MOLECULAR STRUCTUREMOLECULAR STRUCTURE

CHAPTER 14CHAPTER 14

Heteronuclear DiatomicsHeteronuclear Diatomics

• Almost all have polar bonds

• MO of the form: Ψ = cAA + cBB with cA ≠ cB

• where cA and cB are weighted coefficients

• Proportion of AOA = │cA│2 and of AOB = │cB│2

• For pure covalent bond │cA│2 = │cB│2

• For pure ionic bond in A+B− cA = 0 and cB = 1

• e.g., for HF Ψ = cHΨH + cFΨF

H F

electron richelectron richregionregionelectron poorelectron poor

regionregion

FH

e- riche- poor

+ -

Fig 11.36 AOs of H and F atoms with their MOs

Ψ = cHΨH ± cFΨF

1σ

2σ*

Heteronuclear DiatomicsHeteronuclear Diatomics

• Electronegativity - the ability of an atom in amolecule to attract electrons towards itself

Covalent

share e-

Polar Covalent

partial transfer of e-

Ionic

transfer e-

Increasing difference in electronegativity

Classification of bonds by difference in electronegativity

Difference Bond Type

0 Covalent

2 Ionic

0 < and <2 Polar Covalent

The Variation PrincipleThe Variation Principle

• A systematic method for determining thecoefficients in the LCAOs used to build the MOs

e.g., Ψ = cAA + cBB with cA ≠ cB

• Principle is basis for all modern MO calculations

• If an arbitrary wavefunction is used to calculatethe energy, the value calculated can never beless than the true energy

• Coefficients in trial function are varied until lowestenergy is achieved (HΨtrial = EtrialΨtrial)

Example of application of variation principle

Assume Ψtrial = cAA + cBB real but not normalized

d

dHE

*

*

The energy is the expectation value of the energyoperator:

Results are the secular equations

(αA-E)cA + (β-ES)cB = 0

(βA-ES)cA + (αA-E)cB = 0

Solved with the secular determinant

αA-E β-ES

β-ES αA-E= 0

Coulombintegral

Resonance integral

Example of application of variation principle

in HW Ex 11.9(b)

with trial function2ar

trial Ae

1st derivative = slope of line tangent to curve

0da

dEtrial

Fig 11.37 The molecular orbital energy leveldiagram for NO

Ground state configuration:

(1σ)2 (2σ)2 (3σ*)2 (1π)4 (2π*)1

3σ* and 1π primarily of Ocharacter

HOMO

LUMO

N=O••

••

••

•

Molecular orbitals for Polyatomics

• As with diatomics, we construct LCAO-MOs

i

iic

• All diatomics are linear, but polyatomics havea number of different geometries

• To determine molecular geometry, calculate Efor possible nuclear positions

• Lowest E indicates correct conformation

The Hϋckel Approximation

• For conjugated systems

• π orbitals treated separately from rigid molecular

frame formed from σ orbitals

• All C atoms treated identically so all Coulomb

integrals α are set equal

e.g., for H2C=CH2 take σ bonds as fixed

and find energy of π and π*

The Hϋckel Approximation

• Express π orbitals as LCAOs of C2p obitals

• Solve secular determinant

Ψ = cAA + cBB

αA-E β-ES

β-ES αA-E= 0

• Roots of equation: E± = α ± β

Fig 11.38 Hϋckel MO energy levels of ethene

HOMO

LUMO

Frontier orbitals

Fig 11.39 Hϋckel MO energy levels of butadiene

Fig 11.40 The σ framework of benzene formedfrom overlap of Csp2 hybrids

120°

Fig 11.41 Hϋckel MO energy levels of benzene

Bonding character

Antibonding character

Mixture of bonding, nonbonding,and antibondingcharacter

α+2β

α-2β

α+β

α-β