Information from the Electron Density, The Bader Analysis.

In the previous section we have shown how to gain chemical informations from the wave function and their orbitals. Following, we will summarize some of the Bader analysis capabilities that provides us with the tool to extract chemical information from the electron density $\rho (\vec{r})$. The analysis of its gradient $\nabla\rho(\vec{r})$ gives us the possibility to define an atom within a molecule through the zero flux surface condition. In addition the analysis of its critical points ( $\nabla\rho(\vec{r_{c}})$=0) gives us a way to define the molecular structure. Once these $\vec{r_{c}}$ points are located and the curvature with respect to the space directions analyzed, ( i.e., the second derivative), different critical points can be distinguished:

(3,-3) critical points: All curvatures are negative and $\rho$ is a local maximum at $r_{c}$. These points have been found to correspond with the position of the nuclei.

(3,-1) critical points: Two curvatures are negative and $\rho$ is a maximum at $r_{c}$ in the plane defined by their corresponding axes. $\rho$ is a minimum at $r_{c}$ along the third axis which is perpendicular to this plane. When there is a bond between two atoms, it is observed that two (3,-3) critical points are connected by one and only one gradient line that present a (3,-1) critical point in the limit between both basins. It seems a necessary condition for a bond to show such (3,-1) critical point and hence, this point is also called bond critical point, and the gradient connecting both nuclei a bond path

(3,+1) critical points: Two curvatures are positive and $\rho$ is a minimum at $r_{c}$ in the plane defined by their corresponding axes. $\rho$ is a maximum at $r_{c}$ along the third axis which is perpendicular to this plane. Such critical points are detected somewhere within a ring structure, and therefore, they are also called ring critical points.

(3,+3) critical points: All curvatures are positive and $\rho$ is a local minimum at $r_{c}$. These corresponds with cage structures and one (3,+3) point is found inside the cage. We will call then cage critical points.

Further information can be obtained once the critical points of a given molecule are defined. In this thesis we have used the Energy Density at the bond critical point ( $H(\vec{r_{c}})$) to distinguish between covalency and ionicity [63]. $H(\vec{r})$ is defined as,

$$H(\vec{r}) = G(\vec{r}) + V(\vec{r})$$ (1.11)

where $G$ and $V$ stand for the kinetic and potential energies, respectively.
The local virial theorem, in a.u., is defined as,

$$\frac{1}{4}\nabla^{2}\rho(\vec{r}) = 2G(\vec{r}) + V(\vec{r})$$ (1.12)

combining both we get the formula:

$$H(\vec{r}) = \frac{1}{4}\nabla^{2}\rho(\vec{r}) - G(\vec{r})$$ (1.13)

Since the kinetic energy $G(\vec{r_{c}})$ is always $>$0 and $V(\vec{r_{c}})$ always $<$0, with $H(\vec{r_{c}})<$0, the potential energy dominates over the kinetic energy in $\vec{r_{c}}$ , thus the electronic charge stabilizes the system. On the other hand, when the kinetic energy dominates over the potential energy the electronic charge at this point will destabilize the system and $H(\vec{r_{c}})$ will have a positive value. Summarizing, when $H(\vec{r_{c}})<$0, we are dealing with a covalent bond, while when $H(\vec{r_{c}})>$0, that will correspond to a ionic bond.