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The SciTech, Journal of Science & Technology Vol-1, Issue-1, p-40-50, 2012. Article information: Article Received: 16 March 2012 Article Accepted: 18 March 2012 Article Online: 16 April 2012 Journal homepage: www.thescitechpub.com Article No: STJST201211002 DOI:………………………….. Research Article RELATIONSHIP BETWEEN ATOMIC ORBITAL EXPONENT AND ATOMIC HARDNESS N. Islam Department of Basic Science & Humanities/ Chemistry, Techno Global- Balurghat, Balurghat-733103, India. Email: [email protected] Fax: +91-3522271101 Abstract: Considering hardness of atom as the attraction of a nucleus upon the valence shell electron, in this paper a simple approach for the calculation of the global hardness of atoms in terms of their orbital exponents is presented. The basic tenet of the present method is to use our newly computed atomic orbital exponent values to compute the global hardness of the atoms. The express periodicity of periods and groups of periodic table exhibited by the computed atomic hardness, correlation with some other existing hardness data, correlation of the some of the most important physico-chemical properties of atoms etc speak volume of the efficacy of the present method in computing atomic global hardness. Keywords: effective nuclear charge, effective principal quantum number orbital exponents, global hardness, periodicity. η Ghosh Islam(ev) H Li B N F Na Al P Cl K Sc V Mn Co Cu Ga As Br Rb Y Nb Tc Rh Ag In Sb I Cs La Pr Pm Eu Tb Ho Tm Lu Ta Re Ir Au Tl Bi At Fr Ac Pa Np Am Bk Es Md Lr Orbital exponent ξ Global hardness Graphical abstract: Atoms ©The SciTech Publishers, 2012. All Right Reserved 40 The SciTech, Journal of Science & Technology Vol-1, Issue-1, p.40-50, 2012. N. Islam Relationship between atomic orbital exponent & hardness.. Introduction: For the rationalization and prediction of various chemo-physical phenomena scientists are engaged in introducing new concepts. The law of nature is simple but subtle and that one of the main goals of the natural sciences is the formulation of simple models in concepts in terms of which the observed phenomenon can be classified, understood and finally described. The chemical hardness is one of the fundamental chemical properties of atoms and molecule relating to the intrinsic resistance toward the chemical reactivity of molecules, atoms and ions by holding the electron cloud tightly to the species [1,2].The concept of hardness was basically introduced to justify the acid base chemical interactions known as hard-soft acid base theory[2]. Chemical hardness fundamentally signifies the resistance towards the deformation or polarization of the electron cloud of the atoms, ions or molecules under small perturbation of charge cloud [1-12]. For a system of N electrons with ground state energy E [N,v] using the essence of the density functional theory (DFT)[3,13], Parr et al [14] showed that(1) µ=–χ = [∂E(ρ)/∂N]v where µ is the chemical potential, E(ρ) is the energy, N is the number of electrons and v is the external potential. The curvature of E vs. N curve, i.e. [∂2E(ρ)/∂N2] is identified as the hardness of the system[4] 2η = [∂2E(ρ)/∂N2] (2) It is now established that the rigorous evaluation of hardness η in terms of the Eq. (3) is difficult [6-12, 15, 16]. However, Parr and Pearson [4] invoking the calculus of finite difference approximation suggested an approximate and operational formula of hardness as under: (3) η=½(I-A) where I is the ionization potential and A is the electron affinity of atoms, ions or molecules By invoking the Hartree-Fock SCF theory and Koopmans’ theorem, Pearson [17] connected the hardness