pubs.acs.org/JPCL
Electrophilicity Equalization Principle Pratim Kumar Chattaraj,* Santanab Giri, and Soma Duley Department of Chemistry and Center for Theoretical Studies, Indian Institute of Technology, Kharagpur 721302, India
ABSTRACT A new electronic structure principle, namely, the principle of electrophilicity equalization, is proposed. A qualitative rationale as well as numerical support for the same is provided. Equalization of electronegativity and hardness implies that of electrophilicity. Molecular electrophilicity may be expressed roughly as the geometric mean of the electrophilicities of the isolated atoms. SECTION
Molecular Structure, Quantum Chemistry, General Theory
I
n this Letter, we propose a new electronic structure principle, namely, the principle of electrophilicity equalization, and provide a qualitative rationale as well as a numerical support thereof. It states that during molecule formation, like electronegativity and hardness, electrophilicity gets equalized. Conceptual density functional theory1-3 provides quantitative definitions for popular qualitative chemical concepts and theoretical bases for associated electronic structure principles. Chemical potential (μ) (the Lagrange multiplier associated with the normalization constraint and the negative of the electronegativity4 (χ)) and hardness5 (η) are given by the following first- and second-order derivatives, respectively DE ð1Þ μ ¼ -χ ¼ DN υðrÞ and η ¼
D2 E DN2
and electron gain decrease in electronegativity.” Final molecular electronegativity (χGM) may be expressed approximately as the geometric mean of the electronegativities of the isolated atoms,14-16 that is P
χGM ð Π χk Þ1=P k ¼1
if the molecule contains P atoms (same and/or different), and {χk, k = 1, 2, .., P} denote their isolated atom electronegativities. It has been observed17 that the ratio of hardness and electronegativity is roughly a constant for atoms (at least for the atoms belonging to the same group in the periodic table18) and molecules. On the basis of this fact and eq 6, it may be shown19-21 that P
ηGM ð Π ηk Þ1=P k ¼1
ð2Þ υðrÞ
we have η I -A
ð4Þ
ð ωGM
Electrophilicity (ω) of the same system may be defined as7-10 ! ! μ2 χ2 ¼ ð5Þ ω ¼ 2η 2η
¼
P Y
χk Þ2=P
k ¼1 P Y
ηk Þ1=P k ¼1 1=P P P P Y 1 Y χk 2 1 Y ¼ ð2P ωk Þ1=P ¼ ð ωk Þ1=P 2 k ¼1 ηk 2 k ¼1 k ¼1 2ð
¼
Sanderson's electronegativity equalization principle states that,11-13 “When two or more atoms initially different in electronegativity combine chemically, their electronegativities become equalized in the molecule. The equalization of electronegativity occurs through the adjustment of the polarities of the bonds, which is pictured as resulting in a partial charge on each atom. That is, electron loss causes increase,
r 2010 American Chemical Society
ð7Þ
where {ηk, k = 1, 2, ..., P} refer to the corresponding isolated atom hardness values. Therefore, hardness also gets equalized like electronegativity. Equality between local and global hardness values reaffirms this fact, albeit making the definition of the former ambiguous.22-26 In this Letter, we show that the validity of the above two equalization principles tantamounts to that of the electrophilicity equalization principle. By expressing ω of a molecule (ωGM) as χ 2 ð8Þ ωGM ¼ GM 2ηGM
!
