# 4.2: Introduction

- Page ID
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\(\newcommand{\avec}{\mathbf a}\) \(\newcommand{\bvec}{\mathbf b}\) \(\newcommand{\cvec}{\mathbf c}\) \(\newcommand{\dvec}{\mathbf d}\) \(\newcommand{\dtil}{\widetilde{\mathbf d}}\) \(\newcommand{\evec}{\mathbf e}\) \(\newcommand{\fvec}{\mathbf f}\) \(\newcommand{\nvec}{\mathbf n}\) \(\newcommand{\pvec}{\mathbf p}\) \(\newcommand{\qvec}{\mathbf q}\) \(\newcommand{\svec}{\mathbf s}\) \(\newcommand{\tvec}{\mathbf t}\) \(\newcommand{\uvec}{\mathbf u}\) \(\newcommand{\vvec}{\mathbf v}\) \(\newcommand{\wvec}{\mathbf w}\) \(\newcommand{\xvec}{\mathbf x}\) \(\newcommand{\yvec}{\mathbf y}\) \(\newcommand{\zvec}{\mathbf z}\) \(\newcommand{\rvec}{\mathbf r}\) \(\newcommand{\mvec}{\mathbf m}\) \(\newcommand{\zerovec}{\mathbf 0}\) \(\newcommand{\onevec}{\mathbf 1}\) \(\newcommand{\real}{\mathbb R}\) \(\newcommand{\twovec}[2]{\left[\begin{array}{r}#1 \\ #2 \end{array}\right]}\) \(\newcommand{\ctwovec}[2]{\left[\begin{array}{c}#1 \\ #2 \end{array}\right]}\) \(\newcommand{\threevec}[3]{\left[\begin{array}{r}#1 \\ #2 \\ #3 \end{array}\right]}\) \(\newcommand{\cthreevec}[3]{\left[\begin{array}{c}#1 \\ #2 \\ #3 \end{array}\right]}\) \(\newcommand{\fourvec}[4]{\left[\begin{array}{r}#1 \\ #2 \\ #3 \\ #4 \end{array}\right]}\) \(\newcommand{\cfourvec}[4]{\left[\begin{array}{c}#1 \\ #2 \\ #3 \\ #4 \end{array}\right]}\) \(\newcommand{\fivevec}[5]{\left[\begin{array}{r}#1 \\ #2 \\ #3 \\ #4 \\ #5 \\ \end{array}\right]}\) \(\newcommand{\cfivevec}[5]{\left[\begin{array}{c}#1 \\ #2 \\ #3 \\ #4 \\ #5 \\ \end{array}\right]}\) \(\newcommand{\mattwo}[4]{\left[\begin{array}{rr}#1 \amp #2 \\ #3 \amp #4 \\ \end{array}\right]}\) \(\newcommand{\laspan}[1]{\text{Span}\{#1\}}\) \(\newcommand{\bcal}{\cal B}\) \(\newcommand{\ccal}{\cal C}\) \(\newcommand{\scal}{\cal S}\) \(\newcommand{\wcal}{\cal W}\) \(\newcommand{\ecal}{\cal E}\) \(\newcommand{\coords}[2]{\left\{#1\right\}_{#2}}\) \(\newcommand{\gray}[1]{\color{gray}{#1}}\) \(\newcommand{\lgray}[1]{\color{lightgray}{#1}}\) \(\newcommand{\rank}{\operatorname{rank}}\) \(\newcommand{\row}{\text{Row}}\) \(\newcommand{\col}{\text{Col}}\) \(\renewcommand{\row}{\text{Row}}\) \(\newcommand{\nul}{\text{Nul}}\) \(\newcommand{\var}{\text{Var}}\) \(\newcommand{\corr}{\text{corr}}\) \(\newcommand{\len}[1]{\left|#1\right|}\) \(\newcommand{\bbar}{\overline{\bvec}}\) \(\newcommand{\bhat}{\widehat{\bvec}}\) \(\newcommand{\bperp}{\bvec^\perp}\) \(\newcommand{\xhat}{\widehat{\xvec}}\) \(\newcommand{\vhat}{\widehat{\vvec}}\) \(\newcommand{\uhat}{\widehat{\uvec}}\) \(\newcommand{\what}{\widehat{\wvec}}\) \(\newcommand{\Sighat}{\widehat{\Sigma}}\) \(\newcommand{\lt}{<}\) \(\newcommand{\gt}{>}\) \(\newcommand{\amp}{&}\) \(\definecolor{fillinmathshade}{gray}{0.9}\)A conjugated \(\pi\) system is one that contains alternating single and double or triple bonds along a chain of carbon atoms. The most common type of conjugation is alternating double and single bonds. A simple example is the butadiene molecule. \(\ce{H2C=CH-CH=CH2}\)

