hardness equalization principle and molecular hardness

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Re:

Whether There Is a HardnessEqualization Principle Analogous to theElectronegativity EqualizationPrinciple—A Quest

DULAL C. GHOSH, NAZMUL ISLAMAQ2

Department of Chemistry, University of Kalyani, Kalyani, West Bengal 741235, India

Received 5 August 2009; accepted 23 October 2009Published online 00 Month 2009 in Wiley InterScience (www.interscience.wiley.com).DOI 10.1002/qua.22508

ABSTRACT: In this report, we have attempted to explore whether the hardnessequalization principle can be conceived analogous to the well-establishedelectronegativity equalization principle. We have relied upon the fact that the hardness,like the electronegativity, is a qualitative property, and there is commonality in thebasic philosophy of the origin and the operational significance of these twofundamental descriptors—the electronegativity and the hardness of atoms in physicsand chemistry. Starting from the empirical radial dependent formula of computing thehardness of atoms suggested by us, we have derived an ansatz of the molecularhardness assuming that hardness equalization principle is operative and justifiablyvalid. For a validity test, we have applied the suggested ansatz to compute thehardness of as many as four sets of compounds with widely divergent physical andchemical nature. As hardness is not an experimentally measurable property, there is nobenchmark to perform any validity test of our computed data. We have, therefore,computed the hardness data of these four different sets of the compound invoking theapproximate and operational formula of Parr and Pearson, (I � A)/2, to evaluatehardness values and to compute I and A, we have invoked Koopmans’ theorem and anab initio quantum chemical method. We have observed that there is a close correlationbetween the four sets of hardness data computed through the semi-empirical ansatz ofthis work and the quantum mechanical method. Thus, it appears that the ansatz ofcomputing molecular hardness derived on the basis of the hardness equalizationprinciple is efficacious in computing molecular hardness. The detailed comparativestudy suggests that the paradigm of the hardness equalization principle may be anotherlaw of nature like the established electronegativity equalization principle. VC 2009 WileyPeriodicals, Inc. Int J Quantum Chem 000:000–000, 2009

Correspondence to: D.C. GHOSH; e-mail: [email protected]

J_ID: ZQZ Customer A_ID: QUA22508 Cadmus Art: QUAT Ed. Ref. No.: 2009-0414.R2 Date: 17-November-09 Stage: Page: 1

ID: thambikkanue I Black Lining: [ON] I Time: 18:29 I Path: N:/3b2/QUAT/Vol00000/090441/APPFile/C2QUAT090441

International Journal of Quantum Chemistry, Vol. 000, 000–000 (2009)VC 2009 Wiley Periodicals, Inc.

Key words: electronegativity equalization principle; hardness equalization principle;commonality in the fundamental basis of origin of the electronegativity and thehardness; the electronegativity and the hardness are empirical in nature; electrostatictheorem for computing hardness

Introduction

T he hardness is a very old and one of themost useful conceptual constructs of chemis-

try and physics. The notion of hardness was firstintroduced by Mulliken [1] when he pointed outthat the ‘‘Hard’’ and ‘‘Soft’’ behavior of variousatoms, molecules, and ions can be conceived dur-ing acid-base chemical interaction. Soon afterMulliken’s classification, the terms hardness andsoftness were in the glossary of conceptual chem-istry and implicitly signified the deformability ofatoms, molecules, and ions under small perturba-tion. Thereafter, Pearson [2] and Klopman [3]tried to systematize and rationalize this intrinsicproperty of atoms and molecules. Pearson classi-fied molecules, atoms, and ions in three classes:hard, soft, and borderline—known as the HSABprinciple [2]. But the classification was qualitativein nature and its basis was empirical. It was verydifficult to ascertain the relative hardness of themolecules and atoms in terms of Pearson’sclassification.

Klopmann [3], in an attempt to theorize HSABprinciple, had drawn a link to Hard–Soft behaviorwith the HOMO-LUMO gap of the frontier orbitaltheory.

