Delocalization tails

In practice, the NBO program labels an electron pair as a lone pair (LP) on center B whenever cb 2 > 0.95, i.e., when more than 95% of the electron density is concentrated on B, with only a weak (<5%) delocalization tail on A. Although this numerical threshold produces an apparent discontinuity in program output for the best single NBO Lewis structure, the multi-resonance NRT description depicts smooth variations of bond order from uF(lon) = 1 (pure ionic one-center) to bu 10n) = 0 (covalent two-center). This properly reflects the fact that the ionic-covalent transition is physically a smooth, continuous variation of electron-density distribution, rather than abrupt hopping from one distinct bond type to another. 

How do LMOs differ from NBOs Each occupied LMO (Ua) can be written in terms of a parent Lewis-type NBO (Ua) with a weak delocalization tail from other non-Lewis NBOs (fV),... 

The strength of the Fajb interaction and its variations with distance and orientation can be conveniently visualized in terms of the overlap of 7ra and 7tb NBOs, on the basis of a Mulliken-type approximation (cf. Eq. (1.34)). As an example, the top two panels of Fig. 3.38 compare the overlapping 7ta-7tb orbital contours for trails 1 and cis 2 isomers of butadiene. As shown in Fig. 3.38, the overlap in the cis isomer 2 (S = 0.2054) is slightly weaker than that in the trans isomer 1 (S = 0.2209), due to the unfavorable orientation of the 7ta across the nodal plane of the 7tb in the latter case. Consistently with the weaker 7ta-7tb overlap, the JtA F nh ) interaction is less, namely 0.0608 in 1 versus 0.0564 in 2. The delocalization tail of the 7fa NEMO is correspondingly less than its value in the trans isomer... 

Exercise Use perturbation theory to estimate the 7Tb delocalization tail in the 7ta NLMOs of 1 and 2. 

It is prerequisite to define localized, diabatic state wave fimctions, representing specific Lewis resonance configurations, in a VB-like method. Although this can in principle be done using an orbital localization technique, the difficulty is that these localization methods not only include orthorgonalization tails, but also include delocalization tails, which make contribution to the electronic delocalization effect and are not appropriate to describe diabatic potential energy surfaces. We have proposed to construct the locahzed diabatic state, or Lewis resonance structure, using a strictly block-localized wave function (BLW) method, which was developed recently for the study of electronic delocalization within a molecule.(28-3 1)... 

Thus, the Coulson—Fischer representation keeps the simplicity of the covalent picture while treating the covalent—ionic balance by embedding the effect of the ionic terms in a variational way, through the delocalization tails. The Coulson—Fischer idea was subsequently generalized to polyatomic molecules and gave rise to the GVB and SC methods, which were mentioned in Chapter 1 and will be discussed later. 

The resulting LBOs are fairly localized, one on Ci — C2 the other on C3—C4. Elowever, the delocalization tails are significant even though we used Hiickel orbitals. These large localization tails reflect the fact that butadiene has some conjugation between the n-bonds and in terms of VB theory is describable by a linear combination of the major Kekule structure and the minor long bond structure. 

All the VB methods that deal with semilocalized orbitals use a generalization of the Coulson—Fischer idea (12), whereby a bond is described as a singlet coupling between two electrons in nonorthogonal orbitals that possess small delocalization tails resulting from the variational orbital optimization. Albeit formally covalent, this description implicitly involves some optimal contributions of ionic terms, as a decomposition of the wave function in terms of pure AO determinants would show (see Eqs. 3.5 and 3.6). For a polyatomic... 

The BDOs have the advantage of allowing an unambiguous correspondence between the mathematical expression of a VB structure and the associated Lewis structure. On the other hand, owing to the delocalization tails in these orbitals, neither OEOs nor BDOs allow the distinction to be made between the... 

BDO Bond distorted orbital. An orbital that is localized on one atom with a small delocalization tail on an atom with which it shares a bond. 

CF orbitals Coulson-Fisher-type orbitals. These are semilocalized AOs, also called overlap enhanced orbitals (OEOs) which are localized on a given center, but have small delocalization tails on other centers. Special cases of CF AOs are BDOs (see above). 

HAO Hybrid atomic orbitals that are strictly localized on a single atomic center. The HAOs have no delocalization tails. 

An important feature of the BOVB method is that the active orbitals are chosen to be strictly localized on a single atom or fragment, without any delocalization tails. If this were not the case, a so-called "covalent" structure, defined with more or less delocalized orbitals like, e.g., Coulson-Fischer orbitals, would implicitly contain some ionic contributions, which would make the interpretation of the wave function questionable [27]. The use of pure AOs is therefore a way to ensure an unambiguous correspondence between the concept of Lewis structural scheme and its mathematical formulation. Another reason for the choice of local orbitals is that the breathing orbital effect is... 

Still another measure of electron delocalization can be seen in the forms of natural localized molecular orbitals (NLMOs).i By construction, each NLMO remains as close as possible to a parent NBO but includes the weak delocalization tail necessary to preserve exact double occupancy. A determinant of doubly occupied NLMOs is therefore unitarily equivalent to the standard MO determinant, and the Lewis-type NLMOs are able to describe all observable properties of the system just as well as canonical MOs. The perturbative expression (Equation 7.6a) approximates the form of the Lewis-type NLMO a B with parent NBO and weak delo-... 

Adding the ionic structures and transforming the orbitals, one obtains the same in-out and out-in hybrids but each has now a small delocalization tail on the other atom (as shown in Figure 6.6c) ... 

If a vector space representation of electronic states is chosen, that is, a basis-set expansion, two types of basis sets are needed. One for the many-electron states and one for the one-particle states. For the latter, two choices became popular, the molecular orbital (MO) [9] and valence bond (VB) [10] expansions. Both influenced the understanding and interpretation of the chemical bond. A bonding analysis can then be performed in terms of their basic quantities. Although both representations of the wave function can be transformed (at least partially) into each other [11,12], most commonly an MO analysis is employed in electronic structure calculations for practical reasons. Besides, a VB description is often limited to small atomic basis sets as (semi-)localized orbitals are required to generate the VB structures [13]. If, however, diffuse functions with large angular momenta are included in the atomic orbital basis, a VB analysis suffers from their delocalization tails. As a consequence, the application of VB methods can often be limited to organic molecules. 

Similar to (pa and cpb of the Kahn-Briat model, the atomic-like orbitals of the HTH model have the largest amplitudes on the magnetic centers, but in contrast fa and fb show delocalization tails on the ligands to ensure the orthogonality between them. Therefore, in general fa and fb are slightly more delocalized than the nonorthogonal (pa and (pb. 

Fig. 5.5 Magnetic orbitals of gemde (left) and ungerade (right) symmetry with important delocalization tails on the ligands...

Fig. 5.8 Top direct (through space) hopping between two magnetic centers by tat with strongly locaUzed atomic orbittils Bottom effective (through Ugand) hopping by t"fj with self-consistently optimized magnetic orbitals, which have delocalization tails on the (bridging) ligands...

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