J. Phys. Chem. 1995, 99, 10742-10746
10742
Electronegativity and Static Electric Dipole Polarizability of Atomic Species. A Semiempirical Relation’ Sanchita Hati and Dipankar Datta* Department of Inorganic Chemistry, Indian Association for the Cultivation of Science, Calcutta 700 032, India Received: September 12, 1994; In Final Form: January 6, 1995@
A quantitative relation between the static electric dipole polarizability (a)and the electronegativity of an atomic species is developed semiempirically. It emerges that the quantity ( q l)(l/a)II3can be used as a new measure of the electronegativity of an atomic species with charge q 2 0 quite effectively.
+
The static electric dipole polarizability (hereafter referred to simply as the polarizability), a, is a measure of the linear response of the electronic cloud of a chemical species to a weak external electric field. This property, which can be determined theoretically and experimentally in various ways, is used extensively in physics to understand a number of phenomena.’ But its potential use in chemistry has not been exploited properly. Before 1990, the only meaningful application of a in chemistry was probably the development of Rittner’s model for understanding the bonding in some singly bonded diatomic molecules in terms of ionic p~larizabilities.~-~ This is because in the past the chemists have always thought of a as a more physical than chemical q ~ a n t i t y . In ~ 1990, for the first time, Nagle correlated polarizability with one of the most important concepts in chemistry, which is electronegativity.6 Subsequently several reports have appeared exploring the relevance of polarizability in chemistry.’-’ Nagle has demonstrated that for a neutral atom the quantity (n/a)II3where n is the number of valence electrons (only the outermost s and p electrons are counted), can be used as a measure of its electronegativity, He has found that (nla)’/3 correlates very nicely with the other existing atomic electronegativity scales. However, his work is empirical and moreover restricted to neutral atoms only.6 Herein we try to examine the relation between a and for neutral and ionized atoms quantitatively. All the quantities appearing in the various equations here are expressed in au. In the statistical theory of atoms, the Fermi-Amaldi model relates the radius r of a monoatomic ion to its chemical potential p by eq 1.l 2 In eq 1, Z is the atomic number and N the number
x.
x
p=-(2-N+l)/r=-(q+l)/r
(1)
of electrons in the ion under consideration. The charge q on the ion is given by Z - N. From the density functional theory,I3 it emerges that p = -x and hence
A,
+
x
of a chemical species is given as ( I A)/2, which is Mulliken’s definition of electronegativity. I 3 Recently Allen has proposed a new experimental measure of electronegativity ()‘Allen) for a neutral free atom as the “average one-electron energy of the valence-shell electron^".'^,'^ Allen’s definition is quite different from Mulliken’s as it is evident that for H and ~ is simply l l equal ~ to~ I . Still, for neutral the alkali atoms ~ atoms, there does exist a very good linear correlation between ( I -I- A)/2 and xA]len.’’ Since, at present, Mulliken’s definition is very popular among the theoretical chemists, in this article A)/2 as the experimental measure of the we shall use ( I electronegativity. Here we would like to point out that the finite difference approximation of eq 1 using the relation p = also yields Mulliken’s formula. Very recently we have shown semiempirically’ that for an . this expression atomic species r = ( ~ d 0 . 7 9 2 ) ” ~Substituting for r in eq 2, we get eq 3. Using eq 3, we have estimated the
+
-x
’
= 0.925(q
+ 1)( ~ a ) ” ~
electronegativities of some 133 atoms with q ranging from 0 to 7 (Table S; supplementary material), where the polarizability data are and compared these with their known experimental electronegativity values obtained from ionization potential and electron affhity data (by Mulliken’s d e f i n i t i ~ n ) . ’ ~ . ~ ~ The values calculated via eq 3 are found to be uniformly much higher than the actual ones. Details are in Table S; here just to give an idea of the extent of observed deviation, we only mention that in the case of some 55 open shell neutral atoms studied the average value of x&d/xexptl is 1.566 (k0.234). The reasons for this kind of behavior of eq 3 will be clear in a subsequent section. From the density functional theory, the hardness 17 of a system is defined as dp/26N.l3 For a monoatomic ion, from eq 1 it then follows
x
17 = 1/2r An assumption in deriving eq 1 is that the radius of the ion does not vary with the charge q in the range q f 1. With the same assumption if one considers the charging energy of a conducting sphere of radius r classically, as described elsewhere,I’.l4 one obtains = qlr, which is incorrect as this relation would mean that the electronegativity of any neutral atom is zero. In terms of the ionization potential I and electron affinity
x
Dedicated to Professor A. Chakravorty on the occasion of his 60th birthday. @Abstractpublished in Advance ACS Abstracts, June 1, 1995.
