hardness and electronegativity

Determination of Some Descriptors of theReal World Working on the FundamentalIdentity of the Basic Concept and theOrigin of the Electronegativity and theGlobal Hardness of Atoms, Part 1:Evaluation of Internuclear Bond Distanceof Some Heteronuclear Diatomics

DULAL C. GHOSH, NAZMUL ISLAMAQ2AQ3 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.22500

ABSTRACT: In our previous report, we have posited that there is much conceptualcommonality between the two fundamental theoretical descriptors of chemistry andphysics—the electronegativity and the hardness and both the fundamental descriptorsoriginate from the same source—the electron attracting power of the screened nucleusupon the valence electrons. We have discovered the surprising result that if one measureshardness, the electronegativity is simultaneously measured and vice versa. In the presentreport, we have calculated the internuclear bond distances of a series of diatomic moleculesinvoking the ansatz for computing such bond distances in terms of the electronegativities(v) of the atoms. We have just substituted the electronegativities (v) by the global hardness(g) in the well established ansatz for computing bond distances of Ray et al. We havefound that the computed internuclear distances are very close to the experimental bonddistances. This is one validity test of our hypothesis that the electronegativity and globalhardness are two different labels or legends of one and the same fundamental property ofthe atoms. VC 2009 Wiley Periodicals, Inc. Int J Quantum Chem 000:000–000, 2009

Key words: electronegativity and hardness; descriptors of the real world; internucleardistance; commonality between the electronegativity and the hardness; electronegativityequalization principle

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

J_ID: ZQZ Customer A_ID: QUA22500 Cadmus Art: QUAT 22500 Ed. Ref. No.: 2009-0411.R1 Date: 17-November-09 Stage: Page: 1

ID: kumarik I Black Lining: [ON] I Time: 16:12 I Path: N:/3b2/QUAT/Vol00000/090440/APPFile/C2QUAT090440

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

Introduction

I n our previous report [1], we have establishedthe commonality in conceptual structure, phil-

osophical basis, and operational significance ofthe two fundamental theoretical descriptors ofphysics and chemistry—the hardness and theelectronegativity. To justify our statement that‘‘the hardness and the electronegativity are theone and the same in their basic scientific natureof origin and development,’’ in our previousreport, we have substituted the set of evaluatedhardness [1]AQ4 of the atoms of 103 elements of theperiodic table as a scale of electronegativity andfound that such set of pseudo electronegativitydata satisfies the sine qua non of a satisfactoryscale of electronegativity [2–5]. Our presumptionis that if one measures hardness, the electronega-tivity is simultaneously measured and vice versa.We can cite the density functional definition andalgorithm for the evaluation of the electronegativ-ity and hardness by Parr et al. [6, 7] to adduceevidence that supports our presumption. For asystem of N electrons with ground state energyE[N,v], where v is the external potential acting onan electron due to the presence of nucleus, thechemical potential of the electrons l (the negativeof the electronegativity, v).

l ¼ ð@E=@NÞv ¼ �v (1)

The absolute hardness is defined as the sensi-tivity of l for a change in the number of elec-trons;

g ¼ ð@2E=@N2Þv ¼ ð@l=@NÞv (2)

The operational and approximate formulae [7]of these two fundamental descriptors are

v ¼ ðI þ AÞ=2 (3)

and

g ¼ ðI � AÞ=2 (4)

where I and A are the first ionization potentialand electron affinity of the chemical species.

Although very accurate values of I are avail-able, values of A are either very small tending tozero, or in most cases are unknown. It, therefore,transpires that if we neglect A, like many other,

from Eqs. (3) and (4), the simplified equationslook like

v ¼ I=2 (5)

g ¼ I=2 (6)

Thus, we see that the effort of quantification ofthe hardness and the electronegativity in terms ofdensity functional theory degenerates to the givenequation declaring equality of v, g, which is ourcontention from a different viewpoint.

v ¼ g (7)

Since we are after to educe evidence that thebasic nature of the electronegativity and the hard-ness are fundamentally qualitative per se andoperationally the same entity, we may site similarviews expressed by others within the purview ofdensity functional theory. Pearson [8] suggestedthat for donor atoms, the electronegativity can betaken as a measure of the hardness of the base.Putz [9] after rigorous research on systematic for-mulation of electronegativity and hardness,opined out that the hardness and electronegativityare proportional to each other.

v / g (8)

Ayers [10] in his effort to evaluate the electro-negativity and hardness of neutral atoms on thebasis of the energy expression of March andWhite [11], pointed out that the two fundamentalatomic parameters, hardness and electronegativ-ity, have the similar expression, more precisely,proportional to each other.

