On the Validity of the Maximum Hardness Principle and the Minimum

Article pubs.acs.org/JPCA

On the Validity of the Maximum Hardness Principle and the Minimum Electrophilicity Principle during Chemical Reactions Sudip Pan,† Miquel Solà,‡ and Pratim K. Chattaraj*,† †

Department of Chemistry and Center for Theoretical Studies, Indian Institute of Technology, Kharagpur 721302, India Institut de Química Computacional and Department de Química, Universitat de Girona, Campus Montilivi, 17071 Girona, Catalonia, Spain

‡

S Supporting Information *

ABSTRACT: Hardness and electrophilicity values for several molecules involved in different chemical reactions are calculated at various levels of theory and by using different basis sets. Effects of these aspects as well as different approximations to the calculation of those values vis-à-vis the validity of the maximum hardness and minimum electrophilicity principles are analyzed in the cases of some representative reactions. Among 101 studied exothermic reactions, 61.4% and 69.3% of the reactions are found to obey the maximum hardness and minimum electrophilicity principles, respectively, when hardness of products and reactants is expressed in terms of their geometric means. However, when we use arithmetic mean, the percentage reduces to some extent. When we express the hardness in terms of scaled hardness, the percentage obeying maximum hardness principle improves. We have observed that maximum hardness principle is more likely to fail in the cases of very hard species like F−, H2, CH4, N2, and OH appearing in the reactant side and in most cases of the association reactions. Most of the association reactions obey the minimum electrophilicity principle nicely. The best results (69.3%) for the maximum hardness and minimum electrophilicity principles reject the 50% null hypothesis at the 2% level of significance.

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INTRODUCTION In explaining different characteristics of atoms and molecules associated with their chemical bonding and reactivity, the applicability of several popular qualitative chemical concepts, like electronegativity1−4 (χ), hardness5−8 (η), electrophilicity9−12 (ω), etc., is very well appreciated in the literature. The concept of electronegativity as “the power of an atom in a molecule to attract electrons to itself” was first introduced by Pauling,1 whereas Pearson5 put forward the concept of hardness in the context of his renowned hard−soft acids and bases (HSAB) principle;5,8,13,14 “hard acids prefer to coordinate with hard bases and soft acids prefer to coordinate with soft bases for both their thermodynamic and kinetic properties”, which was further refined15 later showing the validity of this principle for the cases of acids and bases having similar strength. Another electronic structure principle related with hardness called maximum hardness principle was proposed by Pearson16 as “there seems to be a rule of nature that molecules arrange themselves so as to be as hard as possible”. A formal statistical mechanical proof of maximum hardness principle (MHP) was also provided by Parr et al.17 under two certain conditions viz., (I) chemical potential (μ) does not change upon deformation of the molecular structure, and (II) external potential (v(r)) also remains unchanged. Although these two terms, μ and v(r), do not appear as constant for none of the chemical processes, still the applicability of MHP18−22 is shown to be valid in many cases. Hence, it seems that these two conditions can be relaxed © 2013 American Chemical Society

to a certain extent. For those chemical reactions in which MHP does not obey, the variation of these two terms might be considered to be responsible. The validity of MHP has been numerically checked for several chemical problems and found to be valid in the cases of molecular vibrations,23−26 internal rotations,27−34 chemical reactions,35,36 isomer stability,37 atomic shell structure,38,39 Woodward−Hoffmann rules,40,41 aromaticity,42−44 electronic excitations,45 time-dependent situations,46 stability of magic clusters,47 chaotic ionizations,48 and several other categories of chemical processes.49−60 Because of the near constancy of chemical potential and external potential during nuclear displacements for most of the nontotally symmetric vibrations, MHP holds quite well although a certain class of nontotally symmetric vibrations61−66 is also reported to violate it. There are also some chemical processes67−73 available in the literature, which do not obey this principle. In addition, a proposed corollary50 to MHP shows a minimum hardness value at the transition state (TS). It is also well-proven that the condition for the occurrence of a minimum value at the TS in the hardness profile is exclusively oriented from the symmetry considerations.74,75 Another electronic structure principle called minimum electrophilicity principle (MEP)73,57,76−83 originates as an extension of MHP. The extrema in ω during chemical Received: December 26, 2012 Revised: February 1, 2013 Published: February 1, 2013 1843

dx.doi.org/10.1021/jp312750n | J. Phys. Chem. A 2013, 117, 1843−1852

The Journal of Physical Chemistry A

Article

criticize or establish these electronic structure principles studying a given set of reactions at a particular level of theory following some particular approximations, but it is also a difficult task for someone to perform a detailed study of a larger set of reactions at various levels of theory using different approximations. However, if someone chooses a particular level and approximation as superior to others, that will definitely accelerate debate in this aspect. Let us allow to perform the larger set of the reactions using Gaussian09 program package91 taking popular B3LYP functional with reasonably large 6-311+ +G(3df,3pd) split valence basis set and with the approximation of ΔSCF technique to calculate I and A and use of geometric and arithmetic means to obtain an average value of η and ω corresponding to reactants and products for the reactions involving more than one reactant and product and see the applicability of MHP and MEP in this context. In the second and third parts, the assessment toward validity of MHP and MEP has been done, respectively. The fourth part contains a large sample test for proportion to check the rejection of 50% null hypothesis.

reactions, molecular vibrations, and internal rotations will occur at points76,77 where 2 μ ⎛ ∂μ ⎞ 1 ⎛ μ ⎞ ⎛ ∂η ⎞ ∂ω = ⎜ ⎟− ⎜ ⎟⎜ ⎟ 2 ⎝ η ⎠ ⎝ ∂y ⎠ ∂y η ⎝ ∂y ⎠

