Up to now we have focused on the electrical properties of organic semiconductors. Electronic devices also involve optical aspects. This is a case, for example, in light emitting diodes and photovoltaic cells. So, today we will describe the interaction of light with conjugated molecules and polymers. In classical physics light is an electromagnetic wave described as a time dependent force field similar to that generated by an oscillating electric dipole. Conversely, electrons in molecules and solids may be set into motion by the oscillating electric field of light. In other words, the interaction between light and matter can be viewed as a process in which energy is exchanged between a radiation field and a collection of oscillating dipoles. Without light the electron is in a molecule holds on average at a distance r from the center of gravity of the molecule. When light is applied the displacing force induced by the electric field will change the distance by an amount Delta r. The new electronic distribution is described by an induced dipole moment called Transition Dipole mu equals minus e time delta r, where e is the the elemental charge. The quantum mechanical description of the transition dipole involves the electrical dipole operator minus e times r. The transition dipole moment is now given by a matrix element between an initial state psi i (without light) and a final state psi f (when light is applied). Another important concept is that of oscillator strength. In the classical theory, the oscillators strength f is a statistical weight indicating the relative number of oscillators bound to a resonant frequency. In quantum mechanics, f measures the relative strength of an electron transition within a molecular system. The oscillator strength is connected to the transition dipole moments by the equation shown in the slide. Here, nu is the resonant frequency, that is, a frequency related to the energy difference between the initial and final states, given by Planck's equation, with Planck's constant h. Other physical quantities used in photochemistry are shown here. The optical density (O.D.) is a dimensionless variable that measures the logarithm of the ratio between the intensity of the incident light and the intensity of the transmitted light. Important to note is the fact that this is a decimal logarithm. The molar extinction coefficient corresponds to the optical density divided by the concentration of the absorbing solution or solid, and its thickness. The wave number, usually measured in centimeters power minus one, is the reciprocal of the wavelength. In quantum mechanics, the wavelength and wave number are correlated to the energy of the light through Planck's constant h. Namely, the energy is h times the speed of light divided by the wavelength, or h times the speed of light times the wave number. Interaction of light with matter takes place in two ways. The first one is absorption. When night impinges on an atom with an energy higher than the energy difference between the ground state and the first excited state, an electron is promoted from the ground state to the first excited state. The second process is emission. The excited electron does not stay long in the excited state and when it decays to the ground state, it emits a photon with the same energy as the energy difference between the ground state and the first excited states. As a consequence, both absorption and emission spectra have the shape of a sharp peak at the same energy. The situation is more complex with a molecule because here, we have to account for the relative movements of the nuclei: vibration, rotation, and collision. As a consequence of the sharp peak broadens into a series of subpeaks. The vibration modes of the molecules are usually represented as an harmonic oscillator where the force between the nuclei is represented by an oscillating spring. The potential energy is proportional to the square of the distance x between the nuclei, with k being the strength of the spring. In quantum mechanics, this leads to quantized states with equally spaced energies. The energy is proportional to the reduced Planck's constants times a frequency Omega, given by the square root of the bond strength divided by the mass capital M of the nuclei. To calculate the wave function and energy of the quantum states, we have to resolve the time independent Schrödinger's equation. The two terms at right hand side correspond to the kinetic energy and the potential energy of the harmonic oscillator. I will not go into details in the equation of the wave number. Suffice to say that the important term is the so-called Hermite's polynomial, the generic form of which is given by the bottom equation multiplied by a Gaussian term. The shape of the first five wave function is shown on the left side hand side. Quantum mechanics tells us that the probability of finding an electron is given by the square of the wave function shown on the right hand side. For the lowest energy states, that is, the ground state, the maximum probability is located at the center of the molecule. However, as the energy increases the highest probability gradually moves to the extremities of the molecules. Because the electron is much lighter than the nuclei, nuclei move much less fast than the electrons. As a consequence, during an electronic transition, nuclei first stay still and then rearrange afterwards. This is known as the Franck-Condon Principle. Also, the distance between nuclei has its minimum value in the ground states because it has a more bonding nature than the excited states. Let's first look at the absorption process. This first parabola represents the potential energy of the ground state centered at R nought, which is also the maximum of the probability density of the vibrational ground state. When the molecule is hit by a photon with an adequate energy the electron is promoted to the first excited state, with a potential energy centered at R nought star, slightly higher than R nought. The excited states also presents vibrational sub-levels. During the electron transition, the distance between the nuclei stays at R nought, so the probability to jump directly to a higher vibrational state maybe higher than a jump to the vibrational ground state. The final vibrational level depends on the geometry of the molecule. Here, the most probable Franck-Condon transition is to the first vibrational state. During emission the distance between nuclei is now R nought star. Before emission, the electron relaxes to the vibrational ground state so its maximum probability density is now also at R nought star. During the transition, the distance between nuclei remains at r note star, so a jump to a state with higher vibrational energy may be more likely than a jump to the vibrational ground state. Here again, the most probability Franck-Condon transition is to the first vibrational level. Let's now go back to the absorption and emission spectra of the molecule. First, note that the spectra presents a mirror shape. Each sub-peak represents a vibrational level of the ground and first excited states. Here, the most probable transition is to the second excited vibrational level, in both absorption and emission. If we define the optical gap as the distance between the vibrational ground level of the electronic ground level and excited state, we see that this energy corresponds to the borderline within the absorption and emission spectra. We finally note that absorption occurs at energy higher than the optical gap and emission at energy lower than the optical gap. In the next lecture we will see that optical gap does not necessarily coincide with energy gap. This is because when promoting an electron from the ground state to the first excited state we actually creates an electron-hole pair called an exciton. Thank you for your attention.
