3: Quantum Mechanics of Some Simple Systems

The simple quantum-mechanical problem we have just solved can provide an instructive application to chemistry: the free-electron model (FEM) for delocalized $\pi$-electrons. The simplest case is the 1,3-butadiene molecule

$$\rho =2\psi _{1}^2+2\psi _{2}^2\label{28}$$

A chemical interpretation of this picture might be that, since the $\pi$-electron density is concentrated between carbon atoms 1 and 2, and between 3 and 4, the predominant structure of butadiene has double bonds between these two pairs of atoms. Each double bond consists of a $\pi$-bond, in addition to the underlying $\sigma$-bond. However, this is not the complete story, because we must also take account of the residual $\pi$-electron density between carbons 2 and 3. In the terminology of valence-bond theory, butadiene would be described as a resonance hybrid with the contributing structures CH₂=CH-CH=CH₂ (the predominant structure) and ºCH_2-CH=CH-CH₂º​ (a secondary contribution). The reality of the latter structure is suggested by the ability of butadiene to undergo 1,4-addition reactions.

The free-electron model can also be applied to the electronic spectrum of butadiene and other linear polyenes. The lowest unoccupied molecular orbital (LUMO) in butadiene corresponds to the n=3 particle-in-a-box state. Neglecting electron-electron interaction, the longest-wavelength (lowest-energy) electronic transition should occur from n=2, the highest occupied molecular orbital (HOMO).

The energy difference is given by

$$\Delta E=E_{3}-E_{2}=(3^2-2^2)\dfrac{h^2}{8mL^2}\label{29}$$

Here m represents the mass of an electron (not a butadiene molecule!), 9.1x10^-31 Kg, and L is the effective length of the box, 4x1.40x10^-10 m. By the Bohr frequency condition

$$\Delta E=h\upsilon =\dfrac{hc}{\lambda }\label{30}$$

The wavelength is predicted to be 207 nm. This compares well with the experimental maximum of the first electronic absorption band, $\lambda_{max} \approx$ 210 nm, in the ultraviolet region.

We might therefore be emboldened to apply the model to predict absorption spectra in higher polyenes CH₂=(CH-CH=)_n-1CH_​2. For the molecule with 2n carbon atoms (n double bonds), the HOMO → LUMO transition corresponds to n → n + 1, thus

$$\dfrac{hc}{\lambda} \approx \begin{bmatrix}(n+1)^2-n^2\end{bmatrix}\dfrac{h^2}{8m(2nL_{CC})^2}\label{31}$$

A useful constant in this computation is the Compton wavelength

$$\dfrac{h}{mc}= 2.426 \times 10^{-12} m.$$

For n=3, hexatriene, the predicted wavelength is 332 nm, while experiment gives $\lambda _{max}\approx$ 250 nm. For n=4, octatetraene, FEM predicts 460 nm, while $\lambda _{max}\approx$ 300 nm. Clearly the model has been pushed beyond range of quantitate validity, although the trend of increasing absorption band wavelength with increasing n is correctly predicted. Incidentally, a compound should be colored if its absorption includes any part of the visible range 400-700 nm. Retinol (vitamin A), which contains a polyene chain with n=5, has a pale yellow color. This is its structure: