Conjugated Systems: Organic Molecules

The Hamiltonian of a molecule containing $n$ electrons and $N$ nuclei can be described according to the Born-Oppenheimer approximation as follows,

$$\hat{H}_{el} = \sum_{i=1}^n \left( -\frac{\hbar^2}{2m_i} \nabla_{... ...=1}^N \frac{z_j e^2}{r_{ji}} \right) +\sum_i^n \sum_{k>i}^n \frac{e^2}{r_{ik}}.$$

This Hamiltonian includes terms that describe both $\pi$ and $\sigma$ electrons. However, the distinctive chemistry of conjugated organic molecules is usually relatively independently of $\sigma$-bonds, and rather correlated with the electronic structure of $\pi$-electrons. For example, the spectroscopy of conjugated organic molecules, as well as ionization potentials, dipole moments and reactivity, can be described at least qualitatively by the electronic structure of the $\pi$-electron model. Therefore, we make the approximation that the solution of the eigenvalue problem of a conjugated system can be factorized as follows,

$$\psi=\hat{A} \psi_{\sigma} \psi_{\pi},$$

where $\hat{A}$ is an antisymmetrization operator upon exchange of $\sigma$ and $\pi$ electrons.

The potential due to the nuclei and the average field due to $\sigma$ electrons, can be described by the following Hamiltonian:

$$\hat{H}_{\pi} = \sum_{i=1}^{n_{\pi}} \hat{h}_{core}(i) +\sum_{i=1}^{n_{\pi}}\sum_{k>i}^{n_{\pi}} \frac{e^2}{r_{ik}},$$ (62)

where $\hat{h}_{core}$ includes kinetic energy of $\pi$ electrons, interaction of $\pi$ electrons with $\sigma$ electrons, and shielding of nuclear charges. An approximate solution can be obtained by disregarding the repulsion between $\pi$ electrons in Eq. (62), and by approximating the Hamiltonian of the system as follows,

$$\hat{H}_{\pi}^{(0)} \approx \sum_{i=1}^{n_{\pi}} H_{\text{eff}}(i... ...) = -\frac{\hbar^2}{2m_j} \nabla_{r_j}^2 - \sum_{k=1}^N \frac{z_k'e^2}{r_{kj}}.$$ (63)

The effective nuclear charge $z_k'$ incorporates the average screening of nuclear charges due to $\sigma$ and $\pi$ electrons.

Since $\hat{H}_{\text{eff}}(j)$ depends only on coordinates of electron $j$, we can implement the separation of variables method and solve the eigenvalue problem,

$$\hat{H}_{\pi}^{(0)} \vert\psi_{\pi}> = E_{\pi} \vert\psi_{\pi}>,$$

according to the factorizable solution $\vert\psi_{\pi}> = \prod_{j=1}^{n_{\pi}} \vert\phi_j>$, where,

$$\hat{H}_{\text{eff}}(j) \mid \phi_j > = \epsilon_j \mid \phi_j >.$$ (64)

The energy $E_{\pi}$ is obtained by using the Pauli exclusion principle to fill up the molecular orbitals, after finding the eigenvalues $\epsilon_j$.

Eq. (63) is solved by implementing the variational method, assuming that $\mid \phi_j >$ can be written according to a linear combination of atomic orbitals,

$$\vert\phi_j> =\sum_{k=1}^N C_{jk} \vert\chi_k>,$$

where $\vert\chi_k>$ represents a $2p_z$ orbital localized in atom $k$ and the sum extends over all atoms in the conjugated system.

Example:

Consider the ethylene molecule represented by the following diagram:

The diagram shows $\sigma$ bonds in the equatorial plane of the molecule, and $\pi$ orbitals 1 and 2 that are perpendicular to such plane.

The LCAO for ethylene is,

$$\mid \phi_j> = c_{j1} \mid \chi_1 > + c_{j2} \mid \chi_2>.$$ (65)

Therefore, the secular equations can be written as follows,

$\left(H_{11} - S_{11} \epsilon_j \right) c_{j1} +\left(H_{12} - S_{12} \epsilon_j \right) c_{j2} = 0$,

$\left(H_{21} - S_{21} \epsilon_j \right) c_{j1} +\left(H_{22} - S_{22} \epsilon_j \right) c_{j2} =0.$

Hückel Method:

The Hückel Method is a semi-empirical approach for solving the secular equations. The method involves making the following assumptions:

1. $H_{kk} = \alpha$, where $\alpha$ is an empirical parameter (vide infra).

2. $H_{jk} = \beta$, when $j = k \pm 1$; and $H_{jk} = 0$, otherwise. The constant $\beta$ is also an empirical parameter (vide infra).

3. $S_{jk} = 1$, when $k=j \pm 1$; and $S_{jk} = 0$, otherwise.

According to the Hückel model, the secular determinant becomes,

$$\left \vert \begin{matrix}\alpha - \epsilon_j & \beta \\ \beta & \alpha - \epsilon_j \\ \end{matrix} \right \vert = 0.$$

Therefore, the eigenvalues of the secular determinant are $\epsilon_j = \alpha \pm \beta$ and can be represented by the following diagram:

The energy difference between ground and excited states is $\Delta E = E_2- E_1 = - 2 \beta$. Parameter $\beta$ is usually chosen to make $\Delta E$ coincide with the peak of the experimental absorption band of the molecule.

An interactive program to perform electronic structure calculations within the "Simple Huckel Molecular Orbital" approximation can be found here.