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 Chemistryhttp://vergil.chemistry.gatech.edu/notes/quantrev/quantrev.html

fixed

Born-Oppenheimer approximation

Constant

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 positions2) Energy curve of the molecule as a function of nuclei

position

Focus on the electronic Schrödinger equation

Born-Oppenheimer Approximation &Potential Energy Surface (Curve)

Potential energy surface

A B

R

Potential Energy Curve (1D = diatomic molecule)

A B

R

E = E(R)

Potential Energy Surface (2D = constrained triatomic)

R R E = E(R,θ)

For molecules, in general, Potential Energy “Hypersurface” (N-Dimensional)

– We cannot draw it!

(R fixed or optimized) (θ fixed or optimized)

Sliced to make 1D curve

Sliced to make 1D curve

1D Slice of Potential Energy Hypersurface

Example: Torsional Energy Curve

Torsion: dihedral angle (for A-B-C-D bond)

fixed or optimized

Stationary point. Minimum

Energy minimizationGeometry optimization

Energy minimum(Equilibrium structure)

for all q

for all q

A stone will roll down.

A stone will stay.

Intrinsic reaction coordinate (IRC)

* Minimum (isomer, confomer, reactant, product)

Transition state (linking two minima)

for all q

for other q’s

for only one q (reaction coordinate)

Stationary point. Transition State

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Molecular Orbitals (MO)Near the equilibrium distance, an electron delocalized over the whole molecule.

Overlap integral

MO energy diagram: E(R) as a function of R

unbound state: antibonding

bound state: bonding

•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

Molecular Orbital (MO) Model – LCAO-MO

LCAO-MO model gives wrong dissociation limit.

MO wave function = VB wave function + ionic terms