(1)

where y is a reaction coordinate (chemical reaction), bond length (stretching), bond angle (bending), or dihedral angle (internal rotation). Now owing to the convexity of the energy, μ is always less than zero and η is always greater than zero; therefore, ω will be an extremum when the slopes of the changes in μ and ω are opposite in signs, which, in turn, may be stated as ω will be a minimum (maximum) when both μ and η are maxima (minima) for a given y. Minimum polarizability principle (MPP)35,84,85 is another electronic structure principle, which may be stated as “the natural direction of evolution of any system is towards a state of minimum polarizability”. Recently, an assessment regarding the validity of MHP has been made at the BLYP/def2-QZVP level of theory with the aid of the NWChem 6.0 program86 by Poater et al.87 taking 34 exothermic chemical reactions, the so-called BH76 set.88 The result is quite a discouraging one for the application of MHP to analyze chemical reactivity. Only 46% of reactions have greater hardness for the products than that of the reactants obeying MHP, and 53% of reactions have greater hardness for the reactants than the transition states. However, they87 have used Koopmans’ theorem89 to evaluate ionization potential (I) and electron affinity (A). It is also mentioned there87 that it will be unfair to these qualitative principles to draw any general conclusion on the basis of the outcome of this small set of reactions rather one should carry out a thorough analysis of a larger set of reactions. In this present study, we have tried to check the validity of MHP taking 101 exothermic reactions and six sets of isomeric molecules to get a general conclusion but we have limited our analysis only for reactants and products excluding the transition states (TSs) since the extension to TS does not come straightway from MHP. We have made an attempt to find out the domains where the MHP will hold fairly and also the reactions where it is likely to fail. The efficacy of MHP in predicting the favorable direction of the chemical reactions, when hardness is expressed in terms of scaled hardness,55 has also been analyzed. Ghosh et al.55 have shown that there occurs a drastic variation of chemical potential in intramolecular proton transfer of water dimer radical cation. Since MHP is valid at constant chemical potential, they55 have defined the scaled hardness as (η/|μ|1/3). In addition, we have also made a thorough study for the assessment of validity of the MEP. Here, it should be mentioned that since these electronic structure principles are qualitative in nature, one should not expect those to be valid in all cases rather should be happy if they provide useful information about the favorable direction of many chemical processes. It may also be noted that Koopmans’ theorem89 is strictly valid within the Hartree−Fock theory. However, in Kohn−Sham computations, one may apply this with the help of Janak’s theorem,90 but still most of the theoreticians prefer ΔSCF technique to calculate vertical I and A. Here, we have chosen this technique to evaluate I and A rather than Koopmans’ theorem89 used in the previous study.87 In the first part, we have shown the effect of the level of theory, basis sets, and approximations used to calculate I and A on the validity of the MHP and MEP taking three reaction processes as reference and found to be very much dependent on them. Hence, it may be a daring attempt for someone to

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COMPUTATIONAL DETAILS Computations on the three processes (1) HNC → HCN, (2) H + CH3F → HF + CH3, and (3) CH3 + H2 → CH4 + H have been carried out at B3LYP, BLYP, and MP2 levels taking 6311++G(3df,3pd), aug-cc-pVQZ, and def2-QZVP as basis sets. For the total 101 reactions and six isomeric molecules, optimizations followed by the frequency calculations have been carried out at the B3LYP/6-311++G(3df,3pd) level of theory excluding those processes involving iodine with the help of Gaussian09 program package.91 The chemical reactions involving iodine have been studied at ECP-corrected B3LYP/ def2-QZVP level of theory. Conceptual density functional theory (CDFT)92,93 based global reactivity descriptors like electronegativity,1−4 hardness,5−8 and electrophilicity9−12 have been calculated as follows: For an N-electron system having total energy E, the electronegativity (χ) and hardness (η) can be defined as follows: ⎛ ∂E ⎞ χ = −⎜ ⎟ = −μ ⎝ ∂N ⎠v(r)

(2)

⎛ ∂ 2E ⎞ η = ⎜ 2⎟ ⎝ ∂N ⎠v(r)

(3)

where v(r) and μ are external and chemical potentials, respectively. Electrophilicity (ω) is defined as

ω=

μ2 χ2 = 2η 2η

(4)

Applying the finite difference approximations to eqs 2 and 3, χ and η can be expressed as I+A 2

(5)

η=I−A

(6)

χ=

and where I and A are the ionization potential and electron affinity, respectively. 1844



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Here, I and A have been computed by applying both Koopmans’ theorem89 and ΔSCF method for the three above-mentioned reactions, and for the rest, only ΔSCF method has been used. According to ΔSCF method, the ionization energy (I) and electron affinity (A) of the system have been computed in terms of the energies of the N and N ± 1 electron systems calculated at the geometry of the N-electron system. For an N-electron system with energy E(N), they may be expressed as follows: I = E(N − 1) − E(N )

Table 1. Hardness (η, eV) Values of Reactant, Transition State, and Product at Different Levels of Theory of the HNC → HCN Rearrangement Reaction

(7)

and A = E(N ) − E(N + 1)

The scaled hardness, η̃ has been calculated as η η η ̃ = 1/3 = 1/3 |χ | |μ|

(8)

55

(9)

The chemical processes having more than one reactant and product, ηavg and ωavg, have been calculated by taking both geometric and arithmetic means. In mathematics, the geometric mean represents the central tendency of a collection of numbers by using the product of their corresponding values, whereas the arithmetic mean shows the central tendency for the same using the sum of the numbers divided by the size of the collection. The geometric mean actually normalizes the ranges, which are averaged, hence it is related to the equalization of the corresponding quantities. Here, the postulate proposed by Sanderson,94,95 as a molecule’s final electronegativity is the geometric mean of the original atomic electronegativities, should be mentioned here. This is also known as electronegativity equalization principle. The usage of arithmetic mean of atomic softness to represent molecular softness is also reported in the literature.96 Here, it is worth mentioning that the test of the MHP by calculating the geometric mean of hardnesses of reactant and product does not really conform to the original proposal of this principle. It introduces some unwanted bias.

