4.3.1: Introduction to Hückel Theory

The molecular orbital (MO) energy level diagrams of a conjugated \( \pi \)- system can be constructed using a set of approximations suggested by Erich Hückel. The Hückel Molecular Orbital (HMO) method is not traditionally approached as an ab-initio quantum calculation (from theory alone), but is usually used as an empirical method (includes experimental data) for doing quantum calculations. This means that some number of adjustable parameters appear in the theory, and these parameters are determined by comparison with experiment (such as the electronic spectra of dye molecules). If the theory is successful, it can be used to predict the spectra and properties of other conjugated systems once the values of the adjustable parameters have been set for some set of reference compounds.

The Hückel model assumes that a \( \pi \)-molecular orbital, \( \Psi \), delocalized over \(n\) atoms can be written as a linear combination of \(n\) atomic orbitals (LCAO) \( \phi_{i} \)

\[ \Psi=\sum_{i}c_{i}\phi_{i} \]

where the \( \phi_{i} \)'s are the atomic p_z orbitals on the atoms \(i\) (z is perpendicular to the molecular plane). The corresponding one-electron \( \pi \) density at atom \(i\) is given by

\[ \rho_{i\pi}=c_{i}^{2} \]

The energy E and the coefficients c_i for the ground state are determined by use of the variation principle, which says that one should choose the coefficients such that

\[ E=\frac{\int \Psi H\Psi d\tau}{\Psi^2d\tau}=minimum \]

Here the Hamiltonian, \(H\), is an effective one-electron energy operator whose explicit form need not be specified in the empirical HMO approach. The variation principle ensures that the lowest-energy state is as close as possible to the true energy, and the coefficients are obtained from the minimization relations

\[ \frac{\partial E}{\partial c_{i}}=0; \;\;\;\; i=1\;...\;n \]

This results in \(n\) equations of the form

\[ \begin{matrix}
c_1\left ( H_{11}-S_{11}E \right )+& c_2\left ( H_{12}-S_{12}E \right )+ & . \; . \; . & c_n\left ( H_{1n}-S_{1n}E \right )=0 \\
.& & & . \\
.& & & . \\
.& & & . \\
c_1\left ( H_{n1}-S_{n1}E \right )+& c_2\left ( H_{n2}-S_{n2}E \right )+ & . \; . \; . & c_n\left ( H_{nn}-S_{nn}E \right )=0 \\
\end{matrix} \label{pi}\]

In these equations, \(H_{ii}=\int \phi_i H\phi_i d\tau \equiv \alpha_i \), called the Coulomb integral, is the energy of an electron in a 2p orbital on atom i, while \( H_{ij}=\int \phi_i H\phi_j d\tau \equiv \beta_{ij} \), the resonance or bond integral, represents the interaction energy of two atomic orbitals on atoms i and j. Both \( \alpha_i \) and \( \beta_{ij} \) are negative energy quantities. \( S_{ij}=\int \phi_i \phi_j d\tau \) is the overlap integral, which, in the simplest approximation, is taken to be 1 if i = j, and 0 otherwise. For a \( \pi \) system involving only carbon atoms, \( \alpha_i \equiv \alpha \) and Equations \ref{pi} take the form

\[ \begin{matrix} c_1\left ( \alpha-E \right )+& c_2 \beta_{12} \; + & . \; . \; . & c_n \beta_{1n}=0 \\ .& & & . \\ .& & & . \\ .& & & . \\ c_1 \beta_{n1}+& c_2 \beta_{n2} \;+ & . \; . \; . & c_n\left ( \alpha-E \right )=0 \\\end{matrix} \label{pi2} \]

These equations have a nontrivial solution only if the corresponding secular determinant vanishes:

\[ \begin{vmatrix} \alpha-E & \beta_{12} \; & . \; . \; . & \beta_{1n} \\ .& & & . \\ .& & & . \\ .& & & . \\ \beta_{n1}& \beta_{n2} \; & . \; . \; . & \alpha-E \\\end{vmatrix} =0 \]

The HMO method makes several simplifying assumptions. The first of these are as follows:

1. The \( \pi \)-orbitals are considered separately from the \( \sigma \)-orbitals. The \( \sigma \)-orbitals only enter indirectly in so far as they determine the geometry of the molecule.

2. Treat all the carbon atoms as identical. This means that all the Coulomb integrals (\( \alpha \)) are equal.

 

The next stage is to express the \( \pi \)-orbitals as LCAOs of the C2p-orbitals, \( \phi \). In ethene (CH₂=CH₂) we would write

\[ \Psi=c_A\left ( \phi_A \right ) +c_B\left ( \phi_B \right ) \]

and in butadiene (\(\ce{H2C=CH-CH=CH2}\))

\[ \Psi=c_A\left ( \phi_A \right ) +c_B\left ( \phi_B \right )+ c_C\left ( \phi_C \right ) +c_D\left ( \phi_D \right ) \]

The optimum coefficients and energies are found by the variation principle. This involves solving the secular determinant, equation (Appendix B.7), which for ethene would be

\[ \begin{vmatrix}
\alpha-E& \beta-ES \\
\beta-ES& \alpha-E \\
\end{vmatrix}=0 \]

and its roots are \( E=\alpha \pm \beta \). The state with energy \( \alpha + \beta \) corresponds to the bonding combination (\( \beta \) is negative) and \( \alpha - \beta \) corresponds to the antibonding combination, Figure \(\PageIndex{1}\).

