o if i = j or (i,j) does not belong to E(G)

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1 Indian Journal of Chemistry Yol. A, June 00, pp. 5-5 Maximum topological distances based indices as molecular descriptors for QSPR. V-Modeling the free energy of hydrocarbons Pablo Duchowicz', Rmulo G Siiiani, Eduardo A Castro' * & Andrey A Toropov 'CEQUINOR, Departamento de Qufmica. Facultad de Ciencias Exactas, Universidad, Nacional de La Plata. Calles y 115, c.c., La Plata 0, Argentina Yostok Innovati on Company, Azimstreet, Tashke nt 000, Uzbekistan.1Departamento de Qufmica, Universidad Mayor de San Andres, Campus Universitario, Calle Cota-Cota, La Paz, Boli via Received August 00 The Gibbs free energy for a set of 0 hydrocarbons is calculated on the basis of topological indices defi ned from the di stance and detour matrices wi th in the realm of the QSPR theory. Linear and noll -linear polynomials fittings are performed and the results demonstrate the need to resort to higher-order regression functi ons in order to obtain better agreements between theoretical results and experimental thermodynamics data. Topological indi ces calculated from detour matrix yield rather satisfactory predictions of Gibbs free energy for hydrocarbons. Introduction The free energy (FE) is often considered to be the most important quantity in thermodynamics. The FE is usually expressed as the Helmholtz function, A, or the Gibbs function, G. The Helmholtz FE is appropriate to a system with constant number of particles, temperature and volume (constant NVT), whereas the Gibbs FE is appropriate to constant number of particles, temperature and pressure (constant NPT). Most experiments are conducted under conditions of constant temperature and pressure, where the Gibbs function is th e appropriate FE quantity. Unfortunately, the FE is a difficult quantity to obtain for systems such as liquids or flexible macromolecules that have many minimum energy configurations separated by low-energy barriers. Associated quantities such as the entropy and the chemical potential are also difficult to calculate t. Considerable progress has been made in modeling and quantifying the thermodynamic properties via Quantitative Structure Property Relationships (QSPR) theory. t. The underlying search for suitable descriptors aims to design them in such a way as to reflect the fundamental molecular properties important in physical chemistry phenomena. Among the huge number of basic topological descriptors usually employed in QSPR are those derived from the application of the distance matrix, such as, for example, Wiener, Balaban and Hosoya indices. Some previous studies have shown the convenience of resorting to maximum distance concept (i.e. detour matrix. ) to obtain improved results relative to the ordinary distance matrix '..0. The ai m of this paper is to continue with such analysis resorting to the calculation of Gibbs FE for a representative set of 0 hydrocarbons. Basic definitions The adjacency matrix The adjacency matrix A = A(G) of a graph G with N vertices is the square Nx N symmetric matrix whose entry in the ith row and jth column is defined as 1 if i i= j and (~,j) belongs to E(G) Aij =... (1 ) o if i = j or (i,j) does not belong to E(G) where E(G) represents the set of edges of G. The sum of entries over row i or column j in A(G) is the degree of vertex i, degj. An example of molecular graph and the adjacency matrix is given below for the methylcyclopropane 0 0 A= The distance matrices The distance matrix D(G) can be defined for G with elements D ij, the distance, or the number of least steps from vertex i to vertex j. Similarly, the detour matrix ~(G) of a labeled connected graph G is a real