## The Particle-In-A-Box approximation

Electrons in the \( \pi \)-electron system of a conjugated aromatic compound are not restricted to specific nuclei but are free to move throughout the system. In a linear conjugated system the potential energy of the electrons will vary along the chain, being lowest near the nuclei and highest between them. To a first approximation, however, it can be assumed that the potential energy of the electrons is constant along the length of the chain and increases to infinity one bond distance beyond the end atoms of the chain. The electrons may be visualized as being contained in a one-dimensional potential well, of length \( \ell \), and occupying orbitals of different energy within the well. The wave functions representing the orbitals and the energy levels of the orbitals can be obtained by solving the Schrödinger equation for this system.

For a one-dimensional system the Schrödinger equation takes the form

\[ \frac{\partial ^2\psi_{(x)}}{ \partial x^2}+\frac{8\pi ^2m}{h^2}\left ( E-V \right )\psi_{(x)}=0 \label{1} \]

where \( \psi_{(x)} \) are wavefunctions that satisfy the equation (eigenfunctions), \(E\) is the energy of the particle of mass \(m\), \(V\) is the potential energy, and \(h\) is Planck's constant.

The potential energy, \(V\), is a constant within the well of length \( \ell \), where \( \psi_{(x)} \) must vanish for \(x≤0\) and \(x≥ \ell \) and \(V = \infty\). This allows us to conveniently set \(V\) equal to zero in the region \(0≤x≥ \ell \) when deriving solutions of the equation. The equation then reduces to

\[ \frac{-h^2}{8\pi ^2m}\frac{\partial^2\psi_{(x)}}{\partial x^2}=E\psi_{(x)} \label{2} \]

Solutions of this equation are \( \psi_{(x)}=A \cdot \sin\left ( \frac{n\pi x}{\ell} \right ) \), where \(A\) is a constant and n = 1,2,3,... etc.

Verification that this wave function is indeed a solution of the Schrödinger equation may be obtained by substitution into the left- and right-hand sides of the above equation.

\[\begin{align*} \text{Left-hand side} &=\dfrac{-h^2}{8\pi ^2m}\left ( \frac{-n^2\pi ^2}{\ell^2} \right )A\cdot \sin\left ( \frac{n\pi x}{\ell} \right ) \\[4pt] &=\frac{-h^2}{8m\ell^2}A \cdot \sin\left ( \frac{n\pi x}{\ell} \right ) \\[4pt] \text{Right-hand side} &=EA\cdot \sin\left ( \frac{n\pi x}{\ell} \right ) \end{align*} \]

The function \( \psi_{(x)} \) is a solution of the Schrödinger equation when the energy \(E\) of the system has the values \( E=\frac{n^2h^2}{8m\ell^2} \). The allowed energy levels are depicted diagrammatically below.

