ELECTRONEGATIVITY: EXPECTATION VALUE OF POWER OF AN ATOM IN A MOLECULE

P Ramakrishnan. Department of Chemical Engineering,NIT,Rourkela.Odisha. ...................................................................................................................... Manuscript Info Abstract ......................... ........................................................................ Manuscript History Received: 20 March 2019 Final Accepted: 22 April 2019 Published: May 2019 Advancing theory of electronegativity , new approach is established by the study of binding (or bonding) state in between two homoatoms or heteroatoms. Electronegativity is a confused as it is sandwiched among three entities such as i) energy ii) force iii) Charge. This paper interprets that Electronegativity (χ) is the expectation value of attracting or holding power of electron an atom in either of homo-atomic or hetro-atomic system. This value has been described in terms of von Neuman -minimax theorem: χ(maxA . min B) =χ (min A. max B) where max A and max B stands for atom‘s maximum ability and min A and min B stands for atom‘s minimum ability . Three structures(i)AB(Covalent structure) ;mini-max theorem,(ii)A+B-(ionic structure);right-side of mini-max theorem, (iii)A-B+(ionic structure);left-side of theorem for giving mathematical formulation electronegativity are established.Hellmann-Feynman force as an expectation value for electronegativity is established.

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( D) arises out of contribution of ionic canonical forms to bonding and it was experimentally verified 11,12 . Mulliken 14,27 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

Mulliken's (1934 and 1935) absolute electronegativity
Where IP A +EA A or IP B +A B is a measure of electronegativity of atom A or B, Mullikan electronegativity can be also termed as negative of chemical potential by incorporating energetic definitions of IP and EA so that Mullikan Chemical Potential will be a finite difference approximation of electronic energy with no of electrons. aa number of order unity hω p /2πplasma energy k s -Thomas Fermi screening radius for a free electron gas This scale is exclusively used for calculation of electronegativity values for tetravalent elements like Carbon, Silicon, Germanium and Tin.

Hinze-Whitehead-Jaffe -formulation to Electronegativity
Hinze et al. 37 defined orbital electronegativity as the first derivative of energy of an atomic orbital (j) with respect to electron occupancy (n j ) of the orbital i.e χA.j(atomic orbital j)=δEA/δnj …….(i) , (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 half-filled orbital(n j =1) before the formation of bond, Sanderson 59 has also defined electronegativity in terms of electron density. G Klopman 33,61,62 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
Klopman 33 defined atomic electronegativity as the derivative of total electronic energy of the valence cell with respect to the charge q i as mentioned below.
(1 ) 2 And also neutral atomic electronegativity is obtained from the above equation when all the values of q j (the occupation number of particular atomic spin orbital by an electron) will be equal to 1 except for participating electrons in the bonds where q j =1/2. R Ponec 11,63 has reported a generalization of the orbital electronegativity concept of Hinze et al. 37  Allen's formula of Spectroscopic Electronegativity Allen 23,24 defines Electronegativity as the average one-electron 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 oneelectron 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 X SPEC and energy level spacing (large X Spec with large spacing) like energy level like energy levels of Fermi-Thomas-Dirac atom and in case of other atoms.

Ponec 's idea of Global electronegativity
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 Eq-, A. Therefore, ectronegativity is termed as a function of oxidation number. Zhang electronegativity is given by, 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 64 was confirmed with Putz's fundamental quantum mechanical arguments which helped in identifying the flaws made by Bergmann and Hinze 106 .

Ionocovalency formula of electronegativity
Yonghe Zhang 101,107,108 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(Iz) , Nuclear charge(Z) by effective nuclear charge(Z*), principal quantum number (n)by effective principal quantum number(n*) . The expression, so obtained, Z*=n*[(Iz)/R] was used to correlate the bond properties to the quantum mechanics and IC model is represented as

Allred and Rochow electronegativity formula
AL Allred and EG Rochow 43 defined the electronegativity of an atom with electrostatic field and presented an equation for its evaluation and electronegativity will be equal to Coulomb force of attraction between the nucleus and an electron at the covalent radius. X (AR) ≡ Z*e^2 / r^2 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 straight -forward as it is evaluated through expression (i). The quantity Z * /r 2 was calculated through Pauling's work and Slater rules for determining the  109,110 which 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-Oppenheimer has suggested that total wave function ( ) can be written as the product of the nuclear wave function ( n ) and electronic wave function Where λ is treated as parameter and it may vary between 0 and 1. The exact solution to electronic 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) can be written as follows 127 .
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, (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(also Hartree-Fock) force because nuclear co-ordinates are simply treated as external parameters. This term -(∂H/∂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).

Electronegativity in terms of Hellmann-Feynman Force
Hellmann -Feynman 91,128-130 theorem is an intuitive topic . This theorem have already been reported by different authors [130][131][132][133][134] . 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 states 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 135 [144][145][146] for force modelling of molecular geometry,(iii)by P.Politzer and K.C.Daiker 147,148 for models of Chemical Reactivity, (iv) by A.J.Coleman [149][150][151] 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 92 gave clear interpretation of this electrostatic force arising out of Hellmann and 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(f i > 1), anti-binding region(f i < 1) and the nonbinding region(f i =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 f i . The net value of f i around 1 helps to assign the electronegativity to the concerned atom in molecule for the diatomic system with Z B. >Z A, the anti-binding region for A is closed while anti-binding region for B in the limit Z B >>Z A 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 91 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 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 x i denotes the product of space co-ordinate (r i )and spin co-ordinate (s i) of the ith electron. The charge density difference distribution being combined with electrostatic HF theorem gives rise to a novel physical model to the chemical binding. 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. By identifying λ as real parameter in H, ψ as a normalizable Eigen function, E as Eigen value, This force arises out of two opposing terms such as one from nuclear-nuclear repulsions and other from electronnuclear attractions. The electron-nuclear attractive force is expressed in terms of three dimensional electron density. H-F force concept follows from the Born-Oppenheimer energy approximation (in turn Hartree-Fock) since the rapid motion of electron allows electronic wave function and probability density will be adjusted immediately to changes in nuclear configuration. The fast motion of electron causes the sluggish nuclei to see electrons as charge cloud rather than discrete particles. The fact that -effective force on nuclei are electrostatic‖ affirms that there is no mysterious quantum mechanical force in mono-atomic, di-atomic 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.

ISSN: 2320-5407
Int. J. Adv. Res. 7(5), 801-820 814 χ=Electronegativity <F A >= Hellman-Feynman force is a sum of classical contribution due to electronic charge density (i)and ii)classical nuclear contribution F A =one electron, momentum-independent operator ρ(r)=electronic charge density (always positive) x i =product of space coordinate r i and spin coordinate s i of the ith electron R A =Distance of nucleus of atom A from electron R B = Distance of nucleus of atom B from electron

Computation of Electronegativity
Electronegativity values of some elements (Table-1

Conclusion:-
Electronegativity is a confused in spite of a vast no of papers published by the various authors. Mathematical formulation is required for reification of this concept. Till today, there exists no unique-mathematical formulation for which there had been scope of many scales of measurement. This attempt to define electronegativity is characterized by specific physical meaning and reliable theoretical basis since it is derived from Hellmann-Feynman theorem and Born-Oppenheimer (in turn conventional Hartree-Fock) approximation and min-max theorem. This various definitions of electronegativity such as in terms of energy or force or intrinsic strategy are logical ones 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. N.B:Symbol For Electronegativity; C and X