Prediction of stability constants of coordination compounds from their molecular graphs

Every molecular structure can be regarded as a graph, and each graph can be transformed into characteristic scalar – topological index. There are many, over 1000 topological indices used by chemists, but they are not equally successful in solving specific problems. For the prediction of stability of coordination compounds valence molecular connectivity index of the 3rd order (χ) proved best. Stability of these compounds was expressed as log K (K stays for stability constant), which is proportional to the Gibbs free energy of reaction between metal (or metal complex) and ligand. Various molecular graphs were checked, from a simple graph of ligand, to the graphs of various metal-ligand complexes (aqua complex, complex with additional bonds etc.) leading to conclusion that chemically sounder graphs generally yield better models. The best results were obtained for the systems of one metal, mesured at the same experimental conditions. However, our graphtheoretical approach enables development of models for the simulateneous prediction of stability constants of few metals. 1. Topological indices in chemistry Application of topological indices in chemistry commenced in 1947 when H. Wiener introduced path number W, later known as Wiener number or Wiener index [1,2]. The path number was defined as the number of bonds between all pairs of atom in an acyclic molecule, i.e.: W(G) = 1⁄2 Σ Dij(G) (1) where Dij(G) are off-diagonal elements of (topological) distance matrix D(G). Wiener correlated his path number with physical parameters of acyclic hydrocarbons, and this problem occupy the chemists and mathematicians till our days; i.e. most of the papers in mathematical chemistry deal with modelling boiling points of hydrocarbons by using a variety of topological indices [3,4]. Later, topological indices (now, there are more than thousand of them) were used for prediction of many physico-chemical properties (QSPR, quantitative structure-properties relationships) [5-7] and biological activity (QSAR, quantitative structure-activity relationship) [8] of many kinds of molecules. Among the hundreds of topological indices [9] very few, however, found its application in chemistry. In our first paper in application of these indices for the estimation of stability of coordination compounds [10], we found the best fit with the Wiener index, W, and three valence connectivity IC-MSQUARE 2012: International Conference on Mathematical Modelling in Physical Sciences IOP Publishing Journal of Physics: Conference Series 410 (2013) 012024 doi:10.1088/1742-6596/410/1/012024 Published under licence by IOP Publishing Ltd 1 indices, χ, χ, and χ. The last one, the valence molecular connectivity index of the 3 order, later proved to be the best of all indices, is defined by relation [11,12]: χ (G) = Σ [δ(i)δ(j)δ(k)δ(l)] (2) where δ(i), δ(j), δ(k), and δ(l) are weights (valence values) of vertices (atoms) i, j, k, and l making up the path of length 3 (three consecutive chemical bonds) in vertex-weighted molecular graph, G. 2. The stability of coordination compounds The stability of coordination compounds is usually measured by log K values, where K is equilibrium constant of reaction between metal (M) and ligand (L):