Galvez Topological Charge Indices[7,8,9]

These indices describe charge transfer between pairs of atoms and therefore global charge transfer in a molecule. To obtain definitions of the indices we first define the matrix $\varmathbb{M}$ as

$$\varmathbb{M} = \varmathbb{A} \cdot \varmathbb{D}^{-2}$$

where $\varmathbb{A}$ is the adjacency matrix and $\varmathbb{D}^{-2}$ is the reciprocal square distance matrix. Note that the diagonal elements of the distance matrix remain the same. $\varmathbb{M}$ is the Galvez matrix and is a square unsymmetric $A \times A$ matrix where $A$ is the number of atoms in the molecule.

$\varmathbb{M}$ gives rise to an unsymmetric charge term matrix, $\varmathbb{CT}$ defined as

$$CT_{ij} = \left\{ \begin{array}{lcl} \delta_i & if & i = j\\ m_{ij} - m_{ji} & if & i \neq j \end{array} \right.$$

where $m_{ij}$ are elements of $\varmathbb{M}$ and $\delta_i$ is the vertex degree of atom $i$.

The diagonal entries of $\varmathbb{CT}$ represent the topological valence of the atoms and the off diagonal entries $CT_{ij}$ represent the amount of charge transferred from atom $j$ to atom $i$. If heteroatoms are to be considered the diagonal entries of $\varmathbb{A}$ can be substituted by Paulings atom electronegativity or valence vertex degree (for an atom $i$ is given by $\delta_{i}^{v} = \sigma_i + \pi_i + n_i - h_i$ where $\sigma_i, \pi_i, n_i, h_i$ are the number of sigma, pi, lone pair electrons and hydrogen atoms respectively).

For each path length $k$ we can define a topological charge index, $G_k$ as

$$g_k = \frac{1}{2} \cdot \sum^{A}_{i=1} \sum^{A}_{j=1} \left\vert CT_{ij} \right\vert \cdot \deltaup(k;d_{ij})$$

where $\deltaup(k;d_{ij})$ is the Kronecker delta,

$$\deltaup(k;d_{ij}) = \left\{ \begin{array}{lcl} 1 & if & d_{ij} = k \\ 0 & if & d_{ij} \neq k \end{array} \right.$$

where $d_{ij}$ are elements of the distance matrix. Thus $G_{K}$ represents the total charge transfer between atoms at a topological distance $k$. The maximum number of $G_k$ terms equals the topological diameter, $D$.

The average topological charge index, $J_k$ is defined as

$$J_k = \frac{G_k}{A-1}$$

and the global topological charge index, $J$, is defined as

$$J = \sum^{5}_{k=1} J_k$$

$G_k$ values (and hence $J_k$ & $J$ values) are set to zero for $k$ values greater than $D$.