concept and the Hartree Fock SCF theory as under η = - [εHOMO - εLUMO] / 2 (4) where the orbital energy of the Highest Occupied Molecular Orbital (HOMO), εHOMO = –I and the orbital energy of the Lowest Unoccupied Molecular Orbital (LUMO), εLUMO = –A The effective nuclear charge: The effective nuclear charge (Zeff.) is the net positive charge experienced by an electron in a multi electronic atom[18]. The effective nuclear charge is somewhat lower than the actual nuclear charge (Z) and it is a parameter which characterizes any atom or ion. Thus, Zeff = Z – point charge at the nucleus representing the average electron repulsion. Or simply, Zeff = (Z-S) (5) where the screening constant or shielding constant, S, accounts for the average inner electronic repulsion on the outer electron of interest. ©The SciTech Publishers, 2012. All Right Reserved 41 The SciTech, Journal of Science & Technology Vol-1, Issue-1, p.40-50, 2012. N. Islam Relationship between atomic orbital exponent & hardness.. The effective nuclear charge, Zeff and the effective principle quantum number, n* are the two very important conceptual ptual terms. Both factors are opposing in nature. The effective nucleus nucleus- electron attraction is proportional to the effective nuclear charge and inversely proportional to the effective principal quantum number. The effective nuclear charge increases gradually lly while we are going across a period where the effective quantum number remains constant. Thus, along a period, the effective attraction depends mainly on the effective nuclear charge as other factor (n*) remains constant. But in the next period, they bo both th jump in number. The second factor, the effective principal quantum number is the dominating factor while we are going across a group. Thus the consideration of only effective nuclear charge to explain the periodicity of periods and groups is erroneous. The ratio of the effective nuclear charge and the effective principle quantum numbers i.e., Zeff : n* is known as the orbital exponent- physically represents the effective attraction power of the nucleus upon the electron. Thus the “effective nucleus –electron” lectron” attraction can be well documented by the orbital exponent. Thus, the orbital exponent, ξ is defined as asξ = Zeff/n* (6) In this work, we have proposed a simple formula of computing hardness of atoms based on the orbital exponent of the atoms. Radial dependent electrostatic formula of computing hardness of atoms: Pearson [19] pointed out that a simple formula of computing hardness of atoms could be derived from simple electrostatic considerations using atomic size. Dutta and Hati [20] have used the formula derived by Pearson to calculate the hardness of a series of atoms. We [6] derived the electrostati electrostatic formula for computing the hardness of atoms asη ∝ e2/2r where e is the electronic charge in esu and r is the ab absolute solute or most probable radius of atom in cm. (7) In atomic unit, this equation becomes η ∝1/2r (8) In a recent nt work, working on the electrostatic definition of the hardness and relying the classic Bohr model of hydrogenic atom, [21] we have explored a new route for calculating the global hardness of atoms using spectroscopy. The suggested spectroscopic atomic ha hardness model is η= e2/{(2e2/ hc )+ (4π2Zeff 2 e4m/h2 n*2)}] (9) where h is the Planck constant, c is the velocity of light, m is the mass of electron. Method of computation: Computation of atomic orbital exponents: Recently considering the pairing energy for p and d orbitals, Reed[15] modified Slater’s[18] grouping of orbital and proposed two set of rules for the computation of the