On the Pi-Electron Approximation and Its Possible Refinement

The pi-electron approximation is defined to be the approximation in which the following two restrictions are imposed upon the total approximate electronic wave functions for some group of molecular states: (I) The wave function for each state satisfies the sigma-pi separability conditions: (A) the wave function has the form Ψ = (Σ) (II), where (Σ) and (II) are antisymmetrized functions describing the so-called sigma and pi electrons, respectively, and the outer brackets connote antisymmetrization with respect to sigma-pi exchange; (B) each of (Σ), (II), and Ψ is normalized to unity; (C) each of (Σ), (II), and Ψ is well-behaved. (II) The sigma description is the same for all states. Imposition of these restrictions is shown to be sufficient to validate the customary procedure in which the pi electrons in a molecule are treated apart from the rest. A formula is given for the pi-electron Hamiltonian to be used when the pi-electron approximation is invoked. Present day pi-electron theories are examined, and lines for carrying out improved calculations are suggested. An iterative procedure is proposed for treating both sigma and pi electrons wherein first a sigma function is assumed (which defines a ``core'' in the field of which the pi electrons move), then a pi function is computed (which defines a ``peel'' in the field of which the sigma electrons move), then a new sigma function is computed, and so on. Certain generalizations of the quantum-mechanical argument are made which give it wider applicability, and several illustrations are drawn from pi-electron theory and elsewhere.