Electronegativity: A Force or Energy

Copyright © 2019 by author(s) and International Journal of Trend in Scientific Research and Development Journal. This is an Open Access article distributed under the terms of the Creative Commons Attribution License (CC BY 4.0) (http://creativecommons.org/licenses/ by/4.) ABSTRACT Electronegativity as force or energy leads to new ansatz at critical point in binding (or bonding) state in between two similar atoms or dissimilar atoms. Electronegativity as a quantum-mechanical entity (energy) or non-quantum entity (force) is yet to be answered. The dual approach to electronegativity has been discussed in this paper. The aim of this paper is to prove that Electronegativity as Hellman-Feynman Force is more accurate and absolute. Electronegativity has been computed using the Hartree-Fock and Rothan-HrtreeFock energy equations and equivalent electrostatic force equation.


INTRODUCTION
Electronegativity is unique and useful concept in the science of chemistry, physics and biology. The historical background of this concept dates back from the beginning of 19 th century. In the year 1811, J.J.Berzelius, a proponent of electrochemical dualism has first introduced the term electronegativity. In the year 1809, Amen do Avogadro has also introduced 'Oxygencity' a correlated topic of electronegativity. In the year 1836, Berzelius has proposed a correlation between evolution of heat and neutralization of charge in a chemical reaction on the basis of caloric theory of heat where caloric was proposed to consist of positive and negative electrical fluid.
He could not exploit the use of correlation to quantify the electronegativity scale by bringing a similar relationship between evolution of heat and difference of electronegativity. In the year1870 Baker had already inserted three atomic parameters like weight (quantity of matter), valence (quantity of an atom's combining power), and electronegativity (quality of an atom's combining power). The caloric theory of heat wasdiscarded completely in 1930s and the birth of thermo-chemistry from the laws of thermodynamics and kinetic molecular theory compelled the scientists to establish a correlation between the heat of a reaction and electronegativity. The probable correlation between electronegativity and heat of reaction was suggested by Van'tHoff 1,2 , Caven & Lander 1,3 and Sackur 1,4 . Electronegativity was defined with help of terminologies such as hetrolytic/homolytic bond dissociation enthalpy data, electron affinity, ionization energy (adiabatic, ground state, ionization, ionization potential and vertical ionization), effective nuclear charge and covalent radius, average electron density, stretching force constants, compactness, configurational energy, dielectric properties, work function, number of valence electrons, pseudopotentials and power. The electronegativity is an intuitive-cum-qualitative construct 5 . This qualitative construct is very difficult to be quantified. The first quantification and assignment of numerical value to electronegativity was given by Linus Pauling 6 . From 1932, a number of qualitative and quantitative scales for electronegativity have been proposed by different researchers across the globe. The quest for new electronegativity scale study is still going on as this concept is confusing 7 . The concept of electronegativity has been used to sketch the distribution and rearrangement of electronic charge in a molecule 8,9 . The fundamental descriptors in chemical science like bond energies, bond polarity, dipole moments, and inductive effects are being conceptualized and modeled for evaluation The scope of this concept is so broad that ionic bond, atom-atom polarizability, equalization of electronegativity, apicophilicity, group electronegativity, principle of maximum hardness, electronic chemical potential, polar effect(inductive effect, effective charge ,pi-electron acceptor/donor group)field effect, conjugative mechanism, mesomeric effect could have been explained. The correlations between electronegativity and superconducting transition temperature for solid elements and high temperature superconductors 10,11 , the chemical shift in NMR spectroscopy 12 , isomer shift in Mossbauer spectroscopy 13 have already been explained. This concept has also been utilized for the design of materials for energy conversion and storage device 14 . The experimental determination of electronegativity of individual surface atoms using atomic force spectroscopy has already been reported 15 . In this article, various concepts of electronegativity are overviewed followed by introduction to a new concept based on Hellmann-Feynman theorem. electronegativity as a virtually constant