.

$$\pi \frac{R^5}{16} \int_1^\infty d\lambda \left ( \frac{2}{3}... ...ac{2}{5}\lambda^2 +\frac{2}{3} \right ) e^{-\alpha \lambda R}$$

or

$$\frac{ \pi }{15\alpha^5}\left ( \alpha^4R^4+2\alpha^3R^3-3\alpha^2R^2-15\alpha R -15\right ) e^{-\alpha R}$$

 

Figure 9: Overlap Integral as a function of Internuclear Distance.

 

Figure 10: Overlap Integral as a function of Internuclear Distance and Exponent in Wave Function.

This term is sometimes positive and sometimes negative depending on $\alpha$ and R. So, the argument, that $\beta^h$ is somehow proportional to the overlap, fails.

Assume the ``horizontal'' $\beta$ is negative

If one follows the adiabatic paths during con- and dis-rotatory ring closure, starting at the ground state, one has the following: For dis-rotatory ring closure, when one adiabatically follows the energy levels as the twist angle changes from zero to $\pi/2$ the four ground state electrons remain in the ground state, switching identity, but no more (see Figure 11).

For the con-rotatory ring closure, one sees that the ground state electrons find themselves in an excited state of the cyclic product (see Figure 12).

Assume the ``horizontal'' $\beta$ is positive

We start with dis-rotatory ring closure, and adiabatically follow the energy levels as the twist angle changes from zero to $\pi/2$. The ground state electrons find themselves promoted to an excited state (see Figure 12).

For the con-rotatory ring closure, we find that the ground state electrons adiabatically remaining in a ground state configuration (see Figure 11). Clearly, given these diagrams, it is very important what the sign of $\beta^h$ is!

  

Figure: Disrotatory ring closure, $\beta ^h<0$<0$"> and Conrotatory ring closure, $\beta ^h>0$0$">. The latter corresponds to the thermally allowed ring closure predicted by Woodward and Hoffmann.

  

Figure: Conrotatory ring closure, $\beta ^h<0$<0$"> Disrotatory ring closure, $\beta ^h>0$0$">