Hückel Molecular Orbital Theory

In the chapter on cyanine dyes, you learned several important concepts.  You saw that a simple model, the particle in a box, could be used to understand the spectra of conjugated molecules.  The model clearly showed the relationship of chain length and absorption wavelength and let you compute longest absorption wavelength by using the energy difference between the highest energy occupied molecular orbital, the HOMO, and the lowest unoccupied molecular orbital, the LUMO.  You also learned that certain transitions were allowed and others not allowed and were able to predict these by using symmetry arguments and the transition moment integral.

In this chapter we will explore another model. This new model is called the Hückel model or Hückel Molecular Orbital (HMO) theory.  Through HMO theory we will extend our understanding of molecular orbital methods.  The HMO theory is an excellent vehicle for this extension as it employs some of the mathematical concepts and techniques that are used in current high-level ab initio molecular orbital calculations, which are considered in the next chapter.

We will use linear combination of atomic orbitals (the LCAO method) to form each molecular orbital.  We will create and solve the secular equation for computing the energies of the molecular orbitals.  Matrix methods, like those employed for vibrational normal mode analysis, will be used to solve the secular equation for large molecular systems.  Charge densities for each atom in the molecule and bond orders for bonded pairs of atoms will be computed.  Also, molecular orbitals will be used for computing chemically interesting properties such as the site of electrophilic or nucleophilic attack in a reaction.

 

 

12.1 Application to Conjugated Molecules

Conjugated double bonds are double bonds that alternate with single bonds along a molecular chain or in a molecular ring.  Conjugated molecules are those that contain this pattern of alternating double and single bonds.  These molecules have equivalent resonance structures and interesting chemical properties.  Benzene is the prime example.  Conjugated compounds were the first class of polyatomic molecules studied by molecular orbital theory because of their extended pi system of electrons.  Pi electrons are those electrons found in pi orbitals.  Pi orbitals are composed of p_z atomic orbitals that are perpendicular to a plane of symmetry in the molecule.  Pi orbitals change sign upon reflection in this plane, whereas sigma orbitals do not.

Several conjugated molecules are shown in Figure 12.1.1.  Notice that they include both non-cyclic and cyclic molecules and heteroatoms such as nitrogen and oxygen.  Hetero means different from carbon and hydrogen.  One of the most important members of this group of molecules is benzene, which is given a special place in organic chemistry because of its stability.

Figure 12.1.1. Sample conjugated molecules.

We can address a number of questions pertaining to benzene and other conjugated molecules using Hückel molecular orbital theory.  What is the cause of benzene's stability?  Can we quantify this stability?  Since all the bonds in benzene have the same length, can we account for the fact that the bonds in benzene are characterized as intermediate between single and double bonds?  Can we write molecular orbitals for benzene and systematically explain some of its chemistry using a quantitative molecular orbital model?  Can we explain the chemical reactivity and spectroscopic properties of other diverse conjugated molecules with the same model?

Although the Hückel model is simple, is it sufficiently robust to predict trends and produce insights that are not possible through the very simple particle-in-a-box model that was used to describe the cyanine dyes.  We will explore these questions in the following sections of this chapter and along the way introduce some of the mathematical tools needed in this molecular orbital approach and the molecular orbital methods used by scientists today.  Just to keep things in perspective, remember that Hückel theory was developed at the time when computers were not available to chemists, and a simple mathematical approach was a very important development.  Although HMO theory includes some drastic assumptions, it enabled the early calculations to be done with mechanical calculators or by hand.

 

12.2. Describing the Sigma and P Electrons Separately.

One of the important ideas used over and over again in Quantum Mechanics is the separation of variables.  In the case of conjugated molecules, we have a chemical system where the separation of variables naturally falls out of our studies in organic chemistry.  When one considers the pi and sigma electrons in a molecule like ethylene or 1,3-butadiene, it is easy to organize the electrons into sigma electrons and pi electrons.  This separation follows from the molecular symmetry and the fact that hybrid atomic orbitals are used to describe bonds in molecules.  The orbitals that form the sigma bonds are the sp²-hybrid orbitals on carbon and the s-orbital on hydrogen as shown in Figure 12.2.1.  The 2p_z orbital on carbon is perpendicular to the plane of these three hybrid orbitals and forms a pi bond through sideways overlap with a p_z-orbital on a neighboring atom.  This picture of bonding is very successful when carbon is bonded to three other atoms because it provides three orbitals that are highly directional.

 

In quantum chemistry, this picture of hybridization and sigma/pi bonding is translated into a mathematical statement about the molecular orbitals in a molecule.  The total wavefunction of a molecule is written as a product of a sigma part and a pi part. 

                                                                               12.2.1
where  is the wavefunction describing the electrons in sigma orbitals and  is the wavefunction describing the electrons in pi orbitals.  Such a product function would be an eigenfunction of the molecular Hamiltonian if the sigma and pi electrons did not interact.

The wavefunction for the pi electrons then is described as a product of all the pi molecular orbitals.

                                                                 12.2.2
Each , with j = 1…N, represents a molecular orbital, i.e. a wavefunction for one electron moving in the electrostatic field of the nuclei and the other electrons.  Two electrons with different spin are placed in each molecular orbital so that the number of occupied molecular orbitals N is half the number of electrons, n, i.e. N = n/2.

