## ❓ Questions about Many-Electron Systems and their Approximation.
1. What are bosons? What are fermions? How do the wavefunctions differ?
1. Write the Schrödinger equation for a many-electron molecule in the Born-Oppenheimer approximation.
1. How does one use the secular equation to approximate the wavefunction for a many-electron system?
1. Write a Slater determinant for the lowest-energy pentuplet state of the Carbon atom, with electron configuration $1s^2 2s^1 2p^3$. Assume all unpaired electrons have $\alpha$ spin.
1. By explicit evaluation, confirm that the 3x3 Slater determinant recovers the result that would be obtained by explicit antisymmetrization.
1. Show that the spin-angular-momentum operators commute with the molecular Hamiltonian? What is the implication of that in terms of the observability of a molecule’s spin.
1. What is the key difference between molecular-orbital and valence-bond theory? Explain one situation where molecular-orbital theory is preferred and one situation where valence-bond theory is preferred.


## ❓ Questions about Many-Electron Atoms and Molecules.
1. Solve the Schrödinger equation for a muon bound to Neon nucleus.
1. How does the energy of a one-electron atom decrease as the atomic number increases?
1. The ground-state term symbol for Samarium (Sm, atomic number 62) is ${}^7F$. (a) How many nominally degenerate spatial and spin states are associated with this term symbol? (b) In addition to J=0, what other J values are possible for this term symbol?
1. Write an expression for the lowest-energy molecular orbital of $\text{H}_3^{+2}$.
1. What is the most probable distance to find an electron from the nucleus in the 4f orbital of a hydrogen atom?
1. What is the expectation value of $r^{-1}$ for Hydrogenic Atoms
1. Write a slater determinant for the excited-state configuration of the Beryllium atom: $1s^2 2s^1 3s^1$.
1. Write expressions for the energies of the harmonic oscillator using creation and annihilation operators.
1. What are the term symbols for the Cobalt atom; $[Ar]4s^2 3d^7$. What is the ground-state term predicted by Hund's rules?
1. What are the key approximations implicit in Hückel theory?

## ❓ Questions about Molecular Spectroscopy
1. Write down and interpret Fermi's Golden Rule.
1. What are spectroscopic selection rules? How can one tell if an excitation is electric-dipole allowed?
1. Explain the long-wavelength approximation? What corrections appear when the long-wavelength approximation is not accurate?
1. Explain the long-time approximation? What corrections appear when the long-time approximation is not accurate?
1. Explain the weak-field approximation? What corrections appear when the weak-field approximation is not accurate?
1. What are electric-dipole selection rules for atoms?
1. What are electric-dipole selection rules for the harmonic oscillator?
1. What are electric-dipole selection rules for the rigid rotor?
1. What ar electric-dipole selection rules for the particle in a box?
1. Given two orbitals, show how to determine whether a transition between them is an electric-dipole-allowed transition.