Frontier orbital, definition

The contribution of the frontier orbitals would be maximized in certain special donor-acceptor reactions. The stabilization energy is represented by Eqs. (3.25) and (3.26). Even in a less extreme case, the frontier orbital contribution maybe much more than in the expression of the superdelocalizability. If we adopt the approximation of Eq. (6.3), the intramolecular comparison of reactivity can be made only by the numerator value. In this way, it is understood that the frontier electron density, /r, is qualified to be an intramolecular reactivity index. The finding of the parallelism between fr and the experimental results has thus become the origin of the frontier-electron theory. The definition of fr is hence as follows ... 

Mulliken s model is entirely compatible with the descriptions given previously in this Section, since charge shifts that result from polarization are already taken into account in forming the cr-complex by subsequent changes in hybridization. Fukui s model, upon which the definition of the superdelocalizability is based, resembles Mulliken s only in the use of a pseudo-TT orbital , and the formulation of the hyperconjugation problem is quite different, since )3 is taken by Fukui to be small, so that 8f can be defined by perturbation formulae. In particular, the bonding of the pseudo-TT orbital in Fukui s model primarily involves the least bound, or frontier orbitals, whereas in Mulliken s model the most bound MO is involved. 

Dnring an electron transfer, the acceptor places its LUMO at the electron disposal and the donor releases an electron that is located on its HOMO. These orbitals are frontier orbitals. In the corresponding ion-radicals, the distribution of an unpaired electron proceeds, naturally, under frontier-orbital control. This definitely reflects in the ion-radical reactivity and not always by a self-obvions manner. Let ns concisely trace peculiarities of ion-radical fragmentation reactions that are very important in organic synthesis. 

As chemists we can pose a simple, focussed question how do the Woodward-Hoffmann rules (WHR) [18] arise from a purely electron density formulation of chemistry The WHR for pericyclic reactions were expressed in terms of orbital symmetries particularly transparent is their expression in terms of the symmetries of frontier orbitals. Since the electron density function lacks the symmetry properties arising from nodes (it lacks phases), it appears at first sight to be incapable of accounting for the stereochemistry and allowedness of pericyclic reactions. In fact, however, Ayers et al. [19] have outlined how the WHR can be reformulated in terms of a mathematical function they call the dual descriptor , which encapsulates the fact that nucleophilic and electrophile regions of molecules are mutually friendly. They do concede that with DFT some processes are harder to describe than others and reassure us that Orbitals certainly have a role to play in the conceptual analysis of molecules . The wavefunction formulation of the WHR can be pictorial and simple, while DFT requires the definition of and calculations with some nonintuitive ( ) density function concepts. But we are still left uncertain whether the successes of wavefunctions arises from their physical reality (do they exist out there ) or whether this successes is merely because their mathematical form reflects an underlying reality - are they merely the shadows in Plato s cave [20]. 

In Table 6.8, atom type-specific subset correlations of AMI, PM3, PM5, and HF with B3LYP are shown for frontier orbital energies and related descriptors, for parameters based on the charge distribution, and for the PPSA and PNSA descriptors. In contrast to Table 6.4 and Table 6.7, however, now all compounds providing a zero descriptor value by definition (e.g., PPSA-1Z = 0 if, for a given method, the compound has no positively charged heavy atom) were included for generating the statistics. 

We have now seen that the effort of Parr and collaborators [8-12] to put Fukui s frontier-orbital concept of chemical reactivity on sound footing in density-functional theory through the definition of the Fukui function and the local and global softness works only for extended systems. This restriction to extended systems raises a sixth issue. In both the local softness and the Fukui function, Eqs. (54) and (53a), the orbitals at the chemical potential represent both the LUMO and the HOMO in the Fukui sense. However, there is a continuum of unoccupied KS states above the chemical potential accessible even to weak chemical perturbations any linear combination of which could in principle be selected as the LUMO, and similarly for states below fi and the HOMO. This ambiguity in the frontier-orbital concept obviously applies as well to localized systems when there is more than one KS state significantly affected by a chemical perturbation. 

Fig. 7. A qualitative MO scheme for ferrocene and other first-row transition-metal metallocenes. The inset highlights the ordering of the (/-orbitals in the frontier orbital region and provides the definition for A, and A (see the text).

It s important to know how many electrons one has in one s molecule. Fe(II) has a different chemistry from Fe(III), and CR3+ carbocations are different from CRj radicals and CR3 anions. In the case of Re2Cl82, the archetypical quadruple bond, we have formally Re(III), d4, i.e., a total of eight electrons to put into the frontier orbitals of the dimer level scheme, 17. They fill the a, two x, and the 6 level for the explicit quadruple bond. What about the [PtHj2] polymer 12 Each monomer is d8. If there are Avogadro s number of unit cells, there will be Avogadro s number of levels in each bond. And each level has a place for two electrons. So the first four bands are filled, the xy, xz, yz, z2 bands. The Fermi level, the highest occupied molecular orbital (HOMO), is at the very top of the z2 band. (Strictly speaking, there is another thermodynamic definition of the Fermi level, appropriate both to metals and semiconductors,9 but here we will use the simple equivalence of the Fermi level with the HOMO.)... 

We are now in a position to reformulate the Lewis definition of acids and bases in terms of frontier orbitals A base has an electron pair in a HOMO of suitable symmetry to interact with the LUMO of the acid (although lone pair orbitals with the wrong geometry may need to be ignored). The better the energy match between the base s HOMO and the acid s LUMO, the stronger the interaction. 

Although the local softness includes, by definition, both the differences of frontier orbitals of the neutral substrate and the differential electron densities between the neutral and ionized states, as expressed in the global softness and Fukui functions, the actual computations of these quantities suffer from some severe practical limitations . 

Definition Two fragments are isolobal if the number, symmetry properties, approximate energy and shape of their frontier orbitals, and the number of electrons occupying them are similar - not identical, but similar. [R. Hoffmann 1982]. 

While DFT does not require orbitals, it is not inconsistent to consider molecular orbitals in the form of frontier orbital theory. This means the electrons must come from definite occupied orbitals in D, and go into definite empty orbitals in C. Usually there is some electron transfer in both directions, as in a+7r-bonding. The nature of these interacting orbitals is of great importance in determining the interaction between C and D. 

We can now express the Lewis definition of acids and bases in terms of frontier orbitals ... 

An acceptable definition of local hardness and subsequent identification of the hard sites in a molecule is, therefore, in demand [298-304], The minimum Fukui function rule [301] asserts that hard reactions, unlike softer ones, would prefer sites with minimum Fukui function values. Although there are applications [305-307] of this rule, subsequent criticisms [303, 308] are also reported. The minimum Fukui function rule is unable to correlate the electrostatic hard-hard interactions that are predominantly charge-controlled with hardly any relevant effect from the associated frontier orbitals. The given rule cannot justify the inadequacy of the frontier orbitals and also misses the role of electrostatic interactions in hard-hard interactions [303, 304, 308]. 

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