1. Lecture 10. Chemical Bonding. H 2 Molecule References Engel, Ch. 12 Ratner & Schatz, Ch. 10 Molecular Quantum Mechanics, Atkins & Friedman (4th ed. 2005), Ch.8 Computational Chemistry, Lewars (2003), Ch.4 A Brief Review of Elementary Quantum Chemistry http://vergil.chemistry.gatech.edu/notes/quantrev/quantrev.html

2. fixed Born-Oppenheimer approximation Constant

3. Born-Oppenheimer Approximation Simplifies further the Schrödinger equation (separation of variables) Difference in the time scales of nuclear and electronic motions Nuclei are much heavier (~1800 times) and slower than electrons. Electrons can be treated as moving in the field of fixed nuclei. A full Schrödinger equation for a molecule can be solved in two steps: 1) Motion of electron around the nuclei at fixed positions 2) Energy curve of the molecule as a function of nuclei position Focus on the electronic Schrödinger equation

4. Born-Oppenheimer Approximation & Potential Energy Surface (Curve) Potential energy surface A B R

5. Potential Energy Curve (1D = diatomic molecule) A B R E = E(R) Potential Energy Surface (2D = constrained triatomic)  RR E = E(R,θ) For molecules, in general, Potential Energy “Hypersurface” (N-Dimensional) – We cannot draw it! (R fixed or optimized)(θ fixed or optimized) Sliced to make 1D curve Sliced to make 1D curve

6. 1D Slice of Potential Energy Hypersurface Example: Torsional Energy Curve Torsion: dihedral angle (for A-B-C-D bond) fixed or optimized

8. Stationary point. Minimum Energy minimization Geometry optimization Energy minimum (Equilibrium structure) for all q A stone will roll down. A stone will stay.

9. Intrinsic reaction coordinate (IRC) * Minimum (isomer, confomer, reactant, product) Transition state (linking two minima) for all q for other q’s for only one q (reaction coordinate) Stationary point. Transition State

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13. Molecular Orbitals (MO) Near the equilibrium distance, an electron delocalized over the whole molecule.

15. Overlap integral

19. MO energy diagram: E(R) as a function of R unbound state: antibonding bound state: bonding

22. Buildup of electron charge around protons & between protons Decrease of charge outside of bonding region Decrease of electron charge around protons & between protons Increase of charge outside of bonding region

26. Molecular Orbital (MO) Model – LCAO-MO

27. LCAO-MO model gives wrong dissociation limit. MO wave function = VB wave function + ionic terms