Читать книгу Quantum Mechanical Foundations of Molecular Spectroscopy - Max Diem - Страница 30

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.

Figure 2.6 (a) Structure of 1,6‐diphenyl‐1,3,5‐hexatriene to be used as an example for the PiB calculations. (b) Energy level diagram, based on the PiB formalism, showing the three lowest energy levels occupied by the π‐electrons.

Example 2.4 Calculation of the energy difference between n = 3 and n = 4 energy levels for the 1,6‐diphenyl‐1,3,5‐hexatriene system, shown in Figure 2.6, assuming that the electrons obey the particle in a box formalism. What is the wavelength of a photon that causes this transition?

Answer:

1 Estimation of the conjugated length. Since the single and double bonds, with bond lengths of 154 pm and 130 pm, respectively, are approximately 120o from each other, one can approximate the length of the box as(E2.4.1)

2 Calculation of the energy difference between n = 3 and n = 4. Use me = 9.1×10−31 [kg] and h = 6.6 × 10−34 [Js] for the electron mass and Planck's constant. Since the length of the box was estimated to 2 significant figures, the entire computation is carried out with 2 significant figures:

Analysis of units:

(E2.4.2)

ΔE = 3.7 × 10⁻¹⁹ [J]

Figure 2.7 Absorption spectra of nanoparticles as a function of particle size. As expected, the larger particles exhibit lower energy (longer wavelength) transitions.

1 ΔE = hc/λ or λ = hc/ΔE(E2.4.3)