Recently, we have noticed another suggestioncorroborating our view that the electronegativityand the hardness are two different appearances ofthe one and the same fundamental property ofatoms and molecules. Xue and coworkers [12]hold that ‘‘electronegativity represents the elec-tron holding power of an atom; the stiffness ofthe atoms can thus be defined as electron holdingenergy of atoms per unit volume.’’

It is a fact that neither electronegativity norhardness is physically observable. Hence, thesetwo fundamental descriptors are neither experi-mentally measurable nor quantum mechanicallydeterminable quantities.

They occur in the domain of the hypothesis.They exist but never seen. We may refer to theopinion of Parr et al. [13] who seem to have



GHOSH AND ISLAM

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

connected the reality of the hardness and the elec-tronegativity with the noumenon of Kantian phi-losophy. The noumenon is an object knowable bythe mind or intellect, not by the senses. Thus, thehardness and/or electronegativity is an object ofpurely intellectual intuition. We feel it pertinentto recall the opinion of Ayers [10] that before anyalgorithm of computing the hardness and theelectronegativity is developed, the reification ofabstract concept into things of the real world isnecessary.

From the previous discussions, it transpires thatthere is no benchmark to perform any validity testof any scale of measurement of the hardness or theelectronegativity. But some well-known chemico-physical descriptors of molecules are conceived interms of the electronegativity. Such descriptors areinternuclear distance, charge distribution, dipolemoment, atomic polar tensors, bond energy, forceconstant, etc. Although both the electronegativityand the hardness are conceptual constructs andoccur in mind, these descriptors occur in the realworld. We, therefore, feel it necessary and expedi-ent to perform some validity test to justify ourassumption relating the electronegativity and thehardness to one unified single principle.

Because these are not the experimental quanti-ties, some electronegativity or hardness depend-ent descriptors of the real world should be com-puted in terms of the hardness and/or theelectronegativity.

It is widely known that such descriptors likethe bond distances, bond energies, bond polar-ities, dipole moments, and force constants arecomputed through ansatz suggested on the basisof electronegativity equalization principle [14]. Inthis report, we have exploited our statement thatg ¼ v to calculate the internuclear bond distanceof a series of diatomic molecules in terms of thehardness. We have exploited the available ansatzderived on the basis of electronegativity equaliza-tion principle [14] by Ray et al. [15].

The diatomic molecules have gained increasedinterest [16] over the past years in both experi-mental and theoretical studies because of theirimportance in astrophysical process and manychemical reactions. Since the internuclear distanceis directly related to the bond dissociation energy,which has prime importance in astrophysicalstudies, we have invoked our assumption that v¼ g to evaluate the internuclear distances of somediatomic molecules of diverse physico-chemicalnature.

Method of Computation

Now, let us consider the formation of a dia-tomic molecule AB from its constituent atoms Aand B as follows:

Aþ B ! AB (9)

Let the equilibrium bond length, the electrone-gativity of the molecule AB, and the electronega-tivities of the corresponding atoms A and B arerAB, vAB, vA, and vB, respectively. Now let usimagine that, after the formation of the molecule,a point charge is located at a distance r1 from Aand r2 from B with r1 þ r2 ¼ rAB.

When atoms approach to form the molecule,the electron density function over the whole spaceundergoes rearrangement. Thus, there is aninteratomic charge transfer and rearrangement inheteronuclear molecules as a consequence of thephysical process of electronegativity equalization.Let the electronegativities of the atom A and B inthe molecule AB are v0A and v0B, respectively. Theelectronegativity equalization principle of Sander-son [14] provides:

vAB ¼ v0A ¼ v0B (10)

Now, on the basis of simple bond chargemodel [17–19], Ray et al. [15] derived the internu-clear bond distances of diatomic molecules usingthe concept of electronegativity equalization [14]and the zero order approximation of Pasternak[20] as follows:

rA ¼ r1 and rB ¼ r2

RAB ¼ ðrA þ rBÞ� fðrArBðv1=2A � v1=2B Þ2g=ðvA rA þ vB rBÞ ð11Þ

where, rA and rB are the covalent radii of theatom A and B, respectively, vA and vB are theelectronegativities of the atoms A and B, respec-tively, and rAB is the internuclear bond distanceof the molecule, AB. We may mention that Kim[21] calculated the internuclear bond distances ofsome selected diatomics invoking the ansatz (11).