atomic property irrespective of the valence states being different. Pauling proposed the difference in electronegativity as a square root of extra ionic resonance energy (∆). Again, Pauling and Sherman 16 The second term in eq. 2 represents energy of covalent bond A-B based on arithmetic mean and geometric mean respectively. DE(AB) = Bond dissociation energy of AB (Actual bond energy) DE(A2) = Bond dissociation energy of A2 DE(B2) = Bond dissociation energy of B2 Pauling's quantum mechanical approach also indicates the dipole moment due to the presence of significant ionic structure A + B -. The extra-ionic resonance energy( Δ) arises out of contribution of ionic canonical forms to bonding and it was experimentally verified 17,18 . Pauling proposed valence bond in terms of covalent part and ionic part. Pauling has established quantitative ionicity scale for molecules and crystals based on electronegativity difference, such as Pauling's thermochemical scale was viewed as the culmination of the 19 th century concept of electronegativity. Pauling's empirical electronegativity values derived from bond energies have been used to correlate between chemical and physical properties of a large number of elements followed by theoretical justification 19-21 .In the year 1932, electronegativity values of ten non-metallic elements was proposed by Pauling 6 where χ(H)=2.1(arbitrary reference to construct a scale) latter changed to2.2, χ(F)=4. Furthermore, electronegativity values of 29 main group elements was proposed by Linus Pauling in 1939 19,22 Mulliken 20,33 developed an alternative definition for the electronegativity shortly after Pauling's definition based on energy concept. He considered three structures (i)AB ,(ii)A+B-, (iii)A-B+ where the two ionic structures (ii) and (iii would be of equal weights in the wave function containing ii and iii and so that the complete covalent structure will be possible under the condition Mullikan suggested the term IPA+AA or IPB+AB is a measure of electronegativity of atom A or B respectively. V is coulomb potential. With IA and IB assumed to be I and AA and AB assumed to be A, Mullikan expressed electronegativity as χM A =(IA+AA)/2 or χM B =(IB+AB)/2 Eq. 6 In general, Where, V=Coulomb Potential IP -ionization potential (in eV or kcal/mol) EA -electron affinity (in eV or kcal/mol) The values of IP and EA can be computed for atoms in either of states such as ground, excited, or valence state. The scientific reports made by Stark 1,34 , Martin 1,35 , and Fajans 1,36 have concluded the co-relation between Electronegativity, ionization energy and electron affinity. The rigorous qualitative derivation has also been examined by Moffitt 37 and Mulliken 33 himself. The half factor included in eq. 7 represents electronegativity as the average binding energy of the electron in the vicinity of the concerned atom. Mulliken's electronegativity is an arithmetic average of ionization potential and electron affinity of an atom in the ground state.
Mulliken electronegativity can be also termed as negative of chemical potential by incorporating energetic definitions of IP and EA so that Mulliken Chemical Potential will be a finite difference approximation of electronic energy with no of electrons. addition of electron to lowest unoccupied atomic orbital(LUAO). Thus, conceptually orbital electronegativity is a measure of the power of bonded atom or molecule (an aggregate of atoms) to attract an electron to a particular atomic orbital or a molecular orbital. The scientific validity of this scale was justified by Pearson 41 . Mulliken electronegativity is absolute, reasonable and in principle dependent on chemical environment of an atom. This scale is independent of an arbitrary relative scale. A bond between two atoms is assumed as competition for a pair of electrons where each atom will lose one electron (i.e. resist to be a positive ion) and simultaneously gain the second electron (i.e. to be a negative ion). Thereby, the two processes can be seen as involving the ionization potential and electron affinity respectively. So, the average of the two values is a measure of the competition and in turn gives value of electronegativity. A series of papers appearing in early 1960s provides with an extensive studies of Mulliken's electronegativity values for non-transition atoms with various valence states 17,42,43  where val , Ia ,Ea stand for a fraction less than 1,ionization energy (ionization potential IP), electron affinity respectively The ionization energy values (Ia) have been adjusted for pairing and exchange interaction. They have reported a set of electronegativity values for elements from hydrogen to Astatine except zero group elements.

