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Journal of the Chilean Chemical Society

versión On-line ISSN 0717-9707

J. Chil. Chem. Soc. v.50 n.3 Concepción sep. 2005 


J. Chil. Chem. Soc., 50, N° 3 (2005), págs: 569-574





Departamento de Química Inorgánica y Analítica, Facultad de Ciencias Químicas y Farmacéuticas, Universidad de Chile, Casilla 233, Santiago , Chile


The second stepwise formation constants of binary and ternary copper(II) complexes with a-amino acidate ligands are modeled by using four-descriptor sets consisting of three indexes defined as linear combinations of the E-states of some skeletal groups of the different species involved in the formation equilibria, and the logarithm of the statistical factor. Hydrophobicity and basicity properties of the ligands are also described. Results indicate that both hydrophobic interactions and ligand basicities, as differential factors over the sequence of copper(II) complexes, would operate mainly through changes in the stability of the copper(II)-carboxylate bonds.


Thermodynamic stability of coordination compounds in aqueous solution has been an important topic to almost all areas of chemistry for a long time. The stability of a metal complex is appropriately characterized by the equilibrium constant for its formation process, i.e. by the stability constant (also referred to as formation constant). The knowledge of the stability constants of complexes with ligands of biochemical interest has been relevant for the experimental modeling of both reactions of metal ions with biomolecules1-4) and noncovalent interactions occurring in biological systems.5-13) Excepting the ligation energy, which can be estimated by means of the ligand field theory,14) factors affecting the stability of metal ion complexes are usually analyzed by correlating the stability constants with some relevant structural and physicochemical properties of metal ions and ligands.15) In accounting for trends in stability with changes in the ligands, significant correlations are obtained only for ligands very close in structure, i.e. having the same donor groups and, in the case of multidentate ligands, having similar chelating properties. Thus, the ligand basicity and the intramolecular noncovalent interactions become of central importance as determining factors for the differences in stability over series of complexes of a same metal ion with different ligands closely similar in structure. Among the intramolecular ligand-ligand noncovalent interactions, the hydrophobic interactions have been recognized to influence significantly the stability and reactivity of binary and ternary metal complexes.5-13) The pKa of the protonated ligand, in aqueous solution, has usually been taken as a measure of the ligand basicity.15) In turn, for binary and ternary metal ion complexes with a-aminoacidates, the sum of free energies for transfer of amino acids side chains from 100% organic solvent to water16) has been taken as a measure of the intramolecular ligand-ligand hydrophobic interactions.11) Correlations of stability constants of metal complexes with these bulk properties, i.e. ligand basicities and intramolecular ligand-ligand hydrophobic interactions, have already been discussed on the basis of molecular structure, but merely from a qualitative point of view.5-9,15) Hence, it would be interesting to explore the possibility to account quantitatively for stability constant trends of metal complexes by means of nonempirical parameters which follow directly from the molecular structure. We have recently studied the modeling of the logarithms of the second stepwise stability constants of binary and ternary copper(II) complexes with a-amino acidates17) by using four- and five-descriptor combinations consisting of the electrotopological state indexes (E-states)18-20) of some skeletal groups, the first-order 1c molecular connectivity index21) and the logarithm of the statistical factor.22) Explicitly, the modeled stability constants correspond to formation equilibria of the type: [CuA] + B = [CuAB], where A and B denote amino acidate ligands, A = B for binary complexes, A B for ternary complexes, and both ionic charges and coordinated water molecules have been omitted for simplicity. The stability constants for the above formation equilibria will be quoted hereafter as KCuABB. Results from such study were satisfactory even though, excepting the statistical factor, the descriptors employed were not intrinsically based upon the formation equilibrium as a whole. Indeed, on following the usual procedures for quantitative structure-property analyses, the E-state and 1c indexes had obviously to be calculated from chemical graphs of the complexes whose stability constants was being modeled, namely, [CuA2] and [CuAB] species. It should be considered, however, that for a ternary complex the second stepwise formation equilibrium would be either [CuA] + B = [CuAB] or [CuB] + A = [CuAB]. Clearly, different equilibrium constant values should be expected for these processes.

