Journal of Quantum Informatio n Science, 2011, 1, 135-141
doi:10.4236/jqis.2011.13019 Published Online December 2011 (http://www.SciRP.org/journal/jqis)
135
The Electronegativity and the Global Hardness Are
Periodic Properties of Atoms
Nazmul Islam, Dulal C. Ghosh
Department of Chemistry, University of Kalyani, Kalyani, India
Received July 20, 2011; revised November 8, 2011; accepted November 22, 2011
Abstract
The electronegativity and the hardness are two popular and useful theoretical descriptors of chemistry and
physics successfully used by both chemists and physicists in correlating chemico-physical properties of at-
oms, molecules and condensed matter physics. We have tried to explore the fundamental nature of the hard-
ness and electronegativity of atoms and have observed that their fundamental nature is basically lying in
electrostatics and manifest as the electron attracting power emanati ng from the nucleus of the atom. We have
tried to correlate the periodic nature of variation of the electronegativity and the hardness to the electron at-
tracting power of the nucleus from which they are originated and developed. We have developed the formu-
lae for evaluating both electronegativity and hardness and found that they have the direct correlation with the
effective nuclear charge of the atoms and hence their periodicity.
Keywords: Effective Nuclear Charge, Electronegativity, Hardness, Chemical Periodicity
1. Introduction
The terms electronegativity and chemical hardness are in
the glossary of chemistry and the students are taugh t that,
along with other periodic properties, both the electro-
negativity and the global hardness of atoms are periodic
in nature. Although the ratio nale of th e period ic n ature o f
electronegativity can be linked to the internal constitu-
tion of atoms, the periodicity of the hardness of atoms is
not that straight forward. In this report we want to ex-
plore simple algorithms that will justify the periodic na-
ture of both electronegativity and hardness of atoms. The
electronegativity and chemical hardness are two different
fundamental descriptors having different fields of appli-
cations. Notwithstanding the erudite discussion of Putz
[1] on the problem of observ ability of the electrone gativ-
ity and chemical hardness, the hardness and the electro-
negativity are fundamentally hypothesis and conceptual
structures and are not physical observables and, therefore,
cannot be evaluated experimentally [2-6].
Thus, both electronegativity and chemical hardness are
qualitative mental construct s and one can suggest or m odel
their semi empirical evaluation only.
Although the periodic table does not follow from quan-
tum mechanics, the periodic law is an indispensable tool
in understanding, rationalizing and correlating the chemi-
cal and physical behaviour of elements. The concept of
shell structure and Pauli Exclusion Principle justifies
chemical periodicity [7] of elements.
It is important to mention here some outstanding fun-
damental works of Putz and his coworkers [8-13] on
electronegativity and hardness and their usefulness for
the theoretical prediction of several physicochemical
properties-like the fundamentals of chemical bonding. It
is shown that the aromaticity of peripheral topological
path may be well described by superior finite difference
schemes of electronegativity and chemical hardness in-
dices in certain calibrating conditions.
Although there are reports [14,15] that the electro-
negativity and hardness are periodic, no rationale has
been put forward justifying their periodicity. We are after
the quest for the origin and development of these two
descriptors with an intention to correlate and ju stify their
periodic nature. We strongly guess that the global hard-
ness and electronegativity originate and develop from the
same fundamental source within the constitution of at-
oms and their periodic nature would be straightforward
in this rationale.
2. The Definition of Electronegativity
The attempt of scientific definition and measurement of

N. ISLAM ET AL.
136
electronegativity was started with the seminal work of
Pauling [16,17] who suggested for the first time a scien-
tific definition of electronegativity as “the power of an
atom in a molecule to attract electrons toward itself”.
Though the electronegativity has been defined in many
different ways after Pauling, the most logically it has
been identified as electrostatic force or energy [2-6,18-20]
—with which an atom holds the valence electrons. In this
model, electronegativity has the origin in the electrostatic
field and interaction within the volume of the atom. Thus,
in the electrostatic model, electronegativ ity has its origin
in the attraction (or influence) of the nucleus on the va-
lence electrons or the electron cloud of the atom. For that
reason, to assign the electronegativity value of any sys-
tem, we have to suggest a model relying upon its funda-
mental nature—the holding power of the electron cloud
by the chemical species for its measurement.
Because our prime motive is to rationalize the perio-
dicity of electronegativity, we lay emphasis on such
scales of measurement that are based upon electrostatic
concept and periodicity can be easily included and justi-
fied. In 1946, the direct relation between electronegativ e-
ity and the effective nuclear charge was suggested by
Gordy [20]. Recently, Ghosh and Chakraborty [3] modi-
fied Gordy’s electrostatic scale of electronegativity. They
[3] suggested that the electronegativity of an atom is
equal to the electrostatic potential felt by one of its va-
lence electrons at a radial distance equal to its absolute
radius or most probable radius. Justifiably, such po tential
is created by the conjoint action of the nucleus and the
remaining electrons in the atom.
Ghosh and Chakraborty [3] argued that the electro-
negativity, χ, is no equal but proportional to the ratio of
effective nuclear charge, Zeff and absolute radius or most
probable radius, r of the atoms and proposed the electro-
negativity equation as follo ws:
eff
Z
a
r
 b