Allen's absolute scale of Spectroscopic
Electronegativity Allen 29,30 defines Electronegativity as the average oneelectron energy of valence shell electrons in ground-state free atom and proposed it as third dimension and also energy dimension of periodic table. So, this type of electronegativity is a Free-atom -ground -state quantity with a single defining number which gains its meaning as an extension of periodic table. Allen has introduced two terms Eenrgy index (in situ Xspec of free atom) and Bond polarity Index (projection operator being applied to a molecular orbital wave function to get in situ average one-electron energies for atoms in molecules i.e in situ ∆×spec).The fractional polarity defined from Bond polarity index is equivalent of Pauiling's dipole moment referenced 'ionic character percent' .Allen has reported a new chemical pattern by mounting a series funnel -shaped potential energy plots(E vs r) along a line of increasing Z i.e along a row of periodic table where a composite curve one-electron energy(vertical axis) vs a part row of periodic table is obtained. This composite curve shows a strong correlation between magnitude of XSPEC and energy level spacing (large XSpec with large spacing) like energy level like energy levels of Fermi-Thomas-Dirac atom and in case of other atoms.
Electronegativity for representative elements is independent of oxidation state because of the fact that the atomic charges carried by representative elements during the formation polar covalent bond are slightly close to their oxidation number there by negligible changes in electronegativity with change in molecular environmental system. For transition elements electronegativity is dependent on oxidation state because of closely spaced energy levels.
Electronegativity-for representative elements i.e. X spec= (a ∈s + b ∈p)/ a+ b equation (i) is occupation weighed average per electron ionization energy of an atom where a,b are occupation number and Is ,Ip are spherically ionization potentials which are determined through multiplet averaging. But for transition elements, I p is replaced by I d and a,b are the valence-shell occupancies of s-orbitals and dorbitals in overlap region.
The main strength of this definition is that necessary spectroscopic energy data are available for many elements and electronegativity of Francium was estimated. The core question of this scalei. "How to determine the valence electrons for d-block and f -block elements'' is still an ambiguity in estimation of electronegativity because no such theory to determine the valency electron has been developed so far. ii. Reason for electronegativity order such as Neon>Fluorine>Helium>Oxygen is yet to be given.
This scale is established with χLithium=1 and χFluorine=4. Also, this scale is quite consistent with Pauling scale and Allred-Rochow scale.

2.7
St. John and Bloch 51 have reported quantum-defect electronegativity scale using ''Pauli force'' model potential 52 .This force model potential represents the pseudo potential of a one-valence-electron ion except in the vicinity of nucleus and is applied in studies of atoms, molecules and solids. Energy of the orbital is represented as Eq. 14 Where Z=core charge Î(l)-l=quantum defect The orbital electronegativity for valence orbital is defined as  61 have reported about direct relation of the total energy of the system with the charges.
3.1 Mulliken-Jaffe 20,33,38,43 electronegativity approach is based on the fact that the first ionization energy and the electron affinity are the simple sum of multiple ionization potential-electron affinity energies which fit a quadratic equation as follows.
α -mulliken electronegativity β -charge coefficient E-Total energy in eV q-ionic charge (+1 for cataion, -1 for anion) IE is IP of sec 2.2 Based on this approach the electronegativity of a few elements of the periodic table can be computed.

(1 )
A A X c c d = + = Where χ represents electronegativity for the molecule or the group, n represents number of atoms of A, N=Σ(n) represents the total number of atoms, δG is the charge in the group. J Mullay 17 has reported the value of 'b' as 1.5 times of 'a'.

Weakness
Huheey's method speaks of total electronegativity equalization but this method has three major demerits i.e. inability to account for differences in isomers, treating groups with multiple bonding and overestimating of the effect of the atoms or groups linked to the bonding atom. In order to avoid the three major deficiencies Huheey 38,63 modified his method for 80% electronegativity equalization.