0022-3654/95/2099-10742$09.00/0
(3)
(4)
Incidentally, in terms of I and A, 17 = ( I - A)/2.l3 From eqs 2 and 4, we have
(q
+ l ) ~ / x= 0.5
(5)
This is a very important result which means that for an atomic species with charge q, r] is proportional to and the proportionality constant is O S ( q 1). Such proportionality between 17 and was first indirectly and empirically assumed by Parr and B a r t ~ l o t t for i ~ ~the neutral atoms and molecules in order to prove Sanderson’s geometric mean principle for the equalization
x
+
0 1995 American Chemical Society
x
Electronegativity and Polarizability of Atomic Species
TABLE 1: The Constancy of (q atodion H Li
B C N 0 F Na A1 Si
P S
c1 K
sc Ti V Cr Mn Fe co Ni cu Ga Ge As Se Br Rb Y Zr Nb
Be Mg Ca Sr
Ba Zn Cd Hg
Li+
Na+ K+ Rb+ CS+
(q
+ l)r]/x 0.895 0.794 0.935 0.797 0.990 0.806 0.673 0.807 0.858 0.709 0.868 0.666 0.564
0.793 0.958 0.977 0.861 0.823 1.ooo 0.938 0.837 0.739 0.725 0.906 0.739 0.849 0.657 0.556 0.791 1.000 0.882 0.750
0.918 1.040 1.818 1.850 1.208 1.110 1.076 1.128 1.733 1.608 1.516 1.465 1.462
J. Phys. Chem., Vol. 99, No. 27, 1995 10743
+ l)t&for Some Open and Closed Shell Neutral and Ionized Atoms!
atodion Mo Ru Rh
Pd Ag In Sn Sb Te I
Cs La Hf Ta W
Re os Ir Pt
Au T1 Pb Bi Li2+
Be3+ B4+ C5+ N6+
07+
Fa+ Ne9+ Na2+
cu+ Ag+ Au+ T1+ Be2+ Mg2+ Ca2+ Sr2+
Zn2+ Ge2+ Cd2+ Sn2+ Hg2+
(q
+ l)r]/x 0.795 0.667 0.735 0.874 0.707 0.903 0.709 0.783 0.641 0.546 0.784 0.839 0.789 0.922 0.814 0.963 0.775 0.704 0.625 0.600 0.906 0.905 0.797 0.709 0.687 0.674 0.666 0.660 0.656 0.653 0.650 0.614 0.896 0.958 0.752 1.079 2.365 2.052 1.866 1.791 1.132 1.094 1.135 1.055 0.872
atodion
( q + l)r]/x
a. Open Shellb Mg3+ 0.615 ~ 1 4 + 0.616 Si5+ 0.618 P+ 0.465 S7f 0.621 ClS+ 0.623 Ar9+ 0.624 K2+ OS47 Ca3+ 0.549 sc4+ 0.551 Ti5+ 0.553 V6+ 0.555 Cr7+ 0.546 Mn8+ 0.545 Fe9+ 0.544 Kr+ 0.540 Kr2+ 0.6 16 Kr7+ 0.506 Rb2+ 0.567 Sr3+ 0.533 Y4+ 0.547 Nb6+ 0.689 Mo7+ 0.749 Zn+ 0.626 Cd+ 0.611 Hg+ 0.570 B a+ 0.630 co+ 0.738 m+ 0.832 Ir+ 0.600 C1' 0.589 Br+ 0.595 b. Closed ShellC Pb2+ 1.080 B3+ 2.979 ~ 1 3 + 2.467 Ga3+ 1.447 1n3+ 1.268 sc3+ 1.984 Y3f 2.000 La3+ 1.781 LU3+ 1.465 ~ 1 3 + 1.032 c4+ 3.587 Si4+ 2.870 Ti4+ 1.963
atodion
( q + l)r]/x