Figure \(\PageIndex{1}\): The Hückel Molecular Orbital Energy Levels of Ethene. Two electrons occupy the lower \( \pi \)-orbital. (CC-NC-BY-DUKE CHEM)

Each carbon atom supplies one electron to the \( \pi \)-system and the bonding orbital is occupied by an electron pair. The \( \pi \)-electronic energy of ethene is therefore \( 2\alpha + 2\beta \). The excited state of the molecule, when an electron is excited into the \( \pi^* \)-orbital, lies about \( 2\beta \) above the ground state.

In the case of butadiene, the approximations result in the determinant

\[ \begin{vmatrix}
\alpha - E & \beta & 0 & 0 \\
\beta & \alpha - E &\beta & 0 \\
0 & \beta & \alpha - E & \beta \\
0 & 0 & \beta & \alpha - E \\
\end{vmatrix}=0 \]

This expands into the quadratic equation

\[ x^4 -3x^2 + 1 = 0; \; \; \; \; x=\frac{\left ( \alpha - E \right )}{\beta} \]

with roots x² = 2.62, 0.38. Therefore, the energies of the four LCAO-MOs are

\[ E= \alpha \pm 1.62 \beta \; \; \; \; and \; \; \; \; E= \alpha \pm 0.62 \beta \]

as shown in Figure \(\PageIndex{2}\). There are four electrons to accommodate, and so the ground state configuration is \( 1 \pi^2 2 \pi^2 \).

Figure \(\PageIndex{2}\): The Hückel molecular orbital energy levels of butadiene and the top view of the corresponding \( \pi \)-orbitals. The four p-electrons (one supplied by each carbon atom) occupy the two lower ( \pi \)-orbitals. Note that the orbitals are delocalized. (CC-NC-BY-DUKE CHEM)

An important point emerges when we calculate the total \(\pi \)-electron binding energy in butadiene and compare it with what we find in ethene. In ethene the total energy is \(2\left ( \alpha +\beta \right )=\left ( 2\alpha +2\beta \right ) \); in butadiene it is \(2\left ( \alpha +1.62\beta \right )+2\left ( \alpha +0.62\beta \right )=\left ( 4\alpha +4.48\beta \right ) \). Therefore, the energy of the butadiene molecule lies lower by \( \left ( 4\alpha +4.48\beta \right )-2\left ( 2\alpha +2\beta \right )=0.48\beta \) (roughly –36 kJ mol^–1) than the sum of two \( \pi \)-bonds. This extra stabilization of a conjugated system is called the delocalization energy.

To obtain the wavefunction for a given energy state, E_j, one substitutes E = E_j into Equation \ref{pi2} and solves for the jth set of coefficients c_j1, c_j2, ..., cj_n. This process actually gives only ratios of coefficients; to determine numerical values, we add the normalization condition

\[ \int\Psi_{j}^{2}d\tau=\sum_{i}c_{ij}^2=1 \]

The four Hückel molecular orbitals for 1,3-butadiene are

\[ \Psi_1=0.372\phi_1 + 0.602\phi_2 +0.602\phi_3 +0.372\phi_4 \\
\Psi_2=0.602\phi_1 + 0.372\phi_2 -0.372\phi_3 -0.602\phi_4 \\
\Psi_3=0.602\phi_1 - 0.372\phi_2 -0.372\phi_3 +0.602\phi_4 \\
\Psi_4=0.372\phi_1 - 0.602\phi_2 +0.602\phi_3 -0.372\phi_4 \\ \]

In addition to \( \pi \)-electron energies and wavefunctions, the HMO approach also allows one to calculate several other quantities of chemical interest, such as electron densities, bond orders and free valence. 

Heteroatoms

  Heteroatoms (X) may be incorporated in the HMO method by appropriate changes in the empirical \( \alpha \) and \( \beta \) parameters associated with each atom and bond. A common example would be the inclusion of the \( \pi \)-electron contribution from the nitrogen atom in pyridine. These changes are incorporated in units of the standard \( \alpha_0 \) and \( \beta_0 \), usually of benzene, by use of the definitions

\[ \alpha_X=\alpha _0 + h_X \beta_0 \]

\[ \beta_{CX}=k_{CX} \beta_0 \]

Calculations involving such changes can be made by solving the secular equations. Unfortunately there remains considerable controversy concerning the specific values to be used for the h and k-parameters of various heteroatoms. Part of the problem is that there are several variations of MO theory and each will yield somewhat different parameter values with a given application.

We should distinguish, for example, the nitrogen in pyridine from a pyridinium nitrogen. In the former case, the heteroatom contributes two electrons to the \( \pi \)-system; in the latter case it contributes one. In the second part of the analysis of your experimental data, you will use a "canned" HMO computer program to determine the energy levels and molecular orbitals of the \( \pi \)-states of the dye molecules (as a function of \( \beta \)). By comparison of these results with your experimental data, you will determine an empirical value for \( \beta \). Note that this program incorporates values for the heteroatom parameters \(h\) and \(k\). You will need only to specify the location and type (e.g., pyridine-type nitrogen) of each heteroatom in the molecule to be studied.