Computational and Theoretical Chemistry 1052 (2015) 42–46
Contents lists available at ScienceDirect
Computational and Theoretical Chemistry
journal homepage: www.elsevier.com/locate/comptc
A new equation based on ionization energies and electron affinities
of atoms for calculating of group electronegativity
Savasß Kaya ⇑, Cemal Kaya
Department of Chemistry, Cumhuriyet University, Sivas 58140, Turkey
a r t i c l e
i n f o
Article history:
Received 25 September 2014
Received in revised form 21 November 2014
Accepted 21 November 2014
Available online 28 November 2014
Keywords:
Group electronegativity
DFT
Electronegativity equalization
A new group electronegativity equation
a b s t r a c t
In particular, in organic chemistry, the electronegativity of functional groups are taken into consideration
significantly to predict reaction mechanism and to explain inductive effects of functional groups. In the
present study, considering the relationship with charge of the electronic energy for atoms, Sanderson’s
electronegativity equalization principle and Density Functional Theory (DFT), we have obtained a new
equation by which group electronegativity can be calculated from ionization energies (I) and electron
affinities (A) of atoms that constitute the group, where the results obtained were compared with equation
of Sanderson who proposed that group electronegativity is the geometric mean of atomic electronegativities. For a large number of groups, it was found to be a very close agreement with a standard deviation of
0.12. The advantage of the present equation is that it can be used for ionic groups. In addition, the derived
equation can be used to calculate Mulliken and Pauling electronegativities of molecules.
Ó 2014 Elsevier B.V. All rights reserved.
1. Introduction
The principal aim of theoretical chemistry is to develop rules to
explain chemical reactions and molecules in a quantitative and
predictable way. The electronegativity is a useful theoretical
descriptor in correlating chemico-physical properties of atoms
and molecules. The idea of group electronegativity is historically
important because the electronegativity concept evolved largely
from the desire of organic chemists to understand reaction mechanisms in terms of the inductive effects of various functional
groups. In recent years, there has been a considerable amount of
work done that deals with the evaluation and use of group electronegativity [1–6].
A significant development in the electronegativity concept has
been provided by Sanderson’s electronegativity equalization principle [7,8]. According to this principle, when two or more atoms
initially different in electronegativity combine chemically, their
electronegativities have become equalized. The equalized value
gives the electronegativity of the formed molecule and is equivalent to the geometric mean of the electronegativity values of the
constituent atoms, as in Eq. (1). Electronegativity was originally
introduced by Pauling as a measure of an atom in a molecule to
attract electronic charge. Considering the Sanderson’s electronegativity equalization principle it can be said that when two or more
⇑ Corresponding author.
E-mail address: [email protected] (S. Kaya).
http://dx.doi.org/10.1016/j.comptc.2014.11.017
2210-271X/Ó 2014 Elsevier B.V. All rights reserved.
different atoms combine to form a molecule, their electronegativities change to a common intermediate value and become equalized. If so, group electronegativity that is a measure of inductive
effects of functional groups can be defined as final electronegativity values of atoms in a molecule.
v¼
N
Y
!1=N
vi
ð1Þ
i¼1
where v is the electronegativity of molecule or combined atom, vi is
the electronegativity of pre-bonded i-th atom, N is the total number
of atoms in the molecule. For the calculation of group electronegativity, apart from geometric mean method of Sanderson, the
arithmetic mean method [9–11] is also used for this purpose.
According to arithmetic mean method, the electronegativity of
any group is equivalent to the arithmetic mean of the electronegativity values of the constituent atoms in the group and can be
calculated via following equation.
vG ¼
PN
i¼1
N
vi
ð2Þ
XG represents the electronegativity of molecule or combined atom,
vi is the electronegativity of pre-bonded i-th atom, N is the total
number of atoms in the molecule.