for an N electron system with energy E and external potential υ(r). Using a finite difference approximation, one may approximate them in terms of the ionization potential (I) and electron affinity (A) as1,6 IþA χ ¼ -μ ð3Þ 2 and
ð6Þ
ð9Þ Received Date: January 27, 2010 Accepted Date: March 3, 2010 Published on Web Date: March 08, 2010
1064
DOI: 10.1021/jz1001117 |J. Phys. Chem. Lett. 2010, 1, 1064–1067
pubs.acs.org/JPCL
Table 1. Energy (E, au), Ionization Potential (I, eV), Electron Affinity (A, eV), Electronegativity (χ, eV), Hardness (η, eV), and Electrophilicity (ω, eV) Values of Some Selected Diatomic Molecules, Calculated at the B3LYP/6-311þG** Level of Theory molecules
E
I
A
χ
χGM
η
ηGM
ω
ωGM
LiF LiCl LiBr NaF NaCl NaBr KF KCl KBr BeO MgO CaO BeS MgS CaS
-107.46822 -467.83380 -2581.75600 -262.22188 -622.60101 -2736.52690 -699.87089 -1060.24770 -3174.17350 -89.93401 -275.26018 -752.81566 -412.93000 -598.29821 -1075.83020
11.755 10.123 9.551 10.367 9.361 8.898 9.959 8.844 8.408 10.204 7.864 7.021 9.279 7.837 6.993
0.490 0.680 0.735 0.680 0.871 0.898 0.463 0.680 0.735 2.231 1.959 0.925 2.367 2.150 1.388
6.123 5.415 5.143 5.524 5.116 4.898 5.197 4.762 4.571 6.231 4.898 3.973 5.823 4.980 4.191
6.204 5.415 5.170 6.123 5.333 5.116 5.578 4.871 4.680 6.612 6.068 5.497 5.769 5.306 4.816
11.265 9.469 8.816 9.687 8.490 8.000 9.497 8.163 7.674 7.973 5.905 6.123 6.939 5.714 5.606
9.524 7.619 7.184 9.306 7.456 7.020 8.435 6.748 6.367 10.095 9.333 8.191 8.218 7.592 6.667
1.660 1.551 1.497 1.578 1.551 1.497 1.415 1.388 1.361 2.422 2.041 1.279 2.449 2.177 1.578
2.014 1.905 1.878 2.014 1.905 1.850 1.850 1.769 1.714 2.150 1.986 1.850 2.014 1.850 1.742
Table 2. Electronegativity (χ, eV), Hardness (η, eV), and Electrophilicity (ω, eV) Values of Some Selected Molecules Using Experimental Ionization Potential (I, eV) and Electron Affinity (A, eV) Values I
A
χ
I2 BrI S2 Br2 Cl2 P2 SO CH O2 OH NH F2
9.400 9.790 9.400 10.560 11.480 9.600 10.000 10.640 12.060 13.180 13.100 15.700
2.420 2.550 1.660 2.600 2.400 0.650 1.130 1.240 0.440 1.830 0.380 3.080
5.910 6.170 5.530 6.580 6.940 5.125 5.565 5.940 6.250 7.505 6.740 9.390
CS2 COS SO2 O3 NH2 N2 O
10.080 11.180 12.340 12.670 12.800 12.890
1.000 0.460 1.050 1.820 0.780 1.470
5.540 5.820 6.695 7.245 6.790 7.180
PBr3 PCl3 POCl3 CH3I SO3 CF3I C2H2 CF3Br CH3 HNO3 SF6 C6H5NO2 C6H4O2
9.850 9.910 11.400 9.540 11.000 10.230 11.410 11.820 9.840 11.030 15.350 9.860 9.670
1.600 0.800 1.400 0.200 1.700 1.400 0.430 0.910 0.080 0.570 0.750 0.700 1.890
5.725 5.355 6.400 4.870 6.350 5.815 5.920 6.365 4.960 5.800 8.050 5.280 5.780
molecules
χGM
η
ηGM
ω
ωGM
6.980 7.240 7.740 7.960 9.080 8.950 8.870 9.400 11.620 11.350 12.720 12.620
7.393 7.916 8.280 8.476 9.395 9.738 10.031 11.325 12.152 12.492 13.664 14.021
2.502 2.629 1.976 2.720 2.652 1.467 1.746 1.877 1.681 2.481 1.786 3.493
3.088 3.245 2.334 3.409 3.677 1.619 2.336 1.984 2.338 2.165 1.909 3.864
9.080 10.720 11.290 10.850 12.020 11.420
8.814 10.017 10.693 12.152 13.384 13.696
1.690 1.580 1.985 2.419 1.918 2.257
2.203 2.204 2.337 2.338 1.940 1.977
8.250 9.110 10.000 9.340 9.300 8.830 10.980 10.910 9.760 10.460 14.600 9.160 7.780
8.775 9.480 9.962 10.935 11.041 11.527 11.325 11.847 12.059 12.736 13.005 11.542 11.221
1.986 1.574 2.048 1.270 2.168 1.915 1.596 1.857 1.260 1.608 2.219 1.522 2.147
2.830 2.995 2.851 2.176 2.337 3.227 1.984 3.291 1.994 2.156 3.596 2.017 2.035
Diatomic Molecules 6.758 7.167 6.217 7.602 8.313 5.615 6.846 6.703 7.538 7.354 7.222 10.410 Triatomic Molecules 6.232 6.645 7.069 7.538 7.206 7.358 Polyatomic Molecules
r 2010 American Chemical Society
7.047 7.536 7.536 6.899 7.184 8.625 6.703 8.831 6.935 7.410 9.671 6.823 6.758
1065
DOI: 10.1021/jz1001117 |J. Phys. Chem. Lett. 2010, 1, 1064–1067
pubs.acs.org/JPCL
Figure 1. Plots of χ, η, and ω values with their respective geometric mean values for the molecules presented in Tables 1 and 2.