screening co constants. ©The SciTech Publishers, 2012. All Right Reserved 42 The SciTech, Journal of Science & Technology Vol-1, Issue-1, p.40-50, 2012. N. Islam Relationship between atomic orbital exponent & hardness.. Now goaded by the periodic law and following Reed, we[22-24] have evaluated the effective nuclear charge and the orbital exponents for the atoms of the118 elements of the periodic Table with some modifications as under: We have considered Reed’s suggestions for the evaluation of screening constants of the s, p and d block elements and extended Reed’s rule for the evaluation of screening constants of f block elements. When electron entire in the 5f, 6p and higher we have used the contribution of 4f as 1. In the same shell f electrons shield each other by a factor 0.3228. It is significant to mention here that in order to evaluate the orbital exponent, we have used the Eq.(4) and the value of n* proposed by Slater 16 for n=1 to n=6 and for n=7, we have used the value of n*= 4.3 proposed by Ghosh and Biswas[25]. Now, the hardness refers to the resistance of the electron cloud of the atomic and molecular systems under small perturbation of electrical field. It is now well established that the hardness originate from the electron attracting power of the screened nuclear charge of the atom [6-12]. The attraction of a nucleus upon the valence shell electron is directly proportional to the effective nuclear charge (Zeff) and inversely proportional to its radius. Now, as the absolute radius or most probable radius is directly proportional to the effective principal quantum number (n*)[26], we can safely and reasonably conclude that the attraction of a nucleus upon the valence shell electron is inversely proportional to the n*. i.e., η ∝ Zeff and η ∝1/ n* At this point, to propose an algorithm to correlate the hardness, the effective nuclear charge and the effective principal quantum number, we may simply consider the expression as η ∝ Zeff /n* (10) or, η ∝ ξ (11) or, η = K (Zeff/n*) = K ξ (12) However, an equation can also be suggested for computing the hardness of the atoms based upon the conjoint action of the two parameters– the effective nuclear charge (Zeff) and the effective principal quantum number as follows– η = a (Zeff) + b (1/ n*) + c (13) where K is the correlation constant. It is constant throughout a period of the periodic table. Now as the aim of the paper is to establish a simple relation between two parameters– the global hardness and the orbital exponent of the atoms, we have relied upon the simple expression, the Eq.(12). At this point it is necessary to discuss here some analogous formulae for the computation of atomic hardness. Following the Slater’s definition[18] that the maximum of the radial density function of the orbital of the valence shell might be considered as a measure of the theoretical atomic radius and using the radial part of the Slater’s one-electron function (STO’s) we [26] derived the formula for calculation of theoretical radii (r) is derived as r = n*/ξ (14) where n* is the effective principal quantum number and n is the orbital exponent. ©The SciTech Publishers, 2012. All Right Reserved 43 The SciTech, Journal of Science & Technology N. Islam Vol-1, Issue-1, p.40-50, 2012. Relationship between atomic orbital exponent & hardness.. If we put the value of r (Eq 14) in to the electrostatic definition of hardness (Eq.10) the formula for hardness looks like η ∝ξ /2n* (15) Putting ξ = Zeff/n* in the above equation, we get η ∝ Zeff/n*2 (16) Based on the commonality in the basic philosophy of the origin and the operational significance of the electronegativity and the hardness, in a recent work [27], we have also proposed a new scale of computing global hardness of atoms asη = a ( Zeff /r ) + b (17) The global hardness data of the 103 elements of the periodic table is computed using the suggested algorithm, Eq.