APPROXIMATE LCAO MOLECULAR ORBITAL THEORY

We use Hückel theory [E. Hückel, Z. Physik, 1931, 70, 204] to introduce the idea of molecular orbitals as linear combinations of atomic orbitals because:

• it is simple enough mathematically to enable us to follow qualitatively what is going on

• elimination of the simplifying assumptions leads directly to a qualitative description of the more sophisticated semi-empirical and ab initio methods

For more extensive description of the method and its applications, the best sources are two classic texts:

• Roberts, J. D., "Notes on Molecular Orbital Calculations", W. A. Benjamin, Menlo Park, CA, 1962

• Streitwieser, Jr., A., "Molecular Orbital Theory for Organic Chemists", Wiley, New York, 1961

Both are out of print, but still available in libraries.

Hückel theory begins with two structural assumptions:

• The electrons of interest initially occupy a system of carbon 2p orbitals having a common nodal plane; that is, with their long axes parallel; they interact to form p-type molecular orbitals;

• The rest of the electrons in the molecule occupy a s-orbital framework that is orthogonal to the 2p orbitals and therefore does not interact with them.

The description of Hückel theory as an LCAO method means that it assumes that molecular orbitals, Y, can be represented as a linear combination of atomic orbitals, j:

where

• j is an index over molecular orbitals (MOs)

• n is an index over atomic orbitals (AOs)

• c is a set of coefficients weighting the contributions of the atomic orbitals to the molecular orbitals

The set of atomic orbitals used to create the molecular orbitals is called the basis set.

The wave functions, Y, are called one-electron wave functions:

• They represent the motion of a single electron in the electric field provided by the nuclei and the averaged distribution of the other electrons.

• Because they take account of the other electrons only in an average, they do not account properly for electron correlation, the tendency of electrons to avoid each other insofar as possible.
  • This problem is particularly severe for paired electrons in the same orbital.

  • We shall see later how more sophisticated MO methods attempt to overcome this difficulty.

According to the Schroedinger Equation [for the derivation of the equation in this form, see Lowry, T. H; Richardson, K. S. "Mechanism and Theory in Organic Chemistry", 3rd ed., Harper and Row, New York, 1987; p 87 ff], the energy of the electron for which Y is the wavefunction is given by

where H is the electronic Hamiltonian (see p 89 of the reference cited just above for a definition of H) and dt is an element of the electron space.

Mother Nature always builds systems in states of lowest energy.

• Therefore, e calculated from the above equation always will be greater than E_o, the true minimum energy, unless we have chosen the correct set of coefficients, c_rj, in representing the MO as a linear combination of AOs.

• So, if we can find the minimum of the energy with respect to the coefficients, we must have the right answer for both the energy and the coefficients. That is,
  

Let's work through this for ethylene, with just two AOs, and then generalize. The molecular wave function can be written as the linear combination of the two carbon 2p atomic wave functions:

Substituting the right side of this expression into the Schroedinger equation, multiplying through, and extracting the coefficients from the integrals (which we can do because they're constants), leads to:

To make the equations a little easier to read, we shall make a couple of substitutions:

• Integrals of the type jHj will be replaced by H

• Integrals of the type j² will be replaced by S.

Thus, for example, we write:

With these substitutions the energy expression becomes:

To make differentiation easier, we multiply through by the denominator:

Now take the partial derivative with respect to c₁, leading to:

Apply the variation principle and set de/c₁ = 0, and we obtain:

which can be rearranged to

Likewise, partial differentiation with respect to c₂ leads to

Thus, the result for a two-AO system is a set of two linear equations. This is a completely general result: taking a linear combination of n AOs of any kind (s, p, d....) leads to a set of n simultaneous equations:

Assumptions of Hückel theory. In Hückel theory, we do not actually evaluate the integrals represented by the Hs and Ss above. Furthermore, we make a set of assumptions to reduce the complexity of the linear equations. These assumptions are as follows:

1. The integrals of type H_ij, where i = j, are related to the energy of an electron in an isolated carbon 2p orbital. They are called Coulomb integrals.

   • we assume they all are equal, regardless of the molecular environment of the particular carbon

   • we use for all of them the symbol a.

2. The integrals of the form H_ij, where i is not equal to j, are related to the energy lowering that occurs upon allowing an electron to occupy both orbitals.
   • this energy is dependent upon the distance between the orbitals

   • Hückel theory assumes that if i and j are on adjacent atoms (| i - j | = 1), the interaction will be the same

   • these integrals are represented by b; they are the resonance integrals.

   • If i and j are not adjacent, we assume there is no energy gain, and set these integrals equal to zero.

3. The S type integrals are called overlap integrals. They are related to the energy of interaction between electrons in i and j. We assume they can be divided into two groups.
   • If i = j, we set the integrals = 1

   • If i does not equal j, we set them = 0

   • This trick, of ignoring differences in interaction between orbitals, is called neglect of differential overlap, or NDO.