It is apparent that the hardness as conceived inchemistry fundamentally signifies the resistancetoward the deformation or polarization of theelectron cloud of the atoms, ions, or moleculesunder small perturbation generated during theprocess of chemical reaction. Thus, the generaloperational significance of the hard–soft chemicalspecies may be understood in the following state-ment—If the electron cloud is strongly held bythe nucleus, the chemical species is ‘‘hard,’’ but ifthe electron cloud is loosely held by the nucleus,the system is ‘‘soft’’ [2, 3].

The scientific world had to wait till the seminalwork of Parr and Yang [4, 5] who using the Den-sity Functional Theory, (DFT), as basis, trans-formed the qualitative HSAB principle into asound quantum mechanical theory. Given theelectron density function q(r) in a chemical sys-

tem (atom or molecule) and the energy functionalE(q), the chemical potential, l of that system inequilibrium has been defined as the derivative ofthe energy with respect to the number of elec-trons at fixed molecular geometry.

Gyftpoulous and Hatsopoulos [6] introduced theconcept of chemical potential, l, and is given by

_l ¼ ½dEðqÞ=dq�v (1)

The differential definition more appropriate toatomic system is

l ¼ ½@E=@N�v (2)

Here E is expressed as function of the numberof electrons, N i.e., energy is a functional of N,E(N).

Then following Iczkowski and Margrave [7],Parr et al. [8] defined the electronegativity as theadditive inverse of the chemical potential:

v ¼ �l (3)

or,

v ¼ �½@E=@N�v (4)

The absolute hardness g of the DFT is defined[9] as

g ¼ 1

2½@l=@N�v ¼

1

2½ð@2E=@N2Þ�v (5)

Although mathematical formulae were sug-gested but rigorous evaluation [10, 11] of hard-ness g in terms of the suggested formulae in Eq.(5) is difficult [12]. However, calculus of finite dif-ference approximation was invoked [9] to suggestan approximate and operational formula of hard-ness as under:

g ¼ ðI � AÞ=2 (6)

Where ‘‘I’’ and ‘‘A’’ are the first ionization poten-tial and electron affinity of the chemical species.Pearson [13] proceeded further to evaluate ‘‘I’’



GHOSH AND ISLAM

2 INTERNATIONAL JOURNAL OF QUANTUM CHEMISTRY DOI 10.1002/qua VOL. 000, NO. 000

and ‘‘A’’ in terms of orbital energies of the highestoccupied molecular orbital (HOMO) and the low-est unoccupied molecular orbital (LUMO) by con-necting it with Hartree-Fock SCF theory andinvoking Koopmans’ theorem. The hardness Eq.(6) is reformulated as

g ¼ ð�eHOMO þ eLUMOÞ=2 (7)

where I ¼ �eHOMO, and A ¼ eLUMO

However, the working formula, g ¼ [1/2](I �A), suggested by Parr and Pearson [9] is approxi-mate, and Reed [14] pointed out some mathemati-cal inconsistency in evaluating global hardness byfinite difference approximation method. However,Perdew et al. [10] and Yang et al.,[11] contrary tothe suggestion of Reed, have rigorously formu-lated the E vs N curve by introducing DFT for‘‘fractional particle number’’ and ‘‘reduced densitymatrix functional theory.’’ In spite of this appa-rent incoherency between the definition and mea-surement of hardness, the quest for the theoreticalbasis of the hard soft acid base behavior of Parrand coworkers has created such a surge of funda-mental research in chemistry that it gave birth ofa new branch of density functional based theoreti-cal science known as ‘‘Conceptual Density Func-tional Theory (CDFT)’’[15]. The conceptual den-sity functional theory has added (i) maximumhardness principle [16–18] and (ii) minimumpolarizability principle [19] to the list of the fun-damental laws of nature. CDFT has been success-fully exploited in elucidating and correlatingmechanistic aspects viz. regio-selectivity, catalysis,aromaticity, intramolecualr rotation, inversion,and isomerization reaction [20–27].

The maximum hardness principle has alsoproved useful for describing hard/soft acid/baseprinciple [2, 28–30], which is especially importantsince the HSAB principle is the driving motiva-tion [31, 32] for the hardness concept.

