## Abstract

A critical study is made of the usual approximation of separating the system of $\pi $- electrons from that of the $\sigma $- electrons, when studying the energy levels of conjugated systems. The usual treatment is reformulated using the valence-bond method. The only necessary assumption, here called the Huckel approximation, is that canonical structures involving $\pi $-$\sigma $ or $\sigma $-$\sigma $ resonance are negligible. It is then shown that $\pi $-$\sigma $ and $\sigma $-$\sigma $ exchange terms do not need to be taken into account explicitly because they enter as an additive constant in all the states. This cancels out by subtraction, and is therefore irrelevant when computing resonance energies. The above reformulation is important, as it shows that a complete Hamiltonian and wave function (including $\pi $, $\sigma $ and hydrogen electrons) must be taken for the molecule, whereas previous ab initio computations of exchange integrals have been conducted with purely $\pi $- wave functions. Otherwise our results coincide with those given by empirical treatments, but the usual name of n-electron approximation for these, $\pi $ being the number of $\pi $- electrons in the molecule, is an inappropriate one. Exchange integrals for ethylene are computed along these lines. The atomic terms (involving ionization energies of the various electrons) are taken as a constant which is determined empirically, to avoid uncertainties, particularly in the ionization energy of the $\sigma $- electrons. The exchange integral between the two $\pi $ orbitals in ethylene at 1$\cdot $34 angstrom is - 2$\cdot $73 eV. That between a $\pi $ and the $\sigma $ orbital in the neighbouring carbon atom is + 0$\cdot $968 eV. A large part of the stability of the ethylene molecule is due to interaction between the $\pi $ and $\sigma $ bonds. The possibilities of screening the nuclei by the in-plane electrons are considered. This provides a guide for the effective nuclear charge to be used in various approximations. The results provide an explanation for the hitherto empirical fact that exchange integrals in different unsaturated hydrocarbons are of the same order. The errors due to the Huckel approximation when computing resonance energies are considered. If the neglected canonical structures are now introduced, an additional stability appears due to $\pi $-$\sigma $ resonance, an abbreviation which we use to denote the resonance between $\pi $ and in-plane electrons. This new effect (which is evaluated in the next paper) may be important because these structures are numerous and the value of the $\pi $-$\sigma $ exchange integral quoted above is rather large. The correlation between the computed and the observed resonance energies shows, however, that there is a partial cancellation of the error. The final error is estimated to be a few tenths of an electron volt. The effect of $\pi $-$\sigma $ resonance is, however, of great importance in computing excited states, as shown in a following paper.