LUMO, definition

Another characteristic of aromatic compounds is a relatively large HOMO-LUMO gap, which can be expressed in terms of hardness, t] (see p. 21 for the definition of hardness) ... 

An HSAB analysis of singlet carbene reactivity based on B3LYP/6-31G computations has calculated the extent of charge transfer for substituted alkenes,122 and the results are summarized in Figure 10.3 The trends are as anticipated for changes in structure of both the carbene and alkene. The charge transfer interactions are consistent with HOMO-LUMO interactions between the carbene and alkene. Similarly, a correlation was found for the global electrophilicity parameter, co, and the ANmax parameters (see Topic 1.5, Part A for definition of these DFT-based parameters).123... 

One more step provides an operational definition. The HOMO level lies, I = ionization energy, below the vacuum level, while the LUMO level lies, A = electron affinity, below it. Thus, the chemical hardness lies midway in the gap and usually is given in units of eV. 

The HOMO-LUMO gap calculated by an ah initio method using an STO-3G basis set26 seems to support these ideas. A more definitive test will be possible after the nature of the lowest excited state in carbon unsaturated systems is understood27). 

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. 

Figure 1.11. Definition of F f, g, a, vac, homo, lumo, e, IF and 4>m. For simplicity the unoccupied bands are represented only up to vac-...

Now, by its very definition, the global hardness rj is a measure of the HOMO-LUMO gap of a compound. Consequently, it seems reasonable to assume that the AEE AE should scale with the global hardness. The <(ag/r)3> term describes the p-orbital expansion or contraction as electrons are added or removed from the shielded nucleus. Using effective atomic numbers Znp and Slater rules ... 

The equations 7, 8 and 9 fail to operate when the HOMO - LUMO energy gap becomes too small and do not consider the influence on chemical properties of other orbitals, besides the HOMO and LUMO s. Moreover, it is not possible to study the site selectivity of a chemical species considering only absolute hardness other than space-dependent (local) versions of hardness/softness concepts [5]. Thus in addition to the global definition of r) and S, the local hardness [4] and local softness [5] have been introduced as follows ... 

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. 

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