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CH2422 Quantum Chemistry; Frontier Orbital Theory

Frontier orbital theory is necessary as it accounts for a better description of reactivity than the vague

application of curly arrows and often ill -informed electronic arguments, the deciding factor in

whether a reaction occurs or not is often one of symmetry. For example, there are Diels-Alder

reactions that appear to involve the reaction of two positive nodes that are explainable with Frontier

Orbital theory but would otherwise seem unlikely.

CH2

OH O

CH2

OH

O

O OH

OH

O

CH2+

O-

OH

CH2+

O-

OH

Figure 1 - An example Diels-Alder reaction that is allowed

by frontier orbital theory yet would seem counter-intuitive

Frontier orbital theory is also useful for explaining a preference of one reaction over another, for

example the combination of cyclopentadiene rings in solution.

Figure 2 - The dimerisation of cyclopentadiene showing a

preference for one reaction, despite both being possible

Frontier Orbitals of Commonly Used Molecular Fragments

Determining the Molecular Orbitals of Conjugated Hydrocarbons

For conjugated systems it is the electrons that are of importance. The movement of electrons

within these systems can be considered analogous to that of the electron in a box and using this

model it is possible to calculate the relative signs and orbital coefficients, c, of the p-orbitals at each

carbon of the conjugated chain.

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Figure 3 - Wavefunctions of conjugated systems

represented as sin waves for molecular orbitals, n=1(--),2(--),3(--),4(--)

Each orbital can contain up to 2 electrons, so for an 8 system n=1 to n=3 would be fully occupied,

with n=4 unoccupied, in this case the highest occupied molecular orbital (the HOMO) is n=3, and the

lowest unoccupied molecular orbital (the LUMO) is n=4. These are the frontier orbitals, and it is the

symmetry allowed overlap of a molecular LUMO with a molecular HOMO that dictates whether a

reaction is forbidden or not.

To calculate the orbital signs and coefficients across the molecule the appropriate wave-form can besplit into natoms+1, the p-orbitals are found at each split, the respective sign being that of the wave,

the coefficient determined by the magnitude of the peak or trough at that point.

The sum of the coefficients squared must be normalised, i.e. c2=1.

Ethene

For the simplest example, a 2 electron molecule, it is necessary to consider the first two waves of

an electron in a box.

Figure 4 - The wavefunctions used for describing

the frontier orbitals in ethene

The 2 electrons occupy the first wavefunction, both have the same sign and magnitude (sin 60 and

sin 120 are equal), since the magnitudes must be normalised c2

+c2

=1 and so c=0.707. This is theHOMO of ethene.

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0 360

0 540

0 720

0 180

0 360

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Since the first waveform is occupied, the LUMO is described by the second function. Here the

magnitudes are the same, however the signs are opposing. This represents a node in the centre of

the ethene molecule.

Figure 5 - The frontier orbitals of ethene

Butadiene

For butadiene there are 4 atoms, and so the wavefunctions are split into 5 parts. There are 4

electrons and so the first two waves are occupied (the second therefore being the HOMO), the third

is empty and so is the LUMO.

Figure 7 - The frontier orbitals of butadiene

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0 360

0 540

Figure 6 - The wavefunctions used for

describing the frontier orbitals in butadiene

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Allyl Cations and Anions

Allyl cations and anions are 2 and 4 systems respectively.

Figure 9 - The frontier orbitals of

the allyl cation and anion

0 180

0 360

0 540

Figure 8 - The wavefunctions used for

describing the frontier orbitals in allyl

cations and anions

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Pentadienyl Cations and Anions

Pentadienyl cations and anions are 4 and 6 respectively.

Figure 11 - The frontier orbitals of the

pentadienyl cation and anion

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0 360

0 540

0 720

Figure 10 - The wavefunctions used for

describing the frontier orbitals in pentadienyl

cations and anions

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Hextriene

Hexatriene is a 6 system.

Figure 12 - The wavefunctions used for describing

the frontier orbitals in hexatriene

Figure 13 - The frontier orbitals of hexatriene

Using Frontier Orbital Theory to Predict the Product of a Diels-Alder Reaction

Consider the reaction of a 1-methoxy butadiene with furan-2,5-dione. There are 3 stereogenic

centres formed, with both the endo and exo product theoretically possible.

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The trajectory of this reaction must be considered, with the diene approaching from out of the plane

there are two possible conformations for attack. The exo product involves less stearic interaction

between the furan and the diene, however the endo product is preferred. This is due to secondary

orbital interactions, if the furan is considered to be electronically analogous to hexatriene (an

assumption that in this case works) then the atoms either side of the reacting double bond possess

orbitals of the same sign as those in the middle of butadiene, this interaction is energetically

favourable and so, despite stearic effects, the endo product is favoured.