 (1)
where “a” and “b” are the constants to be determined by
least square fitting and these are different for different
periods.
We have plotted the electronegativity values of Ghosh
and Chakraborty [3] as a function of their atomic number
in Figure 1 to demonstrate the periodic behaviour of the
atoms of the 103 elements of the periodic table.
3. The Definition of Hardness
It is apparent that the hardness fundamentally signifies
Figure 1. Plot of the electronegativity (eV) values of 103 elements of the periodic table as a function of their atomic number.

137 N. ISLAM ET AL.
the resistance towards the deformation or polarization of
the electron cloud of the atoms, ions or molecules under
small perturbation of chemical response. Thus, the hard-
ness as conceived in chemistry signifies the resistance
towards the deformation of charge cloud of chemical
systems under small perturbation encountered during
chemical processes. Still there is another notion of hard-
ness—the physical hardness’ originated in solid-state
condensed matter physics signifying the resistance of a
structure towards deformation [21]. But in case of atoms,
the chemical hardness and physical hardness have fun-
damentally evolved with time to converge to the one and
single concept—the hardness in general.
Parr and Pearson [22] using the density functional the-
ory [23,24] as basis, defined the term “absolute hardness,
η” as
22
v
12E N 


 
(2)
Although the hardness was rigorously defined by Parr
and Pearson [22], evaluation of accurate hardness value
of atoms through the rigorous theoretical calculation us-
ing Equation (2) is not easy [25], because the numerical
method is the only route of evaluating global hardness of
the atoms. Moreover, Reed [26] has pointed out that
there is inherent mathematical inconsistency in evaluat-
ing global hardness by finite difference approximation
method of Parr and Pearson [22]. Moreover, since hard-
ness is not an observable, the possibility of its quantum
mechanical evaluation is ruled out. Thus there is ample
scope of venturing for semi-empirical methods of evalu-
ating global hardness of atoms. These ventures require
relying upon the fundamental nature of the hardness
again—the holding power of the electron cloud by the
chemical species. We [5] developed a semi-empirical
algorithm relating hardness with the radius of the atom.
We have suggested and evolv ed an algorithm of evaluat-
ing the global hardness of atoms presented below:
7.2
a
r
 b

 (in eV) (3)
where r is the absolute ( most probable ) radius of atoms
in proper un it and a and b are constants.
We have computed the global hardness of 103 ele-
ments of the periodic table through the Equation (3) us-
ing atomic radii computed by us [27]. Since the absolute
of atoms are periodic, the periodic nature of hardness is
follows fr om Equation ( 3).
We have plotted the atomic hardness values, computed
through the Equation (3), as a function of their atomic
number in Figure 2 to demonstrate the periodic behaviour
Figure 2. Plot of the hardness (eV) values of 103 elements of the periodic table as a function of their atomic number.

N. ISLAM ET AL.
138
of the atoms of the 103 elements of the periodic t able.
4. Rationale of the Formulae of Evaluation
of Electronegativity and Hardness and
Their Commonalities and Periodicity
We are trying to posit that the electronegativity and the
hardness originate and develop from the same funda-
mental source in the atom. They must have originated
from the atomic nucleus. It is fundamentally explored
that the electronegativity and the global hardness have
two different labels or legends of the same fundamental
property of atoms. In a recent work, we [6] fo und that the
algorithms for the evaluation of the electronega tivity and
the hardness are identical. Now, before discussing the
periodicity of electronegativity and hardness we have to
establish the relation between these two descriptors with
the fundamental property—the effective nuclear charge
that determines their magnitudes.
Following Slater’s suggestion, the atomic radius is the
value of r for which r f(r) has the extrema, Ghosh et al.
[27,28] calculated the most probable radii (rmax) of the
atoms of 103 elements of the periodic table using sug-
gestion of Slater that r = rmax, using the following Equa-
tions
r = rmax= n/ξ (4)
or
*
max n
rr

