**Electronic transitions**

In electronic spectroscopy, we excite the electronic system of a molecule. That means that (in first approximation) the movement of the nuclei does not concern us much and we care only about the electrons. In order to excite a transition between different electronic states, the energy of the light has to match the energetic gap between the two electronic states. Additionally, the transition has to be quantum-mechanically allowed. This is governed by the selection rules, which often boil down to a question of symmetry.

In electronic spectroscopy, the excitation has to alter the distribution of electrons within the molecule. In other words, it has to induce a change in the electric dipole moment. This is described by the transition dipole moment , which in pure electronic spectroscopy is defined by the sum of the movements of the electrons due to excitation:

The vector *r _{i}* describes the movement of each electron

*i*with the elementary charge

*e*. With this, we can compute the intensity of the transition between the initial state and final state . The probability for a transition is the square of the of the absolute value of the integral

which is called the* matrix element*. This matrix element has to be non-zero for a transition to be allowed.

**Cata-condensed hydrocarbons**

Molecules consisting of a chain of benzene rings are called cata-condensed hydrocarbons, and two examples are shown in Figure 1. These molecules are described by the formula C_{4n+2}H_{2n+4} and each carbon atom belongs to a maximum of two rings. In order to describe the electronic states of these molecules, John Platt introduced a convenient nomenclature based on the free electron orbital method [1]. He postulated that in these systems the electrons travel on a loop of constant potential around the perimeter. In this picture, the energy levels can be described by the simplest model in quantum mechanics, a particle in a one-dimensional box.

The length of the box *l* is the perimeter of the hydrocarbon along which the electrons with mass *m*_{e} may travel and *q* is the so-called *ring quantum number*. The higher this quantum number, the larger the energy of the molecule. As *q* gets squared, the energy levels increase quadratically and do not depend on the relative sign of *q*: electrons with a finite velocity may travel clockwise and counter-clockwise along the perimeter *l*, giving rise to *+q* and *-q*. An exception is the state corresponding to *q=0* as in this state the electrons are standing still.

Each of the 4n+2 C-atoms contributes a single electron. The highest occupied shell has the quantum number *q=n* and is designated as *f*. This is regardless of the actual value of *n* and applies to both molecules in Figure 1 although they differ in the number of electrons. The first unoccupied shell is *g*, the second unoccupied shell *h*, and so on.

**Exciting molecules in cata-condensed hydrocarbons**

When we excite the electronic system we lift one electron from the *f*-shell to the *g*-shell. The energy of the final state depends on the interaction of the excited electron with the other electrons. We can also be described as the interaction of the excited electron with the hole it created in the *f-*level. However, we cannot tell whether the electron before (the hole) and after excitation is going clockwise or anti-clockwise as both options have the same energy. As we cannot distinguish between these two options and both contribute, we have to describe the final state by a *linear combination*.

If an electron is excited from an *f(q=n)* to a *g(q=n+1)* shell, the resulting states are described the linear combination: *Q=n (n+1)*. *Q *is the *total ring quantum number *which is the same as the total angular momentum. It is gained by adding and subtracting the quantum number *q* algebraically. States designed *A, B, C, …* correspond to *Q=0, 1, 2, …*. Higher values of *Q* like *2n, 2n+1* and *2n+2* are given the characters *K, L* and *M*. Thus, two doubly degenerate *B (Q=1)* and *L (Q=2n+1)* states result. Again, the degeneracy stems from the fact that we cannot distinguish between clockwise and anti-clockwise travel of the electrons around the perimeter of the molecule.

**L _{a} and L_{b}**

Except for highly symmetrical molecules like benzene, the degeneracy in the energy of the states is lifted. The resulting sub-levels are denoted *a* and *b*, depending on the orientation of the new nodal plane introduced into the electronic wave function. For the *a* state, the nodal planes are situated in such a manner, that the electron density is located at the **a**toms, while the effective charges go to the **b**onds for the *b* state. This is illustrated in Figure 3 for the states of naphthalene. According to the third of Hund’s rules, the *L* states are energetically always lower than the *B* states.

Compared to naphthalene 2 CH groups are replaced by an NH group in indole. If only the number of -electrons is regarded, indole and naphthalene are iso-electronic (they both have 10 electrons). This is the reason why this nomenclature was adapted to indole by G. Weber [2].

**L _{a} and L_{b} in indole**

As the molecular symmetry of indole is much lower than the one of benzene, the energetic degeneracy is lifted and we get two separated transitions. In the gas phase, the difference between both states amounts to 2000 cm^{-1}, which is around 5% of the total transition energy. The transition dipole moments of the two states are perpendicular to each other like in the case of naphthalene, shown in Figure 4.

**Why is this interesting?**

The molecule indole is an integral part of the aromatic amino acid tryptophan, essentially it is the reason why tryptophan absorbs UV light. After exciting tryptophan electronically, light is re-emitted. This fluorescence light is widely used to study the structure and dynamics of proteins. The reason for this is the two excited states we just discussed. In an apolar surrounding, the L_{b} is lowest in energy and hence the emitting state. As soon as the surroundings of tryptophan get more polar, for instance, because the protein changes its structure and water gets in the vicinity of tryptophan, the L_{a} is strongly stabilized. The reason for this behaviour lies in the electric dipole moments of the excited states. The one of the L_{a} is much larger than the one of the L_{b}, leading to a favourable dipole-dipole interaction which lowers the energy of the L_{a}. Hence, tryptophan can be used as a natural probe to tell us about the structure of the protein or its changes.

**Bibliography**

[1] J. R. Platt, “Classification of spectra of cata-condensed hydrocarbons” *J. Chem. Phys.* **17**, 484-495 (1948)

[2] G. Weber, “Fluorescence-polarization spectrum and electronic-energy transfer in tyrosine, tryptophan and related compounds” *Biochem. J.* **75**, 335-345 (1948)