ARTICLE pubs.acs.org/JPCA
Ruling Out Any Electrophilicity Equalization Principle Laszlo von Szentpaly* Institut f€ur Theoretische Chemie, Universit€at Stuttgart, Pfafffenwaldring 55, D 70569 Stuttgart, Germany ABSTRACT: Two gas-phase electrophilicity indices, ω1 and ω2, introduced by Parr, von Szentpaly, and Liu are tested with respect to the recently proposed “principle of electrophilicity equalization.” Although electronegativity is equalized in many cases, there is no functioning “hardness equalization principle” nor are the electrophilicity indices principally equalized during molecule formation: they cannot be generally expressed as the mean of the corresponding atomic indices. For large metal clusters and [n]fullerenes, both electrophilicity indices increase proportional to n1/3 and n1/2, respectively, as the hardness values converge to zero. Two “principles” are shown to be obsolete: the “geometric mean principle for hardness equalization” and the “principle of electrophilicity equalization”, with the latter somewhat relying on the former. An appeal is made to exercise careful judgment before proposing and publishing new structural principles.
1. INTRODUCTION Postulating and establishing a new structural principle should (i) generate an important progress in basic understanding and (ii) help to achieve some of the much needed data reduction in chemistry. This is implicitly the claim of the recent communication by Chattaraj, Giri, and Duley,1 who propose a global “principle of electrophilicity equalization” based on a qualitative rationale and rather limited empirical evidence. An electrophile is a chemical species able to accept additional electrons by reacting with an electron-rich species. Parr, von Szentpaly, and Liu modeled the “electrophilic power” as the energy gained from saturating the species with the maximal additional negative charge, ΔNmax, which the electrophile can accept in the gas phase.2,3 Several electrophilicity indices, ωL, sharing the form ωL ¼ ðχL Þ2 =2ηL
ð1Þ
have resulted form different models, L.24 Here, χL is the electronegativity and ηL the chemical hardness of the species according to the particular model L. The new proposition by Chattaraj et al. is that “the electrophilicity gets equalized during molecule formation, and the final equalized electrophilicity may be expressed as the geometric mean of the corresponding isolated atomic values”.1 Let the molecule contain P atoms of any kind, k = {1, 2,...P} with their electrophilicity indices ωL,k. According to Chattaraj et al.,1 the electrophilicity index of the molecule is expressed by their geometric mean !1=P P Y ωL;k ð2Þ ωL ≈ωL;GM ¼ k¼1
The present contribution investigates the proposed geometric mean protocol, and further discusses the “electrophilicity equalization principle” in general. It will be shown that there is no reason for suggesting a principle of electrophilicity equalization by arithmetic, geometric, or harmonic averaging of atomic values. r 2011 American Chemical Society
On the contrary, it will become conceptually and empirically evident that electrophilicity indices in general include a term opposing their equalization.
2. METHODS In support of eq 2, Chattaraj et al. argue that both molecular electronegativity and hardness may be calculated as the geometric mean of their atomic values, χk and ηk, respectively.1,58 However, of the three models considered for electrophilicity,2,3 only the ground-state exponential (GSE) model6 expresses molecular electronegativity as the geometric mean of the electronegativities of the isolated atoms. According to Parr and Bartolotti, a geometric mean law of electronegativity equalization requires an exponential dependence of the total energy, E(N), on the number of electrons N.6 The geometric mean equalization of electronegativity, viz, χAB(NAB) = [χA(NA)χB(NB)]1/2, holds for the combined species AB, if the charge-dependent electronegativities, χA(N) and χB(N), are exponential with the constant γ being the same for the species A and B, e. g. χA ðNÞ ¼ χA ðN A Þexp½γðN N A Þ
