Huckel Molecular Orbital (HMO) Theory
consider sigma and pi systems independent - only work with pi
- the assumption works well for planar conjugated pi systems
n atomic orbitals (AOs) must combine to give n molecular orbitals (MOs)
each MO must have the symmetry of the molecule (recall methane MOs)
LCAO-MO method: each MO is a linear combination of the AOs
Variation Principle: the best MO is found by minimizing energy with respect
to each coefficient
- there will be n solutions, i.e., n sets of coefficients, which are the
n MOs
Hij (energy integrals) - Hii = alpha (energy of a p AO) , Hij = beta (if i and j bonded, otherwise Hij = 0)
Sij (overlap integrals) - Sii = 1 (AO normalization), Sij = 0 (poor assumption, simplifies math)
solutions for ethene:
E = alpha + beta , alpha - beta
coefficients c1 = c2 , c1 = - c2
general approach - set up the secular determinant, solve it for the energies, solve for coefficients
Qualitative energy level patterns and MO symmetry patterns
linear polyenes (with n carbon atoms in the pi system)
n distinct energy levels, symmetrically located about alpha
lowest MO has no nodes, each successive MO has an additional node, symmetrically located
HOMO and LUMO are the frontier orbitals (most important)
odd-membered linear pi systems (e.g., allyl)
n distinct energy levels, symmetrically located about alpha, including one at alpha (NBMO)
lowest MO has no nodes, each successive MO has an additional node, symmetrically located
NBMO is most important (site of charge density or unpaired electron)
cyclic polyenes (both even and odd)
Frost's Circle - one lowest MO, then pairs of MOs
Applications of HMO calculations
delocalization energy (DE) -
total pi energy compared to that of a localized reference system
- e.g., for 1,3-butadiene, the reference is two isolated ethenes
DE for 1,3-butadiene is about 0.5 beta
DE for allyl is 0.8 beta, for the cation, anion, or radical
charge density -
for a given carbon atom, coefficient squared gives electron density in each
MO
pi bond order -
for adjacent atoms, product of coefficients gives bonding order for each
MO
free valence -
unused pi bond order, compared to a maximally bonded carbon (pi bond order
1.732)
Alternant and nonalternant hydrocarbons
alternant - you can put stars around alternately so no two stars are adjacent
alternant hydrocarbons - MO energy levels are symmetrically arranged about alpha
odd alternant hydrocarbons - one MO appears at energy alpha (NBMO)
NBMO has zero coefficients (nodes) at the unstarred atoms
for NBMO, the coefficients about a node sum to zero
these concepts allow the calculation of NBMO coefficients
Aromaticity
Huckel's Rule - closed shell of (4n+2) pi electrons in a cyclic pi system
antiaromaticity - open shell of (4n) electrons
examples - small rings with charges, large rings (annulenes)
stability criteria for aromaticity -
Dewar Resonance Energy (DRE) - heat of formation relative to a reference
compound
- unlike delocalization energy (DE), the reference here is a standard delocalized
polyene (nonaromatic)
- e.g., for benzene, the reference for DE would be 3 ethenes, but for DRE
it is 1,3,5-cyclohexatriene
Absolute hardness -
- half the difference between HOMO and LUMO energies
*(Skip section 4.3)* -- Methods using Valence Bond Approach
Perturbational MO Theory (PMO)
consider relative changes in MO energies and shapes caused by some perturbation
effect
- e.g., compare ethene to formaldehyde
- e.g., illustrate butadiene made by interacting two ethylenes
charge transfer interactions