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.

 

 

This method is called the linear variational method because the variable parameters affect the energy linearly unlike the shielding parameter in the wavefunction that was discussed in Chapter 10.  The shielding parameter appears in the exponential part of the wavefunction and the effect on the energy is nonlinear.  A nonlinear variational calculation is more laborious than a linear variational calculation.

Equations 12.3.6 and 12.3.7 represent a set of homogeneous linear equations.  As we discussed for the case of normal mode analysis in Chapter 6, a number of methods can be used for solving these equations to obtain values for the energies, ε's, and the coefficients, the c_r’s.

Matrix methods are the most convenient and powerful.  First we write more explicitly the set of simultaneous equations that is represented by Equation 12.3.6.  The first equation has t=1, the second t=2, etc.  N represents the index of the last atomic orbital in the linear combination.

                                              12.3.8

This set of equations can be represented in matrix notation.

                                                                               12.3.9
Here we have square matrix H and S multiplying a column vector C' and a scalar ε. Rearranging produces:

12.3.10

 

 

The problem is to solve these simultaneous equations, or the matrix equation, and find the orbital energies, which are the ε's, and the atomic orbital coefficients, the c's, that define the molecular orbitals.

 

12. 4. The Hückel Method

In the Hückel method only the atoms that are part of the conjugated bond system and their p_z-π orbitals are used in the LCAO. To greatly simplify the calculation, some approximations are introduced for the Coulomb, resonance, and overlap integrals.  While we could evaluate the overlap integrals in Equation 12.3.7, there is no way we can evaluate the other integrals involving the effective Hamiltonian operator since we cannot write an explicit expression for this operator.  It is only a conceptual tool.  The HMO method treats these integrals as parameters that are evaluated empirically by fitting the theory to experimental results.

The Coulomb integrals like H₁₁, H₂₂, etc represent the energy of an electron on a particular atom in the molecule.  These integrals are all taken to be same because the carbon atoms in a pi system are similar, though not identical, and are represented by the symbol α.

The resonance integrals like H₁₂, H₁₃, H₂₃, etc are taken to be zero unless the atoms are neighbors, i.e. bonded together by a sigma bond.  For these cases, it is assumed that the resonance integrals are all equal and are represented by the symbol β.

Since the atomic orbitals are normalized, the S_ii =1.  Although the overlap between functions on neighboring atoms is around 0.3, all the S_ij with  are taken to be zero.

With these approximations, Equations 12.3.8 can be rewritten in matrix form as

                                                   12.4.1
where β’s only appear for neighboring atoms.

 

 

We now need to say a bit more about the relationship of the matrix

in Equation 12.4.1 to the structure of the molecule that it is describing.  For any molecule, you could label the carbon atoms with integer numbers: 1,2,3,4, etc.  These numbers then correspond to the rows and columns in the H matrix.  The diagonal elements (H₁₁, H₂₂, H₃₃, etc.) refer to particular carbon atoms (1,2,3, etc., respectively).  The off-diagonal elements (H₁₂, H₁₃, H₂₃, etc.) refer to pairs of atoms, 1 and 2, atoms 1 and 3, atoms 2 and3, etc., respectively.  If any two atoms are bonded neighbors, then the corresponding matrix element is β, otherwise it is 0.

It also is common in HMO theory, to divide both sides of Equation 12.4.1 by β and make the following substitution.

          .                                                                    12.4.2
Equation 12.4.1 then becomes

                                                          12.4.3
Rearranging produces

                                                                                        12.4.4

This matrix equation now is in the form of an eigenvalue problem, where x is the eigenvalue and the column vector of coefficients, C', is the eigenvector.  The square matrix of ones and zeros is called the topology matrix, T, because it specifies the topology or connectedness of the molecule.  The elements of the topology matrix are 1 if the corresponding atoms are bonded or connected and 0 otherwise.

Putting Equation 12.4.4 into matrix notation produces

          TC′ = C′ x                                                                12.4.5
where T is the topology matrix and C′ is the eigenvector of coefficients for the x eigenvalue.  More generally we can write, similar to what was done in Chapter 6 for the vibrational normal mode analysis,

          TC=CX                                                                         12.4.6
where C is a square matrix where columns are the eigenvectors, one of which was C′, and X is a diagonal matrix with the eigenvalues x as the diagonal elements.  The columns in matrix C contain the coefficients of the atomic orbitals used in the LCAO approximation.