ηreactant

level

ηtransition state

ηproduct

method

8.594 12.869 8.969 13.956 8.489 12.522 6.663 12.789 6.710 6.65a 14.684 6.649 12.789 14.448 13.405 15.849 15.710 13.967 13.158

B3LYP/6-311++G(3df,3pd) B3LYP/6-311++G(3df,3pd) B3LYP/def2-QZVP B3LYP/def2-QZVP B3LYP/aug-cc-pVQZ B3LYP/aug-cc-pVQZ BLYP/6-311++G(3df,3pd) BLYP/6-311++G(3df,3pd) BLYP/def2-QZVP BLYP/def2-QZVPb BLYP/def2-QZVP BLYP/aug-cc-pVQZ BLYP/aug-cc-pVQZ MP2/6-311++G(3df,3pd) MP2/6-311++G(3df,3pd) MP2/def2-QZVP MP2/def2-QZVP MP2/aug-cc-pVQZ MP2/aug-cc-pVQZ

7.522 13.227 7.552 13.587 7.559 13.046 5.338 12.988 5.356 5.21 13.465 5.345 12.988 14.758 13.963 15.853 14.244 14.286 13.834

9.916 14.458 10.030 16.081 9.892 14.121 7.799 14.433 7.863 7.84c 15.976 7.793 14.905 14.389 14.863 16.345 16.885 13.994 14.581

KT ΔSCF KT ΔSCF KT ΔSCF KT ΔSCF KT KT ΔSCF KT ΔSCF KT ΔSCF KT ΔSCF KT ΔSCF

a

By error, this value is reported in ref 87 as 7.84 eV. The corrected value is 6.65 eV. bFrom ref 87 using NWChem 6.0 program. cBy error, this value in ref 87 is reported as 4.58 eV. The corrected value is 7.84 eV.

studied levels. Therefore, not only individual η values depend on levels of theory, basis set, and approximations used to calculate η but also it further affects the validity of MHP, sometimes changing the trend. For the same reaction, B3LYP/ 6-31+G*//HF/6-31G* results36 obtained with both Koopmans’ theorem and ΔSCF method indicate that the reaction follows the MHP. The results (Table 2) obtained for H + CH3F → HF + CH3 also show the significant dependence upon the levels of theory and approximation used to calculate η. However, in this particular case, using Koopmans’ approximation, all the MP2 level results and B3LYP/aug-cc-pVQZ (arithmetic mean) result show ηproduct > ηreactant; for others at the same approximation, it follows ηproduct < ηreactant. In the case of the result obtained from ΔSCF method, in few cases, the arithmetic mean of ηproduct is greater than ηreactant. A look on the result corresponding to CH3 + H2 → CH4 + H (Table 3) also shows the changes of the trend depending on the approximations, levels, and basis sets. We are sure that many of the studied reactions, which do not obey MHP at a particular level of theory, can hold MHP at some other level, and the other way round, those that follow the MHP at a certain level, may not obey at another level of theory. Since we have calculated ω using eq 4, the dependence of η values on approximations, levels, and basis sets suggests the dependence of ω on to the same. Assessment of the Validity of MHP. Among the 101 reactions (Table S1 in Supporting Information), MHP holds for 62 reactions in which the geometric mean of hardness of product exceeds that of the reactant leading to the 61.4% of the total reactions maintaining MHP, whereas 60 reactions (59.4%) have larger arithmetic mean of hardness of product than that of the reactant (Table 4) at the studied level of theory. However, expression of hardness in terms of the scaled hardness improves the percentage of the total reactions maintaining the MHP. Seventy reactions of the total 101 reactions (69.3%) are found

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RESULTS AND DISCUSSION Dependence of η on Levels of Theory, Basis Sets, and Approximations Used. We have taken three processes studied previously by Poater et al.87 to carry out analysis at different levels of theory. For the first case, we have also considered the TS. The results for the HNC → HCN, H + CH3F → HF + CH3, and CH3 + H2 → CH4 + H processes have been provided in Tables 1, 2, and 3, respectively. We have studied at the BLYP/def2-QZVP level of theory to compare the present result with the previously reported87 one. A look at the individual η terms obtained by using Koopmans’ theorem and ΔSCF method at different levels of theory shows significant change in magnitude. Now, a question arises in mind: does the trend remain constant with the change of Koopmans’ theorem and ΔSCF method or levels of theory in spite of the change in individual η values? Let us see the result for HNC → HCN (Table 1). The results obtained from Koopmans’ theorem follows ηproduct > ηreactant > ηtransition state (obeying MHP) at B3LYP and BLYP levels of theory but does not follow this trend at MP2 level, whereas the results obtained from ΔSCF method follows ηproduct > ηreactant > ηtransition state trend at B3LYP, BLYP, and MP2 levels with def2-QZVP basis set but it becomes ηproduct > ηtransition state > ηreactant (violating MHP) at all other 1845



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Table 2. Individual Hardness (η, eV) Values of the Individual Reactants and Products and Their Average Hardness Values by Using Both Geometric and Arithmetic Mean (ηavg, eV) at Different Levels of Theory for the H + CH3F → HF + CH3 Reaction ηavg

ηavg

ηH

ηCH3F

geo

arith

level

ηHF

ηCH3

geo

arith

method

9.352 12.799 12.510 13.460 9.051 12.758 8.100 12.756 9.707 13.136 7.788 12.697 14.961 13.241 17.160 13.535 14.499 13.138

9.493 13.784 10.863 16.144 9.393 13.503 7.567 13.438 8.184 14.842 7.498 13.172 15.576 14.991 17.448 16.634 15.182 14.741

9.442 13.282 11.657 14.741 9.220 13.125 7.829 13.093 8.917 13.963 7.642 12.932 15.265 14.089 17.303 15.005 14.387 13.916

9.423 13.292 11.687 14.802 9.222 13.131 7.834 13.097 8.946 13.989 7.643 12.935 15.269 14.116 17.304 15.085 14.841 13.940