In a study related to metal oxides, F. Di. Quarto and his colleagues [11] proposed that the results of arithmetic mean method
for calculation of group electronegativity are compatible with
S. Kaya, C. Kaya / Computational and Theoretical Chemistry 1052 (2015) 42–46
43
experimental data. Therefore, arithmetic mean method is also a
valid useful method in the calculation of group electronegativity.
Another important study for the electronegativity concept has
been made by Iczkowski and Margrave [12]. They proposed that
the electronegativity of an atom may be expressed as linear function of the charge on the atom:
v ¼ a þ 2bd
ð3Þ
where a = (I + A)/2 and b = (I A)/2, I and A are the valence-state
ionization energy and electron affinity of the atom, respectively, d
is partial charge. Many workers have been used Eq. (3) for calculating the partial charges of atoms in functional groups and molecules
[13–16].
In estimating group electronegativity and partial charges,
Huheey [13] suggested a simple scheme based on the Iczkowski–
Margrave equation and Sanderson’s principle. In his study, the following expressions were used for the substituent ABn:
Fig. 1. Atomic energy (E) change with charge (d) for oxygen atom. (The energy of
neutral atom is assumed to be zero.)
vABn ¼ vA ¼ aA þ 2bA dA
vABn ¼ vB ¼ aB þ 2bB dB
ð4Þ
ð5Þ
Eq. (9), if d is taken (+1), E will be the energy of the (+1) cation
energy or first ionization energy. Likewise, for d = 1, the energy
will be negative of the first electron affinity (Note that the definition
of electron affinity does not follow the usual thermodynamic convention in that a positive electron affinity is exothermic). This argument leads to the following relations:
dA þ ndB ¼ q
ð6Þ
I ¼aþb
ð10Þ
A ¼ a þ b
ð11Þ
where q is the charge of the substituent.
Another group electronegativity equation was derived by
Bratsch [17],
Nþq
vG ¼ P t
v
ð7Þ
here, vG is the electronegativity as equalized through Sanderson’s
principle, N = R(t) is the total number of atoms in the species formula, q is the charge of the species and v is the Pauling electronegativity of pre-bonded or isolated atoms in the species.
A similar equation was proposed by Mullay [18],
N þ 1:5q
vG ¼ P t
ð8Þ
v
In the present report, it was derived a new equation which
enables us to calculate group electronegativity directly from ionization energies and electron affinities of pre-bonded atoms. In
order to the applicability of the derived equation, our results were
compared with those obtained from the other methods given in the
literature.
2. Methodology
E ¼ ad þ bd
a ¼ ðI þ AÞ=2
ð12Þ
b ¼ ðI AÞ=2
ð13Þ
As suggested by Iczkowski–Margrave, the first derivative of the
energy with respect to the charge may be defined as the electronegativity of the atom:
v ¼ a þ 2bd
v1 ¼ a1 þ 2b1 d1
v2 ¼ a2 þ 2b2 d2
v3 ¼ a3 þ 2b3 d3
..
.
..
.
vN ¼ aN þ 2bN dN
On the basis of Sanderson’s principle, one can write the following
relation:
v1 ¼ v2 ¼ v3 ¼ . . . ¼ vN ¼ vG
ð15Þ
where vG is the electronegativity of the species. Using the condition
for the species to be q charge, namely
d1 þ d2 þ d3 þ þ dN ¼ q
ð16Þ
The partial charge of any atom in the species can be given as
follows;
di ¼
vG ai
2bi
¼
vG
2bi
ai
2bi
ð17Þ
From Eqs. (16) and (17) one can obtained the following equation:
PN ai ð9Þ
where E is the total energy of the atom, that is found from RI or RA,
I and A are ionization energy and electron affinity of the atom. In
ð14Þ
As can be seen from this equation, a is equal to electronegativity
of neutral atom. Eq. (12) is similar to that of Mulliken, but he used
valence-state ionization energy and electron affinity whereas we
employed here those of ground-state. Therefore, the electronegativity concept in this paper may be defined as absolute electronegativity, as proposed by Parr and co-workers [22,23].
