Foundations of Chemical Kinetics Lecture 10: Introduction to - - PowerPoint PPT Presentation

Foundations of Chemical Kinetics Lecture 10: Introduction to potential energy surfaces

Marc R. Roussel Department of Chemistry and Biochemistry

Born-Oppenheimer revisited

◮ In an earlier lecture, we discussed the Born-Oppenheimer

approximation which allows us to compute an effective potential (i.e. effective forces) acting on the nuclei as a function of nuclear positions.

◮ Types of interactions:

Bonded strong, roughly parabolic potentials Nonbonded ⇒ intermolecular forces

◮ A variety of functional forms ◮ May depend on both the distance and relative

• rientations of the molecules

Examples of intermolecular forces

Force V (R) Coulomb q1q2 4πǫ0R Ion-dipole −q1µ2 cos θ 4πǫ0R2 µi: dipole moment Dipole-dipole µ1µ2 4πǫ0R3 (sin θ1 sin θ2 − 2 cos θ1 cos θ2) Dipole-induced dipole −µ2

1α2(3 cos2 θ + 1)

8πǫ0R6 αi: polarizability Dispersion − 3I1I2α1α2 2R6(I1 + I2) Ii: ionization energy

Recall: effective potential for a diatomic

• 8
• 6
• 4
• 2

2 4 6 1 2 3 4 5 6 7 8 E R Effective potential

Potential energy surfaces

◮ Consider a three-atom AB + C → A + BC reaction. ◮ The potential energy surface (PES) is a function of

3N − 6 = 3 coordinates, which can be taken to be RAB, RBC and the A-B-C angle.

◮ This is hard to visualize. ◮ Solution: Vary (e.g.) RAB and RBC at fixed angle. ◮ Commonly, we look at a collinear collision, but other angles

are possible.

A potential energy surface for a collinear reaction

A potential energy surface for ∠ABC = π/4

Reaction coordinate

◮ Although somewhat artificial, we can construct a reaction

coordinate that measures the progress along the lowest-energy reaction path from reactants to products.

◮ The maximum point along this path is a saddle point on the

PES (downhill in either direction along the reaction coordinate, uphill in all other directions).

◮ We have 3N − 6 internal degrees of freedom, of which one is

the reaction coordinate, so there are 3N − 7 vibrational modes.

◮ For the vibrational modes, roughly speaking,

ki = ∂2V /∂q2

i > 0, where qi is the corresponding

normal-mode coordinate, and ωi ∼ √ki.

◮ For the reaction coordinate, ∂2V /∂q2

i < 0 so the

corresponding frequency is imaginary.

Avoided crossings

◮ In diatomics, potential energy curves for electronic states with

the same orbital symmetry and spin do not cross.

E R

◮ In polyatomics, the potential energy curves can touch, but not

cross.

◮ In the region of these avoided crossings, systems can cross

from one electronic state to another. Then, a reaction can involve multiple PESs.