Most delocalized orbitals

From any arbitrary choice of orbitals si and 38 it is possible to construct the most localized and the most delocalized orbitals through an unitary transformation. It is possible to show that the following quantities remain invariant to this transformation14 ... 

The off-diagonal elements of the matrix (3.13) contain quantities yrfa, and yrf39. Since + y2M is invariant to the unitary transformation, it is clear that for any perturbation an appropriate choice of si and 58 can make either or vanish. This will be, of course, neither the most localized nor the most delocalized orbitals. On the other hand, since has no clear physical significance, the most convenient working choice of orbitals is the one for which y m vanishes. As was already mentioned, this is the case of the most localized si = A and 58 = B and the most delocalized orbitals si = a and 58 = b. For these choices, consequently, we have to deal with two independent perturbations that are related to each other in two different basis as... 

The actual sign ("phase") of the molecular orbital at any given point r of the 3D space has no direct physical significance in fact, any unitary transformation of the MO s of an LCAO (linear combination of atomic orbitals) wavefunction leads to an equivalent description. Consequently, in order to provide a valid basis for comparisons, additonal constraints and conventions are often used when comparing MO s. The orbitals are often selected according to some extremum condition, for example, by taking the most localized [256-260] or the most delocalized [259,260] orbitals. Localized orbitals are often used for the interpretation of local molecular properties and processes [256-260]. The shapes of contour surfaces of localized orbitals are often correlated with local molecular shape properties. On the other hand, the shapes of the contour surfaces of the most delocalized orbitals may provide information on reactivity and on various decomposition reaction channels of molecules [259,260]. 

Figure 4.19. Wave functions and energy levels of a perfect biradical (center), constructed from the most localized orbitals x nnd Xi> tind from the most delocalized orbitals < > and (right) (adapted from BonaCiC-Kouteck et al., 1987).

Another useful way to think about carbon electrophilicity is to compare the properties of the carbonyls lowest-unoccupied molecular orbital (LUMO). This is the orbital into which the nucleophile s pair of electrons will go. Examine each compound s LUMO. Which is most localized on the carbonyl group Most delocalized Next, examine the LUMOs while displaying the compounds as space-filling models. This allows you to judge the extent to which the LUMO is actually accessible to an approaching nucleophile. Which LUMO is most available Least available ... 

Sometimes 3(d — n ) and k ) states are said to be derived from delocalized orbitals and d—d) state from localized orbitals. The shift of the chelate emission from that of the free ligand increases in the sequence Rh(III) < Ir(III) < Ru(II) and reflects increasing cf-orbital participation in the emission orbital. The decrease in the chelate emission lifetime from the free ligand values also reflect the contamination of the molecular orbitals with d-character. The role of metal complexes as quenchers of excited states of it-electrons in organic compounds can be rationalized from such considerations. Emission from Cr8+ is the basis of one of the most important solid state laser system, the Ruby laser (Figure 10.14). 

It is easy to show that for both limiting choices (the most localized and the most delocalized choice) vanishes.14 The choice of orbitals has no significance, since we work with the exact solution of the 3x3 model. Nevertheless, it is sometimes more convenient to work with localized and sometimes with delocalized orbitals, as we will see. 

It is worth mentioning that exchange and Coulomb integrals and JM are both minimized and maximized for the most localized and the most delocalized choice of orbitals, respectively. This is understandable since localized orbitals A and B try to avoid each other in the space in contrast to the delocalized orbitals a and b. 

The VB calculations describing the 2B2 state were performed using at most seven atomic orbitals for the pair of structures. To form structures Ei and E2, we need two different orbitals on the central atom (atom 2 in Fig.5). These orbitals are shown in Fig. 12. One of them points toward the top atom and the other is spread inside the lower triangle. In this case, the orbitals are distinct, one being more delocalized than the other. The most diffuse orbital could be a typical "metallic orbital", because atom 2 is performing a metallic bond with atoms 3 and 4 through it. Only one orbital on the first atom ( atom 1 in Fig.5) is enough to form each pair of structures of the planar Li4 shown in Fig. 1 lb. On the other... 

There are at least three types of cluster expansions, perhaps the most conventional simply being based on an ordinary MO-based SCF solution, on a full space entailing both covalent and ionic structures. Though the wave-function has delocalized orbitals, the expansion is profitably made in a localized framework, at least if treating one of the VB models or one of the Hubbard/PPP models near the VB limit -and really such is the point of the so-called Gutzwiller Ansatz [52], The problem of matrix element evaluation for extended systems turns out to be somewhat challenging with many different ideas for their treatment [53], and a neat systematic approach is via Cizek s [54] coupled-cluster technique, which now has been quite successfully used making use [55] of the localized representation for the excitations. 

The delocalized n systems of I are similar to that of N3 (Table 3), while II and III have the non-degenerate delocalized orbitals shown in Table 4. II, which also contains a localized tt orbital, is the real structure of RN3 and so must be the most stable. 

It is not necessary to choose the orthogonal orbitals q> and

simple model we consider all configurations that can be constructed from the two nonbonding orbitals by any permissible electron occupancy, the energies and wave functions of the resulting states are invariant to any mixing of these orbitals. Thus, we may equally well choose them to be the most delocalized orthogonal molecular orbitals, related to the most localized ones by... 

The density functional approach naturally offers the possibility of being realized in an algorithm scaling asymptotically like the cube of the number of inequivalent atoms in the system. Most probably, this is the lower limit for the asymptotic scaling law for any delocalized orbital theory. The significantly more favorable multiplier for the cubic component in semiempirical orbital theories translates therefore to not so impressively larger clusters that can be treated as compared to what can be done using a parameter-free density functional method. 

The most famous example of such a delocalized orbital is the one in benzene, C6H6, in which 2p orbitals on all six carbons overlap forming rings of electron density above and below the benzene plane (Figure 5.7 overleaf). There are also other occupied n orbitals in benzene in which two pairs or two triplets of atoms are... 

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