In the present study an improvement of the modeling procedure for the logarithms of the KCuABB stability constants has been undertaken, so that the above mentioned ambiguity may be removed, and both size and heterogeneousness of the employed descriptor sets may be reduced.


For the sake of depicting more realistic chemical graphs, the stepwise formation equilibria of binary and ternary copper(II) complexes with a-amino acidates were expressed as:

[CuA(H2O)2] + + B- = [CuAB] + 2H2O

where the water molecules axially coordinated to the metal center have been omitted. Logarithms of the stepwise formation constants KCuABB = [CuAB]/[CuA+][B-] at 25C were taken from the literature.11 The E-states of the skeletal groups of the different species involved in these formation equilibria were calculated from their respective hydrogen-suppressed graphs following the general procedure developed by Kier and Hall.18-20) Accordingly, the E-state of an atom of a particular skeletal group i was defined as

Si = Ii + DIi

where Ii is the intrinsic state of atom i and DIi is the perturbation of this atom due to its interactions with the remaining atoms of the molecule. The Ii values were calculated through the expression

Ii = [(2/N)2dV + 1]/d

where N is the principal quantum number, dV is the count of valence electrons in the molecular skeleton and d is the count of s electrons in the molecular skeleton. In turn, dV and d were computed by the equations:

dV = ZV - h and d = s - h

where ZV is the number of valence electrons, s is the count of electrons in s orbitals and h is the number of bonded hydrogen atoms. The nonbonded contributions were evaluated by the expression

DIi = å (Ii - Ij)/ (rij)2

where rij is the count of atoms in the shorter path between atoms i and j, including both i and j (i.e. the graph distance plus one). Thus, a particular Si value encodes both electronic and topological information since the intrinsic-state Ii encodes the valence-state electronegativity of atom i, whereas the term DIi accounts for the influence on such atom by all the remaining atoms in the molecular skeleton.19) In order to label the E-states of the different species involved in the formation equilibria, the Si values were further symbolized by Sk(i), where k stand for the considered species and i for the skeletal group, e.g., S[CuAB](>Cu<). In the case of [CuAB] complexes, Sk(i) values for equivalent skeletal groups were added together. These sums were denoted by SSk(i). Since the E-state indexes or, more exactly, the intrinsic states (Ii) have been defined only for the s- and p-block elements,18-20) the intrinsic state for copper(II) skeletal group had to be estimated semiempirically. The procedure was as follows: Intrinsic states for the elements of group IIA involved in four-coordination were calculated. The resulting I(>M<) values were correlated with the respective qc/rc ratios. Here qc denotes the charge of the metal ion (always 2+) and rc the ionic radius. Spherical potential ion radius values23 were taken for rc. The statistical results were as follows:

I(>M<) = 0.5874(1/rc) - 0.1012

r = 0.999 ; s = 0.010 ; F = 1309 ; n = 5.

where the original slope has been multiplied by the metal ion charge (2+). The intrinsic state value for copper(II) skeletal group was then calculated through this regression equation by using the spherical potential ion radius of copper(II):23) Thus, I(>Cu<) = 0.486 @ 0.49 was obtained.

The E-states of the species involved in the formation equilibrium were combined to define new descriptors, D(i) , by means of the expression

D(i) = SS[CuAB](i) - S[CuA](i) - SB(i)

where the E-state indexes are handled as thermodynamic state functions. In this expression the species [CuA] corresponds indeed to [CuA(H2O)2]+, i refers to equivalent skeletal groups, and the contributions of water molecules displaced from the coordination sphere have been disregarded because they should be constant through the series of formation equilibria studied.