B3LYP/6-311++G(3df,3pd) B3LYP/6-311++G(3df,3pd) B3LYP/def2-QZVP B3LYP/def2-QZVP B3LYP/aug-cc-pVQZ B3LYP/aug-cc-pVQZ BLYP/6-311++G(3df,3pd) BLYP/6-311++G(3df,3pd) BLYP/def2-QZVP BLYP/def2-QZVP BLYP/aug-cc-pVQZ BLYP/aug-cc-pVQZ MP2/6-311++G(3df,3pd) MP2/6-311++G(3df,3pd) MP2/def2-QZVP MP2/def2-QZVP MP2/aug-cc-pVQZ MP2/aug-cc-pVQZ

10.917 17.101 11.599 18.939 10.812 16.749 8.476 17.107 8.832 18.331 8.415 16.762 18.898 17.383 20.445 18.766 18.462 17.088

7.773 9.982 7.527 10.233 7.706 9.911 5.857 9.863 4.872 10.139 5.811 9.763 14.227 9.857 15.783 10.084 13.738 9.760

9.212 13.065 9.344 13.921 9.128 12.884 7.046 12.989 6.560 13.633 6.993 12.792 16.397 13.090 17.963 13.756 15.926 12.914

9.345 13.542 9.563 14.586 9.259 13.330 7.167 13.485 6.852 14.235 7.113 13.263 16.563 13.620 18.114 14.425 16.100 13.424

KT ΔSCF KT ΔSCF KT ΔSCF KT ΔSCF KT ΔSCF KT ΔSCF KT ΔSCF KT ΔSCF KT ΔSCF

Table 3. Individual Hardness (η, eV) Values of the Individual Reactants and Products and Their Average Hardness Values by Using Both Geometric and Arithmetic Mean (ηavg, eV) at Different Levels of Theory for the CH3 + H2 → CH4 + H Reaction ηavg

ηavg

ηCH3

ηH2

geo

arith

level

ηCH4

ηH

geo

arith

method

7.773 9.982 7.527 10.233 7.706 9.911 5.857 9.863 4.872 10.139 5.811 9.763 14.227 9.857 15.783 10.084 13.738 9.760

12.350 17.931 13.381 20.325 12.204 17.474 10.556 17.776 11.046 19.304 10.468 17.325 18.066 18.024 19.812 19.635 17.501 17.522

9.798 13.379 10.036 14.422 9.698 13.160 7.863 13.241 7.336 13.990 7.799 13.006 16.032 13.329 17.683 14.071 15.506 13.077

10.062 13.957 10.454 15.279 9.955 13.693 8.207 13.820 7.959 14.722 8.140 13.544 16.147 13.941 17.798 14.860 15.620 13.641

B3LYP/6-311++G(3df,3pd) B3LYP/6-311++G(3df,3pd) B3LYP/def2-QZVP B3LYP/def2-QZVP B3LYP/aug-cc-pVQZ B3LYP/aug-cc-pVQZ BLYP/6-311++G(3df,3pd) BLYP/6-311++G(3df,3pd) BLYP/def2-QZVP BLYP/def2-QZVP BLYP/aug-cc-pVQZ BLYP/aug-cc-pVQZ MP2/6-311++G(3df,3pd) MP2/6-311++G(3df,3pd) MP2/def2-QZVP MP2/def2-QZVP MP2/aug-cc-pVQZ MP2/aug-cc-pVQZ

10.615 15.018 11.434 16.462 10.524 14.749 8.925 14.673 9.604 16.421 8.864 14.705 15.976 15.357 17.821 16.937 15.604 15.090

9.352 12.799 12.510 13.460 9.051 12.758 8.100 12.756 9.707 13.136 7.788 12.697 14.961 13.241 17.160 13.535 14.499 13.138

9.964 13.864 11.960 14.886 9.760 13.717 8.502 13.681 9.655 14.687 8.309 13.664 15.460 14.260 17.487 15.141 15.041 14.080

9.984 13.909 11.972 14.961 9.788 13.754 8.513 13.715 9.656 14.779 8.326 13.701 15.469 14.299 17.491 15.236 15.052 14.114

KT ΔSCF KT ΔSCF KT ΔSCF KT ΔSCF KT ΔSCF KT ΔSCF KT ΔSCF KT ΔSCF KT ΔSCF

polarizabilities. Since there is an inverse relationship7,55,98,99 between hardness and polarizability, one may expect MHP to be valid in most of those cases. Assessment of the Validity of MEP. In the case of MEP, 70 reactions of the total 101 reactions (69.3%) have been found to obey MEP when an average electrophilicity of reactant and product is calculated using geometric mean, whereas the arithmetic mean of the product of 68 reactions (67.3%) is smaller than that of the reactant (Table 5). Table 5 shows that a large number of exchange reactions proceed to the forward direction violating MEP principle; however, many of the processes involving the associative combination of atoms or molecules obey MEP nicely at the studied level of theory.

to obey MHP using geometric mean of scaled hardness, whereas 66 reactions (65.3%) satisfy MHP when average scaled hardness of reactant and product is calculated by means of arithmetic mean. So, when we express the hardness in terms of scaled hardness, the percentage of reaction obeying MHP somewhat improves. In most of the cases, we have found that reactions do not likely follow MHP at the studied level of theory under the situations: (1) very hard species like F−, H2, CH4, N2, and OH lie in the reactant side, and (2) most cases of associative combination where two molecules or one atom and a molecule produce a single molecule. A very recent study by Thakkar et al.97 shows that minimum polarizability principle (MPP)84,85 is violated in 3.6% cases of the total sample of 2386 1846



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Table 4. Geometric and Arithmetic Mean of Hardness (ηavg(geo) and ηavg(ari), eV) and Scaled Hardness (η̃avg(geo) and η̃avg(ari), eV2/3) Values of Reactants and Products and Reaction Enthalpy (ΔH, kcal/mol at 298 K) of the Forward Reactions at B3LYP/ 6-311++G(3df,3pd) Level for Most Cases and at B3LYP/def2-QZVP Level for the Reactions Involving I ηavg(geo)

ηavg(ari)

Rexn

R

P

ηavg(geo) P > R?

