Lecture 12. Potential Energy Surface

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Presentation transcript:

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

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

Born-Oppenheimer Approximation & Potential Energy Surface

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

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)

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

(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

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)

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