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:
For more extensive description of the method and its applications, the best sources are two classic texts:
Both are out of print, but still available in libraries.
Hückel theory begins with two structural assumptions:
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, F:
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:
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.
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:
To make the equations a little easier to read, we make a couple of substitutions:
Thus, for example, we write:
With these substitutions the energy expression becomes:
To make differentiation easier, 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. Hückel theory does not actually evaluate the integrals represented by the Hs and Ss above.
Furthermore, it makes a set of assumptions to reduce the complexity of the linear equations. These assumptions are as follows:
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:
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:
MO2 has an energy of a
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.
Consider now the following integral expansion:
The last integral gives the probability of finding an electron in r
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; cr2 = 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 was 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.
I have linked in to our main page a Java-based Huckel program, for anyone who wishes to "play" with Hückel calculations. As a Java program, it should run in a browser window on both Mac and PC, if you have Java installed.