(12) and the computed orbital exponent data of the corresponding atoms. In order to evaluate K parameters for each period separately we have just divided the global hardness data of Ghosh and Islam [6] with our computed orbital exponent data of the all the elements present in that period. Result and Discussion: The computed orbital exponents of atoms of 118 elements of the periodic table are presented in Table-1. Table- 1: Computed orbital exponent of atoms of 118 elements of the periodic table Atom ξ Atom ξ Atom ξ Atom ξ Atom ξ H 1.00000 Mn 0.92241 Tb 2.16171 Ac 0.71398 Uut 3.10888 He 1.67720 Fe 0.94557 Dy 2.32295 Th 0.73391 Uuq 3.26637 Li 0.66340 Co 0.96873 Ho 2.48419 Pa 1.02895 Uup 3.42386 Be 1.00200 Ni 0.99189 Er 2.64543 U 1.18644 Uuh 3.58135 B 1.34060 Cu 1.01505 Tm 2.80667 Np 1.34393 Uus 3.73884 C 1.67920 Zn 1.03822 Yb 2.96790 Pu 1.63898 Uuo 3.89633 N 2.01780 Ga 1.22124 Lu 2.98831 Am 1.79647 O 2.35640 Ge 1.40427 Hf 3.00871 Cm 1.95395 1.97388 F 2.69500 As 1.58730 Ta 3.02912 Bk Ne 3.03360 Se 1.77032 W 3.04952 Cf 2.26893 Na 0.76907 Br 1.95335 Re 3.06993 Es 2.42642 Mg 0.99480 Kr 2.13638 Os 3.09033 Fm 2.58391 Al 1.22053 Rb 0.57680 Ir 3.11074 Md 2.74140 Si 1.44627 Sr 0.74610 Pt 3.13114 No 2.89888 P 1.67200 Y 0.75178 Au 3.15155 Lr 2.91881 S 1.89773 Zr 0.78895 Hg 3.17195 Rf 2.93874 Cl 2.12347 Nb 0.81038 Tl 0.92557 Db 2.95867 Ar 2.34920 Mo 0.83180 Pb 1.08681 Sg 2.97860 K 0.62357 Tc 0.85323 Bi 1.24805 Bh 2.99853 Ca 0.80659 Ru 0.87465 Po 1.40929 Hs 3.01847 Sc 0.82976 Rh 0.89608 At 1.56945 Mt 3.03840 Ti 0.85292 Pd 0.91750 Rn 1.73176 Uun 3.05833 V 0.87608 Ag 0.93892 Fr 0.53656 Uuu 3.07826 Cr 0.89924 Cd 0.96035 Ra 0.69405 Uub 3.09819 ©The SciTech Publishers, 2012. All Right Reserved 44 The SciTech, Journal of Science & Technology N. Islam Vol-1, Issue-1, p.40-50, 2012. Relationship between atomic orbital exponent & hardness.. Now, in order to justify the proposed algorithm, the Eq.(12), we have just plotted the computed orbital exponents and the Ghosh and Islam’s global hardness data[6] of atoms in Figure-1 η Ghosh Islam(ev) Orbital exponent Global hardness ξ H B F Al Cl Sc Mn Cu As Rb Nb Rh In I La Pm Tb Tm Ta Ir Tl At Ac Np Bk Md Atoms Figure-1: Comparative study of the computed orbital exponents and the Ghosh and Islam’s global hardness data of atoms A look at the Figure 1 revels that the nature of variation of the two periodic parameters are the same. Hence, the assumption that “the two fundamental atomic periodic parameters– the atomic hardness and the atomic orbital exponent are directly proportional to each other” is justified. The global hardness data for 118 elements of the periodic table, evaluated through the Eq.(12) along with the Ghosh and Islam’s hardness data[6] are presented in Table 2. ©The SciTech Publishers, 2012. All Right Reserved 45 The SciTech, Journal of Science & Technology N. Islam Vol-1, Issue-1, p.40-50, 2012. Relationship between atomic orbital exponent & hardness.. Table-2: Computed hardness in eV of atoms of 118 elements of the periodic table as a function of their atomic number Atom Hardness in eV η Ghosh Islam(ev) Atom Hardness in eV η Ghosh Islam(ev) Atom Hardness in eV η Ghosh Islam(ev) H 6.954876 6.429954 Nb 2.787379 2.825974 Tl 1.469533 1.704349 He 11.66472 12.54491 Mo 2.861073 2.92213 Pb 1.725531 1.941353 Li 2.278412 2.374587 Tc 2.934766 3.018371 Bi 1.98153 2.178492 Be 3.441315 3.496763 Ru 3.00846 3.114598 Po 2.237528 2.415812 B 4.604219 4.619009 Rh 