ð3Þ 6
where NA is the number of electrons in the neutral species A. In the context of electrophilicity indices, the GSE model is discussed in detail and compared to the ground-state parabola (GSP) and valence-state parabola (VSP) models by Szentpaly.3 Using the first ionization energy, I, and the first electron affinity, A, the GSE electronegativity of neutral species is χ0GSE = [IA/(I A)] ln(I/A) and its hardness becomes η0GSE = χ0GSE ln (I/A).3 The relevant point is that the electrophilicity index, ω2 = (χ0GSE)2/ 2η0GSE, resulting from this model yields3 ω2 ¼ IA=ðI AÞ
ð4Þ
Received: April 9, 2011 Revised: June 21, 2011 Published: June 23, 2011 8528
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Table 1. Comparison of Electrophilicity Indices, ω1 and ω2, with Geometric Mean Values, ω1,GM and ω2,GMa A
species
ω1
I
F
3.4012
17.423
3.865
Si
1.3895
8.152
1.680
SiF6
6.81 v
14.59 v
7.36
Pt
2.128
8.9587
2.249
PtF6
6.78 v
15.28
7.16
Mo
0.748
7.0924
1.211
MoF6
5.77
14.7 ad
5.87
Au Au20
2.3086 2.75 ad
9.2257 7.05
2.4042 2.79
Cs
0.4716
3.8939
0.6961
ω1,GM
ω2
ω2,GM
4.2262 1.669 3.431
12.77
3.700
2.791 3.577
12.19
3.982
0.836 3.275
9.50
3.353
2.4042
3.0791 4.51
3.0791
0.5366
Cs11
1.102
2.993
1.109
0.6961
1.744
0.5366
Csn
1.93 1.75 n1/3
1.93 + 2.38 n1/3
eq 8
0.6961
eq 9
0.5366
(n > 11)
ref 11
ref 11
C
1.2621
11.263
1.9604
C84
3.14(6)
7.17(2)
3.30(9)
1.9604
5.59(20)
1.421
[n]fullerene (n > 80)
5.13 18.3 n1/2 ref 13c
5.13 + 18.3 n1/2 ref 13c
0.36(2) n1/2
1.9604
0.72(3) n1/2 9.15 n1/2
1.421
1.421
A and I are the first electron affinity and ionization energy, respectively. Values A and I taken from ref 9. All values in eV; “v” refers to vertical, “ad” to adiabatic data.
a
a form rather different from 1 ω1 ¼ ðχGSP Þ2 =2ηGSP ¼ ðI þ AÞ2 =2ðI AÞ 4
ð5Þ
Note that the index ω2 is also the VSP electrophilicity index, ω2 = 2ω1 1/4(I A) = IA/(I A), introduced by Parr et al.2 and discussed in refs 3 and 9. There is a very important difference between the GSE and VSP models, however. The GSE model erroneously predicts all multiply charged negative ions, Xn, to be stable up to n f ∞. On the other hand, the VSP model is highly successful in rationalizing and predicting the maximal uptake of electrons, ΔNmax, and has generated a universal method to calculate (i) the second electron affinity, A2(X), of atoms, molecules, and clusters and (ii) the electronegativity and hardness of gas-phase dianions of arbitrary size.9 According to eq 4, the VSP and GSE models emphasize the geometric mean even in their electrophilicity index, ω2, where the squared geometric mean, IA, replaces the GSP model’s squared arithmetic mean, 1/4(I + A)2. So far, only the GSP electrophilicity index, ω1, was considered for postulating an electrophilicity equalization principle by geometric averaging.1 For assessing the general likelihood of a principle of electrophilicity equalization, it seems in order to (i) consider both indices ω1 and ω2 on equal footing and (ii) extend the set of the investigated chemical species to strong electrophiles, and larger systems. Note that the linear correlations between ω1 and ω1,GM reported by Chattaraj et al.1 for diatomic and other small molecules are of very weak significance, with the correlation coefficient, r2 = 0.430, only.
