Chemical reasoning based on an invariance property: Bond and lone pair pictures in quantum structural formulas

Chemists use one set of orbitals when comparing to a structural formula, hybridized AOs or NBOs for example, and another for reasoning in terms of frontier orbitals, MOs usually. Chemical arguments can frequently be made in terms of energy and/or electron density without the consideration of orbitals at all. All orbital representations, orthogonal or not, within a given function space are related by linear transformation. Chemical arguments based on orbitals are really energy or electron density arguments; orbitals are linked to these observables through the use of operators. The Valency Interaction Formula, VIF, offers a system of chemical reasoning based on the invariance of observables from one orbital representation to another. VIF pictures have been defined as one-electron density and Hamiltonian operators. These pictures are classified in a chemically meaningful way by use of linear transformations applied to them in the form of two pictorial rules and the invariance of the number of doubly, singly, and unoccupied orbitals or bonding, nonbonding, and antibonding orbitals under these transformations. The compatibility of the VIF method with the bond pair - lone pair language of Lewis is demonstrated. Different electron lone pair representations are related by the pictorial rules and have stability understood in terms of Walsh's rules. Symmetries of conjugated ring systems are related to their electronic state by simple mathematical formulas. Description of lone pairs in conjugated systems is based on the strength and sign of orbital interactions around the ring. Simple models for bonding in copper clusters are tested, and the bonding of O2 to Fe(II) in hemoglobin is described. Arguments made are supported by HF, B3LYP, and MP2 computations.