Discussion
Following trial wavefunctions
\(\phi_{1} = e^{-Z'(r_{1}+r_{2})}\) and \(\phi_{2} = e^{-Z'(r_{1}+r_{2})}(1 + g|\vec{r}_{1} - \vec{r}_{2}|)\)
are used to get a variational estimate of the ground state energy of the helium atom. \(Z'\) and \(g\) are the variational parameters, \(\vec{r}_{1}\) and \(\vec{r}_{2}\) are the position vectors of the electrons. Let \(E_{0}\) be the exact ground state energy of the helium atom. \(E_{1}\) and \(E_{2}\) are the variational estimates of the ground state energy of the helium atom corresponding to \(\phi_{1}\) and \(\phi_{2}\), respectively. Which one of the following options is true?
Answer: D
The correct option is \(E_{1} \ge E_{0},\ E_{2} \ge E_{0},\ E_{1} \ge E_{2}\).
The variational principle says that the expectation value of the Hamiltonian in any trial state is never below the true ground state energy, so both estimates satisfy E1 ≥ E0 and E2 ≥ E0.
The trial function φ2 contains φ1 as the special case g = 0, and adds a factor depending on the electron separation that lets the wavefunction keep the electrons apart. Minimizing over a larger family can only lower (or keep) the minimum, so E2 ≤ E1: the Hylleraas-type function gives a better estimate.
Every option with E1 ≤ E0 or E2 ≤ E0 contradicts the variational principle.
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