Li, "Two-Parameter Wave Function for the Helium Sequence," Journal of Chemical Education, 64(2), 1987 pp. 128–129.
 W.-K. Li, "A Lesser Known One-Parameter Wave Function for the Helium Sequence and the Virial Theorem," Journal of Chemical Education, 65(11), 1988 pp. 963–964.
 K. L. Liu and W.-K. Li, "The Simple Variational Wavefunctions for the Helium Atom and the Virial Theorem," American Journal of Physics, 58(12), 1990 pp. 1202–1203.