Prediction of glass transition temperatures of OLED materials using topological indices

Abstract

The QSPR study was performed between topological indices and glass transition temperatures (T _gs) of organic light-emitting diode materials based on a diverse set of 80 compounds. A five-parameter correlation with the squared correlation coefficient R ² = 0.9304 and an average absolute error of 7.7 K was obtained through step-wise multi-linear regression analysis with leave-one-out cross-validation. The new model proposed is predictive and requires only topological indices in the calculations and has the advantage of the relative ease in calculating the descriptors, which makes it easier to apply. The predicted results of the new model are comparable to those of the existing equation by using the Comprehensives Descriptors for Structural and Statistical Analysis approach.

Appendices

Appendix

The definitions of topological indices used in this work are shown below:

Wiener indices

The Wiener number, W, is the sum of distances in a molecular graph [18]. For a given connected molecular graph G

$$W = \frac{1}{2}{\sum\limits_{i = 1}^N {{\sum\limits_{j = 1}^N {D_{{ij}} } }} }$$

where D _ij is a distance matrix of the shortest paths between any two vertices for N vertices. D _ij = l _ij if i ≠ j, otherwise equal to zero. l _ij is the shortest distance between vertices i and j.

Balaban indices

The Balaban index, J, is the average-distance sum connectivity [19–21]. For a given connected molecular graph G

$$J = \frac{M}{{\mu + 1}}{\sum {(D_{i} D_{j} )^{{ - 0.5}} } }$$

where M is the number of edges in G. μ denotes the ring number of G. In a polycyclic graph, μ is the minimum number of edges that must be removed before G becomes acyclic. $D_{i} = {\sum\limits_{j = 1} {D_{{ij}} } }$ and D _ij is defined as for the Wiener index.

Randic–Kier–Hall subgraph connectivity indices

The χ _t indices may be derived from the adjacency matrix [22–24] and they are defined as

$$^{m} \chi _{{\text{t}}} = {\sum\limits_{j = 1}^{N_{m} } {^{m} S_{j} } }$$

where m is the subgraph order, that is, the number of edges in the subgraph, N _m is the number of type t order m subgraphs within the whole graph, and ^m S _j is a factor defined for each subgraph as

$$^{m} S_{j} = {\mathop \prod \limits_{i = 1}^{m + 1} }(\delta _{i} )^{{ - 1/2}}_{j} $$

where j denotes the particular set of edges that constitutes the subgraph and δ _i is the degree of vertex i, that is, its number of edges.

Valence connectivity indices are defined similarly, substituting δ _i by δ ^v _i , defined as

$$\delta ^{v}_{i} = \frac{{Z^{v} - h_{i} }}{{Z - Z^{v} - 1}}$$

where Z is the atomic number of the atom i,Z ^v the number of valence electrons, and h _i is the number of H atoms attached to it.

Kier–Hall electrotopological state indices

Kier and Hall [25, 26] developed electrotopological state indices (E-state indices), based on the electronegativity of an atom and its local topology. The E-state (S _i ) of an atom i (non-hydrogens) is calculated by evaluating the intrinsic state value (I _i ) and the perturbation arising from all the other skeletal atoms (ΔI _i ):

$$S_{i} = (I_{i} + \Delta I_{i} )$$

where

$$I_{i} = \frac{{[(2/N_{i} )^{2} \delta ^{{\text{v}}} + 1]}} {\delta },\quad \quad \Delta I_{i} = {\sum\limits_{j = 1}^{N_{i} } {(I_{i} - I_{j} )/r^{2}_{{ij}} } }$$

and: N _i is the principal quantum number of atom i,δ^v is the count of valence electrons in the skeleton; δ the count of s electrons in the skeleton (see Randic–Kier–Hall Subgraph Connectivity Indices, earlier), ΔI _i the perturbation on atom i by all other skeletal atoms j,r _ij is the number of atoms in the shortest path between atoms i and j (including i and j). Electrotopological state indices of the type of atoms are obtained by summing the electrotopological states for each present type of atoms in the molecule, S ^T (i).

Kier–Hall Kappa indices

The Kappa indices are the basis of a method of molecular structure quantitation in which attributes of molecular shape are encoded into three indices (Kappa values) [27–29]. These Kappa values are derived from counts of one-bond, two-bond, and three-bond fragments, each count being made relative to fragment counts in reference structures which possess a maximum and minimum value for that number of atoms.

Shape flexibility index or Φ. Flexibility of a molecule is directly related to the degree of linearity and the presence of cycles and/or branching [27–29]. The Kappa alpha indices measure these factors while also taking the effects of atomic identities on shape into account. Hall and Kier found that by combining ¹ κ_α and ² κ_α indices, a further index Φ, which measured flexibility, could be defined:

$$\Phi = (^{1} \kappa _{\alpha } \bullet ^{2} \kappa _{\alpha } )/A$$