NBMO Coefficients

Equations (3.4)-(3.8) represent an extension of the pairing theorem to odd AH ions. 

While the pairing theorem provides a wealth of information about the MOs in AHs without the need for detailed calculation, many of the results require a knowledge of the coefficients of AOs in NBMOs. Normally, the coefficients in any one MO can be found only by a full-scale calculation of the whole set of MOs, but the NBMO is fortunately an exception. As Longuet-Higgins first showed, the coefficients in it can be found independently of the rest by a completely trivial pencil and paper procedure. 

Consider one of the unstarred atoms i in an odd AH. This will be attached by (7 bonds to at most three neighboring atomsy, k, I [see (4)]. Since the compound is alternant and atom i is unstarred, any atom attached to atom i must be starred.  

Longuet-Higgins rule states that if the NBMO coefficients of atoms 7, fc, and / are, respectively, aQj, ao, and and if the resonance integrals of the ij ik, and il bonds are, respectively, and then 

If atom i has only two neighbors or only one neighbor, the corresponding equation has only two terms or only one term, respectively. 

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Simply stated, the rule is that if the molecule can be mentally constructed (see Scheme 5.7) by joining two radical units at positions that have NBMO coefficients of (nominally) zero, as in the hypothetical formation of TME from two allyl radicals 35, then the exchange interaction approaches zero. 

Scheme 2. Calculation of the reactivity number (localization energy) for the 1-position of naphthalene according to the PMO and PMO-F method. (The denominator follows from the normalization condition, i.e. the normalized NBMO coefficients are inversely proportional to the root of the sum of squares of the unnormalized coefficients)...

Returning to the problem of aromatic substitution, we see that this is an example of the special case where R, S are odd AH s. Here R is the transition state, an odd AH, while 5 is methyl methyl can be regarded as the limiting case of an odd AH whose NBMO is a single carbon 2p AO. This of course has the same energy (zero on our scale) as a NBMO the corresponding "coefficient bM will be unity. If the NBMO coefficients of atoms r, s in the transition state are a , a0, respectively,... 

Here a is the NBMO coefficient of the AO of the methylene carbon in ArCH2. We find ... 

Compounds derived from methyl -f- pentadienate (NBMO coefficients a = 1/V3)-... 

In any nonbonding molecular orbital (NBMO), the sum of the coefficients of the atoms s adjacent to a given atom r is zero. Hence the NBMO coefficients can be calculated very easily. The benzyl radical provides a nice example. All of the non-starred atoms have coefficients of zero. We give the para atom an arbitrary coefficient a. The sum of the coefficients of the atoms adjacent to the meta carbons must be zero, so the ortho coefficients are —a. For the sum of the coefficients around the ipso carbon to cancel, the benzylic carbon coefficient must be 2a. The value for a is given by the normalization condition ... 

Find the NBMO coefficients in the following alternant hydrocarbons ... 

The first corollary on p. 36 states that the even-numbered atoms in a linear conjugated radical all have NBMO coefficients of zero. For odd-numbered atoms, the coefficients are the same in size, but alternate in sign. Thus, the coefficients at the termini are the same for radicals having An + 1 atoms, but opposite in sign for radicals containing An — 1 atoms. 

According to Eq. (26) the NBMO coefficients of an allyl radical are equal to... 

To determine the character of an even alternant cyclopolyene, one has to consider an odd alternant system obtained from the initial cycle by deleting one carbon atom. The binding of this odd AS to the methyl group results either in an even cyclic AS or in an even AS with one of its rings open. A cyclic system is more stable than any of the compounds with open chain, i.e. is aromatic, only when the NBMO coefficients of the respective atoms in the odd system have identical signs. Otherwise, the cycle is anti-aromatic. 

The FE-NBMO and HMO-NBMO coefficients of it radicals corresponding to a- and 0-substitutions in naphthalene are derived in Chart I to serve as examples. In the case of a-substitution, the FEMO model offers two possibilities for treating the effective potential at a bridge head atom adjacent to a substitution site, that is, either as a joint or a nonjoint atom. In the first case, an appendage of the free-electron pathway is considered to be pointing toward the a-position. In this chapter, these types of atoms in PAH fragments are taken to be joint positions. 

Chart I. Derivation of FE-NBMO and HMO-NBMO coefficients of tt radicals corresponding to a- and p-subsitutions in naphthalene. 

Cob is the nonbonding MO (NBMO) coefficient on the carbon atom where the oxirane ring is opened. It is calculated according to the Longuet-Higgins zero sum rule for odd alternant aromatic hydrocarbons [36,65], Incidentally, the first authors to try such NBMO coefficients of exocyclic atoms of odd-alternant PAH derivatives were Dipple, Lawley and Brookes [66] in 1968. 

We know (see p. 60) that an alternant hydrocarbon (AH) has a self-consistent field so that = 0 at all atoms therefore if we remove an electron from the NBMO to get a benzyl cation, the pasitive charge will be distributed sole over those atoms whose orbital coefficients are not zero for the NBMO. The same will be true if we add an electron to the radical and make the benzyl anion. The NBMO coefficients are clearly of signal importance since their values determine the calculated distribution of the odd electron in the radical and the charges in the cation and anion. For the benzyl radical the NBMO may be rendered schematically as follows ... 

The NBMO coefficients can be used in approximate calculations of a-electron energies by a method developed by Dewar and by Longuet-Higgins. Consider a conjugated hydrocarbon (RS)with an even number of carbons and a tt-electron system that might be considered to be the result of joining up two odd AH radica (R and S) by one or more a bonds. The a-electron system of butadiene would be the result of linking up the a systems of allyl and methyl, while benzene would result from pentadienyl and methyl or two allyls. 

The of RS might be expected to be related to the product of the coefficients of the atomic orbitals of R and S at the point of joining up the larger the coefficients the more bonding to be expected. Dewar and Longuet-Higgins specifically propose that the NBMO coefficients may be used for this purpose with the aid of the following equation ... 

H ere, and Cq refer to the NBMO coefficients of R and S at the jimction points-of a-electron systems. For butadiene, we have... 

In other words, to a first approximation, the sum of the NBMO coefficients at starred atoms adjacent to a given unstarred atom vanishes. 

This rule enables us to find the NBMO coefficients in an odd AH very quickly and easily. The procedure is indicated for a-naphthylmethyl in Fig. 

The last step is to determine a. To do this, we normalize the MO. Since the square of each NBMO coefficient is the fraction of the MO composed of the particular AO, the sum of the squares must add up to unity. In this case. 

A complication arises if at some step in the calculation we encounter an unstarred atom with three neighbors, two of whose NBMO coefficients are unknown. This situation is illustrated in Fig. 3.4 by a calculation for a-naphthylmethyl using a different initial atom. In Fig. 3.4(b) the unstarred atom has two neighbors with two undetermined NBMO coefficients. When this happens, we simply call one of the unknowns b (Fig. 3.4c) and continue. There will always be enough equations to determine b in terms of a. Here b is determined in Fig. 3.4(e). Applying Longuet-Higgins rule to the indicated unstarred atom, we find... 

FIGURE 3.4. An alternative calculation of NBMO coefficients for a-naphthylmethyl. 

Note that the absolute signs of the NBMO coefficients are as usual arbitrary. The orbitals ij/ and — ij/ are equivalent multiplication of all of the NBMO coefficients by — 1 leaves the wave function unchanged. 

FIGURE 3.5. Calculation of NBMO coefficients in benzo[a]perinaphthenyl. 

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