❓ Questions about Many-Electron Systems and their Approximation.

❓ Questions about Many-Electron Systems and their Approximation.#

  1. What are bosons? What are fermions? How do the wavefunctions differ?

  2. Write the Schrödinger equation for a many-electron molecule in the Born-Oppenheimer approximation.

  3. How does one use the secular equation to approximate the wavefunction for a many-electron system?

  4. 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.

  5. By explicit evaluation, confirm that the 3x3 Slater determinant recovers the result that would be obtained by explicit antisymmetrization.

  6. 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.

  7. 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.

  2. How does the energy of a one-electron atom decrease as the atomic number increases?

  3. 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?

  4. Write an expression for the lowest-energy molecular orbital of \(\text{H}_3^{+2}\).

  5. What is the most probable distance to find an electron from the nucleus in the 4f orbital of a hydrogen atom?

  6. What is the expectation value of \(r^{-1}\) for Hydrogenic Atoms

  7. Write a slater determinant for the excited-state configuration of the Beryllium atom: \(1s^2 2s^1 3s^1\).

  8. Write expressions for the energies of the harmonic oscillator using creation and annihilation operators.

  9. 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?

  10. What are the key approximations implicit in Hückel theory?

❓ Questions about Molecular Spectroscopy#

  1. Write down and interpret Fermi’s Golden Rule.

  2. What are spectroscopic selection rules? How can one tell if an excitation is electric-dipole allowed?

  3. Explain the long-wavelength approximation? What corrections appear when the long-wavelength approximation is not accurate?

  4. Explain the long-time approximation? What corrections appear when the long-time approximation is not accurate?

  5. Explain the weak-field approximation? What corrections appear when the weak-field approximation is not accurate?

  6. What are electric-dipole selection rules for atoms?

  7. What are electric-dipole selection rules for the harmonic oscillator?

  8. What are electric-dipole selection rules for the rigid rotor?

  9. What ar electric-dipole selection rules for the particle in a box?

  10. Given two orbitals, show how to determine whether a transition between them is an electric-dipole-allowed transition.

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