Geometric interpretation of density displacements and charge sensitivities

The “geometric” interpretation of the electronic density displacements in the Hilbert space is given and the associated projection-operator partitioning of the hardness and softness operators (kernels) is developed. The eigenvectors |AE equel{|a)}of the hardness operator define the complete (identity) projector P equel zigma a|a cross a| equel1 for general density displacements, including the charge-transfer (CT) component, while the eigenvectors |i> equel{|i>}of the linear response operator determine the polarizational P-projector,Ppequel zigma|icrossi| Their difference thus defines the complementary CT-projector:PCT equel1-Pp. The complete vector space for density displacements can be also spanned by supplementing the P-modes with the homogeneous CT-mode. These subspaces separate the integral (normalization) and local aspects of density shifts in molecular systems.