Lecture 12. Potential Energy Surface

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Lecture 12. Potential Energy Surface
References
Computational chemistry Introduction to the
theory and applications of molecular and
quantum mechanics, E. Lewars (2004) Chapter
2.4-2.5 Molecular Modeling, A. R. Leach (2nd
ed. 2001), Chapter 5 (pp.250-273) Essentials of
computational chemistry. Theories and Models, C.
J. Cramer, (2nd Ed. Wiley, 2004) Chapter 2.4
Introduction to Computational Chemistry, F.
Jensen (1999), Chapter 14
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For molecules, Born-Oppenheimer approximation
(1927) (Review)

• Simplifies the Schrödinger equation for molecules
  (allows 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
• Allows to focus on the electronic Schrödinger
  equation

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H2, the simplest (one-electron) molecule (Review)
Born-Oppenheimer approximation
fixed
Constant
0
0
Constant
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Born-Oppenheimer approximation Potential
energy surface (curve)
E E(R)
A
B
R
Potential energy surface
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Potential Energy Curve (1-Dimensional)
E E(R)
Simplest form Harmonic Oscillator
Simplified
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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, N-dimensional
potential energy hypersurfaces ? We
cannot plot 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. 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)