3D Balaban index
================

The geometric distance matrix is defined as 

.. math::
    \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 :math:`\mathbf{A}` that contains the elements

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

where :math:`d_0` is a threshold that defines the existence of a bond. Similar to the \
:doc:`Wiener number <wiener>`, the 3D Balaban index 

.. math::
    ^{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, :math:`M` is \
the number of edges in the molecule, :math:`\mu` is the cyclomatic number

.. math::
    \mu = M-N+1

and :math:`d_i` is the sum of all entries in the :math:`i`-th row of the \
:math:`\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.