{ωk = (χk2/2ηk), k = 1, 2, ..., P} being the electrophilicities of the isolated atoms. Therefore, the electrophilicity gets equalized during molecule formation, and the final equalized electrophilicity may be expressed as the geometric mean of the corresponding isolated atom values. When an electrophile interacts with a nucleophile, the electrophilicity of the former is reduced (via electronic charge transfer and/or other related processes, from the nucleophile to the electrophile), and that of the latter is increased until they are equalized to a value somewhere in between the two (roughly the geometric mean). An important outcome of this result is that the local electrophilicity27-29 may alternatively be considered to be constant everywhere and is equal to its global variant. Numerical calculations are performed at the B3LYP/ 6-311þG** level of theory to calculate the energies of some selected atoms and their cations/anions to obtain the I and A values of the atoms using a ΔSCF technique. Geometries of the corresponding diatomic molecules are optimized at the
r 2010 American Chemical Society
same level of theory using a Gaussian suite of program,30 and the energies of the related cations/anions are obtained through single-point calculations using the geometries of the associated neutral molecules. Table 1 presents the energy (E), ionization potential (I), electron affinity (A), electronegativity (χ), hardness (η), electrophilicity (ω), and the last three quantities approximated as the respective geometric means [λGM = (λA 3 λB)1/2, λ = χ, η, ω]. Table 2 reports these quantities for some selected molecules, calculated from the experimental I and A values.17 The geometric mean principle for ω is obeyed very well wherever the corresponding principles for χ and η are obeyed properly. In Figure 1 χ, η, and ω values of various molecules as reported in Tables 1 and 2 are compared with the respective geometric mean values. It is clear that all three geometric mean principles are qualitative in nature. The situation improves (see the inset) in the case where one considers the molecules bearing some common features, say all halocompounds.
1066
DOI: 10.1021/jz1001117 |J. Phys. Chem. Lett. 2010, 1, 1064–1067
pubs.acs.org/JPCL
It is important to note that the proof of the Sanderson's electronegativity equalization principle through the geometric mean recipe stems from an approximate E versus N relationship,1 which is reflected in the corresponding numerical values. Moreover, I is much larger than A, and accordingly, hardness becomes roughly proportional to electronegativity, which, in turn, provides another qualitative structure principle, namely, the hardness equalization principle. Obviously, one cannot expect that hardness (and hence polarizability) is constant all throughout the molecule. However, a specific valid choice of local hardness is that it is equal to the global hardness22-26 and hence is constant everywhere in a molecule, which behaves like a homogeneous electron gas. Any local (condensed) quantity (e.g., density partitioning to give rise to various population analysis schemes) suffers from this drawback. One does not require any further assumption to show the validity of a geometric mean principle for electrophilicity. Thus, one equalization principle more or less implies two more such principles all being qualitative in nature. In conclusion, the electrophilicity gets equalized during molecule formation, like the electronegativity and the hardness. The final electrophilicity is roughly given by the geometric mean of the corresponding isolated atom values. Like local hardness, local electrophilicity may be taken as the same as the global electrophilicity.
(10)
(11) (12) (13)
(14)
(15)
(16) (17)
(18)
(19)
AUTHOR INFORMATION
(20)
Corresponding Author: *To whom correspondence should be addressed. E-mail:
[email protected].
(21)
ACKNOWLEDGMENT We thank CSIR, New Delhi, for financial
(22)
assistance and the Editor and the Reviewers for constructive criticism.
(23)
REFERENCES
(24)
(1) (2)
(3)
(4)
(5)
(6) (7) (8) (9)
Parr, R. G.; Yang, W. Density Functional Theory of Atoms and Molecules; Oxford University Press: Oxford, U.K. 1989 Geerlings, P.; De Proft, F.; Langenaeker, W. Conceptual Density Functional Theory. Chem. Rev. 2003, 103, 1793– 1874. Chattaraj, P. K. Chemical Reactivity Theory: A Density Functional View; Taylor & Francis/CRC Press: Boca Raton, FL, 2009. Parr, R. G.; Donnelly, R. A.; Levy, M.; Palke, W. E. Electronegativity: The Density Functional Viewpoint. J. Chem. Phys. 1978, 68, 3801–3807. Parr, R. G; Pearson, R. G. Absolute Hardness: Companion Parameter to Absolute Electronegativity. J. Am. Chem. Soc. 1983, 105, 7512–7522. Pearson, R. G. Chemical Hardness: Application from Molecules to Solids; Wiley-VCH: Weinheim, Germany, 1997. Parr, R. G.; Szentp aly, L. v.; Liu, S. Electrophilicity Index. J. Am. Chem. Soc. 1999, 121, 1922–1924. Chattaraj, P. K.; Sarkar, U.; Roy, D. R. Electrophilicity Index. Chem. Rev. 2006, 106, 2065–2091. Chattaraj, P. K.; Roy, D. R. Update 1 of: Electrophilicity Index. Chem. Rev. 2007, 107, PR46–PR74.