2 DUCHOWICZ: MODELING THE FREE ENERGY OF HYDROCARBONS 55 symmetric Nx N matrix whose (ij) entry is the length of the longest path from vertex i to vertex j (i.e. the maximum number of edges in G between vertices i and/ ). For example, the matrices D and. associated to the previous graph corresponding to methylcyclopropane are: o D= 0 o.= 0 I Topological molecular descriptors We present the basic definitions related to the topological molecular descriptors chosen in the present study. Wiener index 1-5 w=o.s ~D ij... () IJ H. d arary zn ex H = 0.5 LDu'... () I ~ J Originally, the Harary index was defined as the sum of D ij to the power- MTI index, 0... () where ei (1 :S i :S N) are the elements of the row matrices v[a + DJ = [e" e, e,..., en]. v = (deg" deg,......, deg n ) is the so called valence matrix where degi is the degree of the vertex i and is equal to the number of its incident edges. Balaban index,0 IJ... (5) where di = L Dij, q is the number of edges and J..l the j number of cycles in the graph. The summations in formulae (), () and (5) are over all edges i-j in the hydrogen depleted graph. Z d... d 1 ero or er connectlvlty zn.ex. Ox = Ldeg ~"... () where degi = L Aij j Randic's connectivity index IX = ~(d egi degjr" I.J Generalized connectivity index 1 hx = L (deg yq deg v,... deg vh r" path... ()... () where the summation is taken over all possible paths of lengths 0, I,....., h. Th e Za gre b group. d ' 1I1 Ices. MI = Ldeg ~... () I.J... () The employment of these topological descriptors has proven to be quite useful in QSPR studies giving simple and accurate correlations between the selected molecular properties and the molecular structure,5. Multiple regression analysis is often employed in such studies with the hope that it might point to structural factors that influence a particular property. Naturally, regression analysis results do not establish any sort of causal relationships between structural components and molecular properties. However, it may be helpful in model building and be useful in the design of molecules with prescribed desirable properties. An interesting alternative to the previous definitions based upon distance matrix D is to replace it by the detour matrix., defining the associated topological indices W *, H*, etc., on the basis of this last matrix in a similar fashion as done in previous equations. Results and Discussion Since we want to know the relative merits of topological indices defined on the basis of the two distance matrices (i.e. D and.) we have resorted to two sets of topological descriptors: a) {Nc, ox, IX, X, M I, M, W, H,, MTl}, and b) {Nc, ox, lx, X, MI, M, W, H, J, MTl, W*, H*, J*, MTl* } where Nc stands for the number of carbon atoms. This particular choice does not preclude any other suitable way to assess the relative capabilities of those topological descriptors founded on distance matrices D and..

3 5 INDfAN J CHEM, SEC A, JUNE 00 Table I-Topological parameters for hydrocarbons Hydrocarbon Ox MI M W H MTI W* H* J* MTl* I Ethane Propane Butane -Meth ylpropane 5 Pentane -Methylbutane,-Dimethylpropane Hexane -Methylpentane -Methylpentane II,-Dimethylbutne,-Dimethylbutane I Heptane 1 -Methylhexane 15,-Dimethylpentane,,-Trimethylbutane 1 -Meth ylheptane 1 -Methylheptane 1,-Dimethylhexane 0.-Dimethylhexane 1,-Dimethylhexane,5-Dimethylhexane,-Dimethylhexane 1.-Dimethylbenzene 5 1,-Dimethylbenzene 1,- Dimethylbenzene I-Methyl--ethylbenzene I-Methyl--ethylbenzene 1,,-Tri methylbenzene 0 1,,-Trimethylbenzene 1 1,,5-Triethylbenzene 1,-Triethylbenzene 1,-Diethylbenzene 1,,,-Tetramethylbenzene 5 1,,-Triethylbenzene 1,,-Triethylbenzene 1,,5-Triethylbenzene 1,,,-Tetraethylbenzene 1,,,5-Tetraethylbenzene 0 1.,,5-Tetraethylbenzene 1 I-Methylnaphlhalene -Methylnaphthalene I-Eth ylnaphthalene -Ethylnaphthalene 5 1,-Dimethylnaphthalene 1,- Dimethylnaphthalene 1,-Dimethylnaphthalene 1,5-Dimethylnaphthalene 1,-Dimethylnaphthalene 501,-Dimethylnaphthalene 51,-Dimethylnaphthalene 5,-Dimeth ylnaphthalene 5,- Dimethylnaphthalene 5 I-Propylllphthalene 55 -Propylnaphthalene 5 -Ethyl--met hylnaphthalene 5 -Ethyl--methylnaphthalene 5 -Ethyl--methylnaphthalene 5 I-Butylnaphthalene 0 -Butylnaphthalenc I I I 1.5. I.55 I I I 1.05 I I I I io I I I

