Woodward Hoffmann Rules (I)

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This is an enlarged version of the J. Chem. Ed. article of the same name, J. Chem. Ed., xx,yy(1999). The Hückel theory is applied to electrocyclization reactions of the Woodward-Hoffmann type.

Introduction
The Woodward-Hoffmann (R. B. Woodward and R. Hoffmann, "The Conservation of Orbital Symmetry", Verlag Chemie, Weinhein/Bergstr, 1970) rules for electrocyclization reactions provide an example of Hückel type computations (see also, J. J. Vollmer and K. L. Servis, J. Chem. Ed., 45, 214 (1968)). Recent interest (J. M. Oliva, J. Gerratt, P. B. Karadakov, and D. L. Cooper, J. Chem. Phys., 107, 8917 (1997), R. Tian et al. J. A. C. S., 120, 6187 (1998)) in these kinds of reactions makes this discussion even more germane.

Following Woodward and Hoffmann's example we employ the electrocylic reaction of forming a single bond between termini of a system containing four carbon atoms, each sp2 hydridized, with their $\sigma $ bond skeleton essentially ``frozen'', and each contributing a p-electron, i.e., the example concerns the conversion of butadiene to cyclobutene (see Figure 1). We remove the protons (these protons are often used as markers to distinguish con- and dis-rotatory motions in qualitative discussions.) for clarity and focus entirely on the $\pi$ and p orbital structure. We define the molecule's plane as containing the x and z axes, with the y-axis perpendicular to that plane; carbons 1 and 4 define the z-axis, and the x-y plane bisects the 1-4 line. This means that we are dealing with py orbitals when we do normal Hückel computations.

  
Figure 1: The coordinate system employed.
\begin{figure}\fbox{ \epsfig{width=5in,file=fig1.ps} } \end{figure}

In diatomic molecules, the z-axis is usually the bond axis, but when we move to ethylene treatments, the z-axis is reserved for the p-orbitals which are going to be used to form the $\pi$ bonds. It is important to understand that normal Hückel schemes employ pz orbitals, since these have a simpler form than either px or py. That means that the ethylene molecule exists in the x-y plane.
 
Figure 2: The coordinate systems employed for diatomic and Hückel systems.
\begin{figure}\fbox{ \epsfig{width=5in,file=whfigcoords.ps} } \end{figure}

Here, we change procedures again.

The orbitals in the basis set are going to be py orbitals, rather than the standard pz orbitals employed in most texts.


  
Figure: Naked $\sigma $ framework and p-electron basis set
\begin{figure}\fbox{ \epsfig{width=5in,file=fig2.ps} } \end{figure}

In Figure 3 we have the two molecules we are considering, and for comparison sake, we show stretched out butadiene, which is used in elementary discussions of Hückel MO theory. The basis set shown, for the linearized butadiene molecule, in normal texts are pz, but here are py.

In the Hückel computation, the results are a set of energy levels and coefficients of wave functions. In Figure 4 are schematically shown the resultant butadiene MO's (placed according to their relative energies). Notice that the signs of the coefficients are reflected in the lobe arrangements (blackened in means +, open means -, or vice versa. Notice also that the magnitudes of the coefficients reflect in the relative size of the contributing AO's to the resultant MO's.


  
Figure 4: These are the MO's of butadiene, in increasing energy order, emphasizing both the signs (filled in or empty mirroring + or -). Included also is the Hamiltonian matrix in the standard Hückel form.
\begin{figure}\fbox{ \epsfig{width=3in,file=./fig3.ps} } \end{figure}

Defining the Con- and Dis- Processes
To proceed, we define the twist angles ($\theta$'s) (see Figure 5)
  
Figure 5: Defining a Twist Angle
\begin{figure}\fbox{ \epsfig{width=5in,file=fig4.ps} } \end{figure}

required to follow the reaction in order to define a reaction coordinate ($\omega$). The difference between dis- and con- rotatory motions is seen in the relationship betweeen the concerted changes in these $\theta$'s, i.e., $\theta_{1-2} = \theta_{3-4}$ for conrotatory, and $\theta_{1-2} = - \theta_{3-4}$ for disrotatory. Finally, we define an enlarged basis set using pz for horizontal, and py for vertical normalized p-orbitals. A generalized orbital (for atoms 1 and/or 4) is

\begin{displaymath}p = p_z sin \theta + p_K cos \theta \end{displaymath}


What is K?
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Last edit: 9hrs:41min, Wednesday, June 2, 1999