19808
J. Phys. Chem. 1996, 100, 19808-19811
Polarizability of an Ion in a Molecule. Applications of Rittner’s Model to Alkali Halides and Hydrides Revisited‡ Sanchita Hati, Barnali Datta,† and Dipankar Datta* Departments of Inorganic and Physical Chemistry, Indian Association for the CultiVation of Science, Calcutta 700 032, India ReceiVed: March 12, 1996; In Final Form: September 6, 1996X
An attempt is made to estimate the polarizability of an ion in an ionic diatomic molecule. The method is applied to alkali halide and hydride diatoms. In this connection, the applicability of Rittner’s model to these test molecules is examined critically. A new phenomenological form of the repulsive part of Rittner’s model is suggested. It is argued that the polarizability of an ion can be used as an index of the gas-phase chemical hardness of an ion in a molecule.
Determination of the static electric dipole polarizability (hereinafter referred to simply as polarizability) R of an ion in a molecule is an old and important problem. It was realized long ago that R of an ion in a molecule is not the same as that in the free state. In 1934 Fajans1 introduced a principle that states that R is diminished in the field of positive charge and increased in the field of negative charge. This statement was theoretically quantified later by Ruffa in 1963.2 In the past there have been a number of attempts to estimate the R of an ion in a molecule. For a recent brief review on these efforts, see ref 3. Coker has found that in alkali halide crystals polarizability of an anion decreases while that of a cation increases.4 Coker has also tried to find a relation between the R value of a free ion and that of the ion in a crystal. Here we are not interested in finding the R of an ion in a crystal lattice but that in an isolated molecule. A possible way of checking whether the R of an ion in a molecule has been estimated correctly is to work out Rittner’s model5 for a diatomic molecule AB in detail. In 1951, to explain bonding in specifically the alkali halide diatomics and to estimate their various spectroscopic constants theoretically, Rittner developed a model where an AB molecule is described as a cation-anion pair (polarizable charged spheres) held together by electrostatic attraction but prevented from collapse by a repulsive potential. In this model, the dipole moment µ and the ionic bond dissociation energy W are given by eqs 1 and 2,
µ ) er - [er4(R+ + R-) + 4erR+R-]/(r6 - 4R+R-) (1) W ) -e2/r - e2(R+ + R-)/2r4 - 2e2R+R-/r7 - C/r6 + φ(r) + hν/2 (2) respectively. In eq 2 C is the van der Waals constant, φ(r) is the repulsive energy resulting from the overlap of the electron clouds of the two ions A+ and B-, the last term represents the zero-point energy, and the superscripts + and - designate cation and anion, respectively. Using the derivatives of W with respect to r (of various orders) and some standard relations,3,6 one can calculate the rotational-vibrational coupling constant (Re), vibrational anharmonicity constant (ωexe), and other higher order spectroscopic constants, γe and βe (these notations are standard †
Department of Physical Chemistry. Dedicated to Professor R. J. P. Williams on the occasion of his 70th birthday. X Abstract published in AdVance ACS Abstracts, October 15, 1996. ‡
S0022-3654(96)00945-8 CCC: $12.00
as used by Huber and Herzberg7). This model, which because of its nature is likely to hold in rather ionic diatomics, has been examined by many workers in the past using mainly the alkali halide diatoms as the testing grounds3,6,8-12 (in only one study3 have the hydrides of the alkali metals been considered). As a general result, a simplification of Rittner’s model by omitting the product terms R+R-, which arise out of the interactions between the induced dipoles, has been advocated. Fajans was first to suggest such a simplification,13 and subsequently Brumer and Karplus9 have given a quantum mechanical basis for it. The resulting model is known as the truncated-Rittner (t-Rittner) model. The simplified versions of eqs 1 and 2 are as follows.