Fundamental Nature of Hardness andElectronegativity

It is the experience of chemists and physiciststhat the principles of hardness and electronegativ-ity work in chemistry and physics, but they arenot physical observables. The appearance and sig-nificance of heuristically developed concepts ofelectronegativity and hardness in chemistry and

physics resemble the unicorns of mythical saga[33]. They exist but never seen. Without the con-cept and operational significance of hardness andelectronegativity, chemistry and many aspects ofcondensed matter physics become chaotic, andthe long established unique order in chemico-physical world will be disturbed. However, thehardness and the electronegativity are conceptualstructures and are not physical observables and,therefore, cannot be evaluated experimentally.They occur in the domain of the hypothesis andaccording to the rules of quantum mechanics; thepossibility of quantum mechanical evaluation ofhardness and electronegativity are completelyruled out [34].

Commonality in the Basis of thePhysical Structures of Hardness andElectronegativity

Before we venture the idea that there will be avalid physical process of hardness equalization asa fundamental law of nature similar to the elec-tronegativity equalization, let us ponder over thesubject that there is much commonality in thephysical structure and the philosophical basis ofhardness and electronegativity as regards their or-igin and operational significance. The hardnessrefers to the resistance of the electron cloud of theatomic and molecular systems under small per-turbation of electrical field. An atom or moleculehaving least tendency of deformation are hardand having small tendency of deformation aresoft. In other words, list polarizable means mosthard, and in such systems, the electron clouds aretightly bound to the atoms or molecules. On thecontrary, most polarizable means least hard, andin such systems, the electron cloud is looselybound to the atoms or molecules.

Electronegativity though defined in many dif-ferent ways, the most logical and rational defini-tion of it [33] is the electron holding power of theatoms or molecules. The more electronegativespecies hold electrons more tightly, and the lesselectronegative species hold less tightly.

Thus, if we invoke the qualitative definition ofhardness stated above and compare with thequalitative definition of electronegativity, thecommonality of their conceptual structures andphilosophical basis are self-evident.



HARDNESS PRINCIPLE ANALOGOUS TO ELECTRONEGATIVITY PRINCIPLEAQ1

VOL. 000, NO. 000 DOI 10.1002/qua INTERNATIONAL JOURNAL OF QUANTUM CHEMISTRY 3

We are discovering the commonality in funda-mental nature of hardness and electronegativity—the holding power of the electron cloud by thechemical species. Thus, the qualitative views ofthe origins of hardness and electronegativitynicely converge to the one and single basic princi-ple that they originate from the same source—theelectron attracting power of the screened nuclearcharge [34]. There are also a group of scientistswho believe that the electronegativity and hard-ness are linearly correlated with each other andare (ergo) approximately proportional [35–39]. Asit is a fact that the origin and the operational sig-nificance of the electronegativity and hardness arethe same, we may conjecture that the equalizationof hardness must occur like the electronegativityequalization process.

Sanderson [40] pointed out that on formationof a molecule, the electronegativity of all theatoms in the molecule becomes equal. This isknown as ‘‘Electronegativity Equalization Princi-ple’’ and is now universally accepted and funda-mentally justified [40]. It is now accepted by thescientific community [41–45] that the hardness isthe cardinal index of stability and chemical reac-tivity, and the concept was culminated with theintroduction of the maximum hardness principle[16].

The Search of Probable Algorithm ofHardness Equalization

As we have tried to establish that there aremuch commonalities in the fundamental nature ofthe two fundamental descriptors, the electronega-tivity and the hardness, and the electronegativityequalization principle is well established, onemay conjecture that, like the electronegativityequalization principle, there should be a hardnessequalization principle where hardnesses of indi-vidual atoms are equalized in a molecule.

Although quite a good number of workershave enquired into the fundamental nature of theelectronegativity equalization principle, very fewreports [37, 46, 47] seems to have appeared on thepossible existence and occurrence of the hardnessequalization principle. The hardness as definedby Parr and Pearson is a global constant that doesnot depend on position, so it is obviously equal-ized. On the other hand, that principle is not anal-ogous to the chemical potential equalization prin-

ciple, which requires defining a local chemicalpotential. Defining an analogous local hardnesscan be done,[46, 47] and Parr and Ayers [48, 49]forcibly argue that this hardness should be chosento be equalized. They even argue that there is aninfinite hierarchy of conditions of this type.