1 2 3 4 5 6 7 8 9 10 11 12 13 14 15 16 17 18 19 20 21 22 23 24 25 26 27 28 29 30 31 32 33 34 35 36 37 38 39 40 41 42 43 44 45 46 47 48 49 50 51 52 53 54

13.594 12.869 13.282 14.076 10.958 14.136 10.095 10.073 14.105 12.121 10.704 13.099 16.172 13.864 14.800 13.056 11.569 13.759 17.935 12.814 11.998 14.001 11.889 12.107 11.594 10.779 12.552 13.502 10.274 11.224 9.850 10.321 11.799 13.397 9.415 13.780 11.737 12.583 11.603 9.880 9.608 9.492 9.163 11.453 15.047 11.563 10.310 10.830 14.503 14.244 14.585 14.621 17.236 14.458

16.161 14.458 13.065 17.515 12.590 12.567 13.956 15.139 9.783 9.053 8.672 14.360 13.158 13.379 11.620 12.814 13.142 11.066 14.794 12.066 12.480 13.663 13.365 12.951 11.008 10.483 10.286 10.801 13.007 11.243 12.832 12.350 12.262 14.780 11.118 14.208 12.703 12.742 14.220 10.219 9.808 9.617 9.319 12.078 15.448 13.194 10.389 11.298 15.974 10.470 12.608 13.527 14.006 15.554

Y Y N Y Y N Y Y N N N Y N N N N Y N N N Y N Y Y N N N N Y Y Y Y Y Y Y Y Y Y Y Y Y Y Y Y Y Y Y Y Y N N N N Y

R 13.619 12.869 13.291 14.140 11.006 14.349 10.589 10.073 14.171 12.139 10.730 13.103 16.258 13.908 14.801 13.136 11.695 13.782 17.935 12.975 12.023 14.432 11.921 12.170 11.985 10.965 12.559 13.578 10.640 11.431 10.006 10.702 11.811 13.511 9.498 13.953 11.889 12.611 12.487 10.031 9.609 9.493 9.171 11.595 15.170 11.563 10.310 10.832 14.522 14.264 14.585 15.195 17.287 14.458

η̃avg(geo) P

ηavg(ari) P > R?

R

16.246 14.458 13.541 17.520 12.642 12.621 14.199 15.353 9.783 9.053 8.672 14.716 13.163 13.957 11.754 12.833 13.259 11.290 14.950 12.284 13.308 13.692 13.946 13.043 11.061 10.596 10.370 10.835 13.683 11.766 14.213 12.401 12.946 15.537 11.680 14.799 13.269 12.945 14.604 10.638 9.903 9.739 9.489 12.815 15.620 13.814 10.389 11.482 16.005 10.470 12.608 13.527 14.024 15.708

Y Y Y Y Y N Y Y N N N Y N Y N N Y N N N Y N Y Y N N N N Y Y Y Y Y Y Y Y Y Y Y Y Y Y Y Y Y Y Y Y Y N N N N Y

7.300 7.160 7.043 7.127 6.111 8.189 6.675 6.050 7.434 6.690 6.304 6.968 7.994 7.244 7.455 6.887 6.566 7.118 8.409 6.368 6.603 6.828 6.551 6.125 6.085 5.809 6.817 7.141 5.952 6.511 5.990 5.952 6.541 7.948 5.754 8.145 6.763 7.661 6.450 5.689 5.457 5.448 5.303 6.375 7.962 7.036 5.518 6.427 7.775 8.021 6.999 7.573 8.869 7.740 1847

η̃avg(ari) P

η̃avg(geo) P > R?

R

P

η̃avg(ari) P > R?

ΔH

8.112 7.740 7.115 8.175 6.392 8.231 8.142 7.882 5.790 5.689 5.449 7.050 7.006 7.319 6.602 6.906 6.575 6.501 7.551 6.408 6.687 6.800 6.847 6.172 6.137 6.044 6.210 6.306 7.012 6.398 7.286 6.769 6.635 8.418 6.607 8.069 7.187 7.587 7.504 5.859 5.532 5.487 5.354 6.543 8.204 6.999 5.631 6.668 8.385 6.317 7.202 7.428 7.596 8.727

Y Y Y Y Y Y Y Y N N N Y N Y N Y Y N N Y Y N Y Y Y Y N N Y N Y Y Y Y Y N Y N Y Y Y Y Y Y Y N Y Y Y N N N N Y

7.337 7.160 7.058 7.147 6.116 8.331 6.721 6.050 7.485 6.690 6.320 6.977 8.065 7.275 7.470 6.888 6.607 7.119 8.438 6.524 6.603 7.118 6.551 6.226 6.302 5.950 6.830 7.181 6.085 6.642 6.064 6.085 6.552 8.011 5.793 8.237 6.800 7.691 6.725 5.813 5.458 5.448 5.307 6.439 7.988 7.036 5.518 6.427 7.779 8.021 6.999 7.824 8.883 7.740

8.200 7.740 7.248 8.187 6.475 8.266 8.292 8.005 5.790 5.689 5.449 7.287 7.018 7.499 6.648 6.924 6.692 6.559 7.618 6.433 7.014 6.803 7.132 6.221 6.144 6.055 6.234 6.322 7.258 6.740 7.902 6.798 6.866 8.714 6.929 8.259 7.307 7.679 7.649 6.135 5.565 5.525 5.408 6.796 8.251 7.306 5.631 6.707 8.398 6.317 7.202 7.428 7.599 8.751

Y Y Y Y Y N Y Y N N N Y N Y N Y Y N N N Y N Y N N Y N N Y Y Y Y Y Y Y Y Y N Y Y Y Y Y Y Y Y Y Y Y N N N N Y

−59.3 −13.8 −26.2 −99.4 −48.4 −34.0 −18.1 −0.1 −21.4 −37.7 −20.5 −2.4 −11.4 −1.5 −12.9 −11.1 −0.9 −17.6 −30.3 −64.4 −23.5 −62.9 −15.1 −65.2 −40.8 −45.5 −6.5 −1.8 −55.5 −22.4 −56.4 −148.8 −8.6 −12.8 −43.6 −6.5 −8.2 −8.9 −49.1 −8.8 −8.5 −7.5 −10.0 −14.9 −25.3 −3.4 −10.9 −14.1 −15.4 −31.4 −46.1 −147.1 −35.1 −5.3



Article

Table 4. continued ηavg(geo)

ηavg(ari)

η̃avg(geo)

η̃avg(ari)

Rexn

R

P

ηavg(geo) P > R?