Considering equation (14), for each atom in a chemical species
(molecule or group) that contains N atoms, we can write
..
.
One of the most important works related to electronegativity
has been demonstrated by Iczkowski and Margrave. They defined
electronegativity as the derivative of ionization energy with
respect to charge. Later on, Jaffe and co-workers introduced the
idea of orbital electronegativity considering the study of Iczkowki
and Margrave. Assuming that the energy of neutral atom is zero,
the following curve that gives the relationship between atomic
energy and charge is obtained [19,20]. Such a curve for oxygen
atom is given in Fig. 1. For all atoms, same curve is not the case
but it should be expressed that the correlation between atomic
energy and charge is parabolic as indicated in Eq. (9) [21]. In
Fig. 1, it is seen that the energy of atom is zero at d(charge) = 0
point.
The function of the curve may be described rather accurately by
the quadratic formula for small d values [12],
2
From these equations, the following relations are obtained:
vG ¼
þ 2q
PN 1 i¼1 bi
i¼1 bi
ð18Þ
44
S. Kaya, C. Kaya / Computational and Theoretical Chemistry 1052 (2015) 42–46
Furthermore, by inserting Eqs. (12) and (13) into Eq. (18) we
obtained the following equation which allows direct calculation of
electronegativity from the ground-state ionization energies and
electron affinities of the constituent atoms.
PN Ii þAi vG ¼
i¼1 Ii Ai
PN þ 2q
ð19Þ
2
i¼1 Ii Ai
where Ii and Ai are the ionization energy and electron affinity of i-th
atom, respectively, q is the charge of chemical species considered.
3. Results and discussion
To ascertain whether Eq. (19) was the valid one or not, it was
first used to calculate group electronegativities of a large number
of neutral species (groups or molecules) and then compared with
those determined from Sanderson’s equation [Eq. (1)]. In our calculations, ionization energies and electron affinities of atoms are
taken from Ref. [24] and the vi in the Sanderson’s equation was
estimated via Eq. (1). The results obtained are given in Table 1.
Inspection of Table 1 reveals that there exist a very good agreement between the group electronegativities calculated by the
two equations. The mean error measured as (r2/n)1/2 was estimated to be 0.12. Such an excellent fit indicates that Eq. (19) can
be employed to estimate group electronegativity. In addition to
geometric mean method, the results of new method were also
compared with arithmetic mean method that is given by Eq. (2).
Here as well, the agreement between the results that are given in
Table 1 is remarkable.
In the light of presented new method, both absolute electronegativity and Mulliken and Pauling electronegativities of functional
groups and molecules can be calculated. However, in Table 1, only
absolute electronegativity values of groups and molecules are provided. Absolute electronegativity concept firstly was being used by
Pearson. Absolute electronegativity of any chemical species is
determined considering its ground state ionization energy and
electron affinity and Pearson took into consideration the absolute
electronegativity absolute hardness [25–27]. It should be stated
that absolute electronegativity term is not widely used like Pauling
electronegativity and Mulliken electronegativity but, in recent
times, the presence of some methods that consider the absolute
electronegativity concept in the literature draws attention.
In the conceptual Density Functional Theory (DFT), Parr and coworkers [28–30] suggested that the electronegativity of a species
(atom, molecule or ion) can be estimated via Eq. (20):
Table 1
Comparison of the electronegativities calculated by Eqs. (19), (1) and (2).
