Basin populations and related properties.

From a quantitative point of view a localization basin (core or valence) is characterized by its population, i. e. the integrated one electron density over the basin:

in which $\rho ({\bf r})$ denotes the one electron density at ${\bf r}$ and $%\Omega_i$ is the volume of the basin. It is worthy to calculate the variance of the basin population:

$$\sigma^2(\bar N; \Omega_i) = \int\limits_{\Omega_i} d{\bf r}_...... d{\bf r}_2 - \lbrack\bar N(\Omega_i) \rbrack^2 + N(\Omega_i)$$ (2)

where $\pi({\bf r}_1, {\bf r}_2)$ is the spinless pair function It has been shown that the variance can be readily written as a sum of contributions arising from the other basins (covariance):

$$\sigma^2(\bar N; \Omega_i) = \sum\limits_{j \neq i} \bar N(\O......}_1\int\limits_{\Omega_j} \pi({\bfr}_1, {\bf r}_2) d{\bf r}_2$$ (3)

In this expression $\bar N(\Omega_i)\bar N(\Omega_j)$ is the number of electron pairs classically expected from the basin population whereas $\bar N(\Omega_i,\Omega_j)$ is the actual number of pairs obtained by integration of the pair function over the basins $\Omega_i$ and $\Omega_j$. The variance is a measure of the quantum mechanical uncertainty of the basin population which can be interpreted as a consequence of the electron delocalization whereas the pair covariance indicates how much the population fluctuations of two given basins are correlated. In the AIM framework Fradera et al have introduced atomic localization and delocalization indexes noted $\lambda(\mathrm A)$ and $\delta(\mathrm{A, B})$which are defined by:

$$\lambda(\mathrm A) = \bar N(\Omega_{\mathrm A}) - \sigma^2(\bar N;\Omega_{\mathrm A})$$ (4)

$$\delta(\mathrm{A, B}) = 2\bar N(\Omega_{\mathrm A})\bar N(\Omega_{\mathrm B})-2\int\limi......{\bf r}_1\int\limits_{\Omega_{\mathrm B}} \pi({\bf r}_1,{\bf r}_2) d{\bf r}_2$$ (5)

2002-04-01