Molecular orbital theory frontier

From second-order perturbation theory (Section 4.8) the following equation for the change in energy can be derived.  

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 (xa V xb) integral is basically a resonance type integral between two atomic orbitals, one from 

If we are comparing reactions which have approximatively—the same stericr 

All things being equal, the largest contribution to the double summation over orbital the third term will arise when the denominator is smallest. This con esponds to  

The reaction of a nucleophile involves addition of electrons to the reactant, i.e. 

If we are comparing reactions which have approximatively 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 electrophihc 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 (15.1) will dominate, and the reaction is orbital controlled. 

For the 2-1-2 pathway the FMO sum becomes (ab — ac) = a b — c) while for the 4 -I- 2 reaction it is (ab-I-ab) — a (2b). As (2b) (b — c), 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). Consequently the 4 - - 2 reaction should have the lowest activation energy, and therefore occur easier than the 2-1-2. This is indeed what is observed, the Diels-Alder reaction occurs readily, but cyclobutane formation is not observed between non-polar dienes and dieneophiles. 

FMO theory was developed at a time when detailed calculations of reaction paths were infeasible. As many sophisticated computational models, and methods for actually locating the TS, have become widespread, the use of FMO arguments for predicting reactivity has declined. The primary goal of computational chemistry, however, is not tc 

Fukui described pericyclic reactions in terms of frontier molecular orbital (FMO) theory. If the interaction of the HOMO of one reactant with the LUMO of the other reactant leads to bonding interactions, then the pericyclic 

543 and 553 presented a detailed PMO analysis of the interactions between two conjugated molecules that reaffirms the FMO predictions. 

The FMO model provides insight into one aspect of electrocyclic reactions that we have not yet discussed. In the electrocyclic opening of cyclobutenes to butadienes, the tendency of a substituent at C3 of the cyclobutene to rotate inward to form a cis product (equation 11.42) or outward to form a trans product (equation 11.43) has been termed torcfuoselectivity Steric effects 

FMO analysis of the opening of cyclobutene to 1,3-butadiene by disrotatory (left) and conrotatory (right) pathways. 

For a discussion of pericyclic reactions in terms of frontier MO theory, see Houk, K. N. in Marchand, A. P. Lehr, R. E., Eds. Pericyclic Reactions, Vol. II Academic Press New York, 1977 pp. 181-271. 

Although sophisticated electronic structure methods may be able to accurately predict a molecular structure or the outcome of a chemical reaction, the results are often hard to rationalize. Generalizing the results to other similar systems therefore becomes difficult. Qualitative theories, on the other hand, are unable to provide accurate results but they may be useful for gaining insight, for example why a certain reaction is favoured over another. They also provide a link to many concepts used by experimentalists. Frontier molecular orbital theory considers the interaction of the orbitals of the reactants and attempts to predict relative reactivities by second-order perturbation theory. It may also be considered as a simplified version of the Fukui function, which considered how easily the total electron density can be distorted. The Woodward-Hoffmann rules allow a rationalization of the stereochemistry of certain types of reactions, while the more general qualitative orbital interaction model can often rationalize the preference for certain molecular structures over other possible arrangements. 

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. 

Introduction to Computational Chemistry, Second Edition. Frank Jensen. 2007 John Wiley Sons, Ltd 

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 

TABLE 14.10 Solubility products, calculated formation constants, and calculated and experimental E° values (V vs NHE) for the silver halides at 298 K. 

Acid-base interaction MO diagram for H3O+, showing how the LUMO on H+ and HOMO on HjO interact to form a bonding and antibonding pair of MOs in H3O+. 

The shapes of the t2 MOs for NH4+ can be determined by taking linear combinations of the three 2p AOs on N with the three t2 SALCs in such a way as to maximize the orbital overlap. A qualitative depiction of the degenerate set oft2 MOs is shown below. 

The HOMO is the 3a nonbonding MO derived from the Ip AO of the N atom. When NH3 reacts with a proton to form the tetrahedral NH4+ ion, the 3aI HOMO on NH3 will form a bonding and antibonding combination with the I s LUMO on H+, as shown below. 

The shapes of the two e MOs for NH3 are based on the overlap of two of the Ip AOs on N with the two e SALCs depicted below. The jl, nonbonding lone pair is also shown (at right). 

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Simple Approaches to Quantifying Chemical Reactivity 3.4.2.1 Frontier Molecular Orbital Theory... 

The BDE theory does not explain all observed experimental results. Addition reactions are not adequately handled at all, mosdy owing to steric and electronic effects in the transition state. Thus it is important to consider both the reactivities of the radical and the intended coreactant or environment in any attempt to predict the course of a radical reaction (18). AppHcation of frontier molecular orbital theory may be more appropriate to explain certain reactions (19). 

The problems associated with predicting regioselectivity in quinone Diels-Alder chemistry have been studied, and a mechanistic model based on frontier molecular orbital theory proposed (85). In certain cases of poor regioselectivity, eg, 2-methoxy-5-methyl-l,4-ben2oquinone with alkyl-substituted dienes, the use of Lewis acid catalysts is effective (86). 

Frontier molecular orbital theory correctly rationalizes the regioselectivity of most 1,3-dipolar cycloadditions (73JA7287). When nitrile ylides are used as 1,3-dipoles, the dipole... 

Frontier Molecular Orbital theory is closely related to various schemes of qualitative orbital theory where interactions between fragment MOs are considered. Ligand field theory, as commonly used in systems involving coordination to metal atoms, can be considered as a special case where only the d-orbitals on the metal and selected orbitals of the ligands are considered. 

