H\"uckel--Hubbard-Ohno modeling of $\mathit{\pi }$-bonds in ethene and ethyne with application to trans-polyacetylene

Quantum chemistry calculations provide the potential energy between two carbon atoms in ethane (H${}_3$C$-$CH${}_3$), ethene (H${}_2$C$=$CH${}_2$), and ethyne (HC$\equiv$CH) as a function of the atomic distance. Based on the energy function for the $\sigma$-bond in ethane, $V_{\sigma }(r)$, we use the H"uckel model with Hubbard–Ohno interaction for the $\pi$~electrons to describe the energies $V_{\sigma \pi }(r)$ and $V_{\sigma \pi \pi }(r)$ for the $\sigma \pi$ double bond in ethene and the $\sigma \pi \pi$ triple bond in ethyne, respectively. The fit of the force functions shows that the Peierls coupling can be estimated with some precision whereas the Hubbard-Ohno parameters are insignificant at the distances under consideration. We apply the H"uckel-Hubbard-Ohno model to describe the bond lengths and the energies of elementary electronic excitations of trans-polyacetylene, (CH)${}_n$, and adjust the $\sigma$-bond potential for conjugated polymers.