An Alternative Derivation of the Energy Levels of the "Particle on a Ring" System

Abstract

All acceptable wave functions must be continuous mathematical functions. This criterion limits the acceptable functions for a particle in a linear 1-dimensional box to sine functions. If, however, the linear box is bent round into a ring, acceptable wave functions are those which are continuous at the 'join'. On this model some acceptable linear functions become unacceptable for the ring and some unacceptable cosine functions become acceptable. This approach can be used to produce a straightforward derivation of the energy levels and wave functions of the particle on a ring.These simple wave mechanical systems can be used as models of linear and cyclic delocalised systems such as conjugated hydrocarbons or the benzene ring. The promotion energy of an electron can then be used to calculate the wavelength of absorption of uv light. The simple model gives results of the correct order of magnitude and shows that, as the chain length increases, the uv maximum moves to longer wavelengths, as found experimentally.

On-Surface Synthesis and Characterization of Honeycombene Oligophenylene Macrocycles

“…Examples of individual MOs are shown in Figure b. Such large-wavelength modulations are characteristic for quantum-mechanical “particle-on-a-ring” type systems . As an example for one of the simplest possible model systems, Figure S8c in the Supporting Information shows selected molecular orbitals for an artificial planar hydrogen macrocycle.…”

“…Such large-wavelength modulations are characteristic for quantummechanical "particle-on-a-ring" type systems. 62 As an example for one of the simplest possible model systems, Figure S8c in the Supporting Information shows selected molecular orbitals for an artificial planar hydrogen macrocycle. The six corners of the honeycombene macrocycle consist of meta-substituted phenyl rings.…”

“…The dI/dV spectra in Figure 5g, however, reveal subtle differences: With increasing ring size in the series [20]-HC, [25]-HC, and [30]-HC, the maxima shift to lower bias voltages, as qualitatively expected from a particle-on-the-ring model for conjugated systems. 62,65,66 The observation that the shift is most pronounced between the pentagon and the hexagon possibly reflects effects of ring strain, which should especially affect the square and the heptagon. Note that the heptagon [35]-HC shows a maximum at higher voltage than [30]-HC.…”

Particle on a Ring Spectroscopic Selection Rules Determined by Group Theory

“…Previous articles have covered many aspects of spectroscopic transitions. Such aspects include the definition of the rotational energy levels of the PR system, and the basic derivation of selection rules deduced from the electric dipole transition. , Since the derivation of rotational selection rules can be daunting, some have rationalized the selection rules for student simplicity . Other articles have used group theory to derive these selection rules for three-dimensional rotation and labeled symmetry species for rotational levels within full symmetry groups .…”

Teaching Molecular Applications of the Particle-in-a-Ring Model Using Azulene

“…Its simplicity makes it valuable to bridge theoretical and experimental fields in introductory physical chemistry courses. Reports of particle-in-a-box experiments in this Journal date back to the 1960s (1)(2)(3)(4)(5)(6)(7)(8)(9)(10)(11)(12)(13)(14) and include interesting cases of unconventional geometries (4)(5)(6). Cyanine dyes are common molecular applications for onedimensional boxes, although some experimental uncertainties about the box limits are hard to overcome (1,10,11); linear polyenes can also be used (14).…”