In a previous article, I have shown a simple derivation of the properties of a quantum particle confined in a one-dimensional box. Although the model is straightforward with unrealistic assumptions, such as the infinite walls, it produced qualitative results that paved the way for the development of quantum chemistry. There are several example practical applications in chemistry and nanoscience where the particle in a box model can be applied or serves as a conceptual foundation.

Conjugated polymers and dyes molecules, which are important in organic electronics and optoelectronics, can be modeled using the particle in a box approach. The model helps in understanding the electronic structure of these materials, influencing their conductivity and optical properties. In a paper dated 1949, the German theoretical chemist Hans Kuhn (Kuhn, H., 1949) showed that a free-electron model (based on the particle on the one-dimensional box) for -electrons in polymethine dyes such as the cyanine, could be used to predict the electronic properties of these molecules. A long polymethine chain connects two terminal groups containing charged nitrogen atoms in these dyes.

Consider the symmetric carbocyanine dye in Figure 1, where the positive charge “resonates” between the two nitrogen atoms at the end of the chain. 

While the particle in a box model is a simplified representation, it forms the basis for more advanced quantum mechanical models that better describe the behavior of electrons in real-world systems.

The zigzag polymethine chain along which the electrons are relatively free to move extends along with the conjugated system between the two nitrogen atoms. This extended conjugation plays a crucial role in various organic compounds and materials, as it significantly influences their electronic and optical properties.

Kuhn’s model assumes a box of length L, equal to the path length of the conjugated system, plus one extra bond length on each end. This additional length ensures that the nitrogen atoms are not at the very edge of the box, allowing the -electron charge to distribute more evenly.

To determine the length of the box, Kuhn used the average length of a single and double C-C bond, denoted as . By applying this value, the expression for the length of the box becomes:

where represents the number of repeating units in the conjugated system. This approach allows us to estimate the size of the conjugated system and provides insights into its electronic behavior.

The total number of -electrons in the molecule can be calculated as , where represents the number of repeating units in the conjugated system. This expression enables us to determine the number of electrons involved in the conjugated system, which directly affects its electronic structure and properties.

Assuming that each energy level within the box can accommodate a maximum of two electrons, the transition between energy levels becomes crucial. Specifically, the transition occurs between the highest filled level (occupied by electrons) and the lowest empty level (capable of accommodating additional electrons). This transition energy is denoted as and can be related to the corresponding wavelength through the equation:

where represents Planck’s constant and represents the speed of light. By utilizing this equation, we can calculate the energy difference $latex

In the same paper, Kuhn extends his model to other types of conjugated systems.

Other Applications in Chemistry and Nanoscience

Quantum Dots and Nanoparticles:

• Quantum dots involve three-dimensional confinement. For a cubic box, the wave functions become products of three one-dimensional wave functions, extending the particle in a box model to three dimensions.

Nanostructures and Quantum Wires:

• Quantum wires can be modeled as two-dimensional boxes. The confinement of electrons in two dimensions influences their behavior and electronic properties.

Tunneling Phenomena:

• Tunneling probability ((T)) through a barrier can be modeled using quantum mechanics. For a rectangular potential barrier, the transmission coefficient ((T)) can be calculated, incorporating the particle in a box wave functions.

Semiconductor Quantum Wells:

• Quantum wells in semiconductor devices can be modeled using the particle in a box concept. The energy levels of electrons in the well influence the electronic and optical properties of the semiconductor material.

Molecular Orbitals:

• For simple diatomic molecules, the particle in a box model is a particular case of the more general Schrödinger equation, which describes molecular orbitals in molecules. The particle in a box provides a more straightforward example for describing the molecular bond and understanding the quantization of energy levels.

Nanoelectronics and Quantum Computing:

• The particle in a box model contributes to the understanding of quantum confinement effects in nanoscale electronic devices and quantum computers, where electrons are confined to small spaces.

In these applications, the mathematical formalism of the particle-in-a-box model provides a foundation for understanding the quantum behavior of particles in confined systems. Real-world systems may require more sophisticated models, but the particle in a box serves as a starting point for conceptual understanding.

BIBLIOGRAPHY

1. Kuhn, H., 1949. A quantum-mechanical theory of light absorption of organic dyes and similar compounds. The Journal of Chemical Physics, 17(12), pp.1198-1212.