Hinze-Whitehead-Jaffe -contribution to Electronegativity
Hinze et al. 43 defined orbital electronegativity as the first derivative of energy of an atomic orbital (j) with respect to electron occupancy (nj) of the orbital i.e χA.j(atomic orbital j)=δEA/δnj …….(i) Eq. 22 , (atomic orbital j) The justification for the said definition is obtained from the fact that atomic electronegativity is reasonably considered because of its reference to the atomic orbital which halffilled orbital(nj=1) before the formation of bond. As energy of orbital is assumed to be be a quadratic function of nj, then the definition of atomic electronegativity is reduced to Mullikan's electronegativity. The said definition of electronegativity appears to be valid for nj=0(empty orbital), 1(half-filled orbital), 2(lone pair) and also leads to define 'bond electronegativity' for non-integral values of nj. Again, the concept of bond electronegativity arises in the formation of a bond where electron paring occurs followed by electron transfer between two atoms A and B with energy changes (δEA/δnA) dqA and (δEB/δnB) dqB respectively. At equilibrium, there occurs no further change in energy. Hence, electronegativity values will be equalized during bond formation. Mathematically, The electronegativity value acquired by an atom in bond formation is called 'bond electronegativity' which is not to be confused with Pauling electronegativity integral values of orbital occupation.  17,61 have reported the potential usefulness of group electronegativity which are obtained from the idea of orbital electronegativity in conjunction with electronegativity equalization.

Weakness
The Hinze et al.'s 42,43 work is simple still then it did not meet the criterion for electronegativity. Some authors 69 suggest that the orbital concept of electronegativity never solves the meaning -'Atom in Molecule'. G Klopman 39,55,56 used Rydberg formula for the calculation of the atomic spectra and proposed a modified formula for calculation of atomic electronegativity of the system in the valence state and also for quantitative determination of the diagonal matrix elements in self-consistent field calculation of a molecule .Modified Rydberg formula is represented as

G Klopman's atomic electronegativity
Ry-Rydberg constant n -Principal quantum number σ -Screening constant Z-Atomic number dn-Quantum defect The screening constant (σ) is represented as Where qj is the occupation number of spin orbital j σji is the screening of the electron i by the electron j The value of σ (core electron -valence cell electron) is considered to be 1 because core electrons are not considered. Quantum defect (dn) has been calculated from respective ionization potential i.e

3.687( *) /
Where, n -Principal quantum number Z*-effective nuclear charge IP-Ionization potential Total electronic energy of Valence shell, Further, Total electronic-energy equation of the diatomic system (AB) at barycenter is represented as, Klopman 39 defined atomic electronegativity as the derivative of total electronic energy of the valence cell with respect to the charge qi as mentioned below.
(1 ) 2 And also neutral atomic electronegativity is obtained from the above equation when all the values of qj (the occupation number of particular atomic spin orbital by an electron) will be equal to 1 except for participating electrons in the bonds where qj =1/2.

Strength
Kolpman's procedure helps in calculating Neutral Atomic Electronegativity. This procedure provides theoretical support and clarification for electronegativity suggested by Iczkowski and Margrave, Hinze, Whitehead and Jaffe. Weakness :Kolpman's work has been modified and extended to provide a simple procedure for calculation of atomic or orbital electronegativity and also for group electronegativity 17 R Ponec 17,57 has reported a generalization of the orbital electronegativity concept of Hinze et al. 43 and it is based on the semi empirical Complete Neglect of Differential Overlap (CNDO) approximation. Ponec's basic equation is written as,

Ponec 's idea of Global electronegativity
Where χAj -orbital electronegativity Ej A -one electron energy of orbital j γ A -electron repulsion integral pA -total electron density associated with atom A For neutral atoms the orbital electronegativity is reduced to Mulliken-Jaffe values for isolated atom but in a molecule global electronegativity term can be defined as This also represents (i) the tendency of an atom in a molecule to attract electrons for small charge dislocation during interaction of atoms and (ii) the decrease of energy of moreelectronegative atom than the increase in energy for less electronegative atom. Hence, the energy of molecule is decreased simply by transfer of charge from one tom to another. The energy change in this case is not at all accrued from the electrostatic attraction between ions. Thus, electronegativity characterizes both the internal constitution of atom and the ions which can be formed from it. Again, the electronegativity represents an intensity factor in charge transfer from one atom to the other atom.

Strength
This concept of electronegativity in terms of energy-charge derivative have also been justified through ingenious and laudable efforts of various authors [70][71][72][73] .The scope of this definition is described as i) dE/dN have been calculated for various 1 st row and 2 nd row elements and are in close agreement with Mulliken's electronegativity. ii)The calculations were extended to many elements along with metals by C K Jorgensen 39,74 who used similar equations up to three first terms. iii) The above equation up to first two terms using N=1 leads to the Mulliken's definition of electronegativity i.e. With this approximation Jaffe et al. were able to calculate the group orbital electronegativity (i.e. electronegativity of free orbital of an atom bound to other atom). iv)The principle of electronegativity equalization of Sanderson 75 helped in initiating the calculation of charge distribution. V) The above general principle has been used by Ferreira 76 for calculation of bond energy and charge distribution in many binuclear molecules.

