The simplest nontrivial application of the Schrödinger equation is the one-dimensional particle in a box. Remarkably, this simple system can provide a useful model for a significant chemical problem—the structure of linear polyene molecules. A polyene is a hydrocarbon with alternating single and double carbon-carbon bonds, the simplest example being butadiene
. The series continues with hexatriene, octatetrene, and so on, with the generic structure
, a chain of
conjugated carbon atoms. After all the single bonds are accounted for, each carbon atom contributes one
-electron. Linear combinations of these
-orbitals can form
-molecular orbitals (MOs), delocalized over the entire length of the molecule.
The free-electron model represents these MOs as particle-in-a-box wavefunctions
, with orbital energies
). The length of the box is taken as
is the average
bond length, typically on the order of 1.40 Å. One half a bond length is added to each end of the molecule to simplify this formula. Taking into account the Pauli principle, two electrons (with
spins) are fed into each of the available orbitals of lowest energy. The highest-occupied molecular orbital (HOMO) then has
. In the lowest-energy electronic transition, one HOMO electron is excited to the lowest-unoccupied molecular orbital (LUMO), with
, producing a band with its maximum wavelength in the ultraviolet or visible region. Using
, we find
(a useful constant is the Compton wavelength
). The prediction for butadiene is 207 nm, which agrees closely with the observed absorption band at
in the ultraviolet. The increase of
is correctly predicted; however, quantitative results for the higher polyenes are not very accurate. As the absorption band begins to impinge on the visible region of 400–700 nm a polyene becomes colored. A chain of 10 conjugated carbon atoms, as contained in retinol (vitamin A), shows a pale yellow color.
We propose a modification of the free-electron model to improve results for the longer polyene chains. For the potential energy function within the one-dimensional box, we replace
is an additional empirical parameter (in addition to
). The rationale for this form is the fact that the
bonds are not all equivalent (as they are, for example, in benzene). Actually, the bonds exhibit an alternating character, with the
double bonds in the conventional structural formulas having a slightly greater
-electron density. The sinusoidal perturbation describes a lower potential energy for the mobile electrons in these regions.
It is sufficient to compute the modified energies using first-order perturbation theory, such that
. The wavelength formula thereby generalized to
. With appropriate choices of the empirical parameters, much better agreement can be obtained for the absorption frequencies over a large range of polyenes, as shown in the second snapshot.