3.082154 3.210756 At 2.491825 2.652778 C 5.767122 5.740979 Pd 3.155848 3.306947 Rn 2.749524 2.889955 N 6.930026 6.862467 Ag 3.229542 3.403195 Fr 1.004025 0.988253 O 8.092929 7.985436 Cd 3.303235 3.499376 Ra 1.298722 1.28195 F 9.255833 9.106475 In 3.885562 3.916369 Ac 1.336016 1.349725 Ne 10.41874 10.23034 Sn 4.467889 4.333233 Th 1.37331 1.417526 Na 2.224257 2.444141 Sb 5.050216 4.750079 Pa 1.92541 1.936857 Mg 2.877112 3.014651 Te 5.632543 5.166979 U 2.220108 2.230558 Al 3.529967 3.584907 I 6.21487 5.583887 Np 2.514805 2.52412 Si 4.182821 4.155131 Xe 6.797197 6.000897 Pu 3.066905 3.043613 P 4.835676 4.725804 Cs 0.872178 0.682915 Am 3.361602 3.416868 S 5.488531 5.295979 Ba 1.075102 0.920095 Cm 3.6563 3.404984 Cl 6.141386 5.866186 La 1.160573 1.157089 Bk 3.693594 3.92442 Ar 6.794241 6.436619 Ce 1.640173 1.394276 Cf 4.245694 4.218081 K 2.13075 2.327318 Pr 1.896172 1.631473 Es 4.540391 4.511593 Ca 2.756159 2.758724 Nd 2.15217 1.868439 Fm 4.835089 4.805093 Sc 2.835305 2.858192 Pm 2.408168 2.105658 Md 5.129786 5.098982 Ti 2.914451 2.95783 Sm 2.664166 2.342665 No 5.424483 5.392605 V 2.993597 3.057341 Eu 2.920165 2.579815 Lr 5.461777 5.460699 5.499071 Cr 3.072743 3.156725 Gd 3.175785 2.817026 Rf Mn 3.151889 3.256383 Tb 3.432161 3.054037 Db 5.536365 Fe 3.231034 3.355931 Dy 3.68816 3.291169 Sg 5.573659 Co 3.31018 3.455609 Ho 3.944158 3.528297 Bh 5.610954 Ni 3.389326 3.555013 Er 4.200156 3.765525 Hs 5.648248 Cu 3.468472 3.654418 Tm 4.456154 4.002555 Mt 5.685542 5.722836 Zn 3.547618 3.75416 Yb 4.712153 4.239478 Uun Ga 4.173027 4.18552 Lu 4.744549 4.476583 Uuu 5.76013 Ge 4.798436 4.616627 Hf 4.776946 4.706522 Uub 5.797424 As 5.423845 5.066215 Ta 4.809343 4.950847 Uut 5.817442 Se 6.049255 5.479496 W 4.84174 5.187931 Uuq 6.112139 Br 6.674664 5.9111 Re 4.874136 5.425608 Uup 6.406836 Kr 7.300073 6.341847 Os 4.906533 5.661914 Uuh 6.701533 Rb 1.983971 2.120458 Ir 4.93893 5.900043 Uus 6.996231 Sr 2.566298 2.53737 Pt 4.971326 6.136715 Uuo 7.290928 Y 2.585817 2.633547 Au 5.003723 6.37413 Zr 2.713685 2.729753 Hg 5.03612 6.610266 The K values for each period are given in Table-3. ©The SciTech Publishers, 2012. All Right Reserved 46 The SciTech, Journal of Science & Technology N. Islam Vol-1, Issue-1, p.40-50, 2012. Relationship between atomic orbital exponent & hardness.. Table 3. Computed Kn parameters for each period along with effective principal quantum numbers. Period Effective principal quantum number(n*) Kn values 1 2 3 3.7 4 4.2 4.3 0.2305 0.06213 0.05579 0.065 0.071 0.0305 0.0205 st 1 2nd 3rd 4th 5th 6th 7th To explore whether the evaluated data exhibit periodicity, we have plotted the computed global hardness as a function of atomic number in Figure 2. 14 12 He Ne 10 Hardness in eV F O 8 Kr H N Ar Cl C 6 Al Be Sn Ga In Na K Cl Sc Zn Cu Ni Co Fe Mn VCr Ti Sc Ca Al Mg Li Sb Ge Si 4 Te As P B I Se S Uuo Uus Uuh Uup Uuq Uut Uub Uuu Uun Mt Hs Bh fSg LR rDb No Md Fm Es Cf Xe Br Cd Ag Pd Rh Ru Tc Mo Nb Zr Y Sr Rb 2 tHg OsIrPAu Re HTa fW Lu Yb Tm Er Ho Dy Bk Cm Tb Am Gd Pu Eu Rn Sm Np At Pm Po U Nd Bi Pa Pr Pb Ce Tl Th Ac Ra La Ba Fr Cs t Uu t Uu s b M d D M p Bk N At Ac Ir Tl Ta Tb Tm I La Pm In Rh b Nb R u As n C M F B H 0 Atoms Figure 2: Plot of computed global hardness of atoms as a function of their atomic numbers. The strength of any model is its ability to explain experimental observations. But the atomic hardness is neither experimentally observable quantity nor it can be evaluated quantum mechanically [6-12,27]. We, therefore, in absence of any such benchmark for absolute hardness data, rely upon our experience of hardness behavior goaded by periodic law, and the comparative study of our newly computed hardness data with some other available sets of hardness data as a validity test. To examine the nature of variation of such different sets of hardness data vis-à-vis the result of present effort, we have plotted the two sets of data in Figure 3. ©The SciTech Publishers, 2012. All Right Reserved 47 The SciTech, Journal of Science & Technology N. Islam Vol-1, Issue-1, p.40-50, 2012. Relationship between atomic orbital exponent & hardness.. 