atodion
(q + ~)v/x
I+ sc2+ Ti2+ V2+ Cr2+ Mn2+ Fe2+ co2+ ,Ni2+ cu2+
0.587 0.955 1.016 1.Ooo 0.914 1.097 0.927 0.975 0.956 0.869 0.758 0.819 0.817 0.765 0.777 0.792 0.773 0.713 0.660 0.646 0.767 0.882 1.163 0.892 0.915 0.910 0.830 1.131 0.840 0.880 0.793 0.833
Mo3+
1.043 1.062 0.707 0.864 0.926 0.852 0.845 0.734 1.092 0.646 0.607 0.591 0.582 0.577 0.573 0.652 0.610 0.593 0.586 0.574 0.533 0.572 0.570 0.736 0.695 0.683 0.669 0.628 0.607 0.591 0.578 0.547
Y2+
Zr2+ Nb2+ Mo2+ Ru2+ Rh2+
Pd2+ Ag2+ Hf2+ W2+
os2+ R2+
Ce3+ Ti3+ v3+
CI3+ Mn3+ Fe3+ co3+ Ni3+ Zr3+ Nb3+ Ge4+ Zl.l+
Sn4+ Pb4+ N5+ P5+ V5+ As5+ Nb5+ Sb5+ Bi5+ S6+
+
+
x = (q + 1 ) ~ / 0 . 7 4 2
(6)
Using the relation between r and a described in an earlier
HP+ w3+
Re3+ oS3+ 1~3+
Au3+ Ru3+ Be+ B2+ C3+ N4+ 05+
F+ Mg+ A12+ Si3+
P4+ S5+
C16+ Ar7+
Ne7+ C+ N2+
N' Si+ P2+ s3+
ci4+ Ar5+ Ar+ Se6+ CF+ Mo6+ c17+ Mn7+ Br7+ B+ C2+ N3+
1.716 2.036 1.396 1.191 4.193 3.359 1.952 2.049 2.039 1.902 1.343 3.659
Symbols are explained in the text. Experimental r] and x values are from refs 14 and 23. The average value of (q The average value of (q -I- l)r]/x = 1.718 (3~0.801). of electronegativity. Later this proportionality was used only as an experimental observation in several works in various context^.^^-^^ However, for monoatomic ions the proportionality implied by eq 5 remains to be shown. Here we have tested it for a number of ionized and neutral atoms where the experimental Z and A values are k n ~ w n . ' ~ .For * ~ some 105 open shell monoatomic cations of various charges (q varying from 1 to 9) and some 55 open shell atoms (Table la), the averge value of the quantity (q 1)v/x is found to be 0.742 with a standard deviation (a)of 0.152. Earlier Yang et al. have found that for some 33 open shell neutral atoms the average value of r / x is 0.775 (f0.110).29 On the other hand, for some 63 closed shell neutral and cationic atoms (Table lb) the average value of (q 1)v/x comes out as 1.718 (f0.801). Nevertheless, in view of our results, for an open shell atomic species we assume the working form of eq 5 to be
Rh3+
04+
F5+ Ne6+
2.176 1.962 2.113 4.049 1.956 2.429 1.008 0.976 0.961 0.953 0.949 0.946
+ l)r]/x = 0.742 (k0.152).