Let us consider the formation of a diatomicmolecule AB from its constituent atoms A and Bas represented below:

Aþ B! AB (8)

The reaction may be treated as a general acidbase reaction. Pearson [50] pointed out that theHSAB principle might be exploited in case of het-erolytic as well as homolytic bond energies ofmolecules. Assuming that there is hardness equal-ization principle analogous to the electronegativ-ity equalization principle, we have derived analgorithm for computing molecular hardness interms of the knowledge of atomic hardness asunder:

Let the equilibrium bond length and the hard-ness of the molecule AB are rAB and gAB, respec-tively. The hardnesses of the corresponding atomsare gA and gB. Now let us imagine that, after theformation of the molecule, a point charge islocated at a distance r1 from A and r2 from B onthe bond with the obvious identity r1 þ r2 ¼ rAB.

Following Feynman et al. [51] and relyingupon classical electrostatic theorems, we [52, 53]have derived radial dependent formulae for com-puting atomic hardness relying upon the opera-tional and approximate formula of Parr and Pear-son [9].

Classically, the energy E(N) of charging a con-ducting sphere of radius r with charge q is givenby

EðNÞ ¼ q2=2r ðin C:G:S:UnitÞ (9)

EðNÞ ¼ q2=ð4pe0Þ2r ðin S:I:UnitÞ (10)

In Eq. (9) and Eq. (10), E(N) is in ergs, q is inelectrostatic unit, and r is in cm. Now, for anatom, the change in energy associated with theincrease of q, on removal of an electron (ofcharge, e), would be the ionization energy, I. Simi-larly, the energy evolved on addition of an elec-tron with q would be the electron affinity, A.Hence,

I ¼ EðN þ 1Þ � EðNÞ ¼ ½fðqþ eÞ2=2rg � ðq2=2rÞ�(11)



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and

A ¼ EðNÞ � EðN � 1Þ ¼ ½ðq2=2rÞ � fðq� eÞ2=2rg�(12)

Now, putting the values of I and A from aboveinto the Eq. (6), we get,

g ¼ ð1=2Þ½fðqþ eÞ2=2rg � ðq2=2rÞ � ðq2=2rÞþ fðq� eÞ2=2rg� ð13Þ

or,

g ¼ e2=2r (14)

Where, e is the electronic charge in e.s.u and r isthe most probable or absolute radius of the atomin cm.

Equation (14) clearly shows that the hardnesshas the dimension of energy.

Now as the finite difference formula is approx-imate, we [53] therefore proposed that the hard-ness, g is not exactly equal to e2/2r, rather, in allprobability, proportional to e2/2r.

g / e2=2r (15)

Now, let us visualize the hardness equalizationon the process of the molecule formation from itsconstituent atoms or fragments. With the physicalprocess of charge transfer during chemical reac-tion leading to the bond formation, the hardnesskernel of atoms change, and in the process, itwould increase somewhere and decrease else-where ultimately the hardness values of theatomic fragments will equalize to some intermedi-ate values common to all. Let d be the amount ofcharge transfer from atom B to A, during the for-mation of AB molecule.

Thus, afterAQ8 the charge transfer, the hardness ofatom A and B in the molecule AB becomes

g0A ¼ Cðeþ dÞ2=2rA (16)

and

g0B ¼ Cðe� dÞ2=2rB (17)

respectively. Where, g/A and g/

B are the hard-ness of the atom A and B in the molecule AB.

