Quantum Mechanical Foundations of Molecular Spectroscopy. Max Diem. Читать онлайн. Newlib. NEWLIB.NET

Автор: Max Diem
Издательство: John Wiley & Sons Limited
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Жанр произведения: Химия
Год издания: 0
isbn: 9783527829606
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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.

      2.5.1 Transitions in a Conjugated Polyene

Schematic illustration of the (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)StartFraction left-parenthesis upper J s right-parenthesis squared Over k g normal m squared EndFraction equals StartStartFraction StartFraction k g squared normal m Superscript 4 Baseline Over normal s Superscript 4 Baseline EndFraction normal s squared OverOver k g normal m squared EndEndFraction equals StartFraction k g normal m squared Over normal s squared EndFraction equals normal upper J

      ΔE = 3.7 × 10−19 [J]

Graph depicts the 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)

      2.5.2 Quantum Dots

      Certain quantum dot structures can also be modeled by a 2D particle in a box. Quantum dots may be manufactured by creating small circular or square semiconductor deposits on a substrate that is an electric insulator. The electrons of the semiconductor spots are free to move over the entire size of the dot, and the energy levels of the free electrons follow a 2D PiB model [3]. Consequently, the color of electronic transitions can be tuned by varying the size of the quantum dot.

      2.5.3 Quantum Cascade Lasers