91Ó°ÊÓ

Take a trial function for the helium atom as \(\psi=\) \(\psi(1) \psi(2),\) with \(\psi(1)=\left(\zeta^{3} / \pi\right)^{1 / 2} \mathrm{e}^{-\zeta r_{1}}\) and \(\psi(2)=\left(\zeta^{3} / \pi\right)^{1 / 2} \mathrm{e}^{-\zeta_{2}}, \zeta\) being a parameter, and find the best ground-state energy for a function of this form, and the corresponding value of \(\zeta\). Calculate the first and second ionization energies. Hint. Use the variation theorem. All the integrals are standard; the electron repulsion term is calculated in Example 7.2 Interpret \(Z\) in terms of a shielding constant. The experimental ionization energies are \(24.58 \mathrm{eV}\) and \(54.40 \mathrm{eV}\).

An excited state of atomic calcium has the electron configuration \(1 \mathrm{s}^{2} 2 \mathrm{s}^{2} 2 \mathrm{p}^{6} 3 \mathrm{s}^{2} 3 \mathrm{p}^{6} 3 \mathrm{d}^{1} 4 \mathrm{f}^{1}\) (a) Derive all the term symbols (with the appropriate specifications of \(S, L \text { and } J)\) for the electron configuration. (b) Which term symbol corresponds to the lowest energy of this electron configuration? (c) Consider a \(^{3} \mathrm{F}_{2}\) level of calcium derived from a different electron configuration than that shown above. Which of the term symbols determined in part (a) can participate in spectroscopic transitions to this \(^{3} \mathrm{F}_{2}\) level?