Perimeter model

The technique of X-ray crystallography has been, and will remain, indispensable for the determination of the unusual structures of S—N compounds. A more recent development is the application of N NMR spectroscopy in S—N chemistry. Despite the necessity to employ N-enriched materials for these studies, the judicious application of this technique in both structural determinations and in monitoring the progress of reactions will undoubtedly accelerate the progress of the subject. The advent of MCD spectroscopy and the use of the perimeter model have also enhanced our understanding of the electronic structures of cyclic S—N molecules. Rapid advances in this area are to be expected. 

Both singlet and triplet states are generated by the orbital promotion of an electron, n- -it transitions are totally allowed. These energy values can also be calculated from HQckel molecular orbital (HMO) method. For benzene, the free electron perimeter model has been found to be useful. The energy levels and nodal properties of benzene molecule are given in Figure 2.19. 

Ethylene Vffigjtal65 nm Ciuax l 5 X104 Benzene (perimeter model) Afiu> g2S6 nm emu 1.6x 10 Formaldehyde Amax=304 nm max 18... 

Modified Notation.—The Platt notation is applied mainly to aromatic molecules and based on the conceptually simple perimeter model description of electronic excitations (7). Ground states are labeled A, the excited states involved in certain very high intensity transitions are labeled B and the excited states produced in partially forbidden transitions (i.e., those in which selection rules are violated) are labeled L and C. The notation is derived from selection rules appropriate for imaginary monocyclic aromatic systems. States to which transitions are forbidden because of a large change in angular momentum are L states. Transitions to C states are parity forbidden that is, they violate the g g, u u selection rule. In common aromatics other than benzene these selection rules break down and transitions to L and C states occur but at lower intensities relative to B states. 

Alternatively, the porphyrin ring can be constructed starting from [16]annulene. In the first step, two electrons are added to form the corresponding [16]annulene dianion, which is transformed into porphyrin Cl") by adding bridges and heteroatoms. Unlike the structure derived from [18]annulene, the dianion-based model has a fourfold symmetry, and was considered suitable for the description of metal complexes [23] (see Sect. 2.3.2). In yet another approach [24], based on the so-called perimeter model, the porphyrin macrocycle is derived from the [20]annulene dication (I "). Both the [16]- and [20]annulene models were employed to describe electronic absorption spectra and magnetic circular dichroism of porphyrinoids [24, 25],... 

Several qualitative models, e.g. Platt s ring perimeter model [88], Clar s model [89] and Randic s conjugated circuits model [90-92] have either been or are frequently used for the rationalisation of their properties. All these qualitative models rationalise the properties of aromatic and anti-aromatic hydrocarbons in terms of the Hiickel [4n+2] and [4n] rules. The extra stability of a PAH, due to 7t-electron delocalisation, can also be determined, computationally or experimentally, by either considering homodesmotic relationships [36] or by the reaction enthalpy of the reaction of the PAH towards suitable chosen reference compounds [93],... 

In other physical chemical analyses, the perimeter model [73] has been employed to analyze the spectral intensities and MCD signals for a series of porphyrinoid macrocycles derived from the C20H20 perimeter, including the parent, benz-free analogue of texaphyrin 158 [74], These calculations were then compared with the MCD spectra of a number of substituted cadmium texaphyrins (e.g. 116,143, and 148-152, c.f. Scheme 16) [75]. The results confirmed that the perimeter model accounts in a simple way for the signs of the MCD B terms associated with the low-lying electronic transitions of these metallotexaphyrins. 

The electronic spectra of cyclic conjugated n systems depend inherently on the number of n electrons. Closed-ring systems with AN+l n electrons in the perimeter are aromatic compounds, of which benzene is the most important representative. Benzenoid hydrocarbons constitute a class of compounds whose UV spectra have been investigated most extensively both experimentally and theoretically. The fact that the spectra of aromatic compounds are so characteristic meant that formerly they were of considerable importance in the structure determination of organic compounds. However, these spectra cannot be explained in terms of the simple HMO model. If one seeks a theoretical basis for an understanding, one has the choice between the perimeter model and the Pariser-Parr-Pople or a more complicated numerical method. Before discussing these theoretical models, some empirical relations will be presented. Finally, cyclic systems derived from a perimeter of 4N Jt electrons will be considered. 

Jhe perimeter model introduced by Platt (1949), reformulated in the LCAO Mo form by Moffitt (1954a), and extended by Gouterman (1%1), Heilbron-ner and Murrell (1%3), and Michl (1978), has been very useful in understanding trends in the electronic spectra of cyclic Jt systems. It applies equally to singlet and triplet states and has provided the commonly used nomenclature for both. The following discussion is limited to the singlet states, which are more important in ordinary spectroscopy. 