If we left multiply each side of Equation 12.4.6 by the inverse of C we obtain

          C^-1TC=X.                                                                      12.4.7
The problem then is to find the transformation matrix C that diagonalizes T.  Computer routines generally are available for this purpose.

At this point it becomes informative and possible to consider a real molecule.  As an example take the butadiene molecule.  We need only write down the topology matrix, T, and ask a computer to produce the eigenvalues and eigenvectors.  The eigenvalues are values for x, which we then use in Equation 12.4.2 in order to solve for ε in terms of α and β.

Numbering the 4 carbon atoms in butadiene 1 through 4 from left to right, produces the following T matrix, and Mathcad gives us the eigenvalues and eigenvectors.  The resulting energies and molecular orbitals are given in Table 12.4.1 in order of increasing energy.  The energies are expressed in terms of α and β.  All the information about bonding is in the β parameter.  A numerical value for β is not needed to answer many questions.  When a numerical value is needed, it is obtained from spectroscopic or thermodynamic data.

                                                                                                        12.4.8

 

 

Table 12.4.1. Hückel Energies and Wavefunctions for Butadiene

Energy	Molecular Orbital
ε1 = α + 1.618 β	ψ1 = 0.372 1 + 0.602 2 + 0.602 3 + 0.372 4
ε2 = α + 0.618 β	ψ2 = 0.602 1 + 0.372 2 - 0.372 3 - 0.602 4
ε3 = α - 0.618 β	ψ3 = 0.602 1 - 0.372 2 - 0.372 3 + 0.602 4
ε4 = α - 1.618 β	ψ4 = - 0.372 1 + 0.602 2 - 0.602 3 + 0.372 4

 

 

 

 

 

 

 

12.5 Molecular Properties and Chemical Reactivity.

One of the important things molecular orbital models and chemical computations do is provide the chemist with insights into the reactivity of molecules.  At the most basic level we get insight by just knowing the charge on the atom.  If an atom in a molecule is slightly negative then electrophiles are attracted to it.  Similarly nucleophiles are attracted to positive centers in a molecule.

The charge at an atom in a molecule is computed by summing up the contribution of all of the molecular orbitals to the electron density at that atom.  The contribution of a MO to the electron density at an atom is given by the square of the coefficient of atomic orbital used for the atom in the LCAO for that MO.  The total electron density is then the sum of all contributions from all occupied molecular orbitals.  This sum is written as

                                                                        12.5.1
where q_r is the total charge at atom r and c_rj is the coefficient from molecular orbital j for the atomic orbital of atom r.  The sum is taken over all occupied molecular orbitals and n_j is the number of electrons in an occupied molecular orbital.  The coefficient squared  is called the atomic population (of atomic orbital r from one electron in molecular orbital j), and sum q_r is called the net atomic population (of atomic orbital r from all the electrons in occupied molecular orbitals).

 

 

Azulene and benzene are different in that the former is a nonalternant hydrocarbon and latter is an alternant hydrocarbon.  You can see this difference by placing a star next to every other (or alternating) carbon in benzene and in azulene.  In azulene you can not do this labeling, you must eventually put two stars together.  All alternant hydrocarbons have equal charge distributions like benzene.  All nonalternant hydrocarbons have unequal charge distributions like azulene.  All alternate hydrocarbons have equal charge distributions like benzene.

 

Another property that we can compute from the LCAO coefficients is the bond order.  This property gives the strength of the pi bonding between pairs of adjacent atoms.  The pi bond order, p_rs, between two atoms r and s is computed by summing the product of the coefficients between pairs of atoms.

                                                                      12.5.2
where c_rj is the coefficient of the atomic orbital for atom r in molecular orbital j.

 

Predicting the stability of molecules is often a key in a successful synthesis.  An important step in this direction was taken by Hückel when, by using his theory, he showed that monocylclic conjugated polyenes have the greatest stability and degree of aromaticity when they obey the “4n+2” rule.  In this rule, n is an integer and 4n+2 gives the number of pi electrons in the molecule.  Benzene follows this rule for n=1.  This rule is so powerful and elegant in its simplicity that it successfully predicted the stability of the cyclopentadienyl anion.

