Chemistry 251 » Fall » Full Semester
4 Credits
Physical Chemistry I
Instructor(s): Ignacio Franco

Prerequisites: Physics 113-114 or 121-122 and Math 163 or 165.
Crosslisting: CHM 441

Course Summary:
This course is an introduction to the quantum theory of matter, with particular applications to problems of chemical interest. Our discussion of the subject of quantum chemistry will be based on the Schrödinger equation, the wave equation for matter waves. We will discuss the solutions to the Schrödinger equation for a number of important model systems, including piecewise constant potentials, the simple harmonic oscillator, the rigid rotor, and the Coulomb potential. We will apply these results to chemical bonding and atomic and molecular structure. Chemistry 251 is for undergraduates. There are weekly problem sets. Students also participate in workshops each week. Chemistry 441 is for graduate students who have not had previous coursework in quantum chemistry. Chemistry 441 students will have additional homework assignments. This course uses the Tuesday/Thursday 8:00 - 9:30 am Common Exam time.

Course Topics:
  1. Introduction, Planck distribution, necessity for quantum hypothesis.
  2. Photoelectric effect, heat capacity of solids, line spectra of atoms, Bohr theory of the atom.
  3. deBroglie waves, Davisson-Germer experiment, Heisenberg Uncertainty Principle, two-slit diffraction experiment and wave-particle duality.
  4. Mathematics of waves, wave equations, separation of variables, solving linear second-order differential equations with constant coefficients.
  5. Harmonic oscillator differential equation, clamped string: spatial, temporal solutions, normal modes.
  6. Standing waves as superposition of travelling waves, Schrödinger equation for free particle, particle in 1-D infinite square well.
  7. Quantization in the 1-D infinite square well, spectra of conjugated molecules, Born interpretation of wavefunctions, linear operators.
  8. Operators and eigenvalues, Schrödinger equation as energy eigenvalue problem, expectation values, variance, x px for particle in a 1-D square well.
  9. Postulates of quantum mechanics: maximum information in wavefunction, expectation values, observation of eigenvalues, zero variance of eigenfunctions, operators of quantum mechanics.
  10. Wavefunction not an eigenstate of 1-D square well, time-independent Schrödinger equation, stationary states, superposition states.
  11. Hermitian operators: eigenvalues real, eigenfunctions are orthogonal, complete. Projections of wavefunction onto basis functions.
  12. Completeness, orthogonal expansions, Fourier series: resolution into components; probability of measuring an eigenvalue in terms of Fourier coefficients.
  13. Commuting observables, simultaneous eigenfunctions, Schwartz inequality and Uncertainty Principle.
  14. Relationships with commutators, time dependence of expectation values, Ehrenfest's Theorem, classical harmonic oscillator.
  15. Relative coordinates, Taylor's series expansion of real potentials. Schrödinger equation for harmonic oscillator in reduced coordinates.
  16. Asymptotic form for harmonic oscillator wavefunctions. Power series solution to Hermite differential equation.
  17. Two-term recursion relations and termination of power series, quantized energy levels.
  18. Hermite polynomials, parity, comparison with 1-D particle in a box wavefunctions.
  19. Classically forbidden motion, 3-D systems, separability of Hamiltonian, wavefunction, energy.
  20. Spherical polar coordinates, rigid rotor, molecular bond lengths.
  21. Legendre polynomials, associated Legendre functions, angular momentum commutation relations, eigenfunctions of z-component of angular momentum.
  22. Physical significance of m-quantum number. Vector model, space quantization, introduction to the hydrogen atom.
  23. Radial equation for the hydrogen atom, Laguerre, associated Laguerre polynomials, radial wavefunctions.
  24. Radial functions: functional forms and graphs. Angular functions for p-, d- orbitals. Hydrogen atom in a magnetic field.
  25. Approximate methods: first order perturbation theory, corrections to the energy. Introduction to the Variation Theorem.
  26. Proof of Variation Theorem; Gaussian approximation to the hydrogen atom ground state.
  27. Linear variation method. secular determinant and secular equation.
  28. Atoms: atomic units. Perturbation approach to the helium atom. Variation theorem and effective nuclear charge. Slater-type orbitals. Self-Consistent Field Method.
  29. Hartree, Hartree-Fock method. Electron correlation. Electron spin, Pauli Exclusion Principle, Slater determinant applied to helium atom.
  30. Slater determinants for N-electron systems. Coulomb, exchange integrals, Koopman's theorem. Term symbols.
  31. Examples of term symbols. Spin-orbit coupling, atomic spectroscopy. Born-Oppenheimer approximation.
  32. Heitler-London (Valence Bond) method. Chemical bond arising from exchange integral.
  33. Electron spin and the hydrogen molecule. Introduction to the LCAO-MO method.
  34. MO theory for second row homonuclear diatomics; molecular term symbols.
  35. Semiclassical radiation theory: time-dependent perturbation theory, transition dipole.
  36. The electromagnetic spectrum: pure vibrational and rotational spectroscopy. Boltzmann distribution for initial state population.
  37. Vibrational-rotational spectroscopy: centrifugal distortion and vibration-rotation interaction.
  38. Polyatomic vibrations: degrees of freedom and normal coordinates.
  39. Electronic transitions. Franck-Condon Principle.

Required Text:
Donald A. McQuarrie, Quantum Chemistry, Second Edition, University Science Books, 2008. ISBN 978-1-891389-50-4.