Group/molecule
CH3
CH3CH2
(CH3)2CH
(CH3)3C
C6H6
C6H5NH2
C6H5OH
(CH3)3N
CH2O
CH3CHO
CH3COOH
CH3NO2
CH2F
CHF2
CHFCl
CHClBr
CClBrI
SiH3
SiF3
NF2
NCl2
NH2
N(CH3)2
NHOH
PH2
PCl2
PF2
OF
OCl
BrCl
NF3
NCl3
IF
HF
HCl
HBr
BF2
BCl2
Be(CH3)2
BF3
PH3
(CH3)2O
SCN
COOH
Group electronegativities
Group/molecule
Eq. (19)
Eq. (1)
Eq. (2)
6.90
6.87
6.85
6.84
6.67
6.74
6.73
6.93
6.99
6.92
7.00
7.14
7.63
8.34
7.91
7.37
7.22
6.24
8.10
9.39
8.06
7.22
6.94
7.19
6.56
7.47
8.40
8.88
7.97
7.93
9.65
8.12
8.02
8.70
7.83
7.42
7.56
6.82
6.62
8.16
6.70
6.98
6.50
7.09
6.94
6.90
6.89
6.88
6.71
6.78
6.77
6.96
7.03
6.95
7.00
7.16
7.62
8.35
7.90
7.30
7.19
6.48
8.56
9.25
7.95
7.22
6.98
7.30
6.62
7.29
8.47
8.86
7.91
7.94
9.53
8.04
8.38
8.65
7.72
7.38
7.74
6.66
6.68
8.34
6.75
7.00
6.58
7.11
6.95
6.92
6.91
6.65
6.72
6.81
6.78
6.97
6.99
6.94
6.99
7.12
7.75
8.56
8.03
7.27
7.22
6.57
9.00
9.37
7.96
7.21
6.98
7.23
6.66
7.40
8.82
8.88
7.91
7.94
9.63
8.04
7.11
8.79
7.73
7.38
8.37
6.95
6.70
8.88
6.78
7.01
6.59
7.35
C5H5N
C6H5SH
HCONH2
CH4
CH3COCH3
NaF
NaCl
BeO
MgO
BeS
MgS
BrI
SO
OH
NH
F2
S2
CS2
COS
SO2
SO3
O3
N2O
PBr3
PCl3
POCl3
CH3I
HNO3
SF6
CF3Br
H2O
H2S
NH3
CO2
CH3CN
C2H2
CH3Cl
BCl3
HCN
(CH3)3P
C(CH3)4
(CH3)2S
NO
CO
Group electronegativities
Eq. (19)
Eq. (1)
Eq. (2)
6.71
6.62
7.07
6.96
6.90
4.72
4.64
6.02
5.24
5.59
4.95
7.15
6.75
7.37
7.24
10.41
6.22
6.23
6.59
6.98
7.11
7.54
7.41
7.15
7.67
7.63
6.87
7.43
9.49
8.68
7.30
6.76
7.20
7.06
6.85
6.67
7.25
7.18
6.84
6.79
6.86
6.80
7.43
6.84
6.76
6.67
7.09
6.99
6.92
5.44
4.86
6.08
5.32
5.52
4.83
7.16
6.85
7.36
7.24
10.41
6.22
6.23
6.65
7.07
7.18
7.54
7.38
7.04
7.52
7.53
6.90
7.42
9.67
8.83
7.30
6.84
7.20
7.09
6.88
6.71
7.19
7.04
6.90
6.83
6.90
6.86
7.42
6.88
6.77
6.68
7.105
6.99
6.93
6.62
5.56
6.17
5.81
5.51
5.15
7.17
6.83
7.35
7.23
10.41
6.21
6.23
6.67
7.09
7.20
7.53
7.38
7.09
7.62
7.60
6.90
7.41
9.81
9.01
7.29
6.85
7.20
7.11
6.89
6.71
7.21
7.28
6.91
6.84
6.90
6.86
7.42
6.90
45
S. Kaya, C. Kaya / Computational and Theoretical Chemistry 1052 (2015) 42–46
Table 2
Electronegativities of some molecules.