The stability of a molecule is determined to a first approximation by the number of bonding orbitals that are doubly occupied.  Benzene has six electrons in three bonding pi molecular orbitals.  Cyclobutadiene on the other hand has 2 electrons in one bonding orbital and 2 electrons in two degenerate nonbonding molecular orbitals.  Cyclobutadiene is very unstable and has only transient existence at extremely low temperatures.

 

The HMO method has been surprisingly successful.  It led to many insights about chemical reactivity.  Many calculations were possible to do by hand, and with the arrival of computers large and larger hydrocarbons were studied.  Various corrections were made to the method to accommodate heterocyclic atoms in the conjugated systems.  Other corrections were made to account for the interaction of pi electrons with the sigma framework.  Correlations were made between the reactivity of sites in a molecule and computed parameters based on the coefficients of the molecular orbitals.  The rapid growth in computing power quickly made more extensive and accurate calculations possible.  It became possible to include integrals in the calculations and diagonalize larger matrices.  The Extended Hückel Method is a result of these developments.

 

12.6. The Extended Hückel Method

The Extended Hückel Molecular Orbital Method (EH) grew out of the need to consider all valence electrons in a molecular orbital calculation.  By considering all valence electrons, chemists could determine molecular structure, compute energy barriers for rotation about bonds, and even determine energies and structures of transition states for reactions.  The computed energies could be used to choose between proposed transitions states to clarify reaction mechanisms.

In the EH method one writes the wavefunction as a product of a valence wavefunction and a core wavefunction.

                                                         .                                              12.6.1

The total valence electron wavefunction is described as a product of the one-electron wavefunctions.

                                         12.6.2
where n is the number of electrons and j identifies the molecular orbital as was done for the HMO method.  Each molecular orbital is again is given as an LCAO

                                                                            12.6.3
where now the _r are the valance atomic orbitals chosen to include the 2s, 2p_x, 2p_y, and 2p_z of the carbons and heteroatoms in the molecule and the 1s orbitals of the hydrogens.  The set of orbitals defined here is called a basis set.  Since this basis set contains only the atomic-like orbitals for the valence shell of the atoms in a molecule, it is called a minimal basis set.

In the EH method we use an effective one electron Hamiltonian, h_eff, and then we proceed with determining the energy of a molecular orbital as we did above for the simple HMO method.  Using the same notation as in Section 12.2 we have again

                                                                      12.6.4
where  and .

Minimization of the energy with respect to each of the coefficients again yields a set of simultaneous equations just like Equation 12.3.7.
                                                                               12.6.5
As before, these equations can be written in matrix form
                   .                                                                    12.6.6

Equation 12.6.6 accounts for one molecular orbital.  It has energy ε, and it is defined by the elements in the C' column vector, which are the coefficients that multiply the atomic orbital basis functions in the linear combination of atomic orbitals.

As before (see Equation 12.4.6 and the vibrational analysis problem in chapter 6), we can write one matrix equation for all the molecular orbitals.

          HC=SCE                                                                       12.6.7
where H is a square matrix containing the H_rs, the one electron energy integrals, and C is the matrix of coefficients for the atomic orbitals.  Each column in C is the C' that defines one molecular orbital in terms of the basis functions.  In extended Hückel theory, the overlap is not neglected, and S is the matrix of overlap integrals.  E is the diagonal matrix of orbital energies.  All of these are square matrices with a size that equals the number of atomic orbitals used in the LCAO for the molecule under consideration.

As you should recognize from the discussion of Hückel Molecular orbital theory and the normal coordinate analysis in Chapter 6, Equation 12.6.7 represents an eigenvalue problem.  For any extended Hückel         calculation
we need to set up these matrices and then find the eigenvalues and eigenvectors.  The eigenvalues are the orbital energies, and the eigenvectors are the atomic orbital coefficients that define the molecular orbital in terms of the basis functions.