The ground state configuration for the conjugated system of a molecule is derived by allocating two electrons to each of the energy levels in accord with the Pauli exclusion principle. Thus, a molecule with N\(\pi\) electrons will have the N/2 lowest levels filled and all the higher levels empty. When the molecule absorbs UV-vis light, the energy absorbed is from an electron in the HOMO (where n = N/2) becoming excited to the LUMO where n = (N/2) + 1. The energy change, ∆E, for this transition is

\[\begin{align*} \Delta E &=E_{\left ( \frac{N}{2}+1 \right )}-E_{\left ( \frac{N}{2} \right )} \\[4pt] &=\frac{\left [ \left ( \frac{N}{2} \right )+1 \right ]^2h^2}{8m\ell^2}-\frac{\left ( \frac{N}{2} \right )^2h^2}{8m\ell^2} \\[4pt] &=\frac{\left ( N+1 \right )h^2}{8m\ell^2} \end{align*} \]

Since \( \Delta E=h\nu =\frac{hc}{\lambda } \) where \(c\) is the speed of light and \( \lambda \) is the wavelength, then

\[ \lambda=\frac{8mc}{h} \frac{\ell^2}{\left ( N+1 \right )} \label{4} \]

Thus if \( \ell \) is known, this relation can be used to calculate the wavelength maximum of the band representing the transition from the ground state to the first excited state in a \(\pi \)-electron system.

### PIB model for cyanine dyes:

This Particle-in-th-Box model can be applied to carbocyanine dyes which may be represented, for example, by the resonance forms shown below for the 1,1'- diethyl-4,4'-carbocyanine (cryptocyanine) cation.

The \( \pi \)-electrons in the molecule are regarded as a collection of "free" electrons (independent particles) confined to a box, whose wavefunctions are the square-well functions. Once again, these electrons are distributed among the wavefunctions and energy levels according to Pauli principles, so that the absorption of energy (light) according to this model will follow equation \ref{4}. To apply the model to the absorbance \(\lambda\) of a specific molecule we need to determine N (number of \pi electrons) and \( \ell \) (length of the conjugated chain).

**Number of \(\pi\) electrons in cyanine dyes (N):** In the chain between the two nitrogen atoms, each carbon atom contributes one \(\pi\) electron. The \(\pi\) system also includes a total of three additional electrons contributed from the N atoms. Two of these electrons are contributed from the neutral N, while one is contributed from the cationic nitrogen (assuming the resonance structures shown in Figure \(\PageIndex{2}\)). If we denote the number of carbon atoms in the polymethine chain (between the two nitrogen atoms) by p, then N, the number of \( \pi \)-electrons, is p + 3.

\[N = p +3 \label{N} \]

Where \(N\) is the number of \(\pi\) electrons in the system, and \(p\) is the number of carbon atoms between the nitrogen atoms.

**Approximation of \(\ell\) in cyanine dyes: **The total length of the potential well is taken to be the length of the shortest carbon chain __between__ the nitrogen atoms *plus one bond length at each end*. Assuming that the length of one bond is \(1.39 x 10^{–8} \) cm, an approximation for \(\ell]) is

\[ \ell= (p + 3)a \label{l}\]

where \(a = 1.39 x 10^{–8} \) cm is the mean distance between atoms in the chain.

**PIB model for Cyanine Dyes: **Substituting \ref{N} for N and \ref{l} for \( \ell \) in equation \ref{4} yields:

\[ \lambda=\frac{8mca^2}{h} \frac{\left ( P+3 \right )^2}{P+4} \label{5}\]

If polarizable groups are present at the ends of the chain, then the potential energy of the \( \pi \)-electrons in the chain does not rise so steeply at the ends. This results in lengthening of the box length, \( \ell \), which can be incorporated into equation \ref{5} through the addition of a new "lengthening" parameter, \( \gamma \), resulting in

\[ \lambda=\frac{8mca^2}{h} \frac{\left ( P+3+\gamma \right )^2}{P+4} \label{6}\]

where \( \gamma \) should be a constant for a series of dyes of a given type and should lie between 0 and 2. If such a series is studied experimentally, the empirical parameter \( \gamma \) may be adjusted to achieve the best fit to the data.

## The Hückel Molecular Orbital (HMO) model

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

*, 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*

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

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 (CH_{2}=CH_{2}) 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 \(\PageIndex{1}\).

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 x^{2} = 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 \( 1 \pi^2 2 \pi^2 \).

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.