🤔 Thought-Provoking Questions

1. By drawing a diagram with sequential Stern-Gerlach apparatuses, explain how non-commutation of angular momentum operators around various axes works? Can you think of an example where the apparatus commutes instead?

2. What are good quantum numbers for atoms (in the absence of spin-orbit interactions/relativistic corrections)? How do they relate to term symbols?

3. What is the effective nuclear charge felt by an electron close to the nucleus in a many-electron atom? What about far away from the nucleus? What happens in a specific example like, say, the Argon dication?

4. Write a Slater determinant wavefunction that is an angular-momentum and spin-angular-momentum eigenfunction for the \({}^4 F\) term (\(\text{[Ar]}4s^23d^3\) configuration (Vanadium)). The \({}^6 D\) term is only .0096 a.u. above this state. Why?

5. Explain why the Hartree-Fock method is related to “optimizing the effective nuclear charge of each orbital in a Slater determinant via the variational principle?” (This is rigorous for an atomic orbital without nodes; for an atomic orbital with nodes you can still make a hand-waving argument.)

6. What are the possible \(L\) and \(S\) values for a \(\text{[closed shell]}s^1 p^1 d^1\) electron configuration? What is the ground-state term symbol (assuming Hund’s rules hold) and what are the \(J\) values associated with it?

7. Consider a Boron atom in its ground-state electron configuration. What would be the “spot pattern” if a beam of ground-state Boron atoms were sent through a Stern-Gerlach apparatus with a magnetic field gradient along the \(z\)-axis?

8. Explain why a Slater determinant encodes the Pauli exclusion principles that electrons with the same spin cannot be at the same position and that no two electrons can occupy the same spin orbital.

9. Write down the electronic and nuclear Hamiltonians for a molecule and explain each term. Under what circumstances can one solve these equations “exactly?”

10. Explain the Born-Oppenheimer approximation. What are its assumptions? When does it break down?

11. Why are there \(3N-6\) vibrational modes for a non-linear molecule with \(N\) atoms but only \(3N-5\) vibrational modes for a linear molecule with \(N\) atoms?

12. What key assumptions are made when one treats molecular rotation and vibration with the rigid-rotor harmonic-oscillator approximation? What types of molecules do you expect to be best/worst treated thereby?

13. What are the rotational eigenfunctions and eigenvalues for a rigid linear molecule?

14. Why does one use mass-weighted coordinates when solving the nuclear Schrodinger equation (for vibrations) in a polyatomic molecule?

15. Why can the molecular orbitals of a linear molecule be labelled with \(\sigma\), \(\pi\), \(\delta\), etc. symmetry labels even when the molecule isn’t a diatomic? Under what circumstances can the \(u\) and \(g\) labels also be used? What are the degeneracies of these orbitals?

16. The (2,2,6,6-Tetramethylpiperidin-1-yl)oxyl (TEMPO) radical is quite stable and resistant to radical recombination reactions. Write a reasonable wavefunction for the dimer of this radical, taking inspiration from the molecular-orbital and valence-bond models. Which do you expect to be a better description of the dimer? Why?

17. Name three key approximations that are implicit in Hückel theory.

18. Write down Fermi’s Golden Rule and explain its interpretation.

19. Give an example of an effect that is observed only when the “weak field approximation” implicit in Fermi’s Golden Rule breaks down.

20. Give an example of an effect that is observed only when the “long wavelength approximation” implicit in Fermi’s Golden Rule breaks down.

21. Given an example of an effect that is observed only when the “long time approximation” implicit in Fermi’s Golden Rule breaks down.

22. Why are charge-transfer transitions relatively weak (low intensity)?