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1. Molecular Coordinates

   Linked via "internal coordinate vector"

   A bond length, $r_{ij}$, is the straight-line distance between the nuclei $i$ and $j$ forming a chemical bond. These are defined using the standard Euclidean distance formula applied to their Cartesian counterparts:
   $$r{ij} = \sqrt{(xi - xj)^2 + (yi - yj)^2 + (zi - z_j)^2}$$
   In polyatomic systems, the set of $N_{\text{bonds}}$ distinct bond lengths forms a subset of the total [internal coordinate vector…