Diels-Alder Reactions - Normal And Inverse Electron Demand

Typically in a Diels-Alder the reaction will occur between the HOMO of the diene and the LUMO of

the dienophile, this is said to follow normal electron demand. There are cases in which the opposite

is true, in which the LUMO of the diene interacts with the HOMO of the dienophile, these are

considered to follow inverse electron demand. Inverse electron demand Diels-Alder reactions,

DAINV, occur between electron-rich dienophiles and electron-poor dienes, an example of which is the

reaction between acrolein and methyl vinyl ether (Figure 14). DAINV reactions often involve

heteroatoms and are a useful route to heterocyclic molecules. The mechanism of a DAINV reaction is

believed to be slightly asynchronous, but otherwise the same as that for a typical normal electrondemand reaction. The time-delay between scission and formation of bonds is believed to be

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minimal, however, there is still dispute within the literature as to how significant a difference there

is.

Figure 14 - The reaction between acrolein and methyl vinyl ether

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Fleming Notation and the Orbitals Involved in Diels-Alder Reactions

Fleming describes three types of substituent that can form part of a diene or dienophile for use in a

Diels-Alder reaction. These are allyl anions, ally cations and conjugated carbon systems; X, Z and C

respectively.

For dienophiles, considering only the LUMO;

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For 1-substituted dienes, considering only the HOMO;

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For 2-substituted dienes;

Electrocyclic Reactions

Consider the formation of a 6-membered cycle from a substituted hexatriene molecule (Figure 15).

Figure 15 - The electrocyclic reaction of 1,6-dimethyl hexatriene

Considering the HOMO of this molecule (assuming there is no activation of the hexatriene derivative

first) it is possible to determine the stereochemistry of the product by considering the directions in

which the carbons must rotate to achieve successful orbital overlap. Figure 16 shows that in order to

overlap the two carbon atoms must rotate in opposing direction, this reaction is therefore

disrotatory.

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Figure 16 - The HOMO of 1,6-dimethyl hexatriene and the resulting cis product

When considering the photochemical reaction, in which an electron is first promoted by excitement

with light, it is the LUMO that must be considered, as seen in Figure 17. Here the reaction is

conrotatory and leads to the antiproduct.

Figure 17 - The LUMO of 1,6-dimethyl hexatriene and the resulting anti product

In some cases the distinction between thermal and photochemical electrocyclic reactions may

dictate the reaction scheme chosen. For example, the formation of a 5-membered heterocycle

shown below (Figure 18) can be controlled to give a cis or trans product based on the first reaction

step. By deprotonating the cis product will form as the symmetry of the resulting HOMO encourages

disrotatory motion of the bond-forming carbon atoms. If the trans product is desired it could be

proposed that the anionic intermediate is somehow isolated and photochemically excited, as the

LUMO would have a conrotatory bond forming step. This is not possible, however, and so a different

approach must be adopted, the formation of a cation in the first step alters the HOMO, allowing

conrotatory bond formation and the resulting trans product.

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Figure 18 - The formation of two different products through exploitation of frontier orbital theory

Sigmatropic Reactions

A sigmatropic reaction is one in which a -bond is broken. The simplest involves the movement of a

hydrogen from one carbon to another in a conjugated system. A 1,5 and a 1,7 sigmatropic reaction

are shown below (Figure 19), by considering the HOMO of the -bond with the LUMO of the

conjugated system it is possible to rationalise the face to which the hydrogen will move. Thesereactions are then categorised according to the movement of the hydrogen, with reactions in which

the hydrogen is retained on the same face described as suprafacial and those in which it switches

faces are described as antarofacial.

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Figure 19 - 1,5 and 1,7 sigmatropic reactions

The Cope rearrangement (Figure 20) is a 3,3-sigmatropic reaction in which bonds are broken and

formed between carbon atoms.

Figure 20 - The Cope rearrangement

The Cope rearrangement is reversible, and so controlling the outcome of a reaction in which it is a

necessary step requires intelligent reaction design. The most common method is to make the

product as stable as possible relative to a starting material or intermediate. In Figure 21 two

methods of achieving this are shown, the first being the tautomerisation of an alcohol to a much

more stable product, the second being the incorporation of a cyclopropane moiety in the starting

material, the resulting ring strain discourages the return rearrangement.

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Figure 21 - Controlling the Cope rearrangement