Weakness
The expression of energy in terms net-charge is not a continuous function as net-charge takes only integral values.
The assumption of envisioning 'atom in molecule to have an average fractional number of electrons so as to make energycharge expression continuous and differentiable' has already been criticized by various authors 77-80 .  [124].where v stands for fixed potential due to set of nuclei and external field,ૉ represents for electronic density. Parr et. al. 58 defined electronegativity as,

Parr
by considering the similarity between the above expression for ૄ and electronegativity expression of Iczwoscki and Margrave . The concept of chemical potential has also kept Electronegativity as a Global index to characterize the chemical structure. The geometric mean electronegativity equalization principle holds only when each chemical potential is exponential in the number of electrons and the fall-off parameter γ is same for chemical potentials of neutral atoms. Again from density functional theory studies, it is suggested that for a nearly neutral atom, energy is an exponentially decaying function of the number of electrons but the classical suggestion states that the energy is a quadratic function of number of electrons and the classical suggestion leads to the Mulliken formula of electronegativity in equation number μ = -χ = (IP+EA)/2. Parr and Bartolotti 59 proposed the formula for ૄ as  85 have pointed out that Parr et. al. 58 formula implies the transfer of electron between free-atom or free-molecule and external surroundings whereas initial concept of electronegativity is always referred to redistribution of electrons within a molecule.  94,95 . One of all the three ways considers the electronegativity in terms of electrostatic potential and covalent radius.

4.4
Boyd and Edgecombe 100 defines electronegativity quite different from that of Pauling and Allred -Rochow by determining electronegativity from computed electron density distributions for hydrides of representative elements where atomic radii are determined by a point of minimum charge density along non-metallic hydride bond. Electronegativity is supposed to be direct function of charge density (ρ)at minimum no of valence electrons, non-metal hydride separation(d) and an inverse function of atomic radii(r).  Where, Z * = effective nuclear charge, Z*=Z -σ (slater constant=shielding constant), r =mean radius of the orbital i.e. covalent radius for the atom(considering smaller value as well as outer radial maxima).The Coulomb force is a measure of power of an atom in a molecule with which is electron is dragged towards an atom. Thus electronegativity will be absolute one. X (AR) dimension is not straightforward as it is evaluated through expression (i). The quantity Z * /r 2 was calculated through Pauling's work and Slater rules for determining the effective nuclear charge 96,104,105 . The Pauling's Scale and Allred-Rochow scale can be made to coincide by expressing the electronegativity from the electrostatic approach as the linear function of Z * /r 2 . mean radius is expressed in picometer 106 . Strength of this scale is two-fold such as Introducing the idea of force into electronegativity theory so that it seems quite consistent with Pauling's definition. Emphasizing the idea for simple calculation, because r and Z * are readily available quantities for many elements. The modification and extension of the above ideas were reported by different authors.

Malone 101 suggested in 1933 a rough proportionality between the dipole moment of the bond A-B and electronegativity difference as
Weakness of this scale is also three fold such as independent of electron affinities, bond dissociation energies Slater rules for finding effective nuclear charge are empirical Covalent radii are known for few elements

The second extension of Allred-Rochow scale
The second extension of Allred-Rochow scale by Boyd and Markus 17,107 is based on non -empirical approach where empirical covalent radius is replaced by relative covalent radius which is obtained from the free-atom wave function by density contour technique. The effective nuclear charge is obtained through integration of radial density function from nucleus to relative-distance.
Electrostaticelectronegativity is expressed as,

