Lecture 9Lecture 9

Symmetry III: Molecular Orbital Theory

Reading: Shriver and Atkins 2.7-2.9 and g6.6-6.7, 6.10

The orbitals of molecules

H H

• The electron energy in each H atom is -13.6 eV below vacuum.

• What happens to the energy levels as the H-atoms approach

each other? each other?

• Technique: Solve Schrodinger equation, using the potential q g q g p

energy for the coupled electrons.

Molecular Force

•Net Force FN=FA+FR

E ilib i h F 0•Equilibrium when FN=0•r0=bond length

Potential energy, V(r) •V0: bond energy or cohesive 0 gyenergy (energy required to separate the two atoms)

V

VR • In general: mn rB

rArV )(

V0

VA

Bonding & Antibonding orbitals

Two H atoms, both in their 1s state.As they approach, their wavefunctions begin to overlap.

Bonding & Antibonding orbitals

Formation of molecular orbitals - bonding and antibonding ( and *) when two H atomsapproach each other. The two electrons pair their spins and occupy the bonding orbital .

H-H bond: Electron probability distribution

(a) Electron probability distributions for bonding and antibonding orbitals, and *.

(b) Lines representing contours of constant probability (darker lines represent greater relative probability).

Linear combination of atomic orbitals

Two atomic orbitals 1s on atoms A and B can be combined linearly in two different ways to generate two separate

molecular orbitals and *

and * generated from a li bi i f i bi l (LCAO)linear combination of atomic orbitals (LCAO)

Wavefunction around BWavefunction around A

)()( 11 BsAs rr )()( 11 BsAs

)()( 11* BsAs rr

)()( 11 BsAs

Energies using schrodinger’s equation

Energy of and * found using the time-independent Schrodinger equation (TISE) vs. the interatomic separation R.

Molecular orbital energy level diagram

antibondingantibonding

Bonding 11.4eV

bonding

genergy (dissociation energy) =

.4eV

g energy) = 4.5eV

H atom H atomH2 molecule

He-He bond

antibonding

b dbonding

Two He atoms have four electrons. When He atoms come together, two of the electrons enter the E level and two the E* level, so the overall energy is greater than two isolated He

atoms (since |Eantibonding|>>|Ebonding|).Therefore, He-He does not exist!

Key Concepts

1 # molecular orbitals = # atomic orbitals1. # molecular orbitals # atomic orbitals

2. Three types of molecular orbitals: bonding, antibonding, and b dinonbonding

Nonbonding: a MO that neither raises nor lowers the energy of the system. Typically, it consists of a single orbital on one atom (possibly because there is no atomic orbital of the correct symmetry for it to overlap on a neighboring atom)

3. All components of the MO must behave identically under transformation if they are to have non-zero overlap

4. The Pauli exclusion principle limits the number of electrons that can occupy any molecular orbital to two. Those electrons must be paired.

Symmetry adapted linear combinations

σ Orbitals

Molecular orbitals arise from atomic orbitals of the same symmetry (i.e., overlap of s and s or s and pz or d and d)

Symmetry adapted linear combinations

δ Orbitalπ Orbital

Molecular orbitals arise from atomic orbitals of the same symmetry (i.e., overlap of s and s or s and pz or d and d)

Disallowed SACLs

The constructive interference between parts of atomic orbitals with the The constructive interference between parts of atomic orbitals with the same sign exactly matches the destructive interference between the

parts with opposite sign: zero net overlap

All components of the molecular orbital must behave identically under transformation if they are to have nonzero overlap

Energies of molecular orbitals

G ll d i d b b f d Generally determined by number of nodes. The greater the number of nodes in a MO, the greater the antibonding character and the higher the orbital energy. g g gy

yE

nerg

y

Energies of molecular orbitals

Recall: The label g, u indicates the symmetry of the

Ene

rgy

symmetry of the orbital with respect to inversion (and is

t il E not necessarily related to the MO energy)

Benzene

From the six atomic pz orbitals, we can construct six molecular orbitals

Benzene

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