Application of the free electron theory to catacondensed polycyclic aromatic hydrocarbons

Application of the free electron theory to polycyclic aromatic hydrocarbons and porphyrins

Catacondensed aromatic hydrocarbons have all carbons on the periphery of the ring system (no interior carbons). These molecules include benzene, naphthalene, anthracene, triphenylene etc. but not pyrene, perylene etc.

The free electron model (FEM) can be applied to catacondensed aromatic hydrocarbons using the electron on a circular wire approach. For an electron confined to travel in a circular orbit the wavefunctions are e^imf where m = 0, ±1, ±2, ±3, … All of the energy levels above the zero level are doubly degenerate.

We can imagine that the significance of these states is the electron is moving either clockwise (e.g. positive values of m) or counter-clockwise (e.g. negative values of m) around the circle. This means that there are two angular momentum states for each pair of levels. Since all of the catacondensed polycyclic hydrocarbons are aromatic they follow the 4n + 2 rule. The states are filled in pairs up to n = m. Each HOMO/LUMO pair has two possible sets of transitions shown below in red and blue.

The transitions with Dm = ± 1 are allowed transitions. In Platt's nomenclature these were called B transitions. If the molecule has a long a short axis these transitions are denoted B_a and B_b. The transitions with Dm = ± (2j + 1) are forbidden transitions. The correspond to weak bands. In Platt's nomenclature these are denoted L_a and L_b. The simple prediction that the doubly degenerate states will split into groups is valid to the group of catacondensed aromatics as a whole. The absorption spectra typically have very weak bands as the lowest energy p-p* transitions. The higher energy transitions are typically intense. In benzene, for example, the L_b band is at 256 nm (e = 220), L_a is at 203 (e = 6900), and B is at 186 nm (e = 46,000). The prediction made by this model is that the allowed B transitions will have a change in orbital angular momentum equal to 1. The change in orbital angular momentum of the forbidden L states is 2j + 1. This can be seen in difference the magnetic circular dichroism of these molecules.

Just as was done for polyenes this model can be used to predict the energies of transitions using the particle on a circle energies.

The FE model also predicts the number of nodes in the wavefunction. The number of nodes in the circle is equal to |m|.

Application to the spectroscopy of benzene

The free electron model can be compared to a more detailed picture of the electronic spectroscopy of benzene. The structure of benzene is shown below.

This representation is one of the Kekule resonance structures. The bond lengths are equal and there is a six-fold symmetry axis perpendicular to the plane of the benzene molecule. The point group of benzene is D_6h.

The molecular orbital picture gives the following orbitals. Note that the free electron model does not predict the b_2u orbital, however, it is not occupied and it is not apparent that it is directly involved in the first electronic transition.

It can be shown that this orbital does mix in due to configuration interaction. In fact, the validates a second part of the model since there are two electronic transitions, one vibronically allowed (A_1g à B_2u, the one with the large change in orbital angular momentum also known as the L band ) and one electronically allowed (A_1g à E_1u also known as the B band). According the Herzberg in "The Electronic of Polyatomic Molecules" (p. 555): "Historically, [the vibronic progression in] these bands prsented the first extensive and clear-cur example of an electronic transition forbidden by the symmetry selection rules and of an application of the vibronic selection rules."

The vibronic coupling between these two bands will be promoted by vibrational modes having the appropriate symmetry. The appropriate symmetry occurs when the product

(initial state|radiation field|vibronic mode|final state)

is totally symmetric. In the D_6h point group this corresponds to A_1g.

The initial state is A_1g.

The final state is B_2u.

For x,y polarized radiation the character is E_1u.

Thus, the appropriate modes will have the symmetry of direct product B_2uE_1u.

We consider the direct product of (B_2u)(E_1u). In D_6h this transforms as:

E	2C₆	2C₃	C₂	3C₂'	3C₂''	i	2S₃	2S₆	s_h	3s_d	3 s_v
2	-1	-1	2	0	0	2	-1	-1	2	0	0

This can be determined by group theoretical methods. Only E_2g is non-zero.

G_E2g = (1/24)[(2)(2)+2(-1)(-1)+2(-1)(-1)+(2)(2)+(2)(2)+2(-1)(-1)+2(-1)(-1)+(2)(2)] = 1

An analysis of the vibrational modes of benzene can be found in Wilson, Decius, and Cross "Molecular Vibrations" (Dover) starting on page 240. There are 3N - 6 = 30 normal modes. In the x,y plane there are 2N - 3 = 21 modes and there are N - 3 = 9 out-of-plane modes. Group theory can be used to determine the symmetries of these vibrations. There are 4 vibrations with E_2g symmetry appropriate for vibronic coupling discussed above. Indeed, Herzberg discusses that the E_2gvibration at 608 cm^-1 represents part of the the vibronic progession and is confirmed to by the vibronically active mode in benzene.

Although the free electron model cannot predict configuration interaction or vibronic coupling, it does correctly predict the number of nodes in the wavefunctions involved in the electronic transition and the magnetic properties due to the change in orbital angular momentum.

Application to the spectroscopy of porphyrins

Porphyrins can be thought of as big benzenes. This sounds grossly oversimplified at first and yet there are important similarities. Porphyrins have an aromatic ring system with 18 electrons. The structure of the free base porphyrin ring is shown below.

