Discussion
By applying variation method, the ground state energy of the electron in hydrogen atom is (in a.u.) :
Answer: B
The variational ground-state energy of hydrogen is −0.5 a.u.
With the trial function e−αr, the energy expectation value in atomic units is a function of α that is minimised as follows:
\[ \begin{aligned} E(\alpha) &= \frac{\alpha^{2}}{2} - \alpha \\ \frac{dE}{d\alpha} &= \alpha - 1 = 0 \Rightarrow \alpha = 1 \\ E &= \frac{1}{2} - 1 = -0.5\ \text{hartree} \end{aligned} \]
This equals the exact energy, −13.6 eV, because the trial function has the form of the true 1s function. −2 a.u. is the energy of He+, −2.75 a.u. is the simple variational estimate for helium with an unscreened charge, and 0 is the ionisation limit.
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