Conjugated molecules are molecules that have a network of overlapping p-z orbitals whose electrons are delocalized. Typically these molecules have alternating single and double bonds, but they may also include p-z orbitals from radical centers. The terminology "conjugated" molecules was introduced in 1899 by a German chemist Johannes Thiele.  Examples include 1,3- butadiene, 1,3,5-hexatriene, and cyclobutadiene.The skeletal structures of these molecules can be displayed with the Quantum Chemistry Toolbox.  After loading the commands of the QuantumChemistry package

First, we view 1,3- butadiene

Second, 1,3,5-hextriene

and finally, cyclobutadiene

Note the alternating single and double bonds in the skeletal structures of these molecules, indicating that they are conjugated molecules.

To illustrate Huckel theory, we will consider the molecule cyclobutadiene. We can generate the three-dimensional structure of cyclobutadiene by importing the geometry with the MolecularGeometry command

and subsequently plotting the structure with the PlotMolecule command

Click on the above molecule to rotate it and thereby view it from different perspectives.

Using Huckel's rules, we can define the Hamiltonian matrix for cyclobutadiene as follows

Note that the element H[1,4] is not zero because cyclobutadiene is a ring in which the first carbon atom is adjacent to the fourth carbon atom.

The molecular orbitals and their energies can be computed by solving the following eigenvalue equation

$H c_n \= {\mathrm{\ϵ}}_n c_n$

in which n is the quantum number indicating the molecular orbital, ${\mathrm{ϵ}}_n$ is the energy of the n-th molecular orbital, and $c_n$ is the vector of expansion coefficients for the n-th molecular orbital in terms of the p-z atomic orbitals.

In Maple the eigenvalues of the Hamiltonian matrix can be readily computed from the Eigenvalues command in the LinearAlgebra package.

Assuming that β is negative, (a) order the energies from lowest to highest.  (b) How many molecular orbitals are degenerate?

Similarly the eigenvalues and eigenvectors of the Hamiltonian matrix can be readily computed from the Eigenvectors command in the LinearAlgebra package.

For the n-th eigenvalue in ev the eigenvector is given in the n-th column of the output matrix vs.  For example, for the first eigenvalue in ev 

the eigenvector is given by

Note that the eigenvectors are not normalized, meaning that their inner product is not equal to one.

(c) Normalize each of the four eigenvectors generated by using the following Maple code.

(d) Draw a sketch of each of the molecular orbitals including the relative phases between the p-z orbitals as indicated by the computed eigenvectors.

(e) Label each sketch in (d) by its molecular orbital energy.

While Huckel theory is an approximation for the electronic structure of conjugated molecules, with the Quantum Chemistry Toolbox in Maple we can use more advanced electronic structure methods to compute and visualize the π molecular orbitals of cyclobutadiene. Here we use the Variational2RDM command to compute these molecular orbitals in a manner that includes the statistical correlation of electrons beyond the Huckel molecular orbital theory:

We can plot the bonding π molecular orbital with the DensityPlot3D command

The second π molecular orbital is given by

The third π molecular orbital is given by

Finally, the anti-bonding π molecular orbital  is given by

(f) Do the molecular orbitals computed by the variational 2-RDM method agree with the qualitative features of those predicted by Huckel's theory?

The variational 2-RDM calculation also predicts the number of electrons per molecular orbital, known as the orbital occupations.  The occupations of the four π molecular orbitals are given by

(g) In what ways do these orbital occupations agree with those predicted from Huckel theory, and in what ways do they not agree?

The bond distances between each pair of atoms can be readily computed with the BondDistances command

(h) Based on the computed  bond distances, is the geometry of cyclobutadiene square or rectangular?

(i) Can you use the result from (h) to explain the difference in the occupation numbers from the Variational 2-RDM calculations and those from Huckel's method?