Expanding and neglecting the d2 terms fromthe Eq. (16) and Eq. (17), we get:

g0A ¼ Cðe2 þ 2edÞ=2rA (18)

and

g0B ¼ Cðe2 � 2edÞ=2rB (19)

The hardness equalization principle impliesthat, after the formation of the molecule, the hard-ness of the individual constituents is equalized,i.e.,

gAB ¼ g0A ¼ g0B (20)

gAB ¼ Cðe2 þ 2edÞ=2rA ¼ Cðe2 � 2edÞ=2rB (21)

thus,

ðe2 þ 2edÞ=2rA ¼ ðe2 � 2edÞ=2rB (22)

d ¼ e=2fðrA � rBÞ=ðrA þ rBÞg (23)

and so,

gAB ¼ C½e2=ðrA þ rBÞ� (24)

Appling the approximation [54] that r1 � rA

and r2 � rB, we obtain using Eq. (24)

gAB � C½e2=ðRABÞ� (25)

However, the Eq. (25) computes hardness inesu and in electron volt, the Eq. (25) looks:

TABLE IThe semi-empirically evaluated hardness and theab initio SCF hardness of alkali halides. AQ7

Alkalihalides

g(ab initio)in eV

g(semi-empirical)in (eV)

Spectroscopicinternucleardistance (A)

LiF 6.1544505 6.905907 1.5638785LiCl 5.360365 5.344706 2.0206913LiBr 5.0191555 5.342591 2.021491LiI 4.6044335 4.515156 2.391944NaF 8.035461 5.607566 1.9259692NaCl 8.355347 4.574677 2.3608225NaBr 7.7339905 4.31643 2.5020676NaI 7.399555 4.515156 2.391944KF 4.4285295 4.973561 2.1714824KCl 4.9381875 4.049975 2.666683KBr 4.548844 3.828689 2.820809KI 4.2995565 3.543447 3.0478794RbF 5.0859765 4.756953 2.2703609RbCl 5.884438 3.875456 2.786769RbBr 5.3842715 3.667508 2.9447792RbI 5.189208 3.399521 3.1769183



HARDNESS PRINCIPLE ANALOGOUS TO ELECTRONEGATIVITY PRINCIPLE


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δ= e(rA-rB)/2(rA+rB) (23)

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gAB � Cð14:4=RABÞ (26)

where, the internuclear bond distance. Similarinverse relationship between hardness and inter-nuclear bond distance of molecule was suggestedby Ayers [38] and Ghanty and Ghosh [57] Muchearlier.

The RAB in Eq. (26) must be expressed in theAngstrom unit, C is the constant depending on thefundamental nature of hardness, e.g., the bondtype and steric factor However, we have calculatedstandardized value of C ¼ 0.75 with the help of theHyperchem 8.0 professional program [55]. Theprocedure is that we have computed the gAB of aseries of selected molecules invoking the Eqs. (6)and (7) and computing I and A with the help ofKoopmans theorem and Hyperchem 8.0 professio-nal program [55]. There after, we have divided theevaluated gAB of Eq. (23) by (14.4/RAB) and theRAB’s were taken from the reference [56]. There af-ter, we have taken mean of several C’s and thebest value is obtained for C ¼ 0.75.

Just now, we have a semi-empirical ansatz, Eq.(26) to evaluate the hardness of heteronuclear dia-tomic molecules. We have derived this formulaon the basis of the assumption that, similar to

electronegativity equalization principle, there isscientifically valid hardness equalization princi-ple. The ansatz (26) can be invoked to evaluatethe hardness of diatomic molecule. As hardnesshas no experimental benchmark, we cannot per-form any validity test of the hardness equalizationprinciple and the newly derived algorithm ofcomputing molecular hardness on the basis ofhardness equalization principle. However, wehave performed a validity test of our suggestedansatz by invoking the approximate and

FIGURE 1. Plot of theoretically computed and quantum mechanically computed hardness of alkali halides. [Colorfigure can be viewed in the online issue, which is available at www.interscience.wiley.com.]AQ6

TABLE IIThe semi-empirically evaluated hardness and the abinitio SCF hardness of some inter halogencompounds

Interhalogencompound

g(ab initio)in eV

g(semi-empirical)

in eV


IF 7.07391 5.655097785 1.909781ICl 5.897187 4.653357404 2.320905IBr 5.859374 4.346424523 2.484801BrF 8.182237 6.151227939 1.755747BrCl 8.131913 5.055964376 2.136091ClF 8.793004 6.632552827 1.628332



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There after, we have divided the evaluated ηAB of Eq. (23) by (14.4/RAB). The RAB’s were taken from the reference [56].

