In this paper we elaborate and evaluate the Roby-Gould bond index method (RGBI) for classifying and interpreting the nature of chemical bonds [Gould et al., Theor. Chem. Acc., 2008, 119, 275]. We emphasize that the RGBI method defines atoms as subspaces of a Hilbert space, represented as projection operators, and in accord with basic mathematical requirements concerning the assignment of probabilities to such subspaces. The definition of the bond indices between such atoms follows from simple set theoretical ideas concerning the sharing of, and differences between, atomic electron populations; and from the geometric properties associated with these subspaces—properties which are closely aligned with older chemical notions, which are reviewed. We also introduce the Araki angle percentage ionicity of a bond, which is easily read off from a compact two-dimensional representation of the chemical bond, which we call the bond dial diagram. Additionally we explore the theory with defining atoms by the occupied natural atomic spinorbitals (NAOs) of Reed and coworkers, instead of the atomic natural orbitals (ANOs) previously used. Bond indices from these ANO- and NAO-based atoms are applied to a dataset of 353 molecules, including 27 first row molecules, 280 organic molecules, and 46 semiochemical molecules, comprising 33 A–H bonds, where A are first and second row elements, 1779 C–X, 3415 H–X bonds (X = C, N, O, S), and a few more other types of bonds. The bond indices from the ANO-based atoms are much more in accord with chemists' expectations as compared to those obtained from NAO-based atoms provided that the excited triplet state defines the ANOs for group 2 atoms.