hückel’s mo treatment of benzene

8/13/2019 Hckels MO treatment of Benzene

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Outlines electron energy,

MO coefficients

Delocalization Energy

Electrone Density

Bond order

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Hckel Theory An approximate theory that gives us a very quick picture of the MO

energy diagram and MOs of molecules without doing a lot of work

Basic Assumptions Atomic orbitals contributing to the -bonding in a planar molecule

are treated independently

The Coulomb integrals for all the carbon atoms are assumed to beidentical.

All resonance integrals between directly-bonded atoms are assumedto be the same; whilst those between atoms that are not directlybonded are neglected.

All overlap integrals representing the overlap of atomic orbitalscentred on different atoms are neglected

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electron energy Benzene Atomic orbitals 1, ..., 6.

We denote an MO with the symbol , and write it as a linear combination ofAOs whose coefficients cr have to be determined:

Consider the energy of this MO

Expanding the sum, we get

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electron energy The first-derivative w.r.t cr is equal to zero

The cr which satisfy the above are given by the matrix equation

Written in matrix form we have

we will assume that the overlap matrix is simply the identity matrix:

We also set the off-diagonal elements between nearest-neighbour orbitalsto be , and all others to be zero

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electron energy HMO secular equation is

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MO coefficients For coefficients of corresponding AOs, we have

The normalization condition

Shows that all the coefficients in the jth MO have the same absolute value.Hence

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Delocalization Energy HMO theory can explain delocalization energy. The corresponding VB term is resonance

energy.

Given six -electrons, which doubly occupy the three lowest energy levels, the total -energy is:

The energy of the occupied ethylene HMO

benzene has three isolated double bonds thus electron energy would be three time ofethylene ie

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Delocalization Energy benzene is more stable by 36 kcal/mol

Hckel theory predicts a similar stabilization of other cyclic conjugated

systems with 4N+2 electrons. This energetic stabilization explains in partwhy benzene is so unreactive as compared to other unsaturatedhydrocarbons.

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Electrone Density From HMO normalization condition

it is natural to say that an electron in the MO has the probability |cri|2of being in the vicinity of the rth carbon atom

total -electron charge qr in the region of carbon-atom r is defined as

where the sum is over the it MOs. qr is often called the -electron density";this name is misleading, since qr is neither a charge density (which hasdimensions of charge/ volume) nor a probability density (which hasdimensions of 1/volume). Rather, qr is a pure number that gives theapproximate number of electrons in the vicinity of carbon atom r

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Bond order Coulson defined the -electron (or mobile) bond order prsfor the bond between

bonded atoms r and s as

When the coefficients are all real

Applying this formula to the 1-2 bond in benzene

Bond order

Addition of the electrons' single bond gives the total bond order

For benzene each carbon-carbon bond order is found to be 5/3 = 1.667

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hückel’s mo treatment of benzene

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