Charge Distributions from HMOs

Now that we have a method that provides us with orbitals and orbital energies, it should be possible to get information about the way the rr-electron charge is distributed in the system by squaring the total wavefunction xlr . In the case of the neutral allyl radical, 

For most physical properties of interest, we need to know the probability for finding an electron in a three-dimensional volume element dv. Since the probability for finding an electron in cfu is the sum of the probabilities for finding each electron there, the one-electron density function p for the allyl radical is 

To find out how the tt charge is distributed in the molecule, let us express p in terms of AOs. First, we write (j and separately  

one electron in (pi shows up, upon integration, as being carbon 1, at carbon 2, and at carbon 3. We say that the atomic tt-electron densities due to an electron in d i are j, at Ci, Ci, and C3, respectively. If we accumulate these figures for all the electrons, we arrive at a total tt-electron density for each carbon. For the allyl radical. Table 8-1 shows that each atom has a 7r-electron density of unity. 

Generalizing this approach gives for the total tt-electron density qt on atom i 

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Fig. 11. Net w-charges in thymine calculated by different methods (from top to bottom w-HMO,140 wSCF MO,388 CNDO/2,214 and nonempirical219,220,384). For the nonempirical charge distributions in anionic forms of thymine, see ref.364. The symbols NH, CH, or CH3 indicate that the charge for the whole group of atoms is given.

The ring proton chemical shifts (8 values) of l,2,4-triazolo[l,5-a]pyrimi-dine derivatives are in the order H-7 > H-5 > H-2 > H-6 (64CPB204).The charge densities determined from proton chemical shifts showed a remarkably good correspondence with the charge distributions calculated by the simple Htlckel Molecular orbital (HMO) method (64CPB204). HMO calculations for all possible tautomeric forms of the isomeric triazolopyrimidin-5(7)-ones were also performed (88M341). 

If the transition considered is the HOMO LUMO transition of an alternant hydrocarbon, then first-order theory predicts that inductive perturbation will have no effect at all, because for = fo as a consequence of the pairing theorem. Small red shifts are in fact observed that can be attributed to hyper conjugation with the pseudo-7t MO of the saturated alkyl chain.290 On the other hand, alkyl substitution gives rise to large shifts in the absorption spectra of radical ions of alternant hydrocarbons whose charge distribution is equal to the square of the coefficients of the MO from which an electron was removed (radical cations) or to which an electron was added (radical anions), and these shifts are accurately predicted by HMO theory.291... 

Efforts have been made to improve the HMO method by taking account of molecular 7t charge distribution. Suppose that we carry out an HMO calculation on a nonaltemant molecule and find an electron density of 1.2 at one carbon and 0.8 at another. It is reasonable to argue that a tt electron at the latter carbon is more strongly bound because it experiences less repulsion from other tt electrons there. We can try to account for this by making a at that atom more negative. Thus, we could take... 

These approximations are called the Hiickel approximation. The molecular orbital theory using LCAO and Hiickel approximations is called the Hiickel molecular orbital theory (HMO). HMO gives nearly correct and reasonable results when the distribution of the charge density does not largely deviate from homogeneity. However, HMO calculations cannot... 

One of the deficiencies of the MO methods, especially the simple ones, is that they tend to exaggerate uneven distribution of electrons in a molecule and thus make it more polar (with a higher dipole moment) than it actually is. The result is that dipole moments which are based on charge densities obtained from eigenfunctions of the MO approximations are usually considerably higher than the actual experimental dipole moments. Two old, well-known examples of theoretical dipole moments obtained by the HMO method are fiilvene (11) whose calculated dipole moment is 4.7 D [72-74] and the experimental value is 1.2 D [68], and azulene (15), with a calculated dipole moment of 6.9 D [72] and the experimental value of 1.0 D [68]. 

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