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.

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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.