LCAO MO first order density functions: Partition in monocentric and bicentric terms, reciprocal MO spaces, invariant transformations and Euclidian atomic populations

Here, the first order density function is analyzed under the LCAO MO theory framework, using a partition in two terms, namely monocentric (monoatomic or atomic) and bicentric (diatomic or bond) contributions. The monoatomic term possess a Minkowski norm positive definite, while for the diatomic term just a Minkowski real pseudonorm can be defined. The atomic Minkowski norm can be proven to be less, equal or greater than the number of electrons, while the diatomic Minkowski pseudonorm in every one of these monoatomic cases appears to be positive, null or negative, respectively. Such a behavior cannot provide both norms with some physical sense. Moreover, the same behavior can be described for every MO density contribution, using the unit MO norms instead of the number of electrons. Thus, shape functions behave in the same way as these individual MO density terms. In this work it is also studied the possibility to transform the LCAO basis set by means of some unitary transformation in such a way that the density function remains invariant, while zeroing the diatomic Minkowski pseudonorm. In this zero bicentric pseudonorm case some gross atomic populations in the Mulliken sense become coincident with the Roby definition, as the transformed basis set produces a null diatomic contribution. The role of reciprocal space is presented from the point of view of density function partition. Invariant transformations of the density functions are studied within a general formalism, where the problems previously commented arise as particular cases. Finally, a way to define Euclidian atomic populations is given, based on the alternative metric matrix, associated to reciprocal space, which can be linked to the MO coefficients matrix.