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operational formula of Parr and Pearson vide Eqs.(6) and (7). We have calculated the molecularhardness of as many as four different series ofcompounds eg. (i) alkali halides, (ii) interhalo-gens, (iii) hydrogen halides, and (iv) silver hal-ides, with diverse physical and chemical nature,using the spectroscopic internuclear bond distan-ces [56] of the compounds and the ansatz (26).The hardness of all such molecules is computedby invoking an ab initio Hartree-Fock quantummechanical method. To be very specific, to com-pute the ab initio quantum chemical hardnessesof the alkali halides, silver halides, hydrogen hal-ides, and 1:1 type interhalogen compounds, thegeometries of the corresponding molecules havebeen optimized with minimal STO-3G basis setusing Hyperchem 8.0 professional program [55].The equilibrium hardnesses labeled as ab initiohardness of the compounds are calculated usingPearson’s formula: g ¼ (�eHOMO þ eLUMO)/2.

The computed results of the alkali halides, sil-ver halides, hydrogen halides, and 1:1 type inter-

halogen compounds, and the computed ab initiohardnesses along with their spectroscopic equilib-rium bond distances are presented in Table T1I–IV,respectively.

Results and Discussion

To explore the efficacy of our suggested ansatzof computing molecular hardness, we have com-pared the set of hardness data of various mole-cules computed through our suggested ansatz visa vis a set of hardness data of such molecules

TABLE IIIThe semi-empirically evaluated hardness and the abinitio SCF hardness of hydrogen halides

Moleculeg (ab initio)

in eV

g (semi-empirical)

in eV


HF 14.32595 11.77984773 0.91682HCl 10.8576 8.473434645 1.274572HBr 9.998034 7.634359964 1.414657HI 8.656397 6.711709696 1.609128

FIGURE 3. Plot of theoretically computed and quan-tum mechanically computed hardness of hydrogen hal-ides. [Color figure can be viewed in the online issue,which is available at www.interscience.wiley.com.]

FIGURE 2. Plot of theoretically computed and quan-tum mechanically computed hardness of interhalogens.[Color figure can be viewed in the online issue, which isavailable at www.interscience.wiley.com.]

FIGURE 4. Plot of theoretically computed and quan-tum mechanically computed hardness of silver halides.[Color figure can be viewed in the online issue, which isavailable at www.interscience.wiley.com.]





computed by invoking an ab initio quantumchemical method.

A close look in to the Table I and FigureF1 1reveals that there is a very good correlationbetween the semi-empirically evaluated molecularhardness of the Lithium halides with the resultsof ab initio quantum chemical calculation. Similartrend is observed in the variation of the hardnessof the Potassium halides. But it also transpiresfrom the comparative study that in few cases, inTable I and Figure 1, that the correlation betweenthe two sets of hardness data is poor. But it is dis-tinct from the TableT2,F2,T3 II and Figure 2, Table III andFigureF3,T4,F4 3, and Table IV and Figure 4 that there is avery good correlation between the molecularhardness of a series of interhalogens, hydrogenhalides and silver halides compounds evaluatedthrough the suggested ansatz and the quantummechanical method..

Conclusions

In this work, we have basically launched asearch whether there is hardness equalizationprinciple for molecules similar to electronegativityequalization. We are tempted by the fact that theelectronegativity equalization principle is widelyaccepted and theoretically justified and also wehave noticed that there is much communality inthe basic philosophy of the origin of the two fun-damental descriptors—the electronegativity andthe hardness of atoms. Starting from the empiricalradial dependent formula of computing hardnessof atoms suggested by us, we have derived anansatz of the molecular hardness assuming thathardness equalization principle is operative andjustifiably valid. We have applied the suggestedansatz to compute the hardness of as many asfour sets of compounds with widely divergent

