Photochemistry_of_Organic

tions ci and cj of the unperturbed system, making use of the parameter list given in Equation 4.11 for the elements hmn in the AO basis. For example, for a single inductive perturbation at atom m, dam, we obtain hij ¼ hji ¼ cimcjmdam and for a single resonance

Equation 4.16 First-order perturbation of HMO wavefunctions

Let us look at a simple case to illustrate the predictions of Þrstand second-order perturbation and compare them with the full HMO calculation for the perturbed system. We choose ethylene as the parent system and look at inductive perturbation induced by introduction of a substituent at atom 1 or by replacing atom 1 by a heteroatom such N or O. The exact solution is obtained by solving the 2 2 secular determinant (Equation 4.17).

Equation 4.17 Exact HMO calculation for and inductively perturbed ethylene

To illustrate the accuracy of Þrst- (Equation 4.13) and second-order (Equation 4.15) calculations, their results are compared with the exact solution (Equation 4.17) in Figure 4.7. The perturbation da1 is varied from 0 to 3b. The parameters for N (ethylenimine) and O (formaldehyde) are dam/b ¼ 0.5 and 1.0, respectively. If the term (dam/b)2 in the discriminant of Equation 4.17 is much smaller than 4 and can be neglected altogether, then Equation 4.17 is equal to the result of Þrst-order perturbation theory (Equation 4.13). If (dam/b)2 is smaller than 4 but non-negligible, the square root of Equation 4.17 can be expanded, (1 þ x)½ ¼ 1 þ 1/2x þ . . ., and we have the result of second-order perturbation (Equation 4.15). Thus, the results of Þrst-order perturbation are adequate for even a substantial perturbation such as the replacement of CH2 by O in formaldehyde.

However, it must be realized that this result is due to the large difference in energy

Clearly, Þrst-order perturbation theory for degenerate orbitals cannot be done as described above, because the choice of any linear combination of degenerate

148 Quantum Mechanical Models

4.4 Symmetry Considerations

Recommended textbooks.134,273Ð275

Symmetry considerations go a long way in predicting, inter alia, properties of wavefunctions, electronic states and transitions between molecular states. The mathematical treatment of symmetry operations is part of a special branch of mathematics known as group theory. Only a brief outline of some of its most important applications is given here.

A symmetry operation is a movement performed with a molecule Ð such as the reßection of its nuclear coordinates by a mirror plane s or rotation around an axis of symmetry C Ð that leaves the molecule unchanged. If a molecule is symmetrical, that is, if

performed on ethene (Figure 4.9). Group theory includes an eighth operation, the identity operation E; it does nothing, like the identity operation Ômultiplication by 1Õ in algebra. Any molecule or wavefunction will remain unchanged under the identity operation.

Figure 4.9 Symmetry operations

There are three twofold symmetry axes denoted C2(x), C2(y) and C2(z), which coincide with the coordinate axes x, y and z. The origin of the coordinate system shown in Figure 4.9 is an inversion centre i of the molecule, that is, the molecule remains unchanged when the nuclear coordinates of the atoms m, xm, ym and zm, are replaced by xm, ym andzm. Finally, there are three planes of symmetry, s(xy), s(xz) and s(yz), which coincide with the three planes spanned by a pair of coordinate axes [s(xy) is marked as a shaded plane]. Any molecule exhibiting the same set of symmetry operations belongs to the same point group, which is labelled by the Sch€onflies symbol D2h in the present case. The complete set of symmetry operations of an object deÞnes the point group, to which it belongs. A ßow chart to determine the point group of any molecule on the basis of its known (or assumed) 3D structure is shown in Figure 4.10. The combined operation of rotation and reßection through a plane perpendicular to an axis is an improper rotation; n- fold improper rotational axes are designated by the symbol Sn. Note that S1 s and S2 i.

