Corrected structure count

The difference between the even and odd Kekule structures is called algebraic structure count ASC [Wilcox Jr, 1968 Wilcox Jr, 1969] or corrected structure count CSC [Herndon, 1973a Herndon, 1974b] ... 

To check that the line structures represent the correct substances, count the number of intersections and line ends, which should equal the number of C atoms in the compound. The first line structure has five, the second has three, and the third has five, matching the chemical formulas. 

These are the correct structural formulas. Now it is a simple matter to count the number of atoms of each element and write the chemical formulas of the compounds C2 H4 CI2, C4 Hg Og, and C3 H4 O. ... 

To apply Equation, we need complete inventories of the bonds present in all the reagents that participate in the reaction. We build such inventories from sfructural formulas, from which we count the number of bonds of each type. The procedures for generating sfructural formulas are not introduced until Chapter 9, so we provide the correct structures for examples in this chapter. 

These structures are incorrect. To show why none is a valid Lewis structure, count the electrons around the C and N atoms. In the left structure, neither atom satisfies fhe octet rule. In the center structure, C has eight electrons but N has only four. In the right structure, the opposite is tme. Remember that both atoms must simultaneously satisfy the octet rule. Therefore, the correct arrangement is... 

The resonance structure count (RSC) in general involves a procedure for calculating the correct number of contributing resonance structures without actually writing them all down. We will not detail this procedure here. Once the procedure is completed, the resonance energy is calculated as the sum of four types of resonance transformations, which relate each resonance structure to another ... 

Method of the Pfaffian to a Skew-Symmetric Matrix, Consider [12]annulene (see Chart 2), which is a [4k]—cycle, in contrast to [lOjannulene (Chart 1) being a [4k + 2]-cyde. The Kekule structure count is again K = 2, but the adjacency matrix (as well as its submatrices according to the partitioning shown in Chart 2) appear to be singular. Hence eqn. (13) would give zero. A correction is needed. 

The procedure outlined applies to all nonconvex lattices, the inner dual of which is a nonconvex polygon. In Figure 2.9, we have illustrated the Kekule valence structure counts for peri-condensed benzenoid lattices considered by Clar in his booklet The Aromatic Sextet [29]. This time, we only show the result in the top benzene ring leaving it to the readers to flu in the empty benzene rings and verify, as an exercise, that the count shown is correct In the next section, we will briefly outline the count of Kekule valence structures for a more general class of benzenoid hydrocarbons. 

Step 4 The total count here is 26. This does not match the count in step 1, which is 24. Therefore this cannot be the correct structure, and we need to go on to step 5. 

Step 4 The total count in the drawing in Figure 6.16b is 26. Two electrons must by dropped for the count to equal that in step 1 (24). Therefore the drawing in Figure 6.16b is not a correct structure. 

Step 5 Placing a double bond between the carbon atom and an oxygen atom produces the result in Figure 1.16c. This is the correct structure since the counts now match at 24 electrons. 

Step 4 Forty electrons matches the count in step 1. Thus this is the correct structure. 

Already in the seminal paper by Dewar and Longuet-Higgins [15], it was established that the Ai -regularity does not hold for all polycyclic conjugated n-electron systems, and that corrections for the parity of Kekule structures need to be taken into account. This became the concept of algebraic structure count [22, 23], which was later shown to be not applicable to all non-altemant conjugated hydrocarbons [24, 25]. The fortunate fact is that all Kekule structures of benzenoid hydrocarbons have the same parity [20], which has the consequences that all Kekule-structure-based models work best for benzenoid systems. Also in this... 

Proton counting of the various groups to give the correct structural formula... 

There is some confusion in using Bayes rule on what are sometimes called explanatory variables. As an example, we can try to use Bayesian statistics to derive the probabilities of each secondary structure type for each amino acid type, that is p( x r), where J. is a, P, or Y (for coil) secondary strucmres and r is one of the 20 amino acids. It is tempting to writep( x r) = p(r x)p( x)lp(r) using Bayes rule. This expression is, of course, correct and can be used on PDB data to relate these probabilities. But this is not Bayesian statistics, which relate parameters that represent underlying properties with (limited) data that are manifestations of those parameters in some way. In this case, the parameters we are after are 0 i(r) = p( x r). The data from the PDB are in the form of counts for y i(r), the number of amino acids of type r in the PDB that have secondary structure J.. There are 60 such numbers (20 amino acid types X 3 secondary structure types). We then have for each amino acid type a Bayesian expression for the posterior distribution for the values of xiiry. 

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