 

 

12.7. Extended Hückel Calculation Details.

The elements of the H matrix are assigned using experimental data.  This approach makes the extended Hückel method a semi-empirical molecular orbital method.  The basic structure of the method is based on the principles of physics and mathematics while the values of certain integrals are assigned by using educated guessing and experimental data.  The H_rr are chosen as valence state ionization potentials with a minus sign to indicate binding.  The values used by R. Hoffmann when he developed the extended Hückel technique were those of H.A. Skinner and H.O. Pritchard (Trans. Faraday Soc. 49 (1953), 1254).  These values for C and H are listed in Table 12.7.1.  The values for the heteroatoms (N, O, and F) are taken from Pople and Beveridge (reference).  The H_rs values are computed from the ionization potentials according to

                                                                12.7.1
The rationale for this expression is that the energy should be proportional to the energy of the atomic orbitals, and should be greater when the overlap of the atomic orbitals is greater.  The contribution of these effects to the energy is scaled by the parameter K.  Hoffmann assigned the value of K after a study of the effect of this parameter on the energies of the occupied orbitals of ethane.  The conclusion was that a good value for K is K=1.75.

 

The overlap matrix also must be determined.  The matrix elements are computed using the definition  where _r and _s are the atomic orbitals. Slater-type orbitals (STO’s) are used for the atomic orbitals rather than hydrogenic orbitals because integrals involving STO's can be computed more quickly on computers.  Slater type orbitals have the form

                                                12.7.2
where zeta, , is a parameter describing the screened nuclear charge.  In the extended Hückel calculations done by Hoffmann, the Slater orbital parameter ζ was 1.0 for the H_1s and 1.652 for the C_2s and C_2p orbitals.

 

Overlap integrals involve two orbitals on two different atoms or centers.  Such integrals are called two-center integrals. In such integrals there are two variables to consider, corresponding to the distances from each of the atomic centers, r_A and r_B.  Such integrals can be represented as
                                              12.7.3
but elliptical coordinates must be used for the actual integration.  Fortunately the software that does extended Hückel calculations contains the programming code to do overlap integrals.  The interested reader will find sufficient detail on the evaluation of overlap integrals and the creation of the programmable mathematical form for any pair of Slater orbitals in Appendix B4 (pp. 199 - 200) of the book Approximate Molecular Orbital Theory by Pople and Beveridge.  The values of the overlap integrals for HF are given in Table. 12.7.2.

 

                    Table 12.7.2 Overlap Integrals for HF

	F 2s	F 2px	F 2py	F 2pz	H 1s
F 2s					0.47428
F 2px					0
F 2py					0.38434
F 2pz					0
H 1s

 

 

Our goal is to find the coefficients in the linear combinations of atomic orbitals and the energies of the molecular orbitals.  For these results, we need to transform Equation 12.6.7, HC=SCE, into a form that allows us to use matrix diagonalization techniques.  We are hampered here by the fact that the overlap matrix is not diagonal because the orbitals are not orthogonal.  Mathematical methods do exist that can be used to transform a set of functions into an orthogonal set.  Essentially these methods apply a transformation of the coordinates from the local coordinate system describing the molecule into one where the atomic orbitals in the LCAO are all orthogonal.  Such a transformation can be accomplished through matrix algebra, and computer algorithms for this procedure are part of all molecular orbital programs.  The following paragraph describes how this transformation can be accomplished.

If the matrix M has an inverse M^-1 then MM^-1 = 1, and we can place this product in a matrix equation without changing the equation.  When this is done for Equation 12.6.7, we obtain

          H MM^-1C=SMM^-1CE                                                    12.7.4
Next multiply on the left by M^-1and determine M so the product M^-1SM is the identity matrix, i.e. a matrix that has 1's on the diagonal and 0's off the diagonal is the case for an orthogonal basis set.

          M^-1HMM^-1C=M^-1SMM^-1CE                                          12.7.5
which then can be written as

          H"C"=C"E                                                                   12.7.6
where C"=M^-1C.  The identity matrix is not included because multiplying by the identity matrix is just like multiplying by the number 1.  It doesn’t change anything.  The H" matrix can be diagonalized by multiplying on the left by the inverse of C" to find the energies of the molecular orbitals in the resulting diagonal matrix E. 
                   E = C"^-1H"C"                                                                       
The matrix C" obtained in the diagonalization step is finally back transformed to the original coordinate system with the M matrix, C=M C" since C"=M^-1C.

