For the D2h E2N2 and D4h E42+ (E=S, Se and Te) systems with one valence-shell π-electron atomic orbital per atomic centre, the ground-state resonance between two increased-valence structures with one-electron and fractional electron-pair π bonds is equivalent to resonance between the six Lewis-type valence-bond structures considered in this journal (B. Braïda, A. Lo, P. C. Hiberty, ChemPhysChem 2012, 13, 811–819). Approximate values are deduced for the one-electron bond polarity parameters. Three types of electron transfer processes are used to convert one increased-valence structure into the other. Theory is developed to show that one component of the associated electron transfer matrix element is not dependent on atomic orbital overlap. Brief consideration is also given to other types of VB descriptions for the S2N2 ground state.