54
Where Z -Atomic number r -Relative covalent radius ρ(r) -radial charge density The radial charge density ρ(r) can be obtained from the Hartree Fock atomic orbitals data 108,109 . The computed electronegativity values follow the general pattern of Mulliken ground state electronegativity values with an exception for groups 2 and 3 of periodic table because D(r) decreases as expectation of (IP. r) where IP=ionization potential,r->infinity 5.4.The third extension of the scale was made Mande et al. 17,110 where the value of effective nuclear (Z * ) charge was obtained spectroscopic analysis. So the values are less arbitrary than Slater's. This electronegativity scale is more fundamental and reliable. The correlation of the scale is excellent with that of Pauling's scale. The electronegativity values obtained for 1 st transition metals are more reasonable than Allred-Rochow scale. Zhang 17,111 where electronegativity has been calculated on the basis of electrostatic force [F = n*√(IPz/R) /r^2 ]in terms of ultimate ionization potential for outer electron (Iz=R.Z*^2/n*^2). This type of scale is based on the concept of different electron-attracting power of an element in different valence. Therefore, ectronegativity is termed as a function of oxidation number.

Quantum model of Electronegativity
Putz M.V 112-115 defined electronegativity by a specialized affinity-ionization wave function within Fock Space having fermions(electrons) where quantum mechanical description of electronegativity was made through field perturbation on a valence state for chemical system. Putz electronegativity is termed as quantum electronegativity which is considered as viable quantum concept with observable character. The mathematical expression is represented as 115 , This idea of quantum electronegativity helps in applying affinity-ionization wave function on the valence state of a chemical system to recover the Eigen energy value of that state within density functional chemical potential formulation .The density functional electronegativity of Parr et.al 58 was confirmed with Putz's fundamental quantum mechanical arguments which helped in identifying the flaws made by Bergmann and Hinze 116 .

Ionocovalency model of Electronegativity
Yonghe Zhang 111,117,118 has reported ionocovalency model which is correlated with quantum -mechanical potential. This model describes quantitatively the properties of effective ionic potential, charge density, charge distribution, effective polarizing power and bond strengths. Ionocovalency (IC) was defined as a product of the ionic function I(Z*) and the covalent function C(1/r).The Bohr energy expression(E=-R.(Z)2/(n)2) was modified by replacing energy by ultimate Ionization energy(IPz) , Nuclear charge(Z) by effective nuclear charge(Z*), principal quantum number (n)by effective principal quantum number(n*) . The expression, so obtained, Z*=n*[(IPz)/R] was used to correlate the bond properties to the quantum mechanics and IC model is represented as The electronegativity defined in terms of Ionocovalency is correlated with Pauling's electronegativity values and it is mathematically expressed as

New model of electronegativity
In the presented work the force expression based on Hellmann-Feynman theorem has been proposed as electronegativity. This force must be equivalent to the primary definition of electronegativity such as ability of an atom to attract electron towards itself. We propose a modified primary definition of electronegativity as the inherent ability of an atom to attract and hold electron. The electronegativity in terms of this force is also equal to B-O force for an atom in diatomic system and also equal to Hartree-Fock force of an atom in poly-atomic system.