section, 7 can be expressed in terms of a as (from eq 4)"
9 = '/,(0.792/a)"3
(7)
A combination of eqs 6 and 7 gives us a quantitative relation between x and a (eq 8). Using eq 8, we have determined the
x = 0.623(q+ 1)(l / a p 3 electronegativity of some 55 open shell neutral atoms (Table 2a) and some 22 open shell monoatomic cations with q = 1-7 (Table 2b). Among the neutral atoms, the significant deviants are H and the group IIIA elements; when these deviants are not counted, the average error in reproducing the experimental x is only 8.6% (f6.5%). In the open shell cationic atoms, the x values calculated by eq 8 are found to be lower than the experimental ones on the average by 11.7% (&6.4%). The performance of eq 3 can be understood only with reference to that of eq 7. Earlier we found that eq 7 reproduces the experimental hardness data in a number of open shell atoms
Hati and Datta
10744 J. Phys. Chem., Vol. 99, No. 27, 1995
TABLE 2: Calculation of Electronegativities of Some Open Shell Neutral Atoms and Monoatomic Cations by Eq 8 and Comparison with the Experimental Datao
x
X
a
atodion
H Li B
C N
0 F Na
A1 Si P S
c1 K sc Ti V Cr Mn Fe
co Ni
cu Ga Ge
As Se Br
exptl
from eq 8
4.521 163.981 20.447 11.877 7.423 5.412 3.779 159.257 56.280 36.305 24.496 19.570 14.711 292.872 120.118 98.524 83.678 78.279 63.433 56.685 50.61 1 45.888 45.247 54.795 40.962 29.085 25.441 20.582
0.264 0.111 0.158 0.230 0.268 0.277 0.383 0.105 0.119 0.175 0.206 0.229 0.305 0.089 0.123 0.127 0.132 0.137 0.137 0.149 0.158 0.162 0.165 0.1 18 0.169 0.195 0.216 0.279
0.377 0.1 14 0.228 0.273 0.319 0.355 0.400 0.115 0.163 0.188 0.214 0.23 1 0.254 0.094 0.126 0.135 0.142 0.146 0.156 0.162 0.168 0.174 0.175 0.164 0.181 0.203 0.212 0.227
24.70 7.92 3.50 1.81
0.506 1.159 2.065 3.222 4.63 1 6.290 0.417 0.869 1.445 2.139 2.953
0.428 0.938 1.641 2.556 3.678 5.034 0.380 0.768 1.197 1.887 2.606
8 error
a
exptl
fromeq 8
%error
319.190 153.184 120.793 105.947 86.377 64.783 58.034 32.391 53.176 68.832 51.961 44.538 37.115 33.134 402.193 209.869 109.321 88.401 74.905 65.457 57.360 5 1.286 43.863 39.139 51.286 45.888 49.937
0.086 0.1 17 0.134 0.147 0.143 0.165 0.158 0.163 0.163 0.114 0.158 0.178 0.202 0.248 0.080 0.1 14 0.140 0.151 0.162 0.148 0.180 0.198 0.206 0.212 0.118 0.143 0.172
0.09 1 0.1 16 0.126 0.132 0.141 0.155 0.161 0.195 0.166 0.152 0.167 0.176 0.187 0.194 0.084 0.105 0.130 0.140 0.148 0.155 0.161 0.168 0.177 0.183 0.168 0.174 0.169
5.81 0.85 5.97 10.20 1.40 6.06 1.90 19.63 1.84 33.33 5.70 1.12 7.43 21.77
C16+
2.00
Ar7+
1S O
3.899 4.920 0.655 0.81 1 1.416 0.450 0.917 1.509 2.228 3.051 0.797
3.461 4.354 0.684 0.804 1.357 0.462 0.880 1.409 2.037 2.781 0.660
1 1.23 11.50 4.43 0.86 4.17 2.67 4.03 6.63 8.57 8.85 17.19
atodion
a. Neutral Atomsb 42.80 Rb
2.70 44.30 18.70 19.03 28.16 4.44 9.52 36.97 7.43 3.88 0.87 16.72 5.62 2.44 6.30 7.58 6.57 13.87 8.72 6.33 7.41 6.06 38.98 7.10 4.10 1.85 18.64
Y Zr
Nb Mo
Ru Rh Pd Ag In Sn Sb Te
I cs La Hf
Ta W Re
os Ir Pt Au TI
Pb Bi
5.00
7.89 7.14 7.28 8.64 4.73 10.55 15.15 14.08 13.68 42.37 2 1.68 1.74
b. Monoatomic Cations' BeB?+
C" N4+ 05-
1.05
0.65 35.10 14.40 9.03 4.50 2.95
F6Mg-
At2+ Si3+
P ' s5+
15.41 19.07 20.53 20.67 20.58 19.97 8.87 11.62 17.16 11.78 11.75
C+
6.032 3.718 2.61 1 19.635 9.581 5.533 3.576 2.429 6.714
N+
N?'