Different sets of four descriptors, consisting of the D(i) indexes of some selected skeletal groups and the logarithm of the statistical factor, were used to describe the logarithms of the stepwise formation constants KCuABB. For the sake of comparison, four-descriptor combinations consisting of the SS[CuAB](i) indexes of three skeletal groups of the [CuAB] complexes and log(sf) were also tested as descriptors of log KCuABB. The statistical factor 22) was calculated through the expression

sf = (r+s)!/r!s!

where r and s are the stoichiometric subscripts in the general formula [CuArBs]. Hence, log(sf) takes only the values 0 and log2, for binary and ternary complexes, respectively.

The modeling of some physicochemical properties of the amino acidate ligands by means of D(i) and SS[CuAB](i) indexes was also considered. The subscript [CuAB] will be hereafter omitted in the symbols of the E-states of metal complexes, e. g., SS(>C=) instead of SS[CuAB](>C=), and so on.

Multiple regression analyses were carried out by using the software Statgraphics Plus for Window 4.0 on a Pentium III computer.


The calculated E-state values of some skeletal groups, which take part in the formation of chelate rings, are listed in Tables 1-3. The D(i) descriptors calculated from such E-state values are shown in Table 4. Moreover, the four-descriptor combinations tested in multivariable regression analyses against logKCuABB are collected in Table 5, where some statistical results are also included. From these results it can be noticed that the four-index combination {D(-O-), D(>C=), D(-NH2-), log(sf)} provided the best description of logKCuABB. The corresponding regression equation is as follows:

logKCuABB = 1.4436(±0.2858)D(-O-) + 1.1629(±0.4260)D(>C=) - 1.1092(±0.2153)D(-NH2-) +
  1.1188(±0.1650)log(sf) + 8.8895(±1.2566) (1)
r = 0.9719, s = 0.0671, F = 63.8, Durbin-Watson statistic = 2.379, n = 20  

Table 1. Electrotopological state indexes for some selected skeletal groups of binary and ternary dichelated copper(II) complexes with a-amino acidate ligandsa

Table 2. Electrotopological state indexes for some selected skeletal groups of binary monochelated copper(II) complexes with a-amino acidate ligands

Table 3. Electrotopological state indexes for some selected skeletal groups of a-amino acidate ions

The analysis of variance indicates that there is a statistically significant relationship between the variables at the 99% confidence level (p-value < 0.0001). The r-squared statistic shows that this model accounts for 94.45% of the variability in logKCuABB. Moreover, the Durbin-Watson statistic indicates that there is probably no significant autocorrelation in the residuals.

In Table 4 the logKCuABB values calculated through equation 1 are compared with the experimental data.11) As it can be noticed, differences between the experimental and computed values, expressed as percentage, fall in the range 0.06-1.96, whose upper limit is similar to the standard errors usually reported for the experimental data of formation constants of copper(II) complexes with amino acidate ligands. These results are similar to those previously obtained from the correlation between logKCuABB and the five-descriptor set {SS(-O-), SS(>C=), SS(-NH2-), 1c, log(sf)}.17 However, as shown in Table 5, more homogeneous four-descriptor combinations, consisting of the SS(i) indexes of three skeletal groups of the [CuAB] complexes and log(sf), lead to descriptions of logKCuABB which, though significant, are rather poorer than that provided by the combination {D(-O-), D(>C=), D(-NH2-), log(sf)}. Consequently, D(i) indexes appear to be somewhat better and more specific descriptors for logKCuABB than SS(i). Accordingly, as shown in Table 5, the statistical results for the correlations between logKCuABB and different D(i)-index sets clearly suggest that changes in the electronic and topological states of the skeletal groups -O-, >C= and -NH2-, concomitant to the process of formation of [CuAB] from [CuA(H2O)2]+ and B-, would be the main determining factors for the sequence of logKCuABB experimentally observed. Instead, the statistical results for the correlations between logKCuABB and different sets of SS(i) indexes exhibit greater similarities between each other. Further, the best combination of SS(i) indexes excludes the skeletal group >C=, in disagreement with the fact that the index SS(>C=) occurs in descriptor sets which provide the best modeling for the acid-base properties of the amino acid ligands, as will be shown later.

