Lecture 10. Chemical Bonding. H2 Molecule

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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 Chemistry
• http//vergil.chemistry.gatech.edu/notes/quantrev
  /quantrev.html

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Born-Oppenheimer approximation
fixed
Constant
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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

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Born-Oppenheimer Approximation Potential
Energy Surface (Curve)
A
B
R
Potential energy surface
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Potential Energy Curve (1D diatomic molecule)
Potential Energy Surface (2D constrained
triatomic)
E E(R,?)
E E(R)
Sliced to make 1D curve
Sliced to make 1D curve
For molecules, in general, Potential Energy
Hypersurface (N-Dimensional) We cannot
draw it!
(R fixed or optimized)
(? fixed or optimized)
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1D Slice of Potential Energy Hypersurface
Example Torsional Energy Curve
Torsion dihedral angle (for A-B-C-D bond)
fixed or optimized
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Stationary point. Minimum
Energy minimization Geometry optimization
Energy minimum (Equilibrium structure)
A stone will roll down.
A stone will stay.
for all q
for all q
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Stationary point. Transition State
Minimum (isomer, confomer, reactant, product)
Transition state (linking two minima)
for all q
for only one q (reaction coordinate)
for other qs
Intrinsic reaction coordinate (IRC)
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Molecular Orbitals (MO) Near the equilibrium
distance, an electron delocalized over the whole
molecule.
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Overlap integral
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MO energy diagram E(R) as a function of R
unbound state antibonding
bound state bonding
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• 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

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Molecular Orbital (MO) Model LCAO-MO
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MO wave function VB wave function ionic terms
LCAO-MO model gives wrong dissociation limit.
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