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.
1 answer