Fortunately this process is automated in some computer software.  For example, in Mathcad, the command genvals(H,S) returns a list of the eigenvalues for Equation 12.6.7.  These eigenvalues are the diagonal elements of E.  The command genvecs(H,S) returns a matrix of the normalized eigenvectors corresponding to the eigenvalues.  The i-th eigenvalue in the list goes with the i-th column in the eigenvector matrix.  This problem, where S is not the identity matrix, is called a general eigenvalue problem, and gen in the Mathcad commands refers to general.

 

 

12.8. Mulliken Populations

Mulliken populations (R.S. Mulliken, J. Chem. Phys. 23, 1833, 1841, 23389, 2343 (1955)) can be used to characterize the electronic charge distribution in a molecule and the bonding, antibonding, or nonbonding nature of the molecular orbitals for particular pairs of atoms.  To develop the idea of these populations, consider a real, normalized molecular orbital composed from two normalized atomic orbitals.
                                                                                   12.8.1
The charge distribution is described as a probability density by the square of this wavefunction.
                                                                  12.8.2
Integrating over all the electronic coordinates and using the fact that the molecular orbital and atomic orbitals are normalized produces
                                                                             12.8.3
where S_jk is the overlap integral involving the two atomic orbitals.

Mulliken's interpretation of this result is that one electron in molecular orbital ψ_ι contributes  to the electronic charge in atomic orbital _j,  to the electronic charge in atomic orbital _k, and 2c_ijc_ikS_jk to the electronic charge in the overlap region between the two atomic orbitals.  He therefore call  and , the atomic-orbital populations, and 2c_ijc_ikS_jk, the overlap population.  The overlap population is >0 for a bonding molecular orbital, <0 for an antibonding molecular orbital, and 0 for a nonbonding molecular orbital.

It is convenient to tabulate these populations in matrix form for each molecular orbital.  Such a matrix is called the Mulliken population matrix.  If there are two electrons in the molecular orbital, then these populations are doubled.  Each column and each row in a population matrix is corresponds to an atomic orbital, and the diagonal elements give the atomic-orbital populations, and the off-diagonal elements give the overlap populations.  For our example, Equation 12.8.1, the population matrix is
                   .                                                 12.8.4

Since there is one population matrix for each molecular orbital, it generally is difficult to deal with all the information in the population matrices.  Forming the net population matrix decreases the amount of data.  The net population matrix is the sum of all the population matrices for the occupied orbitals.
                                                                                    12.8.5
The net population matrix gives the atomic-orbital populations and overlap populations resulting from all the electrons in all the molecular orbitals.  The diagonal elements give the total charge in each atomic orbital, and the off-diagonal elements give the total overlap population, which characterizes the total contribution of the two atomic orbitals to the bond between the two atoms.

The gross population matrix condenses the data in a different way.  The net population matrix combines the contributions from all the occupied molecular orbitals.  The gross population matrix combines the overlap populations with the atomic orbital populations for each molecular orbital.  The columns of the gross population matrix correspond to the molecular orbitals, and the rows correspond to the atomic orbitals.  A matrix element specifies the amount of charge, including the overlap contribution, that a particular molecular orbital contributes to a particular atomic orbital.  Values for the matrix elements are obtained by dividing each overlap population in half and adding each half to the atomic-orbital populations of the participating atomic orbitals.  The matrix elements provide the gross charge that a molecular orbital contributes to the atomic orbital. Gross means that overlap contributions are included.  The gross population matrix therefor also is called the charge matrix for the molecular orbitals.  An element of the gross population matrix (in the j^th row and i^th column) is given by
                                                                           12.8.6
where Pi is the population matrix for the i^th molecular orbital,  Pi_jj is the atomic-orbital population and the Pi_jk is the overlap population for atomic orbitals j and k in the i^th molecular orbital.

Further condensation of the data can be obtained by considering atomic and overlap populations by atoms rather than by atomic orbitals.  The resulting matrix is called the reduced-population matrix.  The reduced population is obtained from the net population matrix by adding the atomic orbital populations and the overlap populations of all the atomic orbitals of the same atom.  The rows and columns of the reduced population matrix correspond to the atoms.

