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Journal of the Chilean Chemical Society

versión On-line ISSN 0717-9707

J. Chil. Chem. Soc. v.49 n.4 Concepción dic. 2004 


Chil. Chem. Soc., 49, N 4 (2004): págs: 351-354




Departament of Inorganic and Analytical Chemistry. Faculty of Chemical and Pharmaceutical Sciences, University of Chile. Santiago, Chile. e-mail:


Multiplicative numerical factors (Fc) were calculated to modificated each term (di * dj)-0.5 of 1c v index.

The Fc factors were calculated using: nine linear equations system based on the experimental boiling points (B.points,exper.) of nine saturated hydrocarbons, the type of the molecular two-carbon fragment and the number of hydrogen atoms attached to them, that evaluate the chemical graph vertex of the molecule .

It was shown for a group of fifty saturated hydrocarbons that smaller differences were obtained between experimental boiling points (B.points,exper.) and those calculated by each (di * dj)-0.5 * Fc contribution to 1cv index , than the differences between boiling points (B.points,exper ) and those calculated by means of the linear correlation f: (B.points,exper. ) Æ 1c v


The idea of chemical structure is of great importance to chemical science. Two mathematical tools are frequently used to describe chemical structure, the topological and graph theories1-3)

Various attempt have been reported to express by means a numerical index the connectivity of atoms in a molecule. Among these, the first Zagreb group index M1 is defined as a sum of the squared vertex degrees, where, the second Zagreb group index M2 is defined as the sum of all edges product of the vertex degrees of neighboring vertices4) equation 1 and 2

1 2

Milan Randic5) transformed M2 into an inverse square-root function 1c , equation 3 1c = S (ai *aj )-0.5 3


L. B. Kier and L. H. Hall advanced the used of the valence connectivity index6-8) This is calculated with formulae similar to Randic , but the product of edges endpoints (or path vertex) are no longer of vertex degrees but of weighting (valence value di ) given by formula 4


Where Ziv stands for the number valence electrons in atom i, Zi is the atomic number and Hi is the hydrogen number attached to atom i.

Within QSPR discipline (Quantitative Structure-Property Relationship), the L. B. Kier and L. H. Hall first­order valence connectivity index ( 1c v ), has been used extensively by this authors themselves 9-15) and others investigators16) . This index is considered to be one of the most successful in QSPR discipline17-19)

Very often in QSPR discipline is used a single independent variable in correlation mathematical model and when the objective is to evaluate a new index (A) with a set {ai } formed by a series of individuals molecular structure properties ( by examples boiling points) the set cardinality of individuals invariants index (Ai) must be the same of set {ai } in this condition it is possible to apply a regression relation f: (Ai) ® {ai } Very often one elements in set (Ai) has a correspondence with two or more elements on set {ai } this circumstances implies an index with degenerative grades, this is the case with 1cv index and with modification index proposed by the author in this work.

The thesis proposed in this study is based in two classical set within the QSPR discipline: {L} set represented by experimental boiling points (B.points, exper) of fifty saturated hydrocarbons20) set {ai} contains values of 1cv index determined on each graph corresponding to particular hydrocarbons mentioned. The values of each index were obtained by equation 5.


An optimal linear regression model was adopted corresponding to the mathematical structure


The correlation consistency was evaluated by classical statistical parameters commonly used in QSPR field, these parameters are: correlation index (r), deviation standard of correlation (s.d.) and Fisher ration (F)

The relation described in equation 6 is compared with the modification proposed by the author on (di * dj)-0.5 of 1c v index by means a factor Fc to produced a modificated contribution (di * dj)-0.5 * Fc .

To calculated boiling point of a particular molecular structure is necessary to sum (di * dj)-0.5 * Fc ij over all edges , equation 7


The factor Fc is calculated in function of edge type implicit in di*dj of a particular graph and its experimental boiling point. For obtained all Fc values was necessary to used 9 linear equations to cover all different edged types present in saturated hydrocarbons.

The calculated boiling points obtained by means of 6 and 7 equations are compared by statistical parameters.

Also were compared the differences obtained between experimental boiling point vs. calculated one for both systems.


The evaluation of the numerical factors (Fc).

The total number of different two-carbons fragments found in the 50 saturated hydrocarbons is nine. This number is the maximum possible value for the whole saturated hydrocarbons homologous series.

