Orbital most localized

Figure 2. As in Figure 1, but for the broken symmetry state. Iron "a" is the unique iron, and "b" labels the equivalent pair. The three orbitals marked with an asterisk show the g and u spindown orbitals on the "b" irons (whose energy difference determines B), and the spin-up orbital mostly localized on iron "a". The plot corresponds to the state with the final electron in the lower a orbital marked with an asterisk. 

In molecular orbital language, back donation (or back bonding) meana eonsiderable participation of empty ligand orbitals (usually r ) in the occupied 7r-molecular orbitals mostly localized on the metal. We are fortunate to have the simple words back donation to describe such a state of affairs. 

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 ... 

As in the case of iron chemistry, most valuable information concerning bond properties (anisotropic electron population of molecular orbitals) and local structure may be extracted from quadrupole-split Ru spectra. This has been... 

Further pragmatic moves are described in details in numerous books and reviews of which we cite the most concise and recent Ref. [82], Two further hypotheses are an important complement to the above cited theorems. One is the locality hypothesis, another is the Kohn-Sham representation of the single determinant reference state in terms of orbitals. The locality has been seriously questioned by Nesbet in recent papers [83,84], however, it remains the only practically implemented solution for the DFT. The single determinant form of the reference state in its turn guarantees that all the averages of the electron-electron interaction appearing in this context are in fact calculated with the two-electron density given by the determinant term in Eq. (5) with no cumulant. 

The Hamiltonian of valence electrons (39), in the so-called orthogonal representation (or in the most localized representation, neglecting orbital overlap) can be mapped on a tight-binding form Hamiltonian... 

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... 

If a hydrogen is described by an s orbital, the electron cloud in its neighbourhood will have spherical symmetry. Thus the circled hydrogen can be described reasonably well with s-type orbitals in the reagent or in the product. In the transition state, however, this hydrogen is simultaneously bonded to C, and C5 and loses its spherical symmetry. Its electronic density is mostly localized along the CaH and the C5H axes. Such a broken line is impossible to represent with s orbitals, but poses no problem to p orbitals. In conclusion, to describe correctly the transition state of the ene reaction, polarization orbitals are mandatory, at least for the hydrogen to be transferred. 

MO diagram for the tetrahedral [FeOJ cluster The molecular orbitals calculated for Fe3+ in the tetrahedral cluster [FeOJ-5 indicate that the antibonding e and t2 molecular orbitals, corresponding to the iron 3 d orbitals in a tetrahedral crystal field, are mostly localized on the iron atom (Sherman, 1985a). Furthermore, although allowed by symmetry, there appears to be little Fe 4p character in these orbitals, casting some doubt on the intensification mechanism of absorption bands in crystal field spectra of tetra-hedrally coordinated cations ( 3.7.1). 

In most metals there will be bands derived from the outermost s, p, and d atomic levels, leading to a system of bands, some of which will overlap as described above. Where overlap does not occur, the almost continuous energy levels of the bands are separated by a forbidden zone, or band gap. Only the outermost atomic orbitals form bands the inner orbitals remain localized on the individual atoms and are not involved in bonding. 

In this chapter, we focus on the class of reactive intermediates that bear at least two unpaired electrons diradicals and carbenes. The exact definition of a diradical is somewhat in the eye of the beholder. Salem and Rowland provided perhaps the most general, yet effective, definition—a diradical is a molecule that has two degenerate or nearly degenerate orbitals occupied by two electrons. With this definition, carbenes can be considered as a subcategory of diradicals. In a carbene, the two degenerate molecular orbitals are localized about a single carbon atom. 

Besides the mentioned aperiodicity problem the treatment of correlation in the ground state of a polymer presents the most formidable problem. If one has a polymer with completely filled valence and conduction bands, one can Fourier transform the delocalized Bloch orbitals into localized Wannier functions and use these (instead of the MO-s of the polymer units) for a quantum chemical treatment of the short range correlation in a subunit taking only excitations in the subunit or between the reference unit and a few neighbouring units. With the aid of the Wannier functions then one can perform a Moeller-Plesset perturbation theory (PX), or for instance, a coupled electron pair approximation (CEPA) (1 ), or a coupled cluster expansion (19) calculation. The long range correlation then can be approximated with the help of the already mentioned electronic polaron model (11). 

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