It is important to realize that whenever qualitative or frontier molecular orbital theory is invoked, the description is within the orbital (Hartree-Fock or Density Functional) model for the electronic wave function. In other words, rationalizing a trend in computational results by qualitative MO theory is only valid if the effect is present at the HF or DFT level. If the majority of the variation is due to electron correlation, an explanation in terms of interacting orbitals is not appropriate. 

J 7i-Cycloadducts at the C4 — C5 azepine positions are also formed with 1,2,3,4-tetra-chloro-5,5-dimethoxycyclopentadiene260 261 and with hexachlorocyclopentadiene.261 The reactivity of ethyl l//-azepine-l-carboxylate towards cyclopentadienones has been studied in terms of frontier molecular orbital theory which predicts that dimethyl 2-oxo-4,5-diphenyl-cyclopenta-1 (5),3-diene-1,3-dicarboxylate (20) should be more reactive towards the 1/f-azepine than other more common cyclopcntadienone derivatives.262 In fact, in refluxing benzene, the cyclopentadienone and ethyl 1/f-azepine-l-carboxylate (1) form a mixture of the [4 + 2] n-endo,anti-adduct 22, produced by Cope rearrangement of the initially formed [2 + 4] 7T-adduct 21, and the c.w-adduct 23, a rare example of a [6 + 4] rc-cycloadduct. At room temperature, only the [6 + 4] adduct 23 and a small amount of the [2 + 4] adduct 21 are obtained, the latter rearranging to the [4 + 2] adduct 22 on warming.262 Other [4 + 2] 7r-adducts with cyclopentadienones have been prepared similarly.263... 

An interpretation based on frontier molecular orbital theory of the regiochemistry of Diels Alder and 1,3-dipolar cycloaddition reactions of the triazepine 3 is available.343 2,4,6-Trimethyl-benzonitrile oxide, for example, yields initially the adduct 6.344... 

According to frontier molecular orbital theory (FMO), the reactivity, regio-chemistry and stereochemistry of the Diels-Alder reaction are controlled by the suprafacial in phase interaction of the highest occupied molecular orbital (HOMO) of one component and the lowest unoccupied molecular orbital (LUMO) of the other. [17e, 41-43, 64] These orbitals are the closest in energy Scheme 1.14 illustrates the two dominant orbital interactions of a symmetry-allowed Diels-Alder cycloaddition. 

The period 1930-1980s may be the golden age for the growth of qualitative theories and conceptual models. As is well known, the frontier molecular orbital theory [1-3], Woodward-Hoffmann rules [4, 5], and the resonance theory [6] have equipped chemists well for rationalizing and predicting pericyclic reaction mechanisms or molecular properties with fundamental concepts such as orbital symmetry and hybridization. Remarkable advances in aeative synthesis and fine characterization during recent years appeal for new conceptual models. 

Houk, K.N. "Application of Frontier Molecular Orbital Theory to Pericyclic Reactions", in "Pericyclic Reactions", A.P. 

The Diels-Alder reaction (47t 2ir cycloaddition) is by far the best studied reaction of dienes from both theoretical and experimental viewpoints. Frontier molecular orbital theory predicts three types of Diels-Alder reaction. Structural effects on rate constants show the existence of two types of reaction ... 

These relations highlight the fact that the formalism of DFT-based chemical reactivity built by Parr and coworkers, captures the essence of the pre DFT formulation of reactivity under frontier molecular orbital theory (FMO). Berkowitz showed that similar to FMO, DFT could also explain the orientation or stereoselectivity of a reaction [12]. In addition, DFT-based reactivity parameters are augmented by more global terms expressed in the softness. 

Luther, G. W., (1990), The Frontier-Molecular-Orbital Theory Approach in Geochemical Processes", in W. Stumm, Ed., Aquatic Chemical Kinetics, Wiiey-lnterscience, New York, pp. 173-198. 

A fundamental understanding of the mechanistic and stereochemical aspects of the Diels-Alder reaction was unfolded during the 1970s. Several theoretical approaches are available nowadays from which Fukui s Frontier Molecular Orbital theory (FMO theory) is most frequently used because of its simplicity49- 53. 

Sun Shuiyu, Wang Dianzuo, Long Xiangyun, 1994b. Frontier molecular orbital theory consideration for electron transfer process across sulphide mineral-solution interface. J. BGRIMM, 3(1) 34 - 39 (in Chinese)... 

According to frontier molecular orbital theory, the strongest interactions are between those orbitals that have coefficients with similar magnitudes relative to the unperturbed molecules, t.e. the interaction is between the small coefficient on the dienophile and the small coefficient on the diene [16], [17]. 

The above work has also led to the prediction that the so-called -effect for nucleophiles containing a heteroatom adjacent to the reaction centre should display normal behaviour in the gas phase (Wolfe et al., 1981a,b, 1982). This prediction has been corroborated by recent experimental determinations that show little variation of reactivity between HOO- and HO (DePuy et al., 1983), contrary to what is predicted by the application of frontier molecular orbital theory (Fleming, 1976). 

Hydroxy-l-methylquinolinium iodide is deprotonated by tertiary base giving A-methylquinoIinium-3-olate (283 R = Me) which can be trapped by dienes giving adducts 284. These cycloadditions have been discussed in terms of frontier molecular orbital theory (see Section N,C,2)3° ... 

By Frontier Molecular Orbital theory several important predictions can be made 1. There will be a preferred chair transition-state for the reaction ... 

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