Each molecular orbital, , is written as a linear combination of atomic orbitals (LCAO).

            with  ,                                            12.2.3
where  is the 2p_z atomic orbital on atom r of the conjugated pi system.  The number of molecular orbitals that one obtains by this procedure is equal to the number of atomic orbitals.  Consequently, the indices j and r both run from 1 to N.  The c_jr are the weighting coefficients for the atomic orbitals in the molecular orbital.  These coefficients are not necessarily equal, or in other words, the orbital on each carbon atom is not used to the same extent to form each molecular orbital.  Different values for the coefficients give rise to different net charges at different positions in a conjugated molecule.  This charge distribution is very important when discussing spectroscopy and chemical reactivity.

The energy of the j^th molecular orbital is given by a one-electron Schrödinger equation using an effective one electron Hamiltonian, h_eff, which expresses the interaction of an electron with the rest of the molecule.

                                                                               12.2.4
 is the energy eigenvalue of the j^th molecular orbital, corresponding to the eigenfunction .  The beauty of this method, as we will see later, is that the exact form of h_eff is not needed.  The total pi energy of the molecule is the sum of the single electron energies.

                                                                               12.2.5
where n_j is the number of electrons in orbital j.

The expectation value expression for the energy for each molecular orbital is used to find ε_j and then E_π.

          .                                        12.2.6
The notation , which is called a bra-ket, just simplifies writing the expression for the integral.  Note that the complex conjugate now is identified by the left-side position and the bra notation  and not by an explicit *.

After substituting Equation 12.2.3 into 12.2.6, we obtain for each molecular orbital

                                                           12.2.7
which can be rewritten as

                                                             12.2.8
where the index j for the molecular orbital has been dropped because this equation applies to any of the molecular orbitals.

 

 

To simplify the notation we use the following definitions.  The integrals in the denominator of Equation 12.2.8 represent the overlap between two atomic orbitals used in the linear combination.  The overlap integral is written as S_rs.  The integrals in the numerator of Equation 12.2.8 are called either resonance integrals or coulomb integrals depending on the atomic orbitals on either side of the operator h_eff as described below.

 is the overlap integral.  because we use normalized atomic orbitals.  For atomic orbitals r and s on different atoms,  has some value between 1 and 0: the further apart the two atoms, the smaller the value of .

 is the Coulomb Integral.  It is the kinetic and potential energy of an electron in, or described by, an atomic orbital, _r, experiencing the electrostatic interactions with all the other electrons and all the positive nuclei.

 is the Resonance Integral or Bond Integral.  This integral gives the energy of an electron in the region of space where the functions _r and _s overlap.  This energy sometimes is referred to as the energy of the overlap charge.  If r and s are on adjacent bonded atoms, this integral has a finite value.  If the atoms are not adjacent, the value is smaller, and assumed to be zero in the Hückel model.

In terms of this notation, Equation 12.2.8 can be written as

                                                          .                                                12.2.9

We now must find the coefficients, the c's.  One must have a criterion for finding the coefficients.  The criterion used is the Variational Principle.  Since the energy depends linearly on the coefficients in Equation 12.2.9, the method we use to find the best set of coefficients is called the Linear Variational Method.   This method is similar to the nonlinear variational method described in Chapter 10 for optimizing the parameters in a wavefunction.

 

12.3. The Linear Variational Method.

The Variational Principle tells us that the energy we compute for any trial wavefunction used in equation 12.2.6 will be greater than the exact energy for that Hamiltonian.  In more mathematical terms, any energy we compute is an upper bound to the true energy for the molecular system.  We write this statement as

                                                                                                                          12.3.1

If the trial wavefunction we use has adjustable parameters, then we can vary the parameters in a systematic way to find their optimum values.  The optimum values give the lowest energy for the trial function.  Initial guess values of the parameters will always give energies that are greater than the best energy that one could obtain from a given function unless you are extremely lucky and happen to guess the optimum values.  This problem is equivalent to finding the minimum in a potential energy function or any other mathematical function.  To accomplish this minimization, rather than making trial guesses, it is more efficient to use the calculus approach of taking the derivative of the energy function with respect to each parameter and setting each derivative equal to zero.  If you have two parameters, you then will have two simultaneous equations to solve.  If you have N parameters you will have N simultaneous equations.

The task is to minimize the energy with respect to all the coefficients by solving the N simultaneous equations produced by differentiating Equation 12.2.9 with respect to each coefficient.
                     for  t = 1, 2, 3, … N                                            12.3.2
Actually we also should differentiate Equation 12.2.9 with respect to the , but this second set of N equations is just the complex conjugate of the first and produces no new information or constants.

To carry out this task, rewrite Equation 12.2.9 to obtain Equation 12.3.3 and then take the derivative of Equation 12.3.3 with respect to each of the coefficients.

                                                          12.3.3
Actually we don't want to do this differentiation N times, so consider the general case where the coefficient is c_t.  Here t represents any number between 1 and N.

This differentiation is relatively easy, and the result, which is shown by Equation 12.3.4, is relatively simple because some terms in Equation 12.3.3 don't involve c_t and others depend linearly on c_t.  The derivative of the terms that don't involve c_t is zero (e.g.  ).  The derivative of terms that contain c_t is just the constant factor that multiples the c_t, (e.g.  ).  Consequently, only terms in Equation 12.3.3 that contain c_t contribute to the result, and whenever a term contains c_t, that term appears in Equation 12.3.4 without the c_t because we are differentiating with respect to c_t.  The result after differentiating is

                                                        .                                              12.3.4

If we take the complex conjugate of both sides, we obtain                     
                   .                                                          12.3.5
Since ε = ε*, , and , this equation can be reversed and written as

                                                                       12.3.6
or upon rearranging as

          .                                                           12.3.7
There are N simultaneous equations that look like this general one; N is the number of coefficients in the LCAO.  Each equation is obtained by differentiating Equation 12.3.3 with respect to one of the coefficients.