Descriptors and probability distributions in MO theory:: Weighted Mulliken matrices and molecular quantum similarity measures

Starting from the original Mulliken study on population analysis, it is shown how these initial ideas not only are expressible as expectation values of a projector, but also how they can be extended and generalized into the domain of non-singular Hermitian operators using the same technique. Some examples of this possibility are given. The formalism is studied in deep, proving that the Mulliken populations are a zero-th order approach to the description of condensed density functions and their connection with quantum similarity measures is analyzed. Employing positive definite operators as natural weights to compute atomic condensed density terms, lead to produce elements of vector semispaces. Such vectors can be subject to Minkowski normalization becoming discrete probability distributions, which can be gathered using several weight operators into stochastic arrays as molecular fingerprint descriptors.