Born-Oppenheimer Force and Hartree-Fock Force:-This force concept arises out of Born-Oppenheimer energy approximation as well as
Hartree-Fock energy approximation. M Born and J R Oppenheimer 119,120 have contributed a celebrated paper to science that brings the systematic correspondence of the energy of electronic motion, nuclear vibration and rotation to the terms of power series in the fourth root of electron -nucleus mass ratio. Born and Oppenheimer have suggested that total wave function (ૐ) can be written as the product of the nuclear wave function (ૐn) and electronic wave function (ૐe). This approximation simplifies complicated Schrödinger equation into electronic equation (Heૐe=Eψe) and nuclear equation (Hnૐn=Eeૐe ). The equation devised by them for the rotation represents a generalization of the treatment of Kramer and Pauli. This approximation also justifies Frank-Condon principle 121,122 used in explaining the intensity of band lines. In the last several decades, rigorousmathematical work haS been reported on the validity of the B-O approximation. Quite a more no of papers 66,70-81 contain the study of B-O and also have reported that a reduced Hamiltonian is an appreciable approximation to true molecular Hamiltonian but a few is closely related to works 112,113,135 on semi-classical Schrodinger matrix operators. B-O approximation is based on "assumption of ignoring motions of nearly stationary nuclei with much larger mass and smaller velocity with respect to motion of electron with much smaller mass and larger velocity".
Where λ is treated as parameter and it may vary between 0 and 1.
The exact solution to the electronic Schrodinger equation, obtained from B-O approximation can be reachable for one electron systems. For multi-electronic systems, Hartree-Fock approximation is a good enough to approximate the energies and wave function. The electronic Hamiltonian(i) and energy(ii) based on Hartree-Fock approximation can be written as follows 137 .
Eq. 63 The first term represents a one-electron operator, the second term represents a two electron operator and third term is a constant for the fixed set of nuclei coordinates R.
Where the first term represents one-electron integral, the second as two-electron Coulomb integral, the third term as exchange integral and all the integrals can be computed by existing computer algorithms. The energy difference between non-relativistic energy of the system and Hartree-Fock limit energy is considered as both static and dynamic electronic correlation energy. The derivative (-∂He/∂R) of electronic Hamiltonian operator with respect to distance of nucleus of atom from electron can also be defined in quantum mechanics. Further, within simple Born-Oppenheimer approximation or (Hartree-Fock approximation) Energy (E) plays the role of potential energy for actual motion and also -∂E/ ∂R replaces the above derivative and it is equal to the B-O force (also Hartree-Fock force) because nuclear co-ordinates are simply treated as external parameters. The term -(∂He/∂R ≡ F) is the operator which represents the force on atom A due to electrons and other atom B. This force is better to be termed as B-O force in the steady state. The electronegativity will be equal to B-O force (also Hartree-Fock force).

Hellman-Feynman Force:
The force concept is the consequence of Hellmann -Feynman 86,138-140 theorem .The expression for this theorem have already been reported by different authors [140][141][142][143][144] . This concept dictates that the actual force on any nucleus can be interpreted in terms of classical electrostatics if three dimensional charge distribution in a system of electrons and nuclei were known from quantum mechanical procedure. The force on a nucleus will be equal to charge on that nucleus times the electric field due to all electrons and other nuclei. R Feynman further stated that a three dimensional electron cloud in a molecule is restricted from collapsing as it obeys Schrödinger equation. The force concept explains the nature of chemical bonding, the change in molecular shape on excitation, chemical reaction. Energy concept is not proved to be satisfactory always because they lack the simplicity and elegant nature. A.C.Hurley [145][146][147][148] has given the theoretical justification of the actual use of such electrostatic approach and shown that the force calculations are valid even for approximate wave functions. H-F force concept have been used (i) by R.F.W.Bader [149][150][151][152][153] for interpreting chemical binding, (ii)by Koga T and H.Nakatsuji [154][155][156] for force modelling of molecular geometry,(iii)by P.Politzer and K.C.Daiker 157,158 for models of Chemical Reactivity, (iv) by A.J.Coleman [159][160][161] for calculation of first and second order reduced density matrices and also withstand the critical examination of theoretical physists and chemists as well. This force concept has certain advantage over the concept of total energy even though the calculation of force always involves an approximate charge density function. The advantage of calculating charge density is possible through molecular orbital method and total force on a nucleus is simple sum of orbital contributions but total energy is not sum of orbital energies. The second advantage is that force is an expectation value of one-electron, momentum independent operator which is more sensitive to any change in wave functions than energy. T Berlin 87 gave clear interpretation of this electrostatic force arising out of Hellmann-Feynman theorem. This force will be equivalent to infinitesimal change in energy per change in distance (parameter). Classical physics states that a force is the negative gradient of energy. He proposed a term binding (related force acting on the nucleus) in place of bonding (related to changes in energy) in the picture of chemical bonding. He has proposed the physical partitioning of three dimensional space of electrons of diatomic system into a binding region(fi > 1), anti-binding region(fi< 1) and the nonbinding region(fi =1) . The charge density is positive everywhere and thus the sign of contribution to force to the charge in each volume element depends on the sign of fi. The net value of fi around 1 helps to assign the electronegativity to the concerned atom in molecule for the diatomic system with ZB.>ZA, the anti-binding region for A is closed while antibinding region for B in the limit ZB>>ZA approaches a plane perpendicular to inter-nuclear axis. The idea of closing of anti-binding region is used to justify to assign more electronegativity value to B. Hellmann-Feynman force equation can be written in various forms 86,136,162 Where the first term is independent of the electronic coordinates and is constant during integration over the coordinates. This term gives ordinary columbic force of repulsion between the nuclei. The second term represents charge density distribution due to ith electron.
Where the λ is a parameter which solves two problems. Firstly, it helps to apply simultaneously to all nuclei. Secondly it is a continuous function between 0 and 1 so that differentiation of energy w.r.t. nuclear coordinates is made possible.
The other form of Hellmann-Feynman force equation can be written as the electronic contribution to the force on either nucleus can be written as And also the electronic contribution FA(R) in terms of the quantum mechanical average of the electric field operator is also mathematically represented as, The equivalence of the electron in the above equation is equivalent to N times the average force exerted on an atom by one electron so the above equation can be written in the form of electronic charge density. Where ρ(r) denotes electronic charge density in a stationary state, ρ(r) dr stands for amount of electronic charge in a volume element dv and xi denotes the product of space coordinate (ri)and spin co-ordinate (si) of the ith electron. The interpretation of ρ(r) as a physical model of the electrons in line with the HF theorem includes the possibility of ascribing a value to the electrostatic force exerted at atom A by each and every element ρ(r)dr.