physical and chemical properties. Since hardnessis not an experimentally measurable property,there is no benchmark to perform any validitytest of our computed data. We have, therefore,computed the hardness data of these four differ-ent sets of compounds invoking the formula ofParr and Pearson, (I � A)/2, and to compute Iand A, we have invoked Koopmans’ theorem andan ab initio quantum chemical method to com-pute I and A of such molecules. There is closecorrelation between the four different sets ofhardness data evaluated through the semi-empiri-cal ansatz suggested in present work, andthrough the ab initio quantum chemical method.The detailed study suggests that the hardnessequalization principle is a likely valid theoreticalconstruct similar in nature to the electronegativityequalization principle, and the algorithm derivedfor the evaluation of molecular hardness is scien-tifically acceptable.

References

1. Mulliken, R. S. J Am Chem Soc 1952, 74, 811.

2. (a) Pearson, R. G. J Am Chem Soc 1963, 85, 3533; (b) Pear-son, R. G. Science 1966, 151, 172.

3. Klopman, G. J Am Chem Soc 1964, 86, 1463; (b) Klopman,G. J Am Chem Soc 1968, 90, 223.

4. Hohenberg, P.; Kohn, H. Phys Rev 1964, 136, 864.

5. Parr, R. G.; Yang, W. Density Functional Theory of Atomsand Molecules; Oxford University Press, 1989. AQ3

6. Gyftpoulous, E. P.; Hatsopoulos, G. N. Proc Natl Acad SciUSA 1968, 60, 786.

7. Iczkowski, R. P.; Margrave, J. L. J Am Chem Soc 1961, 83,3547.

8. Parr, R. G.; Donnelly, R. A.; Levy, M.; Palke, W. E. J ChemPhys 1978, 68, 3801.

9. Parr, R. G; Pearson, R. G. J Am Chem Soc 1983, 105, 7512.

10. Perdew, J. P.; Parr, R. G.; Levy, M.; Balduz, J. L., Jr.; PhysRev Lett 1982, 49, 1691.

11. Yang, W. T.; Zhang, Y. K.; Ayers, P. W. Phys Rev Lett2000, 84, 5172.

12. Sen, K. D.; Vinayagam, S. C. Chem Phys Lett 1988, 144,178.

13. Pearson, R. G. Proc Natl Acad Sci USA 1986, 83, 8440.

14. Reed, J. L. J Phys Chem A 1997, 101, 7396.

15. Geerlings, P.; Proft, F. D.; Langenaeker, W. Chem Rev2003, 103, 1793.

16. (a) Pearson, R. G. J Chem Educ 1987, 64, 561; Pearson, R.G. Acc Chem Res 1993, 26, 250.

17. Torrent-Sucarrat, M.; Luis, J. M.; Duran, M.; Sola, M. JChem Phys 2002, 117, 10561.

TABLE IVThe semi-empirically evaluated hardness and the abinitio SCF hardness of silver halides

Moleculeg (ab initio)

in eV

g (semi-empirical)

in eV


AgF 5.385806 5.445736014 1.983203AgCl 5.163199 4.735141193 2.280819AgBr 5.18045 4.512903142 2.393138AgI 5.179225 4.244196945 2.544651



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5. Parr, R. G.; Yang, W. Density Functional Theory of Atoms and Molecules; Oxford University Press, Oxford ,1989.

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Comment on Text

.

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18. Torrent-Sucarrat, M.; Luis, J. M.; Duran, M.; Sola, M. J AmChem Soc 2001, 123, 7951.

19. Chattaraj, P. K.; Sengupta, S. J Phys Chem 1996, 100, 16126.

20. Zhou, Z.; Parr, R. G. J Am Chem Soc 1991, 113, 5720.

21. Parr, R. G.; Chattaraj, P. K. J Am Chem Soc 1991, 113, 1854.

22. Chattaraj, P. K.; Nath, S.; Sannigrahi, A. B. J Phys Chem1991, 98, 9143.

23. Pearson, R. G.; Palke, W. E. J Phys Chem 1992, 96, 3283.

24. Pal, S.; Vaval, N.; Roy, R. J Phys Chem 1993, 97, 4404.

25. Chattaraj, P. K.; Liu, G. H.; Parr, R. G. Chem Phys Lett1995, 237, 171.

26. Ayers, P. W.; Parr, R. G. J Am Chem Soc 2000, 122, 2010.

27. (a) Ghosh, D. C.; Jana, J.; Biswas, R. Int J Quantum Chem2000, 80, 1; Ghosh, D. C.; Jana, J.; Bhattacharyya, S. Int JQuantum Chem 2002, 87, 111