We can summarize the assumptions in the form of a table:

Substituting the values into the set of simultaneous equations yields:

Such a set of equations has a solution (other than the trivial one of all unknowns equal to zero) only if the determinant formed from the coefficients of the unknowns is equal to zero:

This determinant is called the secular equation.

Returning to ethylene, we find that the secular equation comes out to be:

which has two solutions: e₁ = a + b and e₂ = a - b. With a taken as the energy of an electron isolated in a carbon 2p orbital, the energy changes associated with the formation of the p-bond can be shown as:

The two electrons occupy the lower level, and the p-bond energy can be calculated as:

To get the coefficients, c₁ and c₂, we can substitute the values for e in the linear equations, and make use of the additional equation:

This expression is called the normalization condition, and it amounts to saying that the fractional AO contributions to one MO have to add up to one. Solving the expressions for ethylene lead to two sets of coefficients:

c₁ = c₂ = 0.707

and

c₁ = -c₂ = 0.707

so we can write the two wavefunctions corresponding to the two energies above as:

The change of sign in the second expression indicates an inversion of the phase relationship between the two AOs: a node. Pictorially, this can be represented in the familiar diagrams:

Information from Hückel Calculations. To illustrate some of the other kinds of information we can get from a simple Hückel LCAO calculation, let's consider the three allyl species: cation, radical, anion: CH2=CHCH2*, where the * represents a positive charge, an unpaired electron, or a negative charge (unshared pair).

With three 2p orbitals contributing to the MOs, the secular equation will look like this:

The zeros appear because carbons 1 and 3 are not bonded to each other.

Solution of this equation (by methods we need not go into, since a computer will do it for us), leads to the energies and molecular orbitals shown below:

MO 2, has an energy of a

• This is the same as an isolated carbon 2p orbital, meaning this is a non-bonding MO

• Electrons occupying this orbital will not contribute to the energy of the species.

• The zero coefficient for C₂ in MO 2 means that the node implied by the sign change of the coefficients passes through C₂
  • Its 2p orbital makes no contribution to the MO

Again, the pictures, shown below, should be familiar:

The allyl cation, with two electrons occupying Y₁, has a p-energy of 2(a + 1.41b) = 2a + 2.82b.

• An interesting comparison is with the energy of those same two electrons in the p-bonding MO of ethylene, 2a + 2b

• The difference, (2a + 2.82b) - (2a + 2b) = 0.82b is the additional energy lowering from allowing the electrons to associate with three nuclei instead of two.

• It is called the p-delocalization energy, or DE.

• The additional electrons of the radical and anion must go into MO 2, and therefore these species must have the same delocalization energy as the cation.

Consider now the following integral expansion:

The last integral gives the probability of finding an electron in r

• Because the integration is over all space, this integral must = 1.

• Multiplication by the c_r² then describes how the probability is altered by incorporation of the atom into the MO.

• Then c_r² must represent the electron density on the atom in the MO.

• Since the total electron density on an atom will be the sum of the electron densities from all occupied MOs, we can write:
  
  • q_r is the electron density on atom r

  • n_j is the number of electrons in the jth molecular orbital

  • c_rj² is the square of the coefficient of atom r in the jth MO, summed over all occupied MOs

• Finally, since a neutral carbon atom would have one electron, the charge, z, on atom r is simply:
  

For the allyl cation, with orbital 1 doubly occupied, this works out as:

The allyl radical has one additional electron in Y₂, for which c₁ = -c₃ = 0.707; c_r² = 0.5, and we add an additional 0.5 to q for atoms 1 and 3. Since c₂ = 0, atom 2 is unaffected, and we have:

The anion has Y₂ doubly occupied, so we add 2 x 0.5 to q for atoms 1 and 3; atom 2 still is unaffected, and we find:

Notice that the calculations tell us the same things as the resonance structures that we learned in sophomore chemistry: the charges and the odd electron are delocalized over the end carbons, which are equivalent.

Hückel theory no longer is important in the sense of giving useful numbers for molecular properties, although a glance at the Streitwieser book cited above will demonstrate that it could be used to develop some interesting correlations. The importance of the approach now is that it defines the LCAO method in a way that will allow us to understand how more sophisticated calculations are done. Furthermore, the notions of locating nodes in an MO by sign changes in the AO coefficients, and electron densities as related to AO coefficients summed over occupied MOs, also are ideas useful in interpreting more quantitative data.

We have available, for anone who wishes to "play" with Hückel calculations, two programs that will handle molecules having up to 20 or so atoms. One of these is for all-carbon systems; the other allows for heteroatoms. Both run under Windows, and provide printed output, but not graphics.If you would like a copy of either or both, let me know.