 (5)
where ξ is the orbital exponent related to the screening
constant and to the effective nuclear charge; n* is effect-
tive principal quantum number.
The orbital exp onent, ξ is defined as
eff
*
Z
n
  (6)
Hence, putting th e value of ξ in Equation (5), we rear-
ranged the formula for computing the most probable ra-
dii as
*2
eff
n
rZ  (7)
Now, we can rearrange the formula for computing
atomic electronegativity of Ghosh and Chakraborty [3]
using the above Equation (7) as follows:
2
*2
Z
a
n
eff
 b (8)
or,
2
eff
Zx (9)
As a, b and n* are the constant in a period.
Thus the electronegativity is intimately connected to
its originator, the effective nuclear charge.
Again, putting this value in our suggested formula of
hardness, Equation (3) above, we can write:
eff
*2
Z
7.2a b
n


 (10)
As a, b and n* are the constant in a period, from the
above Equation (10) we can say that the hardness is pro-
portional to the effective nuclear charge:
η  Zeff (11)
Above Equation (11) clearly shows that the hardness
is directly related to the electron attracting power of the
nucleus—the effective nuclear charge.
Thus the problem of correlating the periodicity of the
electronegativity and the hardness boils down to the
fundamental nature of the variation of the effective nu-
clear charge.
5. The Effective Nuclear Charge—Is It
Periodic?
In a multi-electronic species, the electrons don’t experi-
ence the full positive charge of the nucleus due to the
shielding of the inner electrons. The effective nuclear
charge is the charge felt by the valence electrons after
taken into account the number of shielding electrons that
surround the nucleu s. It is an empirical parameter, which
depends on both the nuclear charge and the number of
shielding electrons. The nuclear charge keeps increasing.
Meanwhile, the shielding electrons stay constant while
we are going across s and p parts of the period, and in-
crease gradually across the d part of the period. Then in
the next period, they jump in number. Consequently, the
effective nuclear charge drops at that point. Therefore,
the effective nuclear charge increases as we go across a
period and then drops and starts over again at +1 when
we start the next period. Within a period the effective
nuclear charge increases as we go across the periodic
table. As we go d own a gr oup , the increase in the nuclear
charge is cancelled out by the increase in shielding elec-
trons and the effective nuclear charge stays pretty much
the same. Effective nuclear charge is quite o ften referred
to as the kernel charge. The “kernel” includes the nu-
cleus and all shielding electrons but does not includ e the
valence electrons.
Ghosh and Biswas [28] following Slater [29], have
evaluated the screening constant, S and the orbital expo-
nent, ξ, for the topmost electrons of the atoms of the 103
elements of the periodic table. However, there are other
sources [30,31,32] we rely upon the work of Ghosh and

139 N. ISLAM ET AL.
Biswas [28] to explore the periodicity of the effective
nuclear charge of the atoms.
In Figure 3 the physical process of screening is de-
picted.
Figure 4 depicts the periodicity in atomic effective
nuclear charge values [28] plotted against the effective
atomic number (Zeff) taken from reference (28).
On moving form left hand side to right hand side of
any period of the periodic table, charge periodically is
added to the outermost orbital, therefore, it can be as-
sumed that attraction of nucleu s on the outermost shell is
increased periodically as “n” or “n*” remains constant in
a period. As a consequence, Zeff increases monotonically
without any exception in a period. Thus the effective
nucleus charge of elements must be a periodic property.
In Figure 5 we made a comparative study of electro-
negativity an d hardness with their origin ator—the effect-
tive nuclear charge.
6. Results and Discussion
Looking on Figures 1, 2, 4 and 5 reveals that in any pe-
riod the values of electronegativity, hardness and effect-
tive nuclear charge is the lowest for alkali metal and
highest for the noble gas atoms.
As there is repetition of shell structure as one proceed s
down ward in the periodic table, a new shell is started
after it is completely filled up. Of course, some new or-
bital appears in lanthanoids and actinoids but it steadily
happen that the effective nuclear charge increases mono-
tonically in each period without any exception.
When we look at the whole Figure 5 at a glance we
are convinced that in a period the effected nuclear charge,
electronegativity and hardness would increase monotoni-
cally to be maximum at the noble gas elements and in the
pattern is repeated next period.
7. Conclusions
Thus, we reach to a converging point that the electro-
negativity and hardness have the same fundamental na-
ture i.e., the electron attracting power although they are
applied to different fields for the shake of convenience.
Their common property—the electron attracting power
and periodicity are controlled by the atomic nucleus cre-
ating electrostatic field of attraction. Their origin and de-
velopment are unequivocally the same and similar. The
physical process of screening is a reality but the proper
operation and manifestation is mysterious to the common
sense. We hope that the puzzle of inter electronic screen-
ing can be rationalized by invoking the quantum field
theory. Thus, the periodicity of electronegativity and hard-
ness find justification in the periodicity of the electron
Figure 3. The physical process of screening and the effective
nuclear charge.
Figure 4. Plot of the effective nuclear charge values of 103 elements of the periodic table as a function of their atomic number.