Si+
P?'
s3+
C14' Ar5+ Art
x,
(x)
ref 14. The average value of the % a Polarizability (a) and electronegativity values are given in au. Sources of data: a,ref 11; exptl error is, excluding H and the group IIIA elements, 8.6 (4~6.5);including all, 12.0 (~4~11.6). Sources of data: a,refs 8 and 19; exptl ref 23. The average value of the % error is 11.7 (56.4).
with q 2 0 very well. A comparison of eq 7 with eq 3 shows that with the same a data, for an open shell neutral atom eq 3 will produce a value twice its 11 value. For an open shell neutral atom, since experimentally the proportionality constant between 11 and comes out as 0.742 instead of the predicted value of 0.5 (see eq 5), the value is expected to be only 1.35 times its 7 value. Thus eq 3 will generate a value of which is 2D.35, Le., 1.48 times the experimental data for an open shell neutral a t o m . The possible reasons for the departure from eq 8 are now discussed. First, in principle I of an atomic species cannot be treated as constant in the range of charge q f 1, contrary to one of our assumptions in deriving eq 8; but from our previous experience with the evaluation of 7 by eq 7 where the same assumption is involved, we can say that in reality the variation of r with charge is not likely to play any significant role here. Second, the a values used here are not very accurate. For example, the error involved in the polarizability data for most of the neutral atoms is reported to be 50%." Recently Fricke30 has tried to correct these data to 20-30%. Since the approach of Fricke is purely empirical, we have chosen not to use his data. The third and probably the most important reason is the
x x
x
x
x,
+
constancy of the quantity ( q l)v/x; as indicated by the magnitude of 0 (vide supra), this quantity can be said to be constant only within 1 6 1 % (which is 30). It can be shown that when the constant is lower than 0.742, eq 8 will give a value higher than the experimental one and when the constant is higher than 0.742, the calculated will be lower than the actual one. Though eq 8 is specifically designed for an open shell atomic species, we have applied it to some 56 closed shell neutral and positively charged atoms also (Table 3). Excepting Ca, Sr, and the cations with ls2, 2p6, and 3dI0 configurations which are found to deviate most, the experimental values are reproduced on the average within 5.8% (f3.7%). This unexpected matching of the values calculated by eq 8 with the experimental values in the closed shell cases deserves some comments. Earlier we found that for a closed shell species the hardness calculated by eq 7, from which eq 8 is derived using eq 6, is much lower than the experimental data; this deviation has been attributed to the failure of eq 7 to reveal the "extra" thermodynamic stability of a closed shell system." Here we find that the actual electronegativity of a number of closed shell atomic species is related to the theoretical hardness (yielded by eq 7)
x
x
x
x
J. Phys. Chem., Vol. 99, No. 27, 1995 10745
Electronegativity and Polarizability of Atomic Species
TABLE 3: Calculation of Electronegativities of Some Closed Shell Neutral and Cationic Atoms by Eq 8 and Comparison with the Experimental Data"
x
X atodion Be Mg Ca Sr Ba Zn Cd Hg Li+ Na+ K+ Rb+ Cs+ cu+ Ag+ Au+
B+ C2+ Be2+ Mg2+ Ca2+ Sr2+ Zn2+ Cd2+ Hg2+ B3+ ~ 1 3 +
sc3+ Ga3+