Platt s original perimeter model was a free-electron (FEMO) model based on a one-dimensional circular potential along which the jt electrons can move freely. The orbitals of an electron confined to such a circular ring are given by... 

Figure 2.11. The perimeter model of an (4N+2)-electron [njannulene, geometry on the left, energies of the MOs on the right. The angular momentum quantum number is given for each MO. The sense and magnitude of electron circulation and the resulting orbital magnetic moment are shown schematically in a perspective view. Orbital occupancy in the ground configuration and the four one-electron HOMO->LUMO excitations are indicated (by permission from Michl, 1978).

The Generalization of the Perimeter Model for Systems with 4N+2 n Electrons... 

The perimeter model can be extended quite easily to apply to all systems that may be derived from a regular 4N+2 r-electron perimeter by introduc-... 

Applications of the PMO Method within the Extended Perimeter Model... 

Figure 2.17. Energies of excited states of polyacenes as predicted by the perimeter model a) without Cl, and b) including first-order Cl, which results in a splitting of the states described by U and X, together with experimental data (o, , V, and A) (by permission from Moffitt, 1954a).

The location of the absorption bands as well as their intensities can be discussed on the basis of the perimeter model and perturbation theory. In Example 2.6 it was shown that the transition dipole moments of the and... 

The intensity of the Lb band arises essentially from a substituent-induced interaction with the Bb state. This can be envisaged by means of the perimeter model as follows. Purely inductive substituents give rise to even perturbations from Figure 2.16 it is seen that in this case the only interaction is between configurations X and V [see Equation (2.12)] and is given by ( a + l l) = (AHOMO -I- ALUMO)/2. To first order in perturbation theory, this is equal to AHOMO. To a first approximation, the amount of the configuration X that is mixed into the Lb state is therefore proportional to AHOMO. This contribution causes the intensity increase of the Lb band due to purely inductive substituents. Since the intensity is given by the square... 

Sometimes it is possible to predict the sign by means of group theory and in this way to obtain an assignment of state symmetries. This is particularly true for those molecules to which the perimeter model discussed in Section 2.2.2 is applicable. 

Some of the results of the perimeter model, which are essential for the following discussions, are briefly repeated in Figure 3.11, which is basically a portion of Figure 2.11 For a (4N-f2)-eIectron perimeter (N 0) HOMOs 0JV nnd and LUMOs 1 [Pg.161]

The perimeter model for the description of electronic states of aromatic molecules discussed in Section 2.2.2 is also suited for deriving relations between the structure of these compounds and the sign of the energetically lowest MCD bands. 

AHOMO and ALUMO may be derived from a perturbational treatment of the union of the [I l]annulenyl cation and C or of the [I3]annulenide anion and C , respectively. This also shows that these molecules have further excited states in addition to those derived from a (4N + 2)-electron perimeter. This is true for the longest-wavelength transitions of both molecules, which therefore cannot be labeled within the framework of Platt s nomenclature. Hence, a prediction of their MCD sign on the basis of the perimeter model is also impossible. However, the next two bands correspond to the L, and Lj states of a (4Af + 2)-electron perimeter and show the expected behavior in the MCD spectra. In acenaphthylene the order of the B terms is -, + and in pleiadiene -f-, -. These signs are not changed by perturbing substituents since the difference between AHOMO and ALUMO is too large. Both molecules represent hard chromophores. 

Although the description of electronic states by means of the perimeter model is somewhat less satisfactory for molecules that can be derived formally from an antiaromatic AN perimeter than for aromatic molecules, simple statements about MCD signs are still possible. While nothing can be said about the S and D bands, which according to the perimeter model have zero electric transition moments and which experimentally are found to be very weak (the latter is normally inobservable), predictions are possible for the strong absorptions that are referred to as the N, N2, P, and P2 bands according to the nomenclature given in Section 2.2.7. The parameters that are essential for the MCD spectra of systems derived from a 4)V-electron perimeter are... 

All of the examples selected here to illustrate the utility of MCD spectroscopy are based on systems that may be formally derived from a AN3- 2)-electron perimeter. For such systems the MCD effect is most simply and lucidly connected through the quantities AHOMO and ALUMO with the form and the ordering of the molecular orbitals. It can therefore be used to draw conclusions regarding the electronic and chemical structure of the system under investigation. A structural class for which the perimeter model analysis has been particularly fruitful is that of the porphyrins (Goldbeck, 1988) and related macrocycles (Waluk et al., 1991 Waluk and Michl, 1991). 

Since for benzene the MOs are well known and all positions are equivalent the MCD spectra of monosubstituted benzene derivatives are particularly suited for a determination of substituent constants. A more quantitative analysis on the basis of the perimeter model neglecting the fi contributions yields the following result for the B term of the L transition of substituted benzenes ... 

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