The symbol s for symmetry planes is usually characterized by the index n (ÔverticalÕ, coincident with the rotational axis of highest order n) or d (ÔdihedralÕ or ÔdiagonalÕ, coincident with the rotational axis of highest order n and bisecting the angle between

We now return to first-order perturbation theory for degenerate orbitals (Section 4.3). As any linear combination of two degenerate orbitals ci and cj is equally valid, we set up a trial wavefunction function ck ¼ aki ci þ akj cj and we have to solve the secular Equation 4.18. The eigenvalues of the unperturbed system will be equal for all linear combinations,

Equation 4.18 First-order perturbation of two degenerate orbitals

If we choose the linear combination of the degenerate orbitals such that one is symmetric and the other antisymmetric with respect to a symmetry element that is retained in the perturbed molecule, then the off-diagonal elements will vanish, hkl ¼ hlk ¼ 0, because integration of any function over the complete range of its variable x vanishes if the

Ð

function is antisymmetric: f(x)dx ¼ 0 (Figure 4.11). A function is said to be antisymmetric (a) if f( x) ¼ f(x). The product of two antisymmetric functions is symmetric (s); we write a a ¼ s; similarly, s s ¼ s but a s ¼ s a ¼ a.

Figure 4.11 Integral of an antisymmetric function

Having chosen a symmetry-adapted basis set ck and cl, the 2 2 determinant (Equation 4.18) is diagonal. The diagonal elements are the solutions, d«ð1Þ ¼ d«ðk1Þ and d«ðl 1Þ, with the associated symmetric and antisymmetric wavefunctions ck and cl. The Þrst-order perturbations of the energies «ðk0Þ and «l(0) are thus simply those given by the equations for Þrst-order perturbation of non-degenerate orbitals (Equations 4.13 and 4.14). The same result would have been obtained by solving the 2 2 non-diagonal determinant (Equation 4.18) obtained with any other linear combination. The equations for first-order perturbation can be applied to degenerate MOs of the parent system when the linear combination of the degenerate orbitals is chosen such that it is adapted to the symmetry of the perturbed system.

As an example, consider the effect of bond length alternation in cyclobutadiene (3). Let us assume that the molecule relaxes from a quadratic structure with an average bond length of 140 pm to one of the Kekule structures with the double bonds connecting, say, atoms 1Ð2 and 3Ð4 and that the length of the double bonds is reduced to 130 pm and that of the single bonds is increased to 150 pm. From Equation 4.12, we get b12 ¼ 1.22b and b23 ¼ 0.82b, that is, db12 ¼ db34 ¼ 0.22b and db23 ¼ db41 ¼ 0.18b. We choose the set of

Figure 4.12 Effect of bond length alternation in cyclobutadiene (3)

nonbonding orbitals c02 and c03 shown in Figure 4.12 (see also Figure 4.3), because these have the proper symmetry to deal with the perturbation. First-order perturbations are

additive; adding up the contributions (Equation 4.14) by each bond gives d«ðþ1Þ ¼ 0:40b and d«ð1Þ ¼ 0:40b.

Let us apply the same procedure to construct a qualitative MO diagram for the dioxygen molecule starting from the valence shell AOs of the separated O atoms, 2s, 2px, 2py and 2pz. The MOs are constructed by choosing symmetry-adapted linear combinations of the basis AOs, for example s ¼ (2s1 2s2)/H2, pz ¼ (2pz1 2pz2)/H2, and so on. The overlap between the 2pz orbitals is largest, leading to the large energy difference between the 3s and 4s MOs. The well-known diagram illustrated in Figure 4.13 results, showing that the HOMO 2p is twofold degenerate and holds only two electrons. Hence the ground state of dioxygen is a triplet state according to Hund s rule.