Atomic-orbital charges are obtained by adding the elements in the rows of the gross population matrix for the occupied molecular orbitals.  Atomic charges are obtained from the atomic orbital charges by adding the atomic-orbital charges on the same atom.  Finally, the net charge on an atom is obtained by subtracting the atomic charge from the nuclear charge adjusted for complete shielding by the 1s electrons.

 

 

 

Activity 12.1

This Mathcad document (Activity 12.1 Formaldehyde.mcd) summarizes an Extended Hückel calculation on formaldehyde and provides the data you need to complete the following analysis.  The calculation was done using the HyperChem software package.  The geometry for formaldehyde was provided by a HyperChem SCF 6-31G** calculation.  Such ab initio calculations are discussed in the next chapter.  A minimum in the calculated energy for formaldehyde was found at a CH bond length of 109.3 pm, a CO bond length of 118.4 pm, and a HCH bond angle of 115.7 degrees.

(1)        Use the Coordinate Table to determine the orientation of the molecule with respect to the Cartesian coordinate system.  Draw a diagram of the molecule relative to the axes, and label and number the atoms according to the numbering used in the table.

 

(2)        Use the Overlap Matrix to identify the signs of the overlap integrals for the carbon and oxygen p_x and p_y orbitals with the two hydrogen 1s orbitals.  Add the carbon and oxygen p_x and p_y orbitals to your diagram.  Explain in terms of your diagram why these four overlap integrals have the signs that are given in the Overlap Matrix.

 

(3)        In the Overlap Matrix, each number in the first row is either 1, 0, a positive number between 0 and 1, a negative number between 0 and 1, or almost 0.  Explain for each of these matrix elements in the first row specifically why such values are obtained.

 

(4)        From the Hückel Hamiltonian matrix identify the values that are used for the atomic-orbital valence ionization energies for C, O, and H in this calculation.

 

(5)        Explain why all the elements in the column for PzO1 in the Hückel Hamiltonian matrix are 0 except for two, and show that the correct values for those two indeed are -14.80000 and -5.27682 eV.

 

(6)        Use the Eigenvector Matrix to draw sketches of the molecular orbitals numbered 3, 4, and 5.  Base your sketches on the coefficients in the Eigenvector Matrix.  Identify which of these orbitals are CH bonding and which are CO bonding.  Note that the molecular orbitals are numbered 0 through 9 not 1 through 10.

 

(7)        Calculate the atomic-orbital populations and the overlap populations for the molecular orbitals numbered 3 and 5 from data in the Eigenvector Matrix and the Overlap Matrix.  What do the values for these populations tell you about the electronic charge on each atom, in the overlap region, and the bonding, nonbonding, or antibonding characteristics of these orbitals?  What happens to the bond lengths if an electron is removed from molecular orbital number 3?  What happens to the bond lengths if an electron is removed from molecular orbital number 5?

 

(8)        Use the eigenvectors to identify each of the 10 molecular orbitals as π or σ. 

 

(9)        Draw an energy-level diagram of the atomic and molecular orbitals.  Draw dotted lines connecting the molecular-orbital levels to the atomic-orbital levels from which the molecular orbitals are derived.  Are all the π molecular orbitals higher in energy than the σ molecular orbitals?  Is the highest energy molecular orbital σ or π?  Explain why the molecular orbitals have the energy ordering that they do.

 

(10)    Write the ground-state electronic configuration for formaldehyde identifying the molecular orbitals by their symmetry.

 

(11)    How many degenerate molecular orbitals are there?

 

(12)    Determine the partial charge at each atom due to electrons in the highest occupied molecular orbital.

 

(13)    For molecular orbital number 3, verify that the entries in the Gross Population Matrix for the Molecular Orbitals tabulate the gross charge (including the overlap contributions) that a particular molecular orbital contributes to an atomic orbital.

 

(14)    What is the physical significance of the elements in the Reduced Population Matrix.  Use the Reduced Population Matrix to verify that the sum of all the atomic orbital populations and overlap populations is equal to the number of valence electrons.

(15)    Verify that that the first (1.718) and last (0.977) values in the list of atomic orbital charges are correct given the values in the Gross Population Matrix.