Table 4. D(i) values for some selected skeletal groups, observed logKCuABB values, and logKCuABB values computed with equation 1

According to the statistical data listed in Table 5, the role of log(sf) as a descriptor would consist in allowing the feasibility of modeling logKCuABB simultaneously for binary and ternary dichelated complexes. Thus, as can be noticed, the statistical parameters for the correlations between logKABB and the descriptor sets containing log(sf) are much better than those for the correlations between logKABB and descriptor sets lacking of log(sf). In fact, if the logKCuABB values for binary (A = B) and ternary (A ¹ B) complexes are separately correlated with the three-descriptor set {D(-O-), D(>C=), D(-NH2-)}, the resulting statistical parameters are again significant, namely:

r = 0.9888, s = 0.0258, F = 29.2, Durbin-Watson statistic = 2.206, n = 6, for binary complexes,
r = 0.9407, s = 0.0602, F = 25.6, Durbin-Watson statistic = 2.139, n = 14, for ternary complexes.

Table 5. Statistical results for correlations of logKCuABB with four-descriptor combinations

Even though the D(i) indexes only import differences in E-state values on passing from the monochelated to the dichelated metal complex, they also encode, to a certain extent, some physicochemical properties of the ligands which are relevant as determining factors for the stability of the metal complexes. Accordingly, the set {D(-O-), D(>C=), D(-NH2-)} gives a significant linear correlation with the hydrophobicity scale of aminoacid side chains.11,16) The corresponding statistical results are:

SDGt = 54.793(±6.326)D(-O-) + 73.923(±10.383)D(>C=) - 62.789(±5,788)D(-NH2-) - 13.706(±24.486) (2)
r = 0.9509, s = 1.999, F = 50.3, Durbin-Watson statistic = 2.022, n = 20  

where SDGt is the sum (HA plus HB) of the group contributions to the free energy transfer of aminoacid side chains from 100% organic solvent to water at 25oC. SDGt may also be referred to as total hydrophobicity scale of amino acid side chains.11) The three-descriptor set {D(-O-), D(>C=), D(-NH2-)} gives also significant linear correlations with pKa1 and the pH at the isoelectric point, pI = 1/2(pKa1 + pKa2). Both pKa1 and pI may be considered as measures of the ligand basicity towards the metal ion. The corresponding regression equations are:

pKa1av = 1.0645(±0.1292)D(>C=) - 0.2737(±0.0787)D(-O-) + 0.3139(±0.0720)D(-NH2-) + 0.5102(±0.3050) (3)
r = 0.9539, s = 0.0249, F = 53.9, Durbin-Watson statistic = 2.302, n = 20  
pIav = 2.4262(±0.2296)D(>C=) - 0.6195(±0.1399)D(-O-) + 0.6125(±0.1280)D(-NH2-) + 1.6605(±0.5420) (4)
r = 0.9655, s = 0.0442, F = 73.4, Durbin-Watson statistic = 2.003, n = 20  

In these expressions the superscript av stands for the average property of the aminoacids HA and HB, i. e., pKa1av = 1/2[pKa1 (H2A+) + pKa1 (H2B+)] and pIav = 1/2[pI(HA) + pI(HB)], respectively.

Seeing that SDGt, pKa1av and pIav have been defined for free amino acids rather than for the processes [CuA(H2O)2]+ + B- ® [CuAB] + 2H2O, it should be expected that the E-states of [CuAB] complexes, SS(i), give much better descriptions for the above mentioned physicochemical properties than the D(i) indexes. Accordingly, as a single descriptor, the E-state SS(-O-) provides a better description of SDGt than the set {D(-O-), D(>C=), D(-NH2-)}, the corresponding regression equation being:

SDGt = 17.050(±0.782)SS(-O-) - 171.938(±8.198) (5)
r = 0.9816, s = 1.1633, F = 475.1, Durbin-Watson statistic = 1.746, n = 20  

