Validity of the Particle in a 1-D box for Conjugated Polyynes
R. Viswanathan
The particle in a one-dimensional box model is a simple quantum mechanical model that can be used to predict the electronic energy levels for the pi electrons in long-chain conjugated unsaturated compounds. This exercise compares the results predicted by the particle-in-a-box model with semi-empirical quantum mechanical calculations.
According to the particle in a 1-D box model, the energy of the electron in a quantum state n is given by:
En = n2h2/8ml2
where h is the Planck's constant, m is the mass of the electron, and l is the length of polyyne. The length of the box is estimated using the C-C triple, single, and C-H bond lengths. The number of pi electrons is 4 times the number of triple bonds in the conjugated polyyne. The HOMO (n) is one half the number of pi electrons. The energy change corresponding to the HOMO-LUMO transition is given by:
DE = En+1 - En
The wavelength corresponding to the transition is calculated as:
lmax = hc/DE
Compound |
No. of carbon atoms |
Length of chain |
lmax 1D-box |
lmax MOPAC |
| 1,3-butadiyne | 4 | 5.87 | 126 | 196 |
| 1,3,5-hexatriyne | 6 | 8.43 | 180 | 210 |
| 1,3,5,7-octatetrayne | 8 | 10.99 | 234 | 192 |
| 1,3,5,7,9-decapentayne | 10 | 13.55 | 288 | 204 |
| 1,3,5,7,9,11-dodecahexayne | 12 | 16.11 | 342 | 220 |
The structures of these polyyne molecules were minimized using the molecular mechanics method, followed by a semi-empirical method. The wavelength corresponding to the HOMO-LUMO transition were then calculated using ZINDO. The values for the wavelength of maximum absorption obtained by the semi-empirical method are compared with the particle in a box results in the following graph.
Both models predict an increase in the wavelength with increasing chain length. The effect predicted by the particle-in-a-box model is more pronounced than the results obtained using MOPAC.
Viswanathan