In our previous report [1], we have concludedthat the electronegativity and the hardness areconceptually and fundamentally same entity.

Now, in a venture to justify our [1] statement‘‘The electronegativity and the hardness are twodifferent appearances of the one and the same



ELECTRONEGATIVITY AND THE GLOBAL HARDNESS OF ATOMSAQ1

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

fundamental property of atoms’’ we have simplyreplaced the electronegativity (v) from Eq. (11) bycorresponding hardness (g) and the equationlooks:

RAB ¼ ðrA þ rBÞ � ½fðrArBðg1=2A � g1=2B Þ2g=ðgA rA þ gB rBÞ� ð12Þ

Using the above ansatz (12), we have calcu-lated the internuclear bond distances of as manyas 10 different series of compounds, e.g., (i) Alkalihalides, (ii) Interhalogens, (iii) Hydrogen halides,(iv) Silver halides, (v) Thallium halides, (vi) Alu-minum halides, (vii) Alkali metal heteronucleardimers, (viii) A few diatomic double bonded spe-cies, (ix) Some binary metal oxides, and (x) some

Ge and Pb compounds with diverse physical andchemical nature.

The calculated internuclear bond distances of10 different series of compounds vis-a-vis theirexperimental and other theoretically evaluatedinternuclear distances are present in Tables asfollows

Table T1I: alkali halides, Table T2II: hydrogen hal-ides, Table T3III: silver halides, Table T4IV: aluminumhalides, Table T5V: some diatomic double bondedspecies, Table T6VI: some binary metal oxides, Table

T8VII: some group 16 compounds of Pb, Table T7VIII:group 16 compounds, Table T9IX: thallium halidemolecules, and Table T10X: alkali metal dimers.

We made a comparative study of our calcu-lated internuclear distances of some diatomic mol-ecule vis-a-vis those evaluated by Kim [13] AQ5usingEq. (12) and the corresponding spectroscopiccounter parts in Figure F11.

We have calculated the internuclear distancesof all the molecules under study through Eq. (11)using the electronegativities of atoms publishedby Ghosh [22]. Such internuclear distances arecompared with those evaluated through Eq. (12)using the global hardness evaluated by us [1] inTable T11XI and in Figure F22. We have plotted theevaluated bond lengths against the correspondingexperimental bond lengths in to evaluate the

TABLE IComputed internuclear distance (RAB) of alkalihalide molecules vis-a-vis their spectroscopicinternuclear distance (RSpect).

Molecule RAB (A) RSpect (A)

LiF 1.763439 1.5638785LiCl 2.206 2.0206913LiBr 2.348567 2.021491LiI 2.547066 2.391944NaF 1.971822 1.9259692NaCl 2.422102 2.3608225NaBr 2.563565 2.5020676NaI 2.76246 2.391944KF 2.331186 2.1714824KCl 2.79961 2.666683KBr 2.937795 2.820809KI 3.137491 3.0478794RbF 2.421249 2.2703609RbCl 2.900984 2.786769RbBr 3.036612 2.9447792RbI 3.236653 3.1769183

TABLE IIComputed internuclear distance (RAB) of hydrogenhalide molecules vis-a-vis their spectroscopicinternuclear distance (RSpect).


HF 0.953967 0.91682HCl 1.309479 1.274572HBr 1.459547 1.414657HI 1.648661 1.609128

TABLE IIIComputed internuclear distance (RAB) of silverhalide molecules vis-a-vis their spectroscopicinternuclear distance (RSpect).


AgF 1.809338 1.983203AgCl 2.228455 2.280819AgBr 2.374759 2.393138AgI 2.571194 2.544651

TABLE IVComputed internuclear distance (RAB) of Aluminumhalides vis-a-vis their spectroscopic internucleardistance (RSpect).


AlF 1.801466 1.6543883AlCl 2.216236 2.130191AlBr 2.363077 2.294886AlI 2.559053 2.5371326



GHOSH AND ISLAM


correlation coefficients, R2, and presented them inTableT12 XII.

Results and Discussion

In our effort to explore the scientific validity ofour hypothesis that ‘‘the hardness and the electro-negativity are the one and the same in their basicscientific nature of origin and development, i.e., v¼ g,’’ we have computed internuclear bond dis-tances through the suggested ansatz (11) of 10 dif-ferent sets of molecules. We have performed acomparative study of the computed bond distan-ces vis-a-vis the spectroscopic internuclear bonddistance data [23] of such molecules. The com-puted bond distances and their experimentalcounter parts are presented in the Tables I–IX. Inthe case of the alkali metal dimer, due to theunavailability of spectroscopic internuclear bonddistance data, we have compared our calculateddata with the bond distance data published byPyykko and Atsumi [24] in Table X.

