Canonical structures equivalent

The bonding in the parent benzophospholide features an extended r-de-localisation around the ring perimeter which is expressed by dissipation of the excess charge on a larger number of canonical formulae such as IIA-IIE (and their mirror images of which all but IIB were omitted). Phosphonio-substituents in 1- and 3-positions increase the weight of the symmetry equivalent canonical structures IIB, IIB. As before, this implies a partial ti-bond localisation and a reduction of the energy of the isodesmic reaction (2) in Fig. 4 [16]. 

The generalized two-particle HF equations are seen to have a structure equivalent to their single-particle counterparts, exhibiting the presence of a direct term, written in terms of the density, and an exchange term. As the canonical HF equations, the present expressions do not contain spurious self-interaction terms. However, unlike the single-particle equations, they allow the determination of fully correlated two-particle states removing to this extent the most basic objection to the HF method. 

Indeed, canonical structure C (Scheme 1) for 11 is equivalent to the ylide in Eq. (9), and may explain the analogous rearrangement of 11. The rearrangement in Eq. (8) is the only reported route to obtain complexes 10, except 10a. The structure assignment was confirmed by a crystal structure analysis of 10c (Table V).15 40... 

The contribution of the canonical structure A is increased by carbene substituents R with electron-attracting abilities. This is equivalent to a stabilization of the transition state of the reaction. The greater this stabilization, the greater the discrimination of a certain carbene, i.e. the discrimination increases in the above sequence from left to right. 

Aromatic hydrocarbons such as naphthalene (19) also undergo electrophilic substitution, although now not all ring positions of the parent hydrocarbon are equivalent. Nitration occurs almost exclusively in the 1- or a-position of naphthalene. Consideration of the contributing structures to the hybrid carbocation indicates why this is so. For a-attack, the canonical structures include 20, 21 and 22. Whereas in 20 and 21 the stable aromatic sextet is preserved, in 22 the aromaticity is disrupted. However, for attack at the 2- or j3-position, only one structure, 23, can... 

Note that the positively charged oxonium ion intermediate has a canonical structure that is the alkyl equivalent of a protonated carbonyl species. The subsequent attack by the alcohol molecule and the loss of a proton are both fast steps. 

In Chap.6 the atomic-sphere approximation is introduced and discussed, canonical structure constants are presented, and it is shown that the LMTO-ASA and KKR-ASA equations are mathematically equivalent in the sense that the KKR-ASA matrix is a factor of the LMTO-ASA secular matrix. In addition, we treat muffin-tin orbitals in the ASA, project out the i character of the eigenvectors, derive expressions for the spherically averaged electron density, and develop a correction to the ASA. 

When the canonical forms all contribute equally to the bonding, they are called equivalent canonical forms when they contribute unequally, they are nonequivalent canonical forms. Therefore, the structures shown in Figure 6.5 for ozone represent equivalent canonical forms. They are related to each other by a symmetry element—in this case, a twofold rotational axis that passes through the central oxygen. As a result, each canonical structure contributes exactly 50% to the resonance hybrid. The double-headed arrow is used to indicate the concept of resonance. The carbonate ion, shown in Figure 6.6, is another example of equivalent canonical forms only in this case, each canonical structure contributes 33.3% to the resonance hybrid. 

For example, rotation of the ozone molecule by 180 around an imaginary axis that bisects the 0-0-0 bond angle leads to an equivalent configuration for the molecule. In each case, there is one 0-0 single bond and one 0=0 double bond in the contributing canonical structures the only difference between them is the placement of the single and double bonds. Likewise, in the carbonate ion, rotation of each subsequent canonical structure by 120° or 240° leads to a structure that is identical to the original. As we see in a later chapter, symmetry plays an important role in the structure and physical properties of molecules. 

The canonical structures of nonequivalent resonance forms, on the other hand, are not related by a symmetry element. Consider the three possible Lewis structures for SCN shown in Figure 6.7. There is no rotational axis about which the canonical forms can be interchanged to form an equivalent configuration. In the case of nonequivalent canonical structures, the weighting is unequal and some of the canonical structures will make a larger contribution to the resonance hybrid than will others. 

Sketch any feasible canonical structures for each of the following molecules or ions (a) N3-, (b) SO32-, (c) SOCIj, (d) XeOFj, (e) NOj", (f) SOjClj, (g) SCN", (h) POF3, (i) SNF3, And (j) lOjFj". Indicate the FCs on each atom in every canonical structure. Then circle the canonical structure that is the largest contributor to the resonance hybrid. If there are equivalent canonical forms, circle all of them. 

In systems of fused six-membered aromatic rings, the principal canonical forms are usually not all equivalent. Structure 36 has a central double bond and is thus... 

The efficiency of translation driven by IRES elements is often significandy lower than that of canonical translation (e.g., Fig. 6.1C) and, thus, control experiments may be needed to demonstrate that genuine IRES activity is measured. This can be done by inserting a stable stem-loop structure into the mRNA 5 UTR, upstream of the IRES element, or by comparing the translational output of a reporter mRNAs with the wild-type IRES to that of the equivalent transcript harboring a mutated, inactive IRES element (which should be much lower Humphreys et al, 2005 Wilson et al, 2000). 

The many-body perturbation theory [39] [40] [41] was used to model the electronic structure of the atomic systems studied in this work. The theory developed with respect to a Hartree-Fock reference function constructed from canonical orbitals is employed. This formulation is numerically equivalent to the M ler-Plesset theory[42] [43]. 

It is a wide-spread assumption that the electronic structure of I can be rationalized by using either Forster-Coulson-Mofitt orbitals23 24 or Walsh orbitals25, which are considered to represent equivalent orbital sets and to lead to similar descriptions of bonding in the C3 ring of 1. The Walsh orbitals have attracted special attention, in particular among organic chemists, since they seem to be close to the canonical (delocalized) SCF MOs of 1 and seem to explain... 

Now we are ready to start the derivation of the intermediate scheme bridging quantum and classical descriptions of molecular PES. The basic idea underlying the whole derivation is that the experimental fact that the numerous MM models of molecular PES and the VSEPR model of stereochemistry are that successful, as reported in the literature, must have a theoretical explanation [21], The only way to obtain such an explanation is to perform a derivation departing from a certain form of the trial wave function of electrons in a molecule. QM methods employing the trial wave function of the self consistent field (or equivalently Hartree-Fock-Roothaan) approximation can hardly be used to base such a derivation upon, as these methods result in an inherently delocalized and therefore nontransferable description of the molecular electronic structure in terms of canonical MOs. Subsequent a posteriori localization... 

Therefore an entirely different approach is preferred. The aforementioned TRG I is used to elaborate trees of reactions, pathways from EM(B) and EM(E) until they merge. Since the reaction pathways have an orientation that is enforced by the transition tables, the latter must be reflected about their main diagonals when the trees are originating from the product side. The chemical constitution of the species generated is represented in canonical form. Thus the identity of any two species is immediately noticed. This does not only eliminate redundancies, but, more importantly, indicates any nodes at which the two trees merge. In this context the distinct resonance structures of a molecule and its tautomeric forms can be treated as equivalent. 

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