Figure 4.13 MO diagram for dioxygen

4.5 Simple Quantum Chemical Models of Electronic Excitation

Recommended textbook.15

Let us assume that organic molecules in their electronic ground state are reasonably well represented by a closed-shell electronic configuration built by Þlling electrons pairwise

The wavelength of a transition is directly proportional to the inverse of its transition energy; hence HMO theory predicts that the absorption wavelengths of linear polyenes are linearly related to the number of carbon atoms n (Equation 4.22).

n þ 1 l0--0 hc 2bp

Equation 4.22 HMO prediction for the wavelength of the HOMO–LUMO transition of linear polyenes

In Figure 4.15, the observed wavelengths l0–0 for eight all-trans linear polyenes276 ( ) are plotted versus n. Although l0–0 does increase with increasing length of the polyenes, the result is disappointing, because the observed relationship is clearly nonlinear; the absorption wavelengths reach a plateau for very long polyenes. Extrapolating linearly from the first three members of the series, we would predict that carrots are blue (b-carotene is an all-trans polyene with 11 conjugated double bonds).

Figure 4.15 Absorption bands of conjugated linear chains. The compound series are defined in the text together with the corresponding data sources

The deviations from Equation 4.22 can be traced to the alternation of bond lengths of linear polyenes. Even stronger effects of bond length alternation are evident in the series of polyacetylenes277 (þ ). We could use Equation 4.12 to adjust the resonance integrals b to the bond lengths of the essential single and double or triple bonds. Rather than introducing two new parameters using Equation 4.12 Ð which would Þx the problem Ð let us see whether the HMO prediction of Equation 4.22 works for linear conjugated systems that do not exhibit bond length alternation. Indeed it does. The symmetrical cyanines278 (&), which possess two equivalent resonance structures Me2NÐ(CH¼CH)(n/2Ð3) CH¼Nþ Me2

Systematic studies of the absorption spectra of benzenoid aromatic hydrocarbons, mostly done by Clar in the 1930s and 1940s,279 showed that these compounds exhibit four types of UVÐVIS absorption bands, which are shifted in a regular way along a homologous series such as the linear acenes (benzene, naphthalene, anthracene, . . .) (Figure 4.16).

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Figure 4.16 Absorption spectra of the linear acenes280,281

Clar labelled the four bands a, p, b and b0 (in order of increasing energy for naphthalene, the b0 band of which is slightly above 5 mm 1) and empirically assigned them in well over 100 benzenoid hydrocarbons on the basis of their characteristics mentioned below and their regular shifts in homologous series. The a band is weak, log(«/ M 1 cm 1) 2Ð3, with sharp, irregular vibronic structure. Its position is insensitive to temperature, solvent polarity and polarizability. Its intensity, but not its position, increases strongly upon substitution, for example by halogens or alkyl groups. The p band is of moderate intensity, log(«/ M 1 cm 1) 4. It exhibits a distinct vibrational progression of 1400 cm 1 and is strongly red shifted in polar solvents and with increasing size of the hydrocarbon. The strong b band, log(«/M 1 cm 1) 5Ð6, is more diffuse and less sensitive to solvent and substitution. Finally, the b0 band, log(«/M 1 cm 1) 4Ð5, is less prominent and not always clearly discernible, because further transitions appear in the far-UV region. The a band is clearly seen as the Þrst absorption band of benzene and naphthalene, then hidden under the stronger p band in anthracene and tetracene, until it resurfaces, showing minute, relatively sharp features in the window between the p and the b band in pentacene and hexacene.

The p bands correlate well with HOMOÐLUMO energy gaps predicted by H€uckel theory (Equation 4.16) (Figure 4.17). Linear regression gives b/hc ¼ 2.2 0.1 mm 1. The points (*) for azulene (1) and cycl[3.3.3]azine (2) fall completely off the regression line (cf. Case Study 4.1). They are shown to emphasize that the correlation holds only for benzenoid hydrocarbons.

Excellent correlation was also found between Þrst ionization potential I1,v and the HOMO energies obtained by a slightly modiÞed H€uckel calculation for a large series of benzenoid hydrocarbons (Koopmans theorem, Section 4.7). The ionization potentials I1,v are also strongly correlated with the energies of the 1La bands, as would be expected on the basis of the pairing theorem (Section 4.6). A few outliers were noted, but a reinvestigation