Orbital most delocalized

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 ... 

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. 

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]. 

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... 

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).

FIGURE 10.28 (a) The six 2p orbitals on the carbon atoms in benzene, (bj The delocalized molecular orbital formed by the overlap of the 2p orbitals. The delocalized molecular orbital possesses pi symmetry and lies above and below the plane of the benzene ring. Actually, these 2p orbitals can combine in six different ways to yield three bonding molecular orbitals and three antibonding molecular orbitals. The one shown here is the most stable. 

In most of the cyclic (pd) systems which may be formed by use of either outer or inner d orbitals, cyclic delocalization through the d-orbital center is possible and leads to an increase in stability, but in no case will the (pd)n system alone lead to an extraordinary ground-state stabilization which can be classified as aromatic. 

The most intensively studied clusters are alkali-metal clusters [428 31], where the stability and the ionization energies have been measured as have the electronic spectra and their transition from localized molecular orbitals to delocalized band structure of solids [432, 433]. 

We shall consider first the one-electron aspects of the model. The spin space and the orbital space are both two-dimensional. We find an amusing parallel between the three canonical basis sets that can be used to span the former (adapted to the x, y, and z molecular axes) and the three that can be used to span the latter (delocalized, complex, and localized). More general basis sets in either space can be produced by applying the rotation operator for a particle of spin 1/2. Since we shall eventually deal with the exact solutions both in the spin space and in the real space (equivalent to full Cl), the choice of the one-electron basis is immaterial. In practice, as we construct the two-electron wave functions from one-electron wave functions, we have to choose the latter somehow, and this choice is frequently dictated by convenience. For instance, the two-dimensional active space of orbitals may have been defined for us by an open-shell SCF calculation on the triplet state of a series of related biradicaloids, which yielded two singly occupied orbitals in each case, but not necessarily in either the most localized or the most delocalized form, or even in similar forms for the different mole-... 

The formal analogy of the most delocalized "real" orbitals and to the spin functions and quantized with respect to the X axis, and of the most delocalized complex orbitals and to the spin functions and quantized along the y axis, is obvious from Table 1 and is emphasized by the notation chosen. The orbitals, B have the same complex phase in all space and can therefore be referred to as "real" we use quotation marks to indicate the distinction. An electron in such an orbital generates no current nor orbital magnetic dipole moment. The orbitals A, B are essentially complex and carry both a current and an orbital magnetic dipole moment. 

We see now that in the two-electron spin space, the zero-fieldsplitting two-electron terms dictate the principal axes x,y,z in the molecular framework. By the same token, in the two-electron geminal space, the two-electron repulsion terms dictate the principal axes in the space of orbital transformations and thus the "principal" orbital choices. The three directions correspond to the most localized orbital choice A, B, the most delocalized "complex choice A, B, and to the most delocalized "real" choice A defined in Table 1. Our insistence on defining the orbitals A, B as one of the principal choices from the start (the choice of the most localized as opposed to the most delocalized "real" ones was arbitrary), well before electron-repulsion terms were considered, is thus understandable in retrospect. 

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