3. RESULTS AND DISCUSSION The electrophilicity indices of a broad variety of species calculated from established first electron affinity and ionization energy data9 are given in Table 1. We present two sets of evidence against calculating molecular electrophilicity from atomic indices. One discusses specific electrophiles; the other
is a principal observation valid for large clusters in general. Thus: (i) According to the geometric mean postulate in eq 2, the electrophilicity index of a molecule has to be smaller than that of its atom with the highest index, that is, ωL,GM e max(ωL,k). However, strongly electrophilic hexafluoride complexes, for example, SiF6, PtF6, and MoF6, show that the ω1 and ω2 values are significantly higher than the corresponding value of atomic fluorine, ωL(F), and the calculated ωL,GM results (Table1). In the case of SiF6, the ratio ω2/ω2,GM = 3.45 is very high. Given the interest in finding and quantitatively assessing strong electrophiles, it is also gratifying to notice that there is no upper limit to ωL presented by the highest atomic electrophilicity index, max(ωL,k). Although the ω2 values are spread over a wider range than ω1, neither index type supports the geometric equalization principle as formulated in eq 2. The same conclusion is reached for homonuclear gold, cesium, and carbon clusters of intermediate and large sizes, up to solids (Table 1). These specific observations lead to a principal argument ruling out all “electrophilicity equalization principle”. (ii) For large metallic clusters and [n]fullerenes, the electrophilicity indices increase proportional to n1/3 and n1/2, respectively, while the hardness values, ηL, and thus (I A) converge to zero. For the infinitely large variety of such compounds, it cannot be true that ωL = (χL)2/2ηL is the average of atomic electrophilicity indices, because the hardness ηL cannot be the mean of atomic increments. The evidence is demonstrated in detail for two sets of clusters, viz., Csn and Cn. The experimental ionization energies and electron affinities of extended metal clusters, Mn, are proportional to n1/3, representing the reciprocal radius of the cluster.1012 Thus, recent work by Assadollahzadeh et al.11 established simple linear dependencies of both the vertical ionization potential, Iv, and the electron affinity, Av, on the inverse cluster size of cesium 8529
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clusters Csn. In eV units, the evolution of these properties is given for large n I v ðCsn Þ=eV ¼ 1:93 þ 2:38n1=3
ð6Þ
Av ðCsn Þ=eV ¼ 1:93 1:75n1=3
ð7Þ
Equations 6 and 7 have been found by constrained linear regressions, where the intercept 1.93 eV at n1/3 = 0 denotes the experimental work function of bulk cesium metal. Slightly different linear regressions were obtained without this physical constraint.11 Inserting eqs 6 and 7 into eq 5 leads to ω1(Csn) given in eq 8, while their insertion into eq 4 yields ω2(Csn) in eq 9 1 ω1 ðCsn Þ ¼ ð3:86 þ 0:63n1=3 Þ2 =4:13n1=3 8
ð8Þ
ω2 ðCsn Þ ¼ ð1:932 þ 1:216 n1=3 4:165 n2=3 Þ=4:13n1=3
ð9Þ
Because the denominators converge to zero for n f ∞, both ω indices increase to infinity as the cluster size grows to form bulk cesium metal. Equations similar to those of Assadollahzadeh et al.11 have been published for many other metallic elements.10,12The dependence of (I A) as a function of n1/3 has been discussed by several groups.12 Indeed, relationships of the type of eqs 69 are generally valid for metallic elements. It has been pointed out right upon introducing the electrophilicity indices ωL that very soft systems, where (I A) f 0, could accommodate large numbers of electrons and are considered very strong electrophiles.3 Therefore, metals have been attributed an infinite electrophilicity index.3 It is not possible to obtain the electrophilicity indices of large metallic clusters, metals, and alloys, by averaging their atomic indices ωL. The set of [n]fullerenes, Cn, allows some further insight as to why a principle of electrophilicity equalization must be ruled out. The I and A data for large fullerenes, say, n > 80, are well modeled by charged hollow spheres.13 Thus, both I and A are proportional to n1/2 and the slopes are symmetrical13c IðCn Þ=eV ¼ 5:13 þ 18:3n1=2
ð10Þ
AðCn Þ=eV ¼ 5:13 18:3n1=2
ð11Þ
The Mulliken electronegativity becomes nearly independent of size and approximates its value for graphene, as the infinite reference system, χ = 5.13 eV.13c Similar to the metals in the above example, the difference I A, thus, the hardness η converges to zero for n f ∞. Here, however, the decrease of η is proportional to n1/2, not n1/3. For large values of n, the leading terms in both ω1 and ω2 are proportional to n1/2, and the electrophilicity indices grow beyond all limits (Table 1) ω1 ðCn Þ=eV ¼ 0:36ð2Þn1=2
ð12Þ
ω2 ðCn Þ=eV ¼ 0:72ð3Þn1=2 9:15ð5Þn1=2
ð13Þ
For convenience, the inverse indices ωL1 are plotted versus n1/2 in Figure 1. The [n]fullerenes thus form an infinite set of compounds for which Sanderson’s principle of electronegativity equalization5 is approximately valid, whereas any kind of hardness equalization is bound to fail. The reason is that Sanderson’s principle5 is based
Figure 1. Increase of fullerene electrophilicity indices with cluster size. The inverse of electrophilicity indices of large [n]fullerenes, ωL1, plotted against n1/2, the inverse root of the number of carbon atoms, following eqs 12 and 13.
