# Free-Electron Model

- Page ID
- 8852

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

The four \(\pi\)-electrons are assumed to move freely over the four-carbon frame-work of single bonds. We neglect the zig-zagging of the C-C bonds and assume a one-dimensional box. We also overlook the reality that \(\pi\)-electrons actually have a node in the plane of the molecule. Since the electron wavefunction extends beyond the terminal carbons, we add approximately one-half bond length at each end. This conveniently gives a box of length equal to the number of carbon atoms times the C-C bond length, for butadiene, approximately 4 x 1.40 Å. Recall that 1 Å=10^{-10}m. Now, in the lowest energy state of butadiene, the four delocalized electrons will fill the two lowest FEM "molecular orbitals." The total \(\pi\)-electon density will be given (as shown in Figure 4) by

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

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 CH_{2}=CH-CH=CH_{2} (the predominant structure) and ºCH_{2-}CH=CH-CH_{2}º (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 CH_{2}=(CH-CH=)_{n-1}CH_{2}. For the molecule with 2*n* 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:

## Contributors

Seymour Blinder (Professor Emeritus of Chemistry and Physics at the University of Michigan, Ann Arbor)