Scale-invariance of the topological equations of the density per particle

The density per particle σ carries the same information of the electron density ρ. However, σ and ρ exhibit different local behavior and consequently their respective topological functions should be different. Paradoxically, the differential geometry equations supporting their topological behavior are the same. This fact implies, in particular, the invariance of the molecular structure and that of the atomic basin. But other topologically-relevant functions—such as the gradient, the unit-less gradient, the Laplacian, and the one-electron potential, among others—may behave differently. The scale transformation ρ = N σ applied to these functions reveals that the only first-degree invariant function is the logarithmic derivative of the density. This function shows the local symmetry around the stationary points of the density. Present work exemplifies what seems to be a general principle: for determining a well-constructed density function it should be formulated equally for both the electron density and the density per particle.