on general observations, namely, that (i) χ(X+) > χ(X) > χ(X) and (ii) electronic charge is transferred upon molecule formation, while the sole rationale presented for hardness equalization is that, for atoms, we have A , I, and thus χ ≈ η.7 As shown above in eqs 611 and Table 1, this assumption does not hold anymore for larger molecules and clusters. Even for atoms, Nazewajski’s numerical test of the constancy of the ratio χ/η shows standard deviations of up to 40% from the constancy of this ratio.8a It might be, however, that the operational formula determining the chemical hardness as ∂2E/∂N2 ∼ I A14 is less adequate and general than its companion formula for Mulliken’s electronegativity, ∂E/∂N ∼ χ0 = 1/2(I + A), is for neutral species.2,3,1517In any event, the list of chemical species, X, for which ωL ðXÞ > maxðωL;k Þ g ωL;GM ¼
P
Π ωL;k
k¼1
1=P ð14Þ
can be further extended and contains an infinite number of large clusters. In all cases, where ωL(X) > max(ωL,k), the electrophilicity indices cannot be averaged atomic indices. This is a true statement for all mean formation protocols, for example, arithmetic, geometric, and harmonic. It is concluded that the current electrophilicity indices, ω1 and ω2, are not generally equalized during molecule formation. They cannot be calculated from atomic increments, primarily because there is no support for any “hardness equalization principle”, even if the electronegativity is equalized in the process. There is absolutely no reason for assuming a principle of electrophilicity equalization. Thus, the suggested electrophilicity equalization principle is hardly worth the effort of any further investigation. Because of the current “popular”18 use of the term “principles”, their character oscillates between those having a truly basic, quasi axiomatic value, for which no exceptions are known, such as the variation principle, Heisenberg’s uncertainty principle, or Pauli’s exclusion principle, through highly useful qualitative guiding rules, such as Sanderson’s principle,5 or the HardSoft AcidsBases principle,14,19 down to hastily tossed ideas, prematurely labeled “principles”, such as the “geometric mean principle for hardness equalization”7 or the present “electrophilicity equalization principle”.1 However, a very careful investigation is 8530
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The Journal of Physical Chemistry A necessary, before a new principle can be proposed. Thus, the attempted rigorous and general proof20 of the “principle of maximum hardness”14b,19,20 has been found to be erroneous,21 and larger-scale empirical validation tests have shown “that this principle may not be obeyed in most cases”.22,23 In the author’s opinion, it is unfortunate and confusing to use the same term “principle” for statements of very different stringency and significance. For the sake of scientific methods and progress, it would be helpful to keep the terminology and semantics clean, and avoid a progressive dilution of the meaning of principles. Research work comparing the efficiencies of the electrophilicity indices, ω1 and ω2, in rationalizing and predicting the rates of electrophilic reactions is underway.
’ AUTHOR INFORMATION Corresponding Author
*E-mail:
[email protected].
’ ACKNOWLEDGMENT I am grateful to Prof. Hans-Joachim Werner for continued hospitality at the Institut f€ur Theoretische Chemie der Universit€at Stuttgart and to Profs. Michael C. B€ohm and Huw O. Pritchard for discussions.
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Sidorov, L. N.; Seifert, G.; Vietze, K. J. Am. Chem. Soc. 2000, 122, 9745–9749. (14) (a) Parr, R. G.; Pearson, R. G. J. Am. Chem. Soc. 1983, 105, 7512–7516. (b) Pearson, R. G. Chemical Hardness: An Application from Molecules to Solids; Wiley-VCH: Weinheim, 1997. (15) Pritchard, H. O.; Sumner, F. H. Proc. R. Soc. London, Ser. A 1956, 235, 136–143. (16) Bergmann, D.; Hinze, J. Angew. Chem., Int. Ed. Engl. 1996, 35, 150–163 with earlier references quoted therein. (17) von Szentpaly, L. J. Mol. Struct.: THEOCHEM 1991, 233, 71–81. (18) Chattaraj, P. K.; Sengupta, S. J. Phys. Chem. 1996, 100, 16126–16130. (19) (a) Pearson, R. G. J. Chem. Educ. 1987, 64, 561–567. (b) Pearson, R. G. Acc. Chem. Res. 1993, 26, 250–255. (20) Parr, R. G.; Chattaraj, P. K. J. Am. Chem. Soc. 1991, 113, 1854–1855. (21) (a) Sebastian, K. L. Chem. Phys. Lett. 1994, 231, 40–42. (b) Chattaraj, P. K.; Liu, G. H.; Parr, R. G. Chem. Phys. Lett. 1995, 237, 171–176. (22) Zhang, Y. L.; Yang, Z. Z. Int. J. Quantum Chem. 2006, 106, 1723–1735. (23) Noorizadeh, S. Chin. J. Chem. 2007, 25, 1439–1444.
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