4 DUCHOWICZ: MODELING THE FREE ENERGY OF HYDROCARBONS 5 Table - Descriptors First order Set a) (see text) R s F M 0..0 X,MI Nc, ' X, M I ' X, lx, M I, MTI Nc, \, M I, W, MTI Set b) (see text) R s F M 0..0 lx, H* Nc, H*, J* ' X, IX, M I, MTI* Nc, ' X, M I, J, J* Statistical results for the hest fitting polynomials Second order Third order R s F R s F R s F R s F The chosen molecular set contains 0 hydrocarbons having from I t 1 carbon atoms and they are displayed in Table 1 together with their corresponding topological parameters. Although it seems this molecular set is rather specialized because it only contains C and H atoms, this choice does not necessarily imply a lack of molecular variation. In fact, the hydrocarbon set includes examples of planar, non-planar, alternant and non-alternant aromatic hydrocarbons, alkyl- and alkenyl benzene derivatives, acyclic and polycyclic alkanes, strained and unstrained olefins and disparate structures combined with aromatics like benzene and naphthalene, which do not demand separate parameterizations for different types of C and H atoms. This molecular set has been used in previous studies on QSPR theory, (see, for example,.1..). We have performed a complete analysis for searching the best one-, two-,..., five-variables fittings at first, second and third polynomial orders. It is important to point the need to look for higher-order regression formulae, since there are not formal restrictions to used them. In fact, simple linear regressions involving but a single descriptor limits statistical analysis considerably. Many correlations, particularly when involving molecules of different size, need not be linear. But even if there are molecules of the same or similar size, a quadratic orland cubic regression may result in a better description of the relationship than a simple linear model. In general, one should test single and multiple descriptors fitting functions for quadratic dependence and, if warranted, for hi gher order polynomial relationships or other functional dependence Y Statistical results are given in Table for both topological sets, while in Table some theoretical results together with experimental data 50 are displayed. The best one to five variables first, second and third order correlations for the two molecular topological indices a) and b) have been included. The fitting polynomial formulae inserted due to concise reasons, but complete results are available and can be obtained upon request to one of us (E. A. C.) at the above address. Analysis of numerical results displayed in Tables and shows clearly the better predictive capability arising from the regression equations derived on the basis of set b) of topological descriptors than those corresponding to the set a). The statistical parameters (Table ) and the correlations between experimental data and theoretical predictions (Table ) makes clear the advantage of resorting to the employment of the detour matrix instead of the standard distance matrix in order to define the corresponding topological descriptors. Besides, it also seems recommendable to employ higher-order fitting polynomials to get a better degree of prediction capabilities. These conclusions are totally in line with other similar previous ones derived in several studies on the usefulness of the application of the detour matrix.-0. Conclusions Several usual topological indices have been applied to study the Gibbs FE of a set of hydrocarbons comprising 0 structurally diverse molecules. In those cases where the definition demands the use of the distance matrix, other similar topological indices resorting to the detour matrix have been defined, in order to analyze the relative quality of both distance definitions. Present results show that the employment

5 5 INDIAN J CHEM, SEC A, JUNE 00 Molecule# Table - L'lC(exp)50 Experimental and theoretical Gibbs free energy (kj/mol) for a sel of 0 hydrocarbons The molecu lar numbering corresponds to the molecular li sting of Table I L'lC(theor)' L'lC(theor)' Molecule# L'lC(exp)50 L'lC(theor)' L'lC(theor)' I I # Numbering as in Table I ' Five-variables cubic regression, set a) ' Fi ve-variables cubic regression, set b) Average absolute deviation of the maximum distance matrix (i.e. detour matrix) for defining the topological indices produces better correlation equations to predict Gibbs FE. These results are in agreement with other similar ones previously found to study other physical chemistry properties, so that it seems to support the use of detour indices in structure-property modeling instead of topological indices defined via the standard distance matrix.-0,5 I-5. It is concluded that the results presented in thi s article are good enough for the chosen molecular set of 0 hydrocarbons to infer truly that the detour matrix A represents a suitable topological device to be applied in the QSPR analysis and it makes up a valuable molecular descriptor which validly adds to the set of usual topological matrices. In order to get more significant and definite conclusions on this issue, it is necessary to extend the calculations to quite different molecular sets and other physical chemistry properties and biological activities. Work along these lines is being carried in our laboratories. References 1 Leach A R, Molecular modelling Principles and applications, (Prentice Hall, Harlow, England), p 1, Beveridge 0 L & Mazei M, Annu Rev Biophys Biochell1, 1 ( 1) 1 Davis M E & McCammon J A, Chell1 Rev, 0 () 50 Kollman P A, Chem Rev, () 5

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