In Figure 1 the SDGt values calculated from the experimental data16 are plotted against those calculated with equation 5. The occurrence of a good linear correlation between SS(-O-) and SDGt is rather significant because, at least for alkyl ethers, the E-state values for bridging oxygen give also a good linear correlation with the oxygen partial charges calculated by an ab initio quantum mechanical method.18) This fact suggests that the enhancing effects of the intramolecular hydrophobic interactions on the stability of the [CuAB] complexes11 would mainly act through an increase in the coordination tendency of the carboxylate groups. Further, seeing that the Cu(II)-O(carboxylate) bonds exhibit the greatest ionic character, such stability enhancement would be regarded as a water structure-enforced ion-pairing type contribution.24 Amongst the tested two-index combinations, the set {SS(-O-), S(>Cu<)} gives the best description of SDGt. The corresponding statistical results are:

SDGt = 17.856(±0.641)SS(-O-) + 4.5213(±1.2351)S(>Cu<) - 164.81(±6.602) (6)
r = 0.9897, s = 0.8951, F = 407.9, Durbin-Watson statistic = 1.553, n = 20  

Obviously, in the above relationship the index S(>Cu<) encodes electronic and topological information upon the amino acidate ligands through its perturbation term, DI(>Cu<).

Figure 1. Plot of the total hydrophobicity scale of amino acid chains, SDGt = DGt (HA) + DGt (HB) (kJ mol-1 ), against SDGt values calculated with the aid of Equation 5. Numbers refer to compounds in Table 4.

On the other hand, the set {SS(-O-), SS(>C=)} gives also a rather good description of SDGt:

SDGt = 16.059(±0.670)SS(-O-) + 3.6477(±1.0264)SS(>C=) - 158.90(±7.368) (7)
r = 0.9895, s = 0.9067, F = 397.3, Durbin-Watson statistic = 1.556, n = 20  

Equations 5-7 appear to be in agreement with equation 1. According to the latter equation, the greater is SS(-O-), the lessnegative should be D(-O-) and the greater logKCuABB. Thus, both SDGt and logKCuABB increase with SS(-O-), which is in good agreement with the observed enhancing effects of the intramolecular hydrophobic interactions on the stability of the metal complexes.11 In turn, the less negative is SS(>C=), the more positive should be D(>C=) and the greater logKCuABB, which is also in accordance with the observed correlation between SDGt and logKCuABB.11 However, the set {SS(-O-), SS(-NH2-)} gives also a significant linear correlation with SDGt, its statistical parameters being also rather similar to those of equations 6 and 7, namely:

SDGt = 15.039(±0.875)SS(-O-) + 4.0429(±1.2246)SS(-NH2-) - 164.854(±6.925) (8)
r = 0.9888, s = 0.9343, F = 373.6, Durbin-Watson statistic = 1.443, n = 20.  

Here, SDGt increases with SS(-NH2-) whereas, according to equation 1, the greater SS(-NH2-), the less negative D(-NH2-) should be and, hence, the smaller logKCuABB. The ambiguous role of SS(-NH2-) in equations 1 and 8 could be rationalized by assuming a water structure-enforced ion-pairing nature for intramolecular hydrophobic interactions.24) Thus, the greater SS(-NH2-), the more negative the nitrogen partial charges should be and, consequently, the greater the contribution of SDGt. However, the more negative the nitrogen partial charges, the lower the covalent character of the Cu(II)-NH2 bonds should be and, hence, the smaller the ligation energy14), and the smaller logKCuABB. This statement is in good agreement with the role of D(-NH2-) in equation 1. Anyway, the contribution of the E-states of coordinating oxygens, SS(-O-), greatly prevails in equations 6-8.

