1. Lecture 12. Potential Energy Surface
References
Computational chemistry: Introduction to the theory and applications of
molecular and quantum mechanics, E. Lewars (2004) Chapter
Molecular Modeling, A. R. Leach (2nd ed. 2001), Chapter 5 (pp )
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

2. Quantum Mechanical Origins
Schrödinger equation, fundamental to everything
wave function
H = Hamiltonian operator
Time-independent form
Born-Oppenheimer approximation
Electrons relax very quickly compared to nuclear motions.
Nuclei move in presence of potential energy obtained by solving
electron distribution for fixed nuclear configuration, which is
still very difficult to do routinely.
Usually nuclei are heavy enough to treat classically.
Nuclear coordinates
Electronic coordinates

3. Born-Oppenheimer Approximation & Potential Energy Surface

4. Potential Energy Curve (1-Dimensional)B R
E = E(R)
Simplest form:
Harmonic Oscillator
Simplified

5. Potential Energy Surface (2-Dimensional)q R
E = E(R,θ)
Sliced to make
1D curve
Sliced to make
1D curve
(R fixed or optimized)
(θ fixed or optimized)
 Potential Energy Hypersurface (N-Dimensional)

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

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

9. 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 q’s
Intrinsic reaction coordinate (IRC)

10. Stationary points in PES
(Geometry optimization, Transition state search)