N. ISLAM ET AL.
140
Periodic Parameters
Figure 5. Comparative plot of the effective nuclear charge, electronegativity and hardness values of 103 elements of the peri-
odic table as a function of their atomic number.
attracting power originated from the nuclei of the atoms.
8. References
[1] M. V. Putz, “Electronegativity, Quantum Observable,”
International Journal of Quantum Chemistry, Vol. 109,
No. 4, 2009, pp. 733-738. doi:10.1002/qua.21957
[2] D. C. Ghosh, “A New Scale of Electronegativity Based
on Absolute Radii of Atoms,” Journal of Theoretical and
Computational Chemistry, Vol. 4, No. 1, 2005, pp. 21-33.
doi:10.1142/S0219633605001556
[3] D. C. Ghosh and T. Chakraborty, “Gordy’s Electrostatic
Scale of Electronegativity Revisited,” Journal of Molecular
Structure: THEOCHEM, Vol. 906, No. 1-3, 2009, pp. 87-
93. doi:10.1016/j.theochem.2009.04.007
[4] D. C. Ghosh, “The Scales and Concept of Electronegativ-
ity,” Journal of the Indian Chemical Society, Vol. 80,
2003, pp.527-533.
[5] D. C. Ghosh and N. Islam, “Semi-Empirical Evaluation
of the Global Hardness of the Atoms of 103 Elements of
the Periodic Table Using the Most Probable Radii as
Their Size Descriptors,” International Journal of Quan-
tum Chemistry, Vol. 110, No. 6, 2010, pp. 1206-1213.
[6] D. C. Ghosh and N. Islam, “Whether Electronegativity
and Hardness Are Manifest Two Different Descriptors of
the One and the Same Fundamental Property of At-
oms—A Quest,” International Journal of Quantum Che-
mistry, Vol. 111, No. 1, 2011, pp. 40-51.
doi:10.1002/qua.22415
[7] R. G. Parr and Z. Zhou, “Absolute Hardness: Unifying
Concept for Identifying Shells and Sub Shells in Nuclei,
Atoms, Molecules, and Metallic Clusters,” Accounts of
Chemical Research, Vol. 26, No. 5, 1993, pp. 256-258.
doi:10.1021/ar00029a005
[8] M. V. Putz, N. Russo and E. Sicilia, “About the Mulliken
Electronegativity in DFT,” Theoretical Chemistry Ac-
counts, Vol. 114, No. 1-3, 2005, pp. 38-45.
doi:10.1007/s00214-005-0641-4
[9] M. V. Putz, “Systematic Formulations for Electronegativ-
ity and Hardness and Their Atomic Scales within Density
Functional Softness Theory,” International Journal of
Quantum Chemistry, Vol. 106, No. 2, 2006, pp. 361-389.
doi:10.1002/qua.20787
[10] M. V. Putz, “Semi Classical Electronegativity and Che-
mical Hardness,” Journal of Theoretical and Computa-
tional Chemistry, Vol. 6, No. 1, 2007, pp. 33-47.

