# Frontier Molecular Orbital Theory

Frontier Molecular Orbital (FMO) theory attempts to predict relative reactivity based on properties of the reactants. It is commonly formulated in term of perturbation theory, where the energy change in the initial stage of a reaction is estimated and "extrapolated" to the transition state.1 For a reaction where two different modes of reaction are possible, this may be illustrated as shown in Figure 15.1.

The reaction mode that involves the least energy change in the initial stage is assumed also to have the lowest activation energy. FMO theory uses a low-order perturbation expansion with the reactants as the unperturbed reference, and it is clear that such a treatment can only be used to follow the reaction a short part of the whole reaction pathway.

The change in the energy can be derived from second-order perturbation theory (Section 4.8) and is given in equation (15.1).2

Reaction coordinate

Figure 15.1 FMO region of a reaction profile atoms atoms

MO MO+X MO

ieA aeB ieB aeA

Here A and B denote atoms in each of the two interacting molecules. The V operator contains all the potential energy operators from both molecules, and the <£A |V|cB) integral is a "resonance" type integral between two atomic orbitals, one from each molecule. The pA is the electron density on atom A, and the first term in (15.1) represents a repulsion (<£A|V|£B) is a negative quantity) between occupied MOs (steric repulsion). This will usually lead to a net energy barrier for a reaction. The second term represents an attraction or repulsion between charged parts of the molecules, QA being the (net) charge on atom A. The last term is a stabilizing interaction (ei - ea < 0) due to mixing of occupied MOs on one molecule with unoccupied MOs on the other, cjcaa being MO coefficients and ejea MO energies. The summation is over all pairs of occupied/unoccupied MOs.

If we are comparing reactions that have approximately the same steric requirements, the first term is roughly constant. If the species are very polar the second term will dominate, and the reaction is charge controlled. This means for example that an elec-trophilic attack is likely to occur at the most negative atom or, in a more general sense, along a path where the electrostatic potential is most negative. If the molecules are non-polar, the third term in eq. (15.1) will dominate and the reaction is said to be orbital controlled. This means that the reaction will occur where the molecular orbital coefficients are largest.

All other things being equal, the largest contribution to the double summation over orbital pairs in the third term will arise when the denominator is smallest. This corresponds to the Highest Occupied Molecular Orbital (HOMO) and Lowest Unoccupied Molecular Orbital (LUMO) pair of orbitals. FMO theory considers only this one contribution in the whole summation. From a purely numerical consideration this is certainly not a good approximation: the contributions from all the other pairs are much larger than the single HOMO-LUMO term. Nevertheless, it is possible to rationalize many trends in terms of FMO theory and thus the result justifies the means. If we furthermore consider a matrix element <£m|V|£a?) to be non-zero only between atoms where new bonds are being formed (where it is furthermore assumed to be roughly constant), the deciding factor becomes a sum over products of MO coefficients from the HOMO on one fragment with LUMO coefficients on the other. A few examples should help clarify this.

The reaction of a nucleophile involves the addition of electrons to the reactant, i.e. interaction of the HOMO of the nucleophile with the LUMO of the reactant. If there is more than one possible centre of attack, the preferred reaction mode is predicted to occur on the atom having the largest LUMO coefficient. Figure 15.2 shows that the orbital component shows preference for addition to the 4-position of acrolein (as a model for unsaturated carbonyl compounds in general), with the second most reactive position being C2. The net charges, however, prefer position 2, as it is the most positive carbon. Experimentally, it is found that attack at the 4-position is usually favoured (especially with "soft" nucleophiles such as organocuprates), but addition at the 2-position is also observed (and may dominate with "hard" nucleophiles such as organo-lithium compounds).3 This is consistent with the reaction switching from being orbital controlled to charge controlled as the nucleophile becomes more ionic.

Figure 15.2 AM1 LUMO coefficients for acrolein with net charges in parenthesis

Similarly, the reaction of an electrophile will involve the HOMO of the reactant, i.e. the reaction should occur preferentially on the atom having the largest HOMO coefficient. The coefficients for furan shown in Figure 15.3 indicate that electrophilic substitution should preferentially occur at the 2-position, again in agreement with experimental results.4

### Figure 15.3 AM1 HOMO coefficients for furan

Consider now the reaction between butadiene and ethylene, where both 2+2 and 4+2 reaction modes are possible. The qualitative appearances of the butadiene HOMO and ethylene LUMO are given in Figure 15.4. The MO coefficients are given as a, b and c, where a > b > c.

Figure 15.4 FMO theory favours the 4+2 over the 2+2 reaction

For the 2+2 pathway the FMO sum becomes (ab - ac)2 = a2(b - c)2 while for the 4+2 reaction it is (ab + ab)2 = a2(2b)2. As (2b)2 > (b - c)2, it is clear that the 4+2 reaction has the largest stabilization, and therefore increases least in energy in the initial stages of the reaction (eq. (15.1), remembering that the steric repulsion will cause a net increase in energy). The 4+2 reaction should consequently have the lowest activation energy, and therefore occur more easily than the 2+2. This is indeed what is observed: the Diels-Alder reaction occurs readily but cyclobutane formation is not observed between non-polar dienes and dieneophiles.

The appearance of the difference in MO energies in the denominator in eq. (15.1) suggests that a smaller gap between the diene HOMO and dieneophile LUMO in a Diels-Alder reaction should lower the activation energy. If the diene is made more electron-rich (electron-donating substituents), or the dieneophile more electron-deficient (electron-withdrawing substituents), the reaction should proceed faster. This is indeed the observed trend. For the reaction between cyclopentadiene and cya-noethylenes (mono-, di-, tri- and tetra-substituted), the correlation is reasonably quantitative, as shown in Figure 15.5.5

This is of course a rather extreme example, as the reaction rates differ by ~107, and rate differences of over a factor of 100 are observed for quite similar HOMO-LUMO differences. For a more varied set of compounds where the reaction rates are more similar, the correlation is often quite poor.

FMO theory can also be used for explaining the stereochemistry of the Diels-Alder reaction, as can be illustrated by the reaction between 2-methylbutadiene and cya-noethylene. These may react to give two different products, the "para" and/or "meta" isomer.

The MO coefficients for the p-orbitals on the butadiene HOMO and ethylene LUMO (taken from AM1 calculations) are given in Figure 15.6. The FMO sum for the