I2
IBr
S2
Br2
Cl2
O2
F2
CS2
COS
SO2
O3
N2O
PBr3
PCl3
POCl3
CH3I
SO3
C2H2
HNO3
CS
CO
H2
a
Ia (eV)
Aa (eV)
9.40
9.79
9.40
10.56
11.48
12.06
15.70
10.08
11.18
12.34
12.67
12.89
9.85
9.91
11.40
9.54
11.00
11.41
11.03
11.71
14.00
15.40
2.42
2.55
1.66
2.60
2.40
0.44
3.08
1.00
0.46
1.05
1.82
1.47
1.60
0.80
1.40
0.20
1.70
0.43
0.57
0.20
1.80
2.00
Electronegativity
Group
Eq. (20)
Eq. (19)
5.91
6.17
5.53
6.58
6.94
6.25
9.39
5.54
5.82
6.70
7.25
7.18
5.73
5.34
6.40
4.87
6.35
5.92
5.80
5.90
6.10
6.70
6.76
7.15
6.22
7.59
8.30
7.54
10.41
6.23
6.59
6.98
7.54
7.41
7.15
7.67
7.63
6.87
7.11
6.67
7.43
6.24
6.84
7.18
Ionization energies and electron affinities were taken from Ref. [33].
v¼
IþA
2
ð20Þ
where I and A are ionization energy and electron affinity of the species. It is clearly seen that this equation is equivalent to our Eq. (12)
for neutral atoms. As a further check upon the validity of Eq. (19),
the electronegativities of some molecules were calculated by Eqs.
(19) and (20) and the obtained results are presented in Table 2. It
can be seen from data in the table that a considerable fit between
the two calculations was obtained.
The advantage of Eq. (19) with respect to Eq. (1) is that it can be
employed for ionic species. The calculated electronegativities via
Eq. (19) for some ionic groups are given in Table 3.
As can be seen from data in Table 3, the electronegativity of a
given group increases as the charge increases. This is not surprising
taking into consideration that Eq. (19) can be rearranged as;
vG ¼ aG þ bG q
ð21Þ
where
aG ¼
X Ii þ Ai X I i Ai
2
I i Ai
and bG ¼
X
1
I i Ai
This equation is implies that there is a linear correlation between
electronegativity and charge of a particular group. This linearity is
clearly seen in Fig. 2 for CH3, OH, NO2 groups.
In the determination of partial charges of atoms in a chemical
species, we have employed Eq. (17). The calculated values by this
equation are given in Table 4 together with those of [31]. It should
be noted that our values in a good agreement with those of the
other methods. Partial atomic charges are widely used for the
description for charge distributions of molecules. However, there
is no physical basis for atomic charge concept. As distinct from
many other electronic properties that can be determined from a
quantum mechanical wave function, there is no a flawless method
for determining the partial charge of an atom in a molecule [34].
Partial charges are not experimentally observable and cannot be
determined by quantum mechanical methods. For prediction of
partial charges of atoms in a molecule, numerous theoretical and
experimental methods have been suggested. Some of these methods are based on equalization principles related to reactivity
indexes such as electronegativity, chemical hardness like our
OH
CO2
3
SH
AsH
2
ClO
PH
2
SO2
4
NH
2
NCS
NO2
CH
3
ClO
2
NO
3
C2H
3
CH3O
vG
Group
vG
1.12
1.40
1.56
2.65
2.68
2.68
2.74
2.77
3.04
3.19
3.91
4.15
4.32
4.45
4.62
ClO
3
4.94
4.94
5.43
5.90
6.28
7.64
8.82
8.87
9.11
9.91
10.44
12.27
12.60
13.62
14.04
CH3CO
ClO
4
CF
3
BF4
C6H+5
CH3CO+
SiH+3
NH2NH+3
CH+3
H3O+
CF+3
CN+
OH+
NO+
16
14
Group electronegativity
Molecules
Table 3
Electronegativities of some common ionic groups.