R

P

ηavg(ari) P > R?

R

P

η̃avg(geo) P > R?

R

P

η̃avg(ari) P > R?

ΔH

55 56 57 58 59 60 61 62 63 64 65 66 67 68 69 70 71 72 73 74 75 76 77 78 79 80 81 82 83 84 85 86 87 88 89 90 91 92 93 94 95 96 97 98 99 100 101

13.292 13.667 11.233 10.224 16.695 11.863 17.925 13.830 12.608 11.787 12.951 12.704 12.228 10.256 9.314 7.458 7.032 6.595 9.526 7.627 7.192 6.777 8.441 6.758 6.372 6.010 8.047 15.152 12.132 11.440 10.968 9.432 13.663 13.382 10.714 10.103 9.563 7.890 12.799 13.561 17.939 11.500 10.225 8.938 5.138 4.836 3.971

14.712 13.407 10.653 14.852 11.699 10.602 11.687 17.063 12.116 10.621 12.180 11.559 10.744 10.620 9.783 8.530 8.026 7.361 11.276 9.435 8.792 8.101 9.550 8.202 7.701 7.041 7.873 17.101 13.407 12.346 11.763 15.543 13.527 13.784 11.887 11.176 10.659 12.755 17.931 17.907 15.481 10.418 8.950 7.681 4.923 4.843 3.771

Y N N Y N N N Y N N N N N Y Y Y Y Y Y Y Y Y Y Y Y Y N Y Y Y Y Y N Y Y Y Y Y Y Y N N N N N Y N

13.603 14.174 11.350 10.237 16.737 11.964 17.925 13.837 12.608 11.893 13.642 13.466 13.135 10.343 11.387 8.168 7.530 6.902 11.499 8.280 7.642 7.038 10.955 7.736 7.098 6.489 8.929 15.369 12.150 11.512 11.199 9.535 13.692 13.961 10.741 10.104 9.585 9.774 12.799 13.561 17.939 11.500 10.225 8.938 5.138 4.836 3.971

14.712 13.407 10.653 14.987 11.699 10.602 11.687 17.124 12.258 10.621 12.448 11.702 10.782 10.932 9.783 8.530 8.026 7.361 11.276 9.435 8.792 8.101 9.550 8.202 7.701 7.041 7.873 17.101 13.407 12.346 11.763 15.543 13.527 13.784 11.887 11.176 10.659 12.755 17.931 17.907 15.481 10.418 8.950 7.681 4.923 4.843 3.771

Y N N Y N N N Y N N N N N Y N Y Y Y N Y Y Y N Y Y Y N Y Y Y Y Y N N Y Y Y Y Y Y N N N N N Y N

7.143 7.180 6.141 5.788 8.738 6.763 9.198 7.531 7.202 6.487 7.294 7.190 7.024 6.195 5.095 4.271 4.085 3.911 5.187 4.349 4.159 4.004 4.757 3.988 3.814 3.677 4.791 7.154 5.997 5.735 5.654 4.821 6.800 6.741 5.651 5.404 5.219 4.500 6.608 6.830 7.745 5.443 4.978 4.529 3.540 3.352 2.922

7.894 7.347 6.302 8.035 6.820 6.532 6.778 8.837 6.600 6.359 6.932 6.704 6.404 6.335 5.531 4.962 4.733 4.434 6.168 5.385 5.102 4.842 5.511 4.882 4.648 4.345 4.822 8.629 7.347 6.936 7.006 8.363 7.428 7.508 6.798 6.515 6.497 7.662 9.131 9.402 7.686 5.646 4.892 4.317 3.471 3.425 2.836

Y Y Y Y N N N Y N N N N N Y Y Y Y Y Y Y Y Y Y Y Y Y Y Y Y Y Y Y Y Y Y Y Y Y Y Y N Y N N N Y N

7.312 7.388 6.225 5.820 8.747 6.873 9.198 7.532 7.202 6.563 7.478 7.396 7.267 6.220 5.549 4.398 4.165 3.953 5.610 4.459 4.226 4.035 5.334 4.183 3.950 3.757 5.041 7.176 6.025 5.793 5.794 4.829 6.803 6.806 5.655 5.422 5.272 5.392 6.608 6.830 7.745 5.443 4.978 4.529 3.540 3.352 2.922

7.894 7.347 6.302 8.086 6.820 6.532 6.778 8.855 6.689 6.359 6.996 6.739 6.414 6.416 5.531 4.962 4.733 4.434 6.168 5.385 5.102 4.842 5.511 4.882 4.648 4.345 4.822 8.629 7.347 6.936 7.006 8.363 7.428 7.508 6.798 6.515 6.497 7.662 9.131 9.402 7.686 5.646 4.892 4.317 3.471 3.425 2.836

Y N Y Y N N N Y N N N N N Y N Y Y Y Y Y Y Y Y Y Y Y N Y Y Y Y Y Y Y Y Y Y Y Y Y N Y N N N Y N

−177.1 −44.8 −101.7 −166.0 −2.8 −53.8 −25.5 −26.8 −15.7 −82.5 −13.2 −15.4 −12.5 −2.8 −111.6 −93.7 −85.1 −73.8 −136.7 −110.4 −99.5 −84.1 −116.9 −98.2 −89.7 −78.4 −57.3 −135.0 −102.3 −89.3 −75.8 −358.2 −116.1 −108.8 −80.0 −68.8 −55.1 −64.4 −104.7 −353.1 −35.6 −55.1 −49.2 −43.3 −19.2 −17.6 −12.0

Large Sample Test for Proportion. We have studied a total of 101 reactions. Let us take two best situations where the percentage of reactions obeying MHP and MEP are maximum, i.e., the case where 69.3% of reactions obey the principles. Let P be the population proportion of reactions as mentioned above. We have to test the null hypothesis

against the alternative hypothesis H1: P ≠ 0.5.