A close look in to all the Tables I–IX revealsthat there is a very good correlation between theinternuclear bond distances of the seven sets ofmolecules computed through our suggested

ansatz and their spectroscopically evaluated coun-ter parts. It is surprising to note that all the setsof calculated bond distances through empiricalansatz are numerically very close to those eval-uated by sophisticated spectroscopic method.Theory predicts and experiment proves. Here, weget an example of very good correlation betweentheoretical prediction and experimental determi-nation. We further note by a deeper scrutiny ofthe size data in Table X that the computed inter-nuclear bond distance values beautifully correlatewith the corresponding values published byPyykko and Atsumi [24]. Pyykko and Atsumievaluated the bond distances of some selectedmolecules through reportedly accurate method.We should refer to a similar calculation of Kim[13] by invoking the ansatz (11). In Figure 1, wehave extra plotted the distances of some selectedmolecules evaluated by Kim [21] invoking ansatz(11) and the bond distances of such moleculesevaluated through ansatz (12) along with theirspectroscopically determined counter parts. Aclose look into Figure 1 reveals that, in majorityof cases, the bond distance values of the presentcalculation have closer agreement with the sophis-ticated spectroscopic bond distances compared tothose bond distances evaluated by Kim. A look atthe Figure 1 make it transparent that the internu-clear distances of the majority of compounds ofpresent calculation fall upon the experimental/spectroscopic distances. Or in other words,

TABLE VComputed internuclear distance (RAB) of somediatomic double bonded species vis-a-vis theirspectroscopic internuclear distance (RSpect).


CS 1.77929 1.53496NO 1.338079 1.15074ClO 1.610616 1.596NP 1.804524 1.4908839NS 1.732824 1.4941CO 1.370651 1.12833632

TABLE VIComputed internuclear distance (RAB) of somebinary metal oxides vis-a-vis their spectroscopicinternuclear distance (RSpect).


PbO 1.832125 1.9218359BaO 1.960454 1.9397119GeO 1.807071 1.624667SnO 1.985981 1.8325271

TABLE VIIComputed internuclear distance (RAB) of somegroup 16 compounds of Pb vis-a-vis theirspectroscopic internuclear distance (RSpect).


PbS 2.321792 2.2868898PbSe 2.436137 2.4022637PbTe 2.645682 2.595006

TABLE VIIIComputed internuclear distance (RAB) of somegroup 16 compounds of Ge vis-a-vis theirspectroscopic internuclear distance (RSpect).


GeS 2.237369 2.0120982GeSe 2.365658 2.134651GeTe 2.567979 2.3401928



ELECTRONEGATIVITY AND THE GLOBAL HARDNESS OF ATOMS


present theoretical effort appears to be betterpoised compared to the effort of Kim.

Now, let us dwell upon the bond distancesevaluated through Eq. (11) using the electronega-tivity and that through Eq. (12) using the globalhardness. It is distinct that, although the bondlengths are not just equal in magnitude for allcases but both sets of lengths are highly corre-lated. The position is more revealing in Figure 2.

Now, we further analyze our result to explorebetter correlation between experimental and theo-retically evaluated bond lengths in terms of thecomputed correlation coefficients, R2. The correla-tion coefficients, R2, for the sets of moleculesstudied are presented in Table XII. A glance atthe Table XII reveals that, except the molecules in

Tables V and VI, there is beautiful correlationbetween the theoretical and experimental bondlengths. The apparent anomaly in molecules inTables V and VI is not unexpected because mole-cules here are doubly bonded.

Conclusion

In this work, we have basically launched asearch whether our hypothesis that ‘‘The electro-negativity and the hardness are two different

TABLE IXComputed internuclear distance (RAB) of thalliumhalide molecules vis-a-vis their spectroscopicinternuclear distance (RSpect).


TlF 1.753797 2.0844623TlCl 2.214894 2.4848554TlBr 2.353686 2.6182148TlI 2.553232 2.813709

TABLE XComputed internuclear distance (RAB) of alkali metaldimers vis-a-vis their internuclear distance ofPyykko and Atsumi (RPyykko).