The symmetry point group of the porphyrin system (shown here as the dianion) is D_4h.

With 18 number of electrons the free electron model predicts that there will be 4 nodes in the HOMO and that it will be doubly degenerate.

This turns out to be qualitatively correct. The four orbital model presented by Gouterman is accepted as an appropriate description of the first electronic transitions of both free base and metalloporphyrins. Metalloporphyrins are perhaps easiest to understand for filled shell metals such as zinc. The HOMOs and LUMOs of Zn porphyrin were calculated using a semi-empirical model. The four orbital model has been verified by these and higher level calculations. The HOMO (p) is not rigorously degenerate, but does consist of two nearly degenerate orbitals of a_1u and a_2u symmetry. The LUMO (p*) is a doubly degenerate set of orbitals of e_g symmetry. Both the transitions from a_1u à e_g and a_2u à e_g are allowed.

Both of the transitions shown have the same excited state configuration. The configuration of the ground state is always A_1g (totally symmetric) for closed shell molecules. The excited state configuration is the direct product of the a_1ue_g or a_2ue_g depending on which of the transitions above we consider. These are both the same as can be seen below.

The direct product a_1ue_g is:

E	2C₄	C₂	C₂'	3C₂''	i	2S₄	s_h	2s_v	2s_d
2	0	-2	0	0	-2	0	2	0	0

The direct product a_2ue_g is:

E	2C₄	C₂	C₂'	3C₂''	i	2S₄	s_h	2s_v	2s_d
2	0	-2	0	0	-2	0	2	0	0

Both of these direct products have the character of E_u. Thus, the excited state configuration is E_u for two transitions. Moreover, the transition moment for each of these states are nearly equal. We consider the y polarized states. There are two transition moments

R₁ = e áA_1g|y|a_2ue_gyñ

R₂ = e áA_1g|y|a_1ue_gxñ

Thus, these transition moments can mix by configuration interaction.

In this representation the B⁰_y and Q⁰_y are coupled by a term V.

We can diagonalize the transition moment matrix

To obtain

for

The y polarized states are now

For small n the transition moments are:

The important point to emerge from this analysis is that there is one band which is allowed and one band that is essentially forbidden due to configuration interaction. The four orbital model leads to a picture that resembles the situtation in benzene (and is found in general to apply to 4n + 2 polycyclic aromatic hydrocarbons). There is a weakly allowed vibronic band (the Q band) and a strong band known as the B band or the Soret band. These are shown in the spectrum below.

This is the spectrum of ferrous carbonmonoxy heme. Since the iron is low spin in this spectrum it has many features in common with spectrum of porphyrins adducts of closed shell metals. Note that the Soret band has some structure. In particular there appears to be a bump to lower wavelength (higher wavenumber). If we convert the Soret band to wavenumbers we see the following.

Note that the peak of the band is at 23,600 cm^-1. The small bump to higher energy is part of a Franck-Condon progression. We suggest this because the Soret band is an allowed band thus totally symmetric modes will be Franck-Condon active. It turns out that the bump is at almost 1360 cm^-1 higher energy than the peak. This corresponds to a totally symmetric breathing mode of the ring known as n₄. The relative size of this feature is about 0.2 times as intense as the main band. If the main band represents 0-0 and the small bump represents 0-1 then S = 0.2 for this band. The Q-band has structure due to vibronic coupling. In wavenumbers it has the following appearance.

The large feature at 18,500 cm^-1 is the first vibronic band. Note that this band is larger than 0-0 at 17,300 cm^-1. The energy difference between the band corresponds to a wavenumber of ca. 1200 cm^-1 for the vibronically active mode. Given the high symmetry of metalloporphyrins the symmetries of vibronic modes can be determined. As we saw for benzene, the symmetries require that the product

(initial state|radiation field|vibronic mode|final state)

be totally symmetric.

For (x,y) polarization the radiation field has E_u symmetry.

The final state has E_u symmetry.

Thus, vibronically active modes will be those modes whose symmetry is spanned by the direct product E_uE_u.

This direct product is:

E	2C₄	C₂	C₂'	3C₂''	i	2S₄	s_h	2s_v	2s_d
4	0	4	0	0	4	0	4	0	0

This can be decomposed into four irreducible representations, A_1g, A_2g, B_1g, and B_2g. The A_1g modes are the Franck-Condon active modes, but they can also participate in vibronic coupling. Resonance Raman spectroscopy is particularly important for determining the symmetries of modes that are coupled to a vibronic transition. The depolarization ratio of various modes can be used to identify their symmetry.

The free electron model predicts that there will be one intense (allowed) and one weak forbidden band based on the difference in the orbital angular momentum change for the transition. For the Soret or B band the free electron model predicts Dm = ±1, just as for benzene. However, for the Q band the free electron model predicts Dm = ±9 for a porphyrin. This is an exceedingly large change in orbital angular momentum. Although beyond the scope of this course it is worth mentioning that the magnetic circular dichroism signal is proportional to the orbital angular momentum and it is significantly larger for the Q band than for the Soret band. Thus, the free electron model is correct in several important respects.