Helium Atom - The Variational Method

The Variational Method

To obtain a more accurate energy the variational principle can be applied to the electron-electron potential V_ee using the wave function

:

After integrating this, the result is:

This is closer to the theoretical value, but if a better trial wave function is used, an even more accurate answer could be obtained. An ideal wave function would be one that doesn't ignore the influence of the other electron. In other words, each electron represents a cloud of negative charge which somewhat shields the nucleus so that the other electron actually sees an effective nuclear charge Z that is less than 2. A wave function of this type is given by:

Treating Z as a variational parameter to minimize H. The Hamiltonian using the wave function above is given by:

After calculating the expectation value of and V_ee the expectation value of the Hamiltonian becomes:

The minimum value of Z needs to be calculated, so taking a derivative with respect to Z and setting the equation to 0 will give the minimum value of Z:

This shows that the other electron somewhat shields the nucleus reducing the effective charge from 2 to 1.69. So we obtain the most accurate result yet:

Where again, E₁ represents the ionization energy of hydrogen.

By using more complicated/accurate wave functions, the ground state energy of helium has been calculated closer and closer to the experimental value -78.95 eV. The variational approach has been refined to very high accuracy for a comprehensive regime of quantum states by G.W.F. Drake and co-workers as well as J.D. Morgan III, Jonathan Baker and Robert Hill using Hylleraas or Frankowski-Pekeris basis functions. It should be noted that one needs to include relativistic and quantum electrodynamic corrections to get full agreement with experiment to spectroscopic accuracy.