For the nine different type of two-carbon fragments, a specific numerical value corresponding to (di*dj )-0.5 was assigned in accordance with L. B. Kier index (equation 5) and also was assigned nine specific unknown ( values of Fc) see Table 2. The evaluation of these unknown was carry out using a vector whose elements are the experimental boiling points of nine saturated hydrocarbons included in group of analysis, see Table 1 and a quadratic matrix whose elements are the (di*dj )-0.5 values. The solution of the linear system equations was obtained by classical linear algebra procedures by means of HP 48GX software. The mathematical expression is expressed by equation 8:


Where exponent ­1 implies the inverse matrix of the original (di*dj )-0.5 quadratic matrix. The different two-carbon fragments, the L. B. Kier (di*dj )-0.5 value, the names of the unknowns values, the Fc factors values and the (di*dj )-0.5 *Fc product are indicated in Table 2.

Obtained the Fc factors, the boiling point were calculated for 50 saturated hydrocarbons by adding the different fractions (di*dj )-0.5 *Fc in function of the bound type of each saturated hydrocarbon.

The following example using 2,2-dimethyl pentane shows how one linear equation is set out, equation 9:


The process of configuring the rest (eight equations) of the linear equations is done in a similar way.

The structure of 2,2-dimethyl pentane, boiling point 79.2 C, is shown below.


Designating f: Experimental boiling point ® 1c v as theory I and the thesis proposed in this study as theory II, it is possible using Otto Exner criteria21) to show that in relation to the former , theory II is valid and superior. The procedure is as follows: the differences between experimental and calculated boiling points were calculated for both theories (DB.point )I , (DB.point )I I These differences were placed in ascending order and the absolute values compared at N/2 in this case correspond to 25.

If T represent the two theories, it was found that:
T (I;II) :Theory II is the best valid theory, given that:
F-1 II (N/2) < F-1 I (N/2) for N = 50

This relation will be related on equation 12 and 13

The linear regression, f: Experimental boiling point ® 1c v is expressed by equation 11

B.point, experimental = -94.3896(± 6.3609) + 57.0672 (± 1.7681)* 1c v
000000000000000000000r = 0.9777

0000000000000000000s.d = 6.645
00000000000000000000 F = 1041.7876
00000000000000000000N = 50

The linear regression between experimental boiling points an the boiling points calculated by means of equation 11 is expressed by equation 12

B. point calculated = 4.8043(±3.3271) + 0.9561(± 0.0296) * B. point, experimental
000000000000000000000r = 0.9778

0000000000000000000s.d = 5.5333
00000000000000000000 F = 1043.9049
00000000000000000000N = 50

F-1 I (N/2) = 3.50 (Otto Exner test21)
The values of (DB.point )I calculated by theory I are presented in Table 3


The boiling points calculated by the thesis II are presented in Table 3 and its were calculated adding all (di*dj )-0.5 *Fc over all two-carbon fragments of the molecular graph.

Linear regression between the experimental boiling points and calculated boiling points by means of thesis II is represented by equation 13.

B.point, experimental = -0.4229(± 1.8044) + 0.9934(± 0.163)
000000000000000000000r = 0.9936

0000000000000000000s.d = 3.0541
00000000000000000000 F = 3700.85645
00000000000000000000N = 50
F-1 II (N/2) = 1.22 (Otto Exner test21)
The differences (DB.point )II are presented in Table 3

All the regression procedures were conducted using Statgraphic Plus22) software. In all given linear regression the size of statistical parameters ( r, s.d., F ) allow a single conclusion to be reached, in all of them, with the value P of ANOVA < 0.01 there are statistically significant relationships between the dependent and independent variable at the confidence level of 99%. The r size are such that in all cases examined, there is a strong relationship between dependent and independent variable.

Nevertheless, the values of r, s.d. and F of equation 13 are better than equation 12, this implies the better quality of thesis II over thesis I, confirmed by the F-1I > F-1II relation values obtained to evaluated both thesis.


Þ The thesis presented in this study with reference to (1cv) as a model, has high statistical reliability.
Þ Using this model, the idea presented makes possible optimization of the different indices used in the field of QSPR by means of the calculation and application of specific factors concerning the dif ferent types of two-carbon fragments and the number of bounds with hydrogen atoms.
Þ The calculation of specific factors was based on a small number of substances. Their application to significantly larger groups of ho mologous substances is statistically valid.
Þ It is suggested that the method be applied to different organic ho mologous series, using other indices.



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(Received: March 1, 2004 - Accepted: October 21, 2004)


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