a 37.790 71.531 161.282 186.250 267.903 42.851 48.587 38.465 0.192 1.002 5.339 9.076 15.81 5.36 8.829 12.705 11.3 3.9 0.052 0.486 3.193 5.813 2.296 4.971 8.401 0.019 0.265 2.129 1.24
exptl 0.180 0.138 0.081 0.073 0.088 0.163 0.159 0.180 1.489 0.963 0.661 0.579 0.533 0.515 0.534 0.548 0.615 1.328 3.162 1.749 1.153 1.003 1.060 1.000 0.974 5.463 2.727 1.805 1.727
fromeq8 0.186 0.150 0.114 0.109 0.097 0.178 0.171 0.185 2.160 1.245 0.7 13 0.597 0.496 0.7 12 0.603 0.534 0.555 1.187 5.007 2.377 1.269 1.039 1.417 1.095 0.919 9.339 3.880 1.937 2.320
%error 3.33 8.70 40.74 49.31 10.23 9.20 7.55 2.78 45.06 29.28 7.87 3.11 6.94 38.25 11.44 2.55 9.76 10.62 58.35 35.91 10.06 3.59 33.68 9.50 5.65 70.95 29.72 7.31 34.34
a
atodion Y3+ 1n3+ La3+
exptl 1.514 1.507 1.270 1.481 8.389 3.894 2.618 2.558 2.128 2.077 2.042 11.942 5.429 3.553 3.495 2.814 3.013 2.65 1 6.780 4.357 4.624 3.579 8.498 5.801 5.435 0.396 0.290 0.257 0.223
4.048 3.22 7.613 5.857 0.009 0.162 1.482 0.763 2.977 2.264 4.170 0.005 0.106 1.078 0.510 2.275 1.68 3.077 0.072 0.360 0.811 1.789 0.051 0.627 0.266 2.672 11.074 16.762 27.290
~ 1 3 +
c4+
Si4+ Ti4+ Ge4+ Z#+ Sn4+ Pb4+ N5+ P5+ V5+
AsS+ Nb5+ SbS+ Bi5+ S6+
Se6+ C$+ Mo6+ c17+
Mn7+ Br7' Ne Ar Kr Xe
from eq 8 1.564 1.688 1.263 1.382 14.975 5.714 2.732 3.409 2.165 2.372 1.935 21.860 7.898 3.646 4.679 2.842 3.144 2.570 10.483 6.130 4.616 3.592 13.440 5.823 1.750 0.449 0.279 0.243 0.207
% error
3.20 12.01 0.55 6.68 78.51 46.74 4.35 33.27 1.74 14.20 5.24 83.05 45.48 2.62 33.88 0.99 4.35 3.05 54.62 40.69 1.12 0.36 58.15 0.38 42.59 13.38 3.79 5.45 7.17
Polarizability (a)and electronegativity (x) values are given in au. Sources of data: a, refs 11 and 20-22; exptl x,refs 14, 16, and 23. The average value of the % error is, with Ca, Sr, and the cations with Is2, 2p6, and 3dI0 configurations excluded, 5.8 (h3.7); with all included, 20.7 (f22.0). via eq 6. This only means that the extra thermodynamic stability imparted to an atomic species by a closed shell c o d ~ g u r a t i o n ' ~ ~ ~ ' is not reflected in its electronegativity. Such a conclusion is probably new. It is imperative to compare our electronegativity values calculated by eq 8 with XA~I~,,for the neutral atoms. For Hg and the first 56 elements (excluding He and Tc) of the Periodic ~ values 1 are 1 available,I6 ~ ~ a strong linear Table where the ~ correlation is obtained between the two (Figure 1) when H is not considered via eq 9 with a correlation coefficient of 0.974.
0.623( l / a f 3 = 0.026
+ 0 . 3 8 4 ~ ~ ~ ~( 9~) ~
As expected, eq 9 is numerically essentially similar to that described in ref 17. We now try to justify Nagle's work6 within the framework of our approach to the extent possible. We have observed that for some 33 main group elements (including the noble gases) there exists some sort of a linear correlation between ( l / ~ ) ' / ~ and (n/a)]l3when H is excluded (eq 10; correlation coefficient = 0.943). Substitution of eq 10 in eq 8 leads to a relation
( ~ a )="0.067 ~ + 0.489(n/a)1'3
x
between ( n / ~ )and ~ / ~(eq 11). Thus, our approach shows that
at least for the main group elements (n/a)Il3 can be a measure of electronegativity. Equation 11 allows us to extend Nagle's definition, originally designed for the neutral atoms, to ionized atoms as well. However, the values produced by eq 11 for the main group atoms with q 2 0 are less accurate than those
x
0
3-
1 0.00 , 0.00
, / , , I
,,,, , 0.20
I , ,
I
\
,,
,,
0.40
I , , . ,
, , , , , I , , ,
