The One Electron Hamiltonian Matrix

The one-electron Hückel Hamiltonian is

$$\left ( \begin{array}{cccc} \alpha&\beta cos \theta_{1-2} & 0... ...4-1} & 0 & \beta cos \theta_{4-3} & \alpha \end{array}\right )$$ (11)

where, when $\theta_{3-4} = \theta_{1-2}=0$, we have maximum overlap. $\alpha$ is the standard Hückel <i|H|i> matrix element. Notice that as $\theta_{1-2}$ approaches 90^o, the overlap between orbitals on atoms 1 and 4 increases to a maximum, and the overlap between orbitals on atoms 1 and 2 as well as between 3 and 4 decrease to zero. Notice further that we need the $\beta^h$ (previously defined) to distinguish between the nascent $\sigma$ bond forming (1-4) and the destroyed $\pi$ bonds, (1-2 and 3-4, there is no twist angle for 2-3).

Technically, we can now plot the energy as a function of the reaction coordinate ($\omega$). The con- and dis-rotatory aspects are obtained by choosing the previously noted relation between the relevant angles, i.e.,

$$\theta_{1-2} = \theta_{3-4} \equiv \omega$$

or

$$\theta_{1-2} = -\theta_{3-K} \equiv \omega$$