28. Chattaraj, P. K.; Lee, H.; Parr, R. G. J Am Chem Soc 1991,113, 1855.

29. Chattaraj, P. K.; Ayers, P. W.; Melin, J. Phys Chem ChemPhys 2007, 9, 3853.

30. Chattaraj, P. K.; Ayers, P. W. J Chem Phys 2005, 123,086101.

31. Ayers, P. W. J Chem Phys 2005, 122, 141102.

32. Ayers, P. W.; Parr, R. G.; Pearson, R. G. J Chem Phys 2006,124, 194107.

33. Frenking, G.; Krapp, A. J Comput Chem 2007, 28, 15.

34. Ghosh, D. C.; Islam, N. Int J Quantum Chem, in press.AQ4

35. Yang, W.; Lee, C.; Ghosh, S. K. J Phys Chem 1985, 85,5412.

36. Nalewajski, R. F. J Phys Chem 1985, 89, 2831.

37. Dutta, D. J Phys Chem 1986, 90, 4211.

38. Ayers, P. W. Faraday Discuss., 2007, 135, 161.

39. Putz, M. V. Absolute and Chemical Electronegativity andHardness; Nova Science Publishers Inc.: New York, 2008.

40. Sanderson, R. T. Science 1951, 114, 670.

41. De Proft, F.; Langenaeker, W.; Geerlings, P. J. Mole Struct(Theochem) 1995, 339, 45.

42. Zhou, Z.; Parr, R. G. J Am Chem Soc 1990, 112, 5720.

43. Chattaraj, P. K.; Nath, S. Proc Indian Acad Sci (Chem Sci)1994, 106, 229.

44. Cardenas-Jiront, G. I.; Toro-Labbe, A. J Phys Chem 1995,99, 12730.

45. Chattaraj, P. K.; Ayers, P. W. J Chem Phys 2005, 123,86101.

46. Berkowitz, M.; Ghosh, S. K.; Parr, R. G. J Am Chem Soc1985, 107, 6811.

47. Ghosh, S. K.; Berkowitz, M.; Parr, R. G. Proc Natl Acad SciUSA 1984, 81, 8028.

48. Ayers, P. W.; Parr, R. G. J Chem Phys 2008, 128, 184108.

49. Ayers P. W.; Parr, R. G. J Chem Phys 2008, 129, 054111.

50. Pearson, R. G. J Am Chem Soc 1988, 110, 7684.

51. Feynman, R. P.; Leighton, R. B.; Sands, M. The FeynmanLecture on Physics, Vol 2; Addison-Wesley: Mass, 1964.

52. Ghosh, D. C.; Biswas, R. Int J Mol Sci 2002, 3, 87.

53. Ghosh, D. C.; Islam, N. Int J Quantum Chem, in press.DOI: 10.1002/qua. 22202.

54. Pasternak, A. Chem Phys 1977, 26, 101.

55. Hypercube, Inc. Hyperchem, 8.0.6; Hypercube, Inc, 2008.

56. Lovas, F. J.; Tiemann, E. J Phys Chem Ref Data 1974, 3, 609.

57. (a) Ghanty, T. K.; Ghosh, S. K. J Phys Chem 1996, 100,12295; (b) Ghanty, T. K.; Ghosh, S. K. J Phys Chem 97,1993, 4951.

58. Dutta, D. J Phys Chem 1986, 90, 4211. AQ5





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53. Ghosh, D. C.; Islam, N. Int J Quantum Chem,109,2009, in press. DOI: 10.1002/qua. 22202.

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34. Ghosh, D. C.; Islam, N. Int J Quantum Chem,109,2009, in press. .

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hardness equalization principle and molecular hardness

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