14 η cal(eV) η Ghosh Islam(ev) 12 Hardness in eV 10 8 6 4 2 M t U ut U us D b M d Bk N p A t Ac Ir Tl Ta Tm Tb Pm I La In R h N b As R b C u Sc M n A l C l F B H 0 Atoms Figure 3: Comparative study of the computed hardness data vis-à-vis the hardness data computed by Ghosh and Islam. We have presented the result of DFT calculation of Sen and Vinayagam [16] and the corresponding data of present calculation and Ghosh and Islam[6] in Table 4. For a better visualization of the comparative study, we have plotted the above three sets of hardness data in Figure 4. Table-4: Comparative study of the hardness of the present calculation vis-à-vis the DFT calculation of Sen and Vinayagam and Ghosh Islam Atom η calculated(eV) η DFT(eV) η Ghosh Islam(ev) Li 2.278411609 2.33 2.374586656 B 4.604218576 4.07 4.619008972 C 5.767122059 5 5.740978922 N 6.930025542 5.91 6.862466529 O 8.092929026 6.8 7.985435701 F 9.255832509 7.66 9.106475372 Na 2.224256852 2.23 2.44414136 Al 3.529966578 2.69 3.584907074 Si 4.182821442 3.33 4.1551309 P 4.835676305 3.91 4.725803974 S 5.488531168 4.45 5.295979241 Cl 6.141386031 4.98 5.866186484 ©The SciTech Publishers, 2012. All Right Reserved 48 The SciTech, Journal of Science & Technology N. Islam Vol-1, Issue-1, p.40-51, 2012. Relationship between atomic orbital exponent & hardness.. 10 η calculated(eV) η DFT(eV) η Ghosh Islam(ev) 9 8 7 Hardness 6 5 4 3 2 1 0 Li B C N O F Na Al Si P S Cl Atoms Figure 4: Comparative study of the computed hardness data vis-à-vis the hardness data computed by Ghosh and Islam and Sen and Vinayagam From Figures 2 and 3 and Table 2 it is evident that computed hardness values of the atoms of 118 elements exhibit perfect periodicity of periods and groups. Each period begins with the hardness of a representative element and ends with a noble gas atom. The hardness of the inert gas atoms occur at the top of the curve. It is evident from Table 4 and Figure 4 that the trend of variation of the hardness data of the twelve elements is similar to the results of Sen and Vinayagam [16] and Ghosh and Islam [6]. It is further observed that the result of present calculation is free from the anomalous pattern of variation of the hardness values of these elements evident in the various finite difference approximation calculations[19,28]. It transpires from a comparative study of the numerical values of the hardness data in Table 4 that the results of present work have close agreement with the results of Sen and Vinayagam[16] . From Figure 5 it is more transparent that at least for 12 cases, results of the present semi empirical calculation have close agreement with a sophisticated DFT calculation[16]. It is the rule of nature that high hardness means less deformability under small perturbation. It is also well known that the lanthanide elements are soft and easily deformable. It is well known fact [19] that size of atoms of f-block elements undergoes a steady but slow contraction and the effect is well reproduced in the radii of such elements. It is distinct from Table 2 and Figure 2 that the magnitude of hardness of all the lanthanide elements is small. It is distinct from Table 2 and Figure 2 that the global hardness values of the inert gas elements are highest in each