  This paper aims at introducing the formal foundations of the application of
reduced density-matrix theory and Green's function theory to the analysis of
molecular electronic transitions. For this sake, their mechanics, applied to
specific objects containing information related to the passage and the
interference between electronic states - the difference and the transition
density operators - are rigorously introduced in a self-contained way. After
reducing the corresponding $N$-body operators (where $N$ is the number of
electrons in the system) using an operator partial-trace procedure, we derive
the kernel of the reduced one-body difference and transition density operators,
as well as the matrix representation of these operators in a finite-dimensional
one-particle-state basis. These derivations are done in first and second
quantization for the sake of completeness - the two formulations are
equivalently present in the literature - and because second quantization is
extensively used in a second part of the paper. Natural orbitals are introduced
as appropriate bases for reducing the dimensionality of the problem and the
complexity of the analysis of the transition phenomenon. Natural-orbital
representation of density operators are often used as a tool to characterize
the nature of molecular electronic transitions, so we suggest with this
contribution to revisit their theoretical foundations in order to better
understand the origin and nature of these tools.
