Final Review Questions

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This is not comprehensive, but gives a flavor of the sort of questions that students are expected to answer on Exam 2. More questions are forthcoming. No solutions will be given.

Go over the Exam1 and Exam2 overviews too. Below is a topical overview of material to study.

Model Systems

How to calculate energy levels of various simple quantum systems:
- Free particle
- Particle in a Box
- Harmonic Oscillator
- Rigid Rotor
- Hydrogen atom
the degeneracies of the energy levels of various simple quantum systems
calculate the numbers of vibrational and of rotational degrees of freedom of a molecule

Vibrational spectroscopy

how to calculate a reduced mass for a diatomic molecule
the meaning of the force constant and its relation to the fundamental vibrationalfrequency•
the meaning of transition dipole moment
the selection rule for a harmonic oscillator, i.e., which transitions are allowed and which are forbidden
the various meanings of the term ν as used in discussing vibrational spectroscopy

Rotational Spectroscopy

the degeneracies of the energy levels
the general expression for the moment of inertia in the reference frame of the principal axes.

Electronic wave functions for one-electron (H, He⁺, Li²⁺ etc.) and many-electron (He, Li, Be...etc.) atoms.

Apply the five basic postulates of quantum mechanics (wavefunctions and probability; operators and observables; results of an individual measurement; average values and superposition) to atoms and ions
calculate the average and the most probable distance between the electron and the nucleus in a one-electron atom
work with angular momentum operators (both orbital and spin angular momentum ) and their eigenfunctions
apply the Pauli principle
write the electronic Hamiltonian for an at om or ion
locate radial and angular nodes in electronic wave functions

Electronic wavefunctions of simple molecules

Write a full Hamiltonian for a simple diatomic molecule
Apply the Born-Oppenheimer approximation
Write approximate ground state MO (LCAO) wavefunctions
Normalize MO wavefunctions
MO configurations of simple diatomic molecules
Write electron configurations for simple diatomic molecules
Estimate bond orders
Identify and sketch the HOMOs and LUMOs in the LCAO approximation

Misc. Questions

Sketch the MO filling diagram for homonuclear diatomic molecules
Give the bond order for each of the following molecules and indicate whether each one is paramagnetic: \(He_2^+\), \(Li_2\), \(Be_2\), \(C_2^{-1}\), and \(N_2^+\)
The J=2→ 3 rotational transition for the ground vibrational state of ³⁹K³⁷Cl occurs at ~2×10⁴ MHz.
1. What is the reduced mass ofthis molecule in kg?
2. Write a general expression for the rotational energy levels of a rigid rotor.
3. If ³⁹K³⁷Cl is a rigid rotor, predictthe frequency of the J=0→1 transition.
4. Would corrections for the non-rigidity of ³⁹K³⁷Cl increase, decrease, or leave unaltered your result in part C? Explain.
What are selection rules and why are they important?