period. The global hardness of Cs is significantly small compared to those of other elements. The strong chemical reactivity of the element Cs is well documented. The chemical inertness of Hg and its state of aggregation is attributed to its small size and least deformability under small perturbation. Table 2 and Figure 2 reveal that the hardness value of Hg atom is quite high placing it in the ©The SciTech Publishers, 2012. All Right Reserved 49 The SciTech, Journal of Science & Technology Vol-1, Issue-1, p.40-51, 2012. N. Islam Relationship between atomic orbital exponent & hardness.. group of inert gas elements. Thus, the present work can well correlate the significant and characteristic properties of elements in terms of its computed global hardness values. Conclusion: Relying upon the general relationship between global hardness of atoms and the electrostatic force of attraction of the screened nucleus upon the valence shell electron, we have suggested an orbital exponent dependent ansatz of computing the global hardness of atoms and have evaluated the global hardness of the atoms of 118 elements of the periodic table. The express periodic behavior and correlation of the most important physicochemical properties of elements suggest that the present semi empirical approach is a meaningful venture of evaluating global hardness of atoms. Reference: [1]R.S. Mulliken, J. Am. Chem. Soc.74, p 811, 1952. [2]R.G. Pearson, J. Am. Chem. Soc..85, p 3533, 1963. [3]R.G. Parr, and W. Yang, Density Functional Theory of Atoms and Molecules, Oxford University Press,1989. [4]R.G. Parr, and R.G. Pearson, J. Am. Chem. Soc. 105, p 7512,1983. [5]R.G. Pearson, J.Mole. Struct.: THEOCHEM, 255, p 261, 1992,. [6]D.C. Ghosh and N. Islam, Int. J. Quant. Chem. 109,p 1206,2009. [7]D. C. Ghosh, and N. Islam, Int. J. Quant. Chem, 111, p 1931,2011 [8]D. C. Ghosh, and N. Islam, Int. J. Quant. Chem, 111, p 1961,2011. [9](a)N. Islam and D. C. Ghosh , Mol. Phys.109, p 917,2011. (b) N. Islam and D. C. Ghosh, Hardness Equalization in the formation poly atomic carbon compounds, ADVANCED IN CARBON PHYSICS AND CHEMISTRY, Ed:M. V. Putz, Carbon Materials: Chemistry and Physics, Volume 5, p 301, 2011, Springer. [10]N. Islam and D. C. Ghosh, Eur. Phys. J. D. 61,p 341,2011. [11]D. C. Ghosh, and N. Islam, Int. J. Quant. Chem, 111,p 2811,2011. [12]N. Islam and D. C. Ghosh, On the method of the determination of the global hardness of atoms and Nanoscience and Advancing molecules, Computational Methods in Chemistry: Research Progress, Eds: E.A. Castro and A. K. Haghi, ch.10,p 247. IGI global(USA). [13]P Hohenberg, and H. Kohn, Phys. Rev. 136, p 864,1964. [14] R. G. Parr, R. A. Donnelly, M. Levy and W. E. Palke, J. Chem. Phys. 68, p 3801, 1978. [15]J.L. Reed , J. Phys. Chem. A. 101, p 7396,1997. [16]K.D. Sen, and S.C. Vinayagam, Chem. Phys. Lett. 144, p 178,1988. [17]R.G. Pearson, Proc. Natl. Acad. Sci. 83, p 8440,1986. [18]J.C. Slater, Phys Rev 36, p 57,1930 [19] R.G. Pearson, Inorg. Chem. 27, p 734, 1988. [20] S.Hati and D Dutta..J.Phys.Chem. 98, p 10451,1994. [21]N.Islam and D. C. Ghosh, Mol. Phys.,109, p 1533, 2011. [22]N. Islam, and A. Jana, J.Theoret.Comput. Chem, 9(3), p 637, 2010. [23]N.Islam, Eur.J. Chem, 2(4),p 448, 2011. [24]N.Islam, A. Jana, M. Das and B. Bhaumik. Int.J. Chem. Model, 2(4), 2011. [In press]. [25]Ghosh D.C., Biswas R., Int. J. Mol. Sci., 3, p 87, 2002. [26]D. C. Ghosh,R.Biswas, T. Chakraborty, N. Islam and S. K. Rajak, J.Mole. Struct.: THEOCHEM, 865, p 60, 2008. [27]N.Islam and D. C. Ghosh, Eur.J. Chem, 1 (2), p 83,2010. [28] J. Robles and L.J.Bartolotti, J.Am.Chem.Soc, 106,p 3723, 1984. ©The SciTech Publishers, 2012. All Right Reserved 50