Lecture 10. Chemical Bonding. H2 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 Chemistryhttp://vergil.chemistry.gatech.edu/notes/quantrev/quantrev.html
fixed
Born-Oppenheimer approximation
Constant
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 positions2) Energy curve of the molecule as a function of nuclei
position
Focus on the electronic Schrödinger equation
Born-Oppenheimer Approximation &Potential Energy Surface (Curve)
Potential energy surface
A B
R
Potential Energy Curve (1D = diatomic molecule)
A B
R
E = E(R)
Potential Energy Surface (2D = constrained triatomic)
R R 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
1D Slice of Potential Energy Hypersurface
Example: Torsional Energy Curve
Torsion: dihedral angle (for A-B-C-D bond)
fixed or optimized
Stationary point. Minimum
Energy minimizationGeometry optimization
Energy minimum(Equilibrium structure)
for all q
for all q
A stone will roll down.
A stone will stay.
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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Molecular Orbitals (MO)Near the equilibrium distance, an electron delocalized over the whole molecule.
Overlap integral
MO energy diagram: E(R) as a function of R
unbound state: antibonding
bound state: bonding
•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
Molecular Orbital (MO) Model – LCAO-MO
LCAO-MO model gives wrong dissociation limit.
MO wave function = VB wave function + ionic terms