The Exchange Integral (Between Carbons 1-4)

We artificially constrained the geometry to a square, so that the forming bond would not be canted relative to the emerging bond line ( Had we left the sp² hybridization of atoms 2 and 3, the bond angles 1-2-3 and 2-3-4 would be 120^o, which would have forced the p-orbitals on atoms 1 and 4 to approach each other in canted fashion. Instead, we force them to 90^o so that the forming bond is $p_\sigma$ or $p_\sigma^*$. This substitutes for the 109.471^o expected if atoms 1 and 4 re-hybridized into sp³. ) . The 1-4 Hamiltonian matrix element then is

$$H_{1-4} = \int ( p_{z_1} sin \theta_{1-2} + p_{y_1} cos \the... ...H ( p_{z_4} sin \theta_{3-4} + p_{y_K} cos \theta_{3-4})d \tau$$