On the other hand, the E-states of the [CuAB] complexes, SS[CuAB](i), give also a much better modeling of pKa1av and pIav than the D(i) indexes. Among the tested two-index combinations, the set {SS(-O-), SS(>C=)} provides the best descriptions for such properties. The regression equations are as follows:

pKa1av = 0.3617(±0.0238)SS(>C=) - 0.1100(±0.0155) SS(-O-) + 3.6579(±0.1707) (9)
r = 0.9652, s = 0.0210, F = 116.0, Durbin-Watson statistic = 1.978, n = 20  
pIav = 0.7471(±0.0438)SS(>C=) - 0.2156(±0.0286) SS(-O-) + 8.6619(±0.3144) (10)
r = 0.9720, s = 0.0387, F = 145.6, Durbin-Watson statistic = 1.694, n = 20  

According to equations 9 and 10, the ligand basicities should increase as SS(-O-) decreases and SS(>C=) becomes less negative. Seeing that these indexes correspond indeed to the [CuAB] complexes, a decrease in SS(-O-) together with an increase in SS(>C=) would be related with a decrease in polarity of the coordinated oxygen-carbon bonds. In turn, this would be consistent with an increase in the covalent character of the coordinating oxygen-copper(II) bonds and, hence, with an increase in the ligand basicities. Anyways, equations 9 and 10 suggest that the contributions of the ligand basicities, as determining factors for the differences in logKCuABB over the series of binary and ternary complexes, may be connected mainly with changes in the basicity of the carboxylate groups.

From the above discussion it can be concluded that the D(i) indices appear to be more appropriate than the E-states SS[CuAB](i) for describing logKCuABB because they encode the changes in electronic and topological properties of the equivalent skeletal groups on the formation of [CuAB] complexes. Instead, the E-states SS[CuAB](i) encode the electronic and topological properties of the skeletal groups of [CuAB] complexes only. Indeed, the modeling of logKCuABB deals with reactions between two different chemical species to generate a third one, i. e., with processes of the type a + b ® c, where a, b, and c belong to different sets of chemically related species. Therefore, it could be expected that linear combinations of the E-states of equivalent skeletal groups of the species participating in the formation equilibria behave as better descriptors of logKCuABB than the SS[CuAB](i) indexes. In fact, if the set of formation equilibria [CuA(H2O)2]+ + Gly- = [CuA(Gly)] +, which consists of a number of reactions between different monochelated complexes and the same species (glycinate), is separately considered, both D(i) and SS[CuAB](i) yield very similar descriptions of logKCuABB, e.g., the sets {D(-O-), log(sf)}and {SS(-O-), log(sf) }, respectively:

logKCuABB = 1.1917(±0.0931)D(-O-) + 0.7811(±0.0877)log(sf) + 11.9748(±0.3978)  
r = 0.9967, s = 0.0212, F = 227.5, Durbin-Watson statistic = 2.967, n = 6  
logKCuABB = 0.5008(±0.0391)SS(-O-) + 0.7529(±0.0887)log(sf) + 1.9047(±0.3898)  
r = 0.9967, s = 0.0212, F = 227.7, Durbin-Watson statistic = 2.953, n = 6  

As can be noticed, the sets of statistical parameters for the above quoted regression equations are closely similar to each other. Thus, the E-state indices appear to be more appropriate for the modeling of physicochemical processes of the type a ® c, or a + (b) ®c, where (b) is a system or chemical species common to all of the considered processes. The above remarks are supported by the successful modeling provided by the E-states for some physical properties and physicochemical processes, namely, boiling points, 17O NMR chemical shift for ethers and carbonyl compounds, binding of compounds to a given receptor, binding of inhibitors to a particular enzyme, and so on.18, 19) Almost all these examples may be fitted to the above given reaction schemes. In future studies, the use of D(i) indices will be extended to other types of reactions. Nevertheless, due to the E-states index characteristics, it is advisable to be applied to all type of molecules25-27, for that reason it is recommendable to be considered like a reference index.


1. Both hydrophobic interactions and ligand basicities, as differential factors over the sequence of copper (II) complexes, would operate mainly through changes in the stability of the copper (II)-carboxylate bonds.
2. A new index D(i) is described that is more appropriate than E-states SS[CuAB](i) for describing log KCuABB because they encode the changes in electronic and topological properties of the equivalent skeletal groups on the formation of [CuAB] complexes. Instead, the E-states.

SS[CuAB](i) encode the electronic and topological properties of the skeletal groups of [CuAB] complexes only.



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