Variational methods: the helium atom

2.3 Variational methods: the helium atom

Summary: The most famous example of the variational principle is the ground state of the two-electron helium atom.

If we could switch oﬀ the interactions between the electrons, we would know what the ground state of the helium atom would be: $Ψ (r_{1}, r_{2}) = ϕ_{100}^{Z = 2} (r_{1}) ϕ_{100}^{Z = 2} (r_{2})$ , where $ϕ_{n l m}^{Z}$ is a single-particle wave function of the hydrogenic atom with nuclear charge $Z$ . For the ground state $n = 1$ and $l = m = 0$ (spherical symmetry). The energy of the two electrons would be $2 Z^{2} E_{Ry} = - 108.8$ eV. But the experimental energy is only $- 78.6$ eV (ie it takes $78.6$ eV to fully ionise neutral helium). The diﬀerence is obviously due to the fact that the electrons repel one another.

The full Hamiltonian (ignoring the motion of the proton - a good approximation for the accuracy to which we will be working) is

- \frac{ℏ^{2}}{2 m} (\nabla_{1}^{2} + \nabla_{2}^{2}) - 2 ℏ c α (\frac{1}{|r_{1}|} + \frac{1}{|r_{2}|}) + ℏ c α \frac{1}{|r_{1} - r_{2}|}

where $\nabla_{1}^{2}$ involves diﬀerentiation with respect to the components of $r_{1}$ , and $α = e^{2} ∕ (4 π ε_{0} ℏ c) \approx 1 ∕ 137$ . (See here for a note on units in EM.)

A really simple guess at a trial wave function for this problem would just be $Ψ (r_{1}, r_{2})$ as written above. The expectation value of the repulsive interaction term is $(5 Z ∕ 4) E_{Ry}$ giving a total energy of $- 74.8$ eV. (Gasiorowicz demonstrates the integral, as do Fitzpatrick and Branson.)

It turns out we can do even better if we use the atomic number $Z$ in the wave function $Ψ$ as a variationalparameter (that in the Hamiltonian, of course, must be left at 2). The best value turns out to be $Z = 27 ∕ 16$ and that gives a better upper bound of $- 77.5$ eV – just slightly higher than the experimental value. (Watch the sign – we get an lower bound for the ionization energy.) This eﬀective nuclear charge of less than 2 presumably reﬂects the fact that to some extent each electron screens the nuclear charge from the other.

Gasiorowicz ch 14.2, 14.4 (most complete)
Mandl ch 7.2, 8.8.2 (clearest)
Shankar ch 16.1 (not very clear; misprints in early printings)

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