Quantum Mechanical Foundations of Molecular Spectroscopy. Max Diem

Figure 2.5 (a) Particle in a box with infinite potential energy barrier. (b) Particle in a box with infinite potential energy barrier.

The solutions of this equation will be of the form

      (2.58)

      (2.59)

      where

      (2.60)

      For is an exponential decay function, and for is an exponential growth function. This is shown in Figure 2.5b to the right and left of the potential energy box, respectively. This represents the probability of finding the electron outside the box, a process that is known as “tunneling.” Inside the box, the solutions of Eq. (2.23) resemble the bound wavefunctions of the particle in a box, except that the amplitude at the boundary is no longer zero, but must meet with the wavefunction outside the box. This is depicted in Figure 2.5. Bound states exist for energies E(n) < V₀ only; for E(n) > V₀, the electron exists as a traveling wave as discussed before for unbound states.

The concept of tunneling may seem esoteric at first, but it has interesting consequences. For example, a technique exists that is known as “tunneling electron microscopy (TEM)” where a very sharp metal tip is moved very close (within fractions of a nanometer) to the surface of the analyte, which is at a positive potential with respect to the metal tip. A tunneling current is observed between the tip and the analyte that is due to electrons tunneling from the tip to the analyte. As the substrate is moved laterally under the tip, the tip is lowered or raised to keep the tunneling current constant. In this way, an “image” of the morphology of the analyte can be obtained. Tunneling may also play a role in certain chemical reactions that depend on electron transfer from a donor to a receptor; some of these reactions are faster than expected from computations of the reaction rate from the activation energy. It is thought that in these reactions, the electron may tunnel from donor to receptor at a very fast rate. Finally, in the last example of “real‐world PIBs” in the section below, tunneling plays a major role.

2.5 Real‐World PiBs: Conjugated Polyenes, Quantum Dots, and Quantum Cascade Lasers

      2.5.1 Transitions in a Conjugated Polyene

Although the PiB was introduced here as a model to demonstrate quantum mechanical principles in a mathematically manageable system, there are several physical examples that can be treated adequately using the PiB formalism. One of these is frequently incorporated as an experiment in physical chemistry laboratories [2] and involves a conjugated dye such as 1,6‐diphenyl‐1,3,5‐hexatriene, shown in Figure 2.6a. In this molecule, the Lewis structure suggests three double and four single bonds in the link between the two phenyl groups. From the viewpoint of the PiB formalism, one may consider the polyene framework the length of the box, indicated by the straight line connecting the two phenyl groups, and the six π‐electrons to be delocalized over the entire conjugated length and constituting six electrons in a box in three electron pairs. A schematic of the π‐bonding scheme is shown in Figure 2.6b. The three electron pairs would, in this model, occupy the n = 1, n = 2, and n = 3 levels, as indicated by the up/down arrows in each of these levels. The absorption spectrum in the visible range shows one absorption peak that is, in this approximation, assigned as a PiB transition of one electron from the highest occupied molecular orbital (HOMO) with n = 3 to the lowest unoccupied molecular orbital (LUMO) with n = 4, as indicated by the heavy up arrow. In Example 2.4, the wavelength of this absorption will be calculated. This example may be a bit premature in this chapter, because it introduced aspects of transitions between energy levels, principles of bonding orbitals, and so forth. These subjects will be taken up in later chapters in more detail, but this example shows quite nicely that the PiB formalism can be applied to real systems. In some experiments in physical chemistry laboratory, the dependence of the absorption wavelength on the “length of the box,” that is, the conjugated length, has been described.