141 N. ISLAM ET AL.
doi:10.1142/S0219633607002861
[11] M. V. Putz, N. Russo and E. Sicilia, “Atomic Radii Scale
and Related Size Properties from Density Functional
Electronegativity Formulation,” Journal of Physical Che-
mistry A, Vol. 107, No. 28, 2003, pp. 5461-5465.
doi:10.1021/jp027492h
[12] L. Tarko and M. V. Putz, “On Electronegativity and Che-
mical Hardness Relationships with Aromaticity,” Journal
of Mathematical Chemistry, Vol. 47, No. 1, 2010, pp. 487-
495. doi:10.1007/s10910-009-9585-6
[13] M. V. Putz, “Chemical Action and Chemical Bonding,”
Journal of Molecular Structure: THEOCHEM, Vol. 900,
No. 1-3, 2009, pp. 64-70.
doi:10.1016/j.theochem.2008.12.026
[14] P. K. Chattaraj and B. Maity, “Electronic Structure Prin-
ciples and Atomic Shell Structure,” Journal of Chemical
Education, Vol. 78, No. 6, 2001, pp. 811-812.
doi:10.1021/ed078p811
[15] P. K. Chattaraj, D. R. Roy and S. Giri, “Electronic Struc-
ture Principles in Static and Dynamic Situations,” Com-
puting Letters, Vol. 3, No. 2-4, 2007, pp. 223-230.
doi:10.1163/157404007782913336
[16] L. Pauling, “The Nature of the Chemical Bond. IV. The
Energy of Single Bonds and Relative Electronegativity of
Atoms,” Journal of the American Chemical Society, Vol.
54, No. 9, 1932, pp. 3570-3582.
doi:10.1021/ja01348a011
[17] L. Pauling, “The Nature of the Chemical Bond,” 3rd Edi-
tion, Cornell University: Ithaca, New York, 1960.
[18] L. Allred and E. G. Rochow, “A Scale of Electronegativ-
ity Based on Electrostatic Force,” Journal of Inorganic
and Nuclear Chemistry, Vol. 5, No. 4, 1958, pp. 264-268.
doi:10.1016/0022-1902(58)80003-2
[19] R. S. Mulliken, “A New Electroaffinity Scale; Together
with Data on Valence States and on Valence Ionization
Potentials and Electron Affinities,” Journal of Chemical
Physics, Vol. 2, No. 11, 1934, pp. 782-793.
doi:10.1063/1.1749394
[20] W. Gordy, “A New Me thod of Determining Electronega-
tivity from Other Atomic Properties,” Physical Review,
Vol. 69, No. 11-12, 1946, pp. 604-607.
doi:10.1103/PhysRev.69.604
[21] J. Gilman,“ Chemical and Physical Hardness,” Materials
Research Innovations, Vol. 1, No. 2, 1997, pp. 71-76.
doi:10.1007/s100190050023
[22] R. G. Parr and R. G. Pearson, “Absolute Hardness, Com-
panion Parameter to Absolute Electronegativity,” Journal
of the American Chemical Society, Vol. 105, No. 26, 1983,
pp. 7512-7516. doi:10.1021/ja00364a005
[23] P. Hohenberg and W. Kohn, “Inhomogeneous Electron
Gas,” Physical Review, Vol. 136, No. 3B, 1964, pp. 864-
871. doi:10.1103/PhysRev.136.B864
[24] R. G. Parr and W. Yang, “Density Functional Theory of
Atoms and Molecules,” Oxford University Press, Oxford,
1989.
[25] K. D. Sen and S. C. Vinayagam, “Absolute Hardness
Parameter: Finite Difference versus Density Functional
Theoretic Definition,” Chemical Physics Letters, Vol. 144,
No. 2, 1988, pp. 178-179.
doi:10.1016/0009-2614(88)87112-4
[26] J. L. Reed, “Electronegativity, Chemical Hardness I,”
Journal of Physical Chemistry A, Vol. 101, No. 40, 1997,
pp. 7396-7400. doi:10.1021/jp9711050
[27] D. C. Ghosh, R. Biswas, T. Chakraborty, N. Islam and S.
K. Rajak, “The Wave Mechanical Evaluation of the Ab-
solute Radii of Atoms,” Journal of Molecular Structure:
THEOCHEM, Vol. 865, No. 1-3, 2008, pp. 60-67.
doi:10.1016/j.theochem.2008.06.020
[28] D. C. Ghosh and R. Biswas, “Theoretical Calculation of
Absolute Radii of Atoms and Ions. Part 1. The Atomic
Radii,” International Journal of Molecular Sciences, Vol.
3, No. 2, 2002, pp. 87-113. doi:10.3390/i3020087
[29] J. C. Slater, “Atomic Shielding Constants,” Physical Re-
view, Vol. 36, No. 1, 1930, pp. 57-64.
doi:10.1103/PhysRev.36.57
[30] E. Clementi and D. L. Raimondi, “Atomic Screening
Constants from SCF Functions,” Journal of Chemical
Physics, Vol. 38, No. 11, 1963, pp. 2686-2689.
doi:10.1063/1.1733573
[31] E. Clementi, D. L. Raimondi and W. P. Reinhardt,
“Atomic Screening Constants from SCF Functions. II.
Atoms with 37 to 86 Electrons,” Journal of Chemical
Physics, Vol. 47, No. 4, 1967, pp. 1300-1307.
doi:10.1063/1.1712084
[32] C. Mande, P. Deshmukh and P. Deshmukh, “A New
Scale of Electronegativity on the Basis of Calculations of
Effective Nuclear Charges from X-Ray Spectroscopic
Data,” Journal of Physics B: Atomic and Molecular Phy-
sics, Vol. 10, No. 12, 1977, pp. 2293-2301.
doi:10.1088/0022-3700/10/12/008