r 2010 American Chemical Society
(25)
(26)
(27)
(28)
(29)
(30)
1067
Chattaraj, P. K.; Giri, S. Electrophilicity Index within a Conceptual DFT Framework. Annu. Rep. Prog. Chem., Sect. C 2009, 105, 13–39. Sanderson, R. T. An Interpretation of Bond Lengths and a Classification of Bonds. Science 1951, 114, 670–672. Sanderson, R. T. Partial Charges on Atoms in Organic Compounds. Science 1955, 121, 207–208. Donnelly, R. A; Parr, R. G. Elementary Properties of an Energy Functional of the First-Order Reduced Density Matrix. J. Chem. Phys. 1978, 69, 4431–4439. Politzer, P.; Weinstein, H. Molecular Electronegativity in Density Functional Theory. J. Chem. Phys. 1979, 71, 4218– 4220. Parr, R. G.; Bartolotti, L. On the Geometric Mean Principle for Electronegativity Equalization. J. Am. Chem. Soc. 1982, 104, 3801–3803. Nalewajski, R. A Study of Electronegativity Equalization. J. Phys. Chem. 1985, 89, 2831–2837. Yang, W.; Lee, C.; Ghosh, S. K. Molecular Softness as the Average of Atomic Softnesses: Companion Principle to the Geometric Mean Principle for Electronegativity Equalization. J. Phys. Chem. 1985, 89, 5412–5414. Chattaraj, P. K. Atomic and Molecular Properties from the Density-Functional Definition of Electronegativity. Curr. Sci. 1991, 61, 391–395. Datta, D. Geometric Mean Principle for Hardness Equalization: A Corollary of Sanderson's Geometric Mean Principle of Electronegativity Equalization. J. Phys. Chem. 1986, 90, 4216–4217. Chattaraj, P. K.; Nandi, P. K.; Sannigrahi, A. B. Improved Hardness Parameters for Molecules. Proc. Indian Acad. Sci., Chem. Sci. 1991, 103, 583–589. Chattaraj, P. K.; Ayers, P. W.; Melin, J. Further Links between the Maximum Hardness Principle and the Hard/Soft Acid/ Base Principle: Insights from Hard/Soft Exchange reactions. Phys. Chem. Chem. Phys. 2007, 9, 3853–3856. Ghosh, S. K. Energy Derivatives in Density Functional Theory. Chem. Phys. Lett. 1990, 172, 77–82. Berkowitz, M.; Ghosh, S. K.; Parr, R. G. On the Concept of Local Hardness in Chemistry. J. Am. Chem. Soc. 1985, 107, 6811–6814. Harbola, M. K.; Chattaraj, P. K.; Parr, R. G. Aspects of the Softness and Hardness Concepts of Density Functional Theory. Isr. J. Chem. 1991, 31, 395–402. Chattaraj, P. K.; Roy, D. R.; Geerlings, P.; Torrent-Sucarrat, M. Local Hardness: A Critical Account. Theor. Chem. Acc. 2007, 118, 923–930. Ayers, P. W.; Parr, R. G. Local Hardness Equalization: Exploiting the Ambiguity. J. Chem. Phys. 2008, 128, 184180(1)– 184180(8). P erez, P.; Toro-Labb e, A.; Aizman, A.; Contreras, R. Comparison between Experimental and Theoretical Scales of Electrophilicity in Benzhydryl Cations. J. Org. Chem. 2002, 67, 4747– 4752. Chamorro, E.; Chattaraj, P. K.; Fuentealba, P. Variation of the Electrophilicity Index along the Reaction Path. J. Phys. Chem. A 2003, 107, 7068–7072. Chattaraj, P. K.; Maiti, B.; Sarkar, U. Philicity: A Unified Treatment of Chemical Reactivity and Selectivity. J. Phys. Chem. A 2003, 107, 4973–4975. Gaussian 03W, revision B.03; Gaussian, Inc.: Pittsburgh, PA, 2003.
DOI: 10.1021/jz1001117 |J. Phys. Chem. Lett. 2010, 1, 1064–1067