12
CH 3
10
OH
NO 2
8
6
4
2
0
-1,5
-1,0
-0,5
0,0
0,5
1,0
1,5
Charge of Group
Fig. 2. Electronegativity of CH3, OH and NO2 groups plotted as a function of partial
charge.
method. As a result, it can be said that the approaches and rules
for determining partial atomic charges can include some
uncertainties.
In the current literature, chemical potential and electronegativity are frequently used interchangeably. Chemical potential (l) is
defined as the negative of electronegativity (v) and the relation
between these two properties is provided by the following equation [35,36].
l ¼ v ¼
@E
@N
ð22Þ
tðrÞ
here, v, l, ˆ(r) are electronegativity, chemical potential and external
potential, respectively. This relation given by Eq. (22) has been criticized by Pearson [37] and Allen [38]. Both Pearson and Allen have
proposed that Pauling’s electronegativity and the chemical potential should be regarded as two separate and distinct properties.
Later on, similar criticisms related to this topic were made by Politzer and his co-workers [39–41] and they stated that the validity of
v = l is not the case. Thus, we recommend that the idea of Politzer
should be considered when used Eq. (18) and our equation should
be used in the calculation of electronegativities of functional groups
and molecules rather than chemical potential calculations.
In conclusion, the method here makes it possible to predict
group electronegativity from ground-state ionization energies
and electron affinities of constituent atoms. The fact that our
results are agreement with those obtained from other methods
shows that our method is applicable one. It should also be noted
that Eqs. (18) and (19) can be employed for estimating the
46
S. Kaya, C. Kaya / Computational and Theoretical Chemistry 1052 (2015) 42–46
Table 4
Comparisons of partial charges.
Molecule
Atom
Present work
Jaffe/Huheey [32]
NaF
Na
F
K
Cl
Ca
Cl
Sr
Br
B
F
Si
I
N
F
H
O
H
O
H
S
H
F
I
F
Br
Cl
C
H
O
C
F
+0.41
0.41
+0.45
0.45
+0.48
0.24
+0.48
0.24
+0.23
0.31
+0.23
0.06
0.16
0.05
0.25
0.24
0.48
0.52
0.03
+0.06
+0.12
0.12
+0.17
0.17
+0.04
0.04
0.16
0.20
0.24
+0.30
0.07
+0.43
0.43
+0.46
0.46
+0.58
0.29
+0.58
0.29
+0.14
0.29
+0.07
0.02
+0.09
0.03
+0.39
0.17
0.46
0.54
+0.02
0.04
+0.17
0.17
+0.15
0.15
+0.05
0.05
0.22
0.16
0.30
+0.24
0.06
KCI
CaCl2
SrBr2
BF
4
SiI4
NF3
H3O+
OH
H2S
HF
IF
BrCl
CH3O
CF4
Mulliken and Pauling electronegativities of a group. For the
Mulliken electronegativity, it is necessary to utilize valence-state
ionization energy and electron affinity in Eq. (19). Such a calculation leads to the results the same as those obtained by Huheey.
With the use of the substitutions, ai = vi and bi = vi/2, one can find
Pauling electronegativity via Eq. (18). The calculated values in this
way are equal to those obtained from Bratsch equation.
4. Conclusions
From the results presented in this study, it is observed that the
results of present equation are agreement with both the results of
Sanderson’s geometric mean equation and experimental data. The
group electronegativity can be calculated from ionization energies
(I) and electron affinities (A) of atoms that constitute the group
using the new equation. The advantage of the present equation is
that it can be used for ionic groups. In addition, the derived
equation can be used to calculate Mulliken and Pauling electronegativities of molecules.
Acknowledgement
This research was supported by department of chemistry of
Cumhuriyet University.
References
[1] J. Mulay, Calculation of group electronegativity, J. Am. Chem. Soc. 107 (1985)
7271–7275.
[2] S.G. Bratsch, Revised Mulliken electronegativities: I. Calculation and
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