The topological substructural molecular design (TOPS-MODE) approach is formulated as a tight-binding quantum-chemical method. The approach is based on certain postulates that permit to express any molecular property as a function of the spectral moments of certain types of molecular and environment-dependent energies. We use several empirical potentials to account for these intrinsic and external molecular energies. We prove that any molecular property expressed in terms of a quantitative structure-property and structure-activity relationships (QSPR/QSAR) model developed by using the TOPS-MODE method can be expressed as a bond additivity function. In addition, such a property can also be expressed as a substructural cluster expansion function. The conditions for such bond contributions being transferable are also analyzed here. Several new statistical-mechanical electronic functions are introduced as well as a bond-bond thermal Green's function or a propagator accounting for the electronic hopping between pairs of bonds. All these new concepts are applied to the development and application of a new QSAR model for describing the toxicity of polyhalogenated-dibenzo-1,4-dioxins. The QSAR model obtained displays a significant robustness and predictability. It permits an easy structural interpretation of the structure-activity relationship in terms of bond additivity functions, which display some resemblances with other theoretical parameters obtained from first principle quantum-chemical methods.