Molecule RAB (A) RPyykko (A)

NaLi 2.879851 2.885KLi 3.28992 3.331915RbLi 3.427362 —LiCs 3.317776 3.6681NaK 3.50948 3.499035NaRb 3.645462 3.6435NaCs 3.506469 3.85063KRb 4.057802 4.0685

FIGURE 1. Comparative plot of the bond distances evaluated through the present method, method of Kim, and thecorresponding spectroscopic counter parts. [Color figure can be viewed in the online issue, which is available atwww.interscience.wiley.com.]AQ9



GHOSH AND ISLAM


appearances of the one and the same fundamentalproperty of atoms’’ can be justified by applicationin the real world? We have computed the internu-clear distances of a number of diatomics as onedescriptor of the real world. We rely upon theansatz for the evaluation of the equilibrium inter-nuclear bond distance by Ray et al. in terms ofelectronegativity. We have just replaced the elec-tronegativity by the hardness of the atoms andrearranged the equation in terms of global hard-ness of the atoms. We have applied the ansatz so

rearranged to compute the internuclear bond dis-tances of as many as 10 sets of compounds withwidely divergent physical and chemical proper-ties. We found surprising results that the bonddistances evaluated through an ansatz obtainedby replacing electronegativity by hardness havefairly close agreement with those determined bysophisticated spectroscopic methods. The detailedcomparative study suggests that the assumptionthat the electronegativity and the hardness aremanifest two different descriptors of the one and

TABLE XIComparative study of the calculated internucleardistance (RAB) of some diatomic molecules vis-a-visKim (RKim) and spectroscopic internuclear distance(RSpect).

Molecule RAB (A) RKim (A) RSpect (A)

LiF 1.76344 1.857 1.56388LiCl 2.206 2.188 2.02069NaF 1.97182 2.025 1.92597NaCl 2.4221 2.36 2.36082KCl 2.79961 2.684 2.66668HF 0.95397 1.069 0.91682HCl 1.30948 1.363 1.27457HBr 1.45955 1.511 1.41466HI 1.64866 1.704 1.60913

FIGURE 2. Internuclear distance between heteronuclear diatomic molecules computed through electronegativityand hardness, respectively. [Color figure can be viewed in the online issue, which is available at www.interscience.wiley.com.]

TABLE XIIComputation of the correlation coefficients, R2, byplotting theoretically evaluated bond lengths againstthe experimentally evaluated bond lengths.

System ofmolecules in table

The correlationcoefficients, R2

1 0.95912 0.99983 0.99954 0.99975 0.76726 0.25547 0.99948 0.99979 0.99910 0.813



ELECTRONEGATIVITY AND THE GLOBAL HARDNESS OF ATOMS


the same fundamental property of atoms isjustified.

References

1. Ghosh, D. C.; Islam, N. Int J Quantum Chem (in press);DOI: 10.1002/qua. 22202.AQ6

2. Ghosh, D. C. J Indian Chem Soc 2003, 80, 527.

3. Ghosh, D. C.; Chakraborty, T. R. J Mol Struct (THEO-CHEM) 2009, 906, 87.

4. Murphy, L. R.; Meek, T. L.; Allred, A. L.; Allen, L. C. JPhys Chem A 2000, 104, 5867.

5. Allen, L. C. J Am Chem Soc 1992, 114, 1510.

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

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

8. Pearson, R. G. Chem Commun 1968, 2, 65.AQ7

9. Putz, M. V. Absolute and Chemical Electronegativityand Hardness; Nova Science Publishers, Inc.: New York,2008.

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

11. March, N. H.; White, R. J. J Phys B 1972, 5. AQ8

12. Li, K.; Wang, X.; Zhang, F.; Xue, D. Phys Rev Lett 2008,100, 235504.

13. Parr, R. G.; Ayers, P. W.; Nalewajski, R. F. J Phys Chem A2005, 109, 3957.

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

15. Ray, N. K.; Samuels, L.; Parr, R. G. J Chem Phys 1979, 70,3680.

16. Noorizadeh, S.; Shakerzadh, E. J Mol Struct 2009, 920,110.

17. Parr, R. G.; Borkman, R. F. J Chem Phys 1967, 46,3683.

18. Borkman, R. F.; Parr, R. G. J Chem Phys 1968, 48,1116.

19. Parr, R. G.; Borkman, R. F. J Chem Phys 1968, 49, 1055.

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

21. Kim, K. Bull Chem Soc 1987, 8, 432.

22. Ghosh, D. C. J Theor Comput Chem 2005, 4, 21.

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

24. Pyykko, P.; Atsumi, M. Chem Eur J 2009, 15, 186.



GHOSH AND ISLAM


hardness and electronegativity

Documents