Let us fix the level of significance of the test as 2% (so that the level of confidence is 98%). It is to be noted that the distribution of the sample proportion (p) is binomial with mean P and variance P(1 − P)/n. The sample size is large (here it is 101), and the distribution of the test statistic

H 0 : P = 0.5 1848



Article

Table 5. Geometric and Arithmetic Mean of Electrophilicity Values (ωavg(geo) and ωavg(ari), eV) of Reactant and Product at B3LYP/6-311++G(3df,3pd) for Most Cases and at B3LYP/def2-QZVP Level of Theory for the Reactions Involving I ωavg(geo) Rexn

R

P

1 2 3 4 5 6 7 8 9 10 11 12 13 14 15 16 17 18 19 20 21 22 23 24 25 26 27 28 29 30 31 32 33 34 35 36 37 38 39 40 41 42 43 44 45 46 47 48 49 50 51

1.533 1.310 1.693 2.109 1.518 0.936 0.593 1.058 1.654 1.459 1.120 1.685 2.119 1.772 2.067 1.777 1.293 1.896 2.624 2.591 1.500 2.656 1.502 2.464 2.064 1.893 1.552 1.692 1.287 1.169 1.004 1.317 1.460 0.856 1.019 0.851 1.164 0.780 1.460 1.388 1.549 1.474 1.453 1.468 1.514 0.852 2.062 1.057 1.453 1.101 2.808

1.935 1.469 1.467 2.761 2.318 0.504 0.909 1.659 1.189 0.897 0.937 2.488 1.668 1.394 1.280 1.592 2.426 1.099 1.912 1.847 1.692 2.407 2.069 3.296 1.512 1.299 1.004 1.169 1.566 1.310 1.163 1.494 1.624 0.991 1.021 1.049 1.200 0.880 1.628 1.377 1.583 1.507 1.492 1.637 1.442 1.700 1.898 1.047 1.496 0.990 1.142

N N Y N N Y N N Y Y Y N Y Y Y Y N Y Y Y N Y N N Y Y Y Y N N N N N N N N N N N Y N N N N Y N Y Y N Y Y

Z=

ωavg(geo)

ωavg(ari) ωavg(geo) P < R?

R

P

ωavg(ari) P < R?

1.601 1.310 1.726 2.110 1.555 0.966 0.821 1.058 1.695 1.548 1.124 1.720 2.204 1.793 2.165 1.966 1.296 2.044 2.952 2.966 1.577 3.005 1.579 2.893 2.161 2.039 1.581 1.702 1.290 1.204 1.005 1.322 1.472 0.861 1.020 0.856 1.182 0.796 1.487 1.488 1.550 1.475 1.453 1.471 1.532 0.852 2.062 1.058 1.454 1.159 2.808

2.070 1.469 1.493 3.037 2.629 0.786 0.931 1.698 1.189 0.897 0.937 2.734 1.706 1.407 1.282 1.614 2.695 1.122 1.917 2.012 1.695 2.436 2.138 3.338 1.548 1.389 1.010 1.170 1.566 1.436 1.176 1.502 1.630 0.991 1.062 1.059 1.292 0.881 1.628 1.492 1.585 1.513 1.500 1.642 1.450 1.751 1.898 1.081 1.496 0.990 1.142

N N Y N N Y N N Y Y Y N Y Y Y Y N Y Y Y N Y N N Y Y Y Y N N N N N N N N N N N N N N N N Y N Y N N Y Y

p−P P(1 − P) n

ωavg(ari)

Rexn

R

P

ωavg(geo) P < R?

52 53 54 55 56 57 58 59 60 61 62 63 64 65 66 67 68 69 70 71 72 73 74 75 76 77 78 79 80 81 82 83 84 85 86 87 88 89 90 91 92 93 94 95 96 97 98 99 100 101

1.771 1.563 1.469 1.562 1.741 1.666 1.486 1.456 1.228 1.528 1.387 1.142 1.526 1.210 1.197 1.138 1.004 2.003 1.899 1.850 1.743 2.014 1.909 1.860 1.734 1.847 1.752 1.707 1.585 1.394 2.980 2.825 2.753 2.429 2.973 2.407 2.287 2.168 2.113 1.979 1.843 2.063 2.259 4.303 3.868 3.673 3.303 0.910 0.932 0.793

1.348 1.404 1.030 1.425 1.377 1.095 1.343 1.089 0.862 1.124 1.519 1.579 1.022 1.208 1.136 1.038 1.045 1.566 1.513 1.482 1.422 1.656 1.533 1.489 1.353 1.418 1.371 1.344 1.285 1.203 1.771 1.377 1.288 0.952 1.326 1.348 1.389 1.203 1.141 0.914 0.834 1.600 1.333 2.156 1.895 2.094 2.066 0.828 0.825 0.733

Y Y Y Y Y Y Y Y Y Y N N Y Y Y Y N Y Y Y Y Y Y Y Y Y Y Y Y Y Y Y Y Y Y Y Y Y Y Y Y Y Y Y Y Y Y Y Y Y

R

P

ωavg(ari) P < R?

