3D Balaban index¶

The geometric distance matrix is defined as

$\mathbf{D} = \begin{pmatrix} 0 & d_{01} & d_{02} & ... & d_{0N} \\ d_{10} & 0 & d_{12} & ... & d_{1N} \\ d_{20} & d_{21} & 0 & ... & d_{2N} \\ ... & ... & ... & 0 & ... \\ d_{N0} & d_{N1} & d_{N2} & ... & 0 \end{pmatrix}.$

and leads to the adjacency matrix $\mathbf{A}$ that contains the elements

$a_{ij} = \begin{cases} 0 & \text{if } d_{ij} > d_0 \\ 1 & \text{if } d_{ij} < d_0 \end{cases},$

where $d_0$ is a threshold that defines the existence of a bond. Similar to the Wiener number, the 3D Balaban index

$^{3D}J = \frac{M}{\mu+1} \sum_i^N\sum_j^N \left( d_i d_j \right)^{-1/2}$

represents a topographical descriptor related to the shape of the molecule. Here, $M$ is the number of edges in the molecule, $\mu$ is the cyclomatic number

$\mu = M-N+1$

and $d_i$ is the sum of all entries in the $i$ -th row of the $\mathbf{D} \otimes \mathbf{A}$ matrix.

The use of the Balaban index is invoked by the following block in the meta-config.json file:

"collective variables": [
  {
    "type": "balaban",
    "name": "2d balaban index",
    "exclude_h": False
    "update_adjacency": False
  },
  ...
]

The additional keyword "exclude_h" allows for the calculation of the distance matrix without considering distances from and to Hydrogen atoms. If "update_adjacency" is set to True, the adjacency matrix is recalculated in every dynamics step and therefore accounts for changes in the molecular connectivity.