Corelation among Electronegativity , Hellman-Feynman and Hartree -Fock Force
This electrostatic force leads two opposing terms such as one from nuclear-nuclear repulsions and other from electron-nuclear attractions. The electron-nuclear attractive force is expressed in terms of three dimensional electron density. This force can be termed as charge-equivalent force and this follows from the energy (Born-Oppenheimer approximation (in turn Hartree-Fock approximation) because the fast motion of electron allows electronic wave function and probability density for immediate adjustment to changes in nuclear configuration. The fast motion of electron causes the sluggish nuclei to see electrons as charge cloud rather than discrete particles. This fact affirms the force as electrostatic by nature thereby ruling out mysterious quantum mechanical force in mono-atomic, diatomic as well as poly-atomic systems.
Electronegativity of an atom (A) in a molecule A-B may be defined as HF (Hellmann-Feynman) force which is also Hartree-Fock force in steady state and also in non-steady state. In steady state, p(r) may be interpreted as a number or charge density and p(r)dr as amount of electronic charge in the volume element. ability of an atom to attract electron.
Based on the basis of Hartree-Fock approximation Where First terms in Eq 75 AND 76 above represent classical nuclear contribution Second terms in Eq 75 AND 76 above represent electronic contribution χ=Electronegativity <FA>= Hellman-Feynman force is a sum of classical contribution due to classical nuclear contribution and electronic contribution FA=one electron, momentum-independent operator ρ(r)=electronic charge density (always positive) xi =product of space coordinate ri and spin coordinate si of the ith electron RA=Distance of nucleus of atom A form electron RB= Distance of nucleus of atom B from electron

Computation of Electronegativity
In this paper, energy was computed by using Hartree-Fock procedure for most of the elements of the periodic In this case, 1 a.u of force=e^2/a^2 where e=charge of electron(in coulomb) and a=Bohr radius(pm). Electronegativity values based on energy and force from Hydrogen to Lawrencium have been computed through the above equations and are mentioned as follows.

Conclusion
The exact status of electronegativity might be attributed as dual concept of force and energy. The attempt to measure electronegativity needs reification of this concept for which mathematical formulation is required. Till today, there exists no unique-mathematical formulation of this reified noumenon for which there had been scope of many scales o measurement. The new attempt to define electronegativity is characterized by specific physical meaning and reliable theoretical basis since it is derived from two famous mathematical formulation i.e Hellmann-Feynman theorem and Born-Oppenheimer (in turn conventional Hartree-Fock) approximation. This definition will be acting like a bridge in between two parallel definitions of electronegativity (either in energy or force) and also it will be logical to consider electronegativity equalization in a diatomic as well as polyatomic system. This new approach will be helpful to assign the more accurate electronegativity values to various elements of the periodic table and also more valuable in different areas of chemical science for example to predict the structure and property of materials and also to design efficiently new electrode materials, electrocatalysts with novel properties for energy conversion devices like Fuel cell, Solar cell etc.