1.789 1.565 1.469 1.607 1.747 1.724 1.593 1.463 1.345 1.533 1.388 1.142 1.576 1.257 1.248 1.205 1.010 2.618 2.400 2.302 2.111 2.623 2.405 2.307 2.107 2.548 2.331 2.233 2.032 1.503 3.183 2.966 2.868 2.545 3.207 2.436 2.759 2.542 2.444 2.244 1.943 2.063 2.259 4.303 3.868 3.673 3.303 0.910 0.932 0.793

1.348 1.406 1.065 1.425 1.377 1.095 1.343 1.089 0.862 1.124 1.523 1.621 1.022 1.240 1.153 1.041 1.079 1.566 1.513 1.482 1.422 1.656 1.533 1.489 1.353 1.418 1.371 1.344 1.285 1.203 1.771 1.377 1.288 0.952 1.326 1.348 1.389 1.203 1.141 0.914 0.834 1.600 1.333 2.156 1.895 2.094 2.066 0.828 0.825 0.733

Y Y Y Y Y Y Y Y Y Y N N Y Y Y Y N Y Y Y Y Y Y Y Y Y Y Y Y Y Y Y Y Y Y Y Y Y Y Y Y Y Y Y Y Y Y Y Y Y

rejected at 2% level of significance if the absolute value of the calculated Z is greater than Z0.01 (= 2.33). Since the observed value of p is given to be 0.693, the value of Z is calculated as 3.879, which is greater than Z0.01. Hence, the null hypothesis is rejected at the 2% level of significance and it is concluded that

(10)

when the null hypothesis is true, can be approximated by the standard normal distribution. The null hypothesis will be 1849



Article

the proportion of reactions cannot be considered to be equal to 0.5. Now, let us see the case where 61.4% of the total reactions obey MHP. By using eq 10 and 5% level of significance, the critical value is Z0.025 (=1.96). Here we have Z (=2.291), which is greater than Z0.025 (=1.96). Therefore, the null hypothesis is rejected at 5% level of significance. Six Isomeric Molecules. We have considered different isomers of SF4, SF4O, PCl4F, PCl3F2, PCl2F3, and PClF4 molecules (Figure 1), which were also studied earlier.71

Table 6. Total Energy (E, au), Hardness (η, eV), Scaled Hardness (η̃, eV2/3), and Electrophilicity (ω, eV) Values of the Studied Systems at B3LYP/6-311++G(3df,3pd) Level of Theory isomers

E

η

η̃

ω

SF4 (1A) SF4 (2A) SF4O (1B) SF4O (2B) PCl4F (1C) PCl4F (2C) PCl3F2 (1D) PCl3F2 (2D) PCl3F2 (3D) PCl2F3 (1E) PCl2F3 (2E) PCl2F3 (3E) PClF4 (1F) PClF4 (2F)

−797.69389 −797.65726 −872.91683 −872.87978 −2282.26493 −2282.25832 −1921.93985 −1921.93337 −1921.92730 −1561.60865 −1561.60252 −1561.59693 −1201.27808 −1201.27227

12.763 12.877 13.387 12.855 9.432 9.138 11.402 10.120 10.128 11.836 11.154 11.638 13.034 12.749

6.878 7.149 7.141 6.764 5.133 4.990 6.188 5.551 5.579 6.491 6.156 6.465 7.100 7.010

1.600 1.326 1.621 1.833 2.041 2.062 1.717 1.813 1.767 1.552 1.587 1.462 1.468 1.420

although for 2D and 3D isomers, they do not hold MHP and MEP. The most stable isomers of PCl2F3 and PClF4 molecules have the maximum hardness compared to the other respective isomers following MHP, but they do not obey MEP.

■

CONCLUSIONS It has been shown that the hardness and electrophilicity values change drastically as one changes the levels of theory, basis set quality, and approximations, which, in turn, may change the qualitative trends in those values for reactants and products in a reaction. Although the maximum hardness and minimum electrophilicity principles are not valid in certain situations, they do provide a guide in understanding the stability of a system as well as the favorable direction of a chemical reaction. Reactions more likely to not obey the maximum hardness principles are those that have very hard species in the reactant side or correspond to the associative combination of two fragments producing a single molecule. However, this latter group of reactions in most cases follows the minimum electrophilicity principle. A 50% null hypothesis is found to reject at the 2% level of significance for the case where 69.3% of reactions obey MHP and MEP; however, it is rejected at the 5% level of significance for the case where 61.4% of reactions obey MHP.

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ASSOCIATED CONTENT

* Supporting Information S

Different studied reactions and their individual hardness (η, eV), scaled hardness (η̃, eV2/3), and electrophilicity (ω, eV) values at the B3LYP/6-311++G(3df,3pd) level for most cases and at the B3LYP/def2-QZVP level for the reactions involving I. This material is available free of charge via the Internet at http://pubs.acs.org.

Figure 1. Different isomers of SF4, SF4O, PCl4F, PCl3F2, PCl2F3, and PClF4 molecules.

■

According to the Bent rule,100 1A isomer of SF4, 1B isomer of SF4O, 1C isomer of PCl4F, 1D isomer of PCl3F2, 1E isomer of PCl2F3, and 1F isomer of PClF4 are energetically more stable than the other respective isomers. Total energy (E, au) as given in Table 6 also shows this trend. Let us see in Table 6 whether hardness and electrophilicity trends follow the same. Most stable isomers are expected to have maximum hardness and minimum electrophilicity. Isomers 1A and 2A of SF4 do not obey both MHP and MEP, but SF4O and PCl4F satisfy nicely both MHP and MEP. In the case of PCl3F2, the most stable 1D isomer has maximum hardness and minimum electrophilicity

AUTHOR INFORMATION

Corresponding Author

*E-mail: [email protected]. Notes

The authors declare no competing financial interest.

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ACKNOWLEDGMENTS We thank Professor A. K. Nanda, IISER Kolkata, for help in statistical analysis. Financial assistance from CSIR, New Delhi, 1850



Article

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is gratefully acknowledged. P.K.C. would like to thank DST, New Delhi, for the J. C. Bose National Fellowship. M.S. thanks the following organizations for financial support: the Spanish MICINN (project CTQ2011-23156/BQU), the Catalan DIUE (projects 2009SGR637 and XRQTC), and the FEDER fund for the grant UNGI08-4E-003. Support for the research of M.S. was received through the ICREA Academia 2009 prize of the Catalan DIUE.

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On the Validity of the Maximum Hardness Principle and the Minimum

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