 

(16)    What is the physical meaning of the net atomic charge?  What is the net charge at each atom calculated for formaldehyde?  Discuss the chemical reactivity of formaldehyde in terms of this charge distribution.

 

(17)    Calculate the dipole moment for formaldehyde predicted by this calculation in units of debye (D), which are the traditional units for dipole moments.  Which end of the molecule is negative?  One debye is equal to 10^-18 esu cm where esu is a non-SI unit for electric charge.  The charge of a proton is 4.803x10^-10 esu.  What is the conversion factor between D and C m (coulomb meters, which is the SI unit for a dipole moment)?

 

(18)    How many molecular orbitals are produced in an Extended Hückel calculation on acetone?

 

Problems

Problem 12.1. Use the results in Table 12.5.1 for butadiene and symmetry arguments to determine the electric dipole allowed transitions between electronic energy levels in the molecule.

 

Problem 12.2. Compute the resonance energy for stilbene. Compare your results to the resonance energy for benzene. What is the significance of this with respect to the presence of the double bond in the center of the molecule? Include the structure of stilbene.

 

Problem 12.3. The compound shown below has never been synthesized. Use HMO theory to explain why.

Problem 12.4. Use a computational chemistry software package of your choice and draw the structure of ethane. Compute the Extended Hückel energy for ethane as you rotate one methyl group in increments of 10°.  Plot the results.  How large is the potential energy barrier to rotation about the CC single bond in ethane.  Repeat the calculations for n-butane rotating about the central CC bond.  Interpret your data in terms of potential energy barriers and identify the most stable conformations.

 

Problem 12.5. Draw cyclohexane in both the chair and boat conformation. Use an Extended Hückel calculation to determine the most stable conformation. How does this result compare to what you learned in organic chemistry?  Calculate the energy of the planar geometry, which should approximate the transition state.  What temperatures correspond to the energy differences between the stable conformations and the transition state?  In the liquid state, do you expect cyclohexane to convert freely between the two forms at some temperature?

 

Problem 12.6. Benzo-[a]-pyrene has been extensively studied because of the connection between this compound and cancer caused by soot and smoke. A proposed intermediate step in the metabolic conversion of this compound to a highly mutagenic metabolite is through epoxidation. Use the Hückel method to predict potential sites of epoxidation of benzo-[a]-pyrene and the likely hood of reaction at each potential site.

 

Problem 12.7. Consider the formic acid molecule and the internal H-bond shown here as a suitable model for the transition state for proton transfer from one oxygen to the second oxygen in the gas phase.  Compute the energy of this transition state and compare it to the energy for the ground state structure of formic acid.  Above what temperature should such proton transfer occur freely?

 

 

Problem 12.8.  Carry out an extended Hückel calculation on the Series A cyanine dyes listed in Activity 5.1.  Compare the experimental transition energies with your results and determine which model (particle-in-a-box or extended Hückel) best accounts for the electronic energy level structure of these dyes.

 

Problem 12.9.  Write the topological matrices for the following molecules: ethylene, cyclobutadiene, and the cyclopentadienyl anion.  Diagonalize the topological matrices to determine the Hückel energies and wavefunctions, construct an energy level diagram, sketch pictures of the molecular orbitals, and identify the bonding, antibonding, and non-bonding orbitals for each molecule.  Discuss the stability of these moleulules using resonance energy as a criterion in comparison to benzene.

 

 

 

Self-study and Review Questions

·     How are values in the Hückel Hamiltonian matrix determined, and how are the eigenvalues and eigenvectors derived from this matrix?

·     How are the atomic orbital populations and the overlap populations calculated for a molecular orbital, and what is their physical meaning?

·     How are values in the Mulliken population matrices obtained?

·     How are values in the Net Population Matrix obtained?

·     How are values in the Charge Matrix (aka Gross Population Matrix) obtained?

·     How are values in the Reduced Population Matrix obtained?

·     How are Atomic Orbital Charges obtained?

·     How are the Atomic Charges and the Net Charges determined?

·     How does the overlap population characterize a molecular orbital as bonding, antibonding or nonbonding?

·     How can you determine the amount of charge that a particular molecular orbital contributes to a bond between two atoms?