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5. The Li atom has three electrons. Measured values of the first and second ionization potentials are 5.3917 eV and 75.6400 eV, respectively.
(i) Calculate the exactly correct value of the third ionization potential. Hint: What kind
of species is Li+2? Give the result in atomic units (hartree).
(ii) Use the above information to determine the exact ground-state energy of Li+. Again,
give the answer in atomic units.
Next you will calculate the ground-state energy of Li+ variationally, using the trial function
psi(r1,r2) = psi1s(Z)(r1)psi1s(Z)(r2), with psi1s(Z)(r) a 1s orbital for nuclear charge Z.
(iii) Write out W(Z), the expectation value of the Li+ Hamiltonian with the trial function. Your answer will contain various integrals, which you can evaluate with information
from either of your textbooks. Cite book and page/eqn number for integrals that you use.
(iv) Plot W(Z) versus Z and determine Z and W(Z) at the minimum. Determine the
variational estimate of the ground-state energy of Li+. Compare this result to the correct
value in (ii). Do the approximate and correct energies show the expected relationship to
each other? Discuss the physical signi cance of your value of Zmin.