µ ) er - e(R+ + R-)/r2 W ) - e2/r - e2(R+ + R-)/2r4 - C/r6 + φ(r) + hν/2
(3) (4)
However, so far, in all the attempts, Pauling’s free ion polarizability (R+ or R-) values14 have been used as such5,6,8,9 or with some corrections.3,10,11 But the original approach of Pauling to calculate ionic R is based on the second-order Stark effect on H-like systems where the many-body effects are not included explicitly.14 It is now realized that correlation must be included.15-20 While for alkali metal cations correlation does not play any significant role, for anions the effect can be dramatic; for example for F-, the presently accepted best value of R (2.814 Å3; obtained by Bartlett and co-workers15) is almost 2.5 times that calculated by Pauling (1.050 Å3).14 In general, the R values of the anions are expected/shown to be larger than those yielded by Pauling’s method (see Table 1). Here for the first time we work with the recent theoretical free ion R values21-27 of the halide and hydride ions and the alkali metal cations (Table 1) in an attempt to estimate the R of an ion in a molecule and to examine the applicability of Rittner’s model to alkali halides and hydrides. Equation 1 and thereby eqs 2-4 actually hold good when r6 . 4R+R-. The free ion polarizabilities given in Table 1 show that the quantity 4R+R-/r6 for the hydrides of K, Rb, and Cs lies in the range 0.76-1.19. This means that one has to work with the polarizability of an ion in a molecule. For this purpose, we take recourse to the method of Shanker and co-workers.3,10 Since R is related to a number of physical properties, it can be evaluated theoretically in various ways.28 A convenient approach relates R to the mean excitation energy Ef of the free ion,2 eq 5, where e and m are respectively the charge and mass
R ) e2h2n/4π2mEf2 © 1996 American Chemical Society
(5)
Applications of Rittner’s Model to Alkali Halides
J. Phys. Chem., Vol. 100, No. 51, 1996 19809
TABLE 1: Polarizability and Ionic Radii Data for Some Monovalent Ions Considered in the Present Studya ionic polarizability (R() free state ion
ionic radiusb (R()
Pauling’s valuesc
values used in this workd
in a molecule (from eq 7)
Li+ Na+ K+ Rb+ Cs+ FClBrIH-
0.60 0.95 1.33 1.48 1.69 1.36 1.81 1.95 2.16 1.40e
0.029 0.181 0.840 1.415 2.438 1.050 3.687 4.813 7.163 10.168
0.028 0.148 0.791 1.345 2.343 2.814 4.369 6.357 8.283 30.526
0.053 0.218 1.289 1.998 3.396 1.192 2.314 3.783 5.307 1.305
( RM ) e2h2n/4π2m(Ef2 - e2/R()2
a Units used: R(, Å; R, Å3. b Pauling ionic radii (except for H-); from ref 29. c From ref 14. d For the sources of the polarizability data, see ref 21. e From Gibb, T. R. P. Prog. Inorg. Chem. 1962, 3, 315.
of an electron, h is Planck’s constant, and n is the number of electrons. Shanker and co-workers have made use of eq 5 to calculate the polarizabilities of the ions in an alkali halide or hydride molecule via ( RM ) e2h2n/4π2m(Ef2 ( eφc)2
in Table 1, it is found that for alkali halides and hydrides the t-Rittner model, i.e. eq 3, reproduces the available experimental µ values better than the full-Rittner model, i.e. eq 1 (see Table 2). In formulating eq 6, Shanker and co-workers have treated the ions as point charges. If we consider an ion to be a charged sphere of a certain radius R in the spirit of Rittner’s model, then we can rewrite eq 6 as
(6)
where the average excitation energy Ef in the free ion is corrected by the Coulomb potential φc at the ionic site due to the counterion. In eq 6, the subscript M indicates “in a + , φc ) -e/r (due to the anion), molecule”. For calculating RM and for RM, φc ) e/r (due to the cation), where r is the internuclear distance, are substituted in eq 6. It may be noted ( values that eq 6 is in line with Fajans’ principle. With the RM yielded by eq 6 starting from the ion polarizability data listed
(7)
where for a cation the Coulombic correction to the free state Ef because of the electric field of the counteranion is -e/R+ (R+ ionic radius of the cation; it should be noted that now the effective distance through which the anion exerts its electric field at the center of the cation becomes simply R+), and the Coulombic correction for an anion is e/R- (R- ionic radius of the anion). The R( values used here are mainly Pauling’s ionic ( values generated by our eq 7 radii29 (see Table 1). When RM are used, the t-Rittner model gives better matching (compared to the approach of Shanker and co-workers) of the dipole moments for the alkali halides and hydrides with the experimental data. Thus eq 7 seems to estimate the R of an ion in an ionic diatomic molecule reasonably well. Next we proceed to compute W and the various spectroscopic constants of the alkali halides and hydrides. In eq 4, all terms (not counting the zero-point energy term) except φ(r) are attractive in nature. As pointed out in an earlier section, incorporation of the repulsive term φ(r) is essential to ensure an equilibrium stability. Unfortunately its exact nature is not yet known. Over the years, almost 50 phenomenological and a few quantum mechanical forms of φ(r) have been considered (see refs 3, 5, 6, 8-11 and Table 3). The lesson is that the choice of φ(r) is very crucial in reproducing the various spectroscopic constantssRe, ωexe, βe, and γesof the alkali halide
TABLE 2: Comparison of Calculated and Experimental Dipole Moment (µ; in Debyes) for Some Diatomic Halides and Hydridesa dipole moment (µ) calculated with free molecule
expt µ
LiF LiCl LiBr LiI LiH NaF NaCl NaBr NaI NaH KF KCl KBr KI KH RbF RbCl RbBr RbI RbH CsF CsCl CsBr CsI CsH
6.28 7.12 7.23 7.43 5.88 8.16 9.00 9.09 9.21 6.96 8.56 10.24 10.60 11.05
1.64 4.43 3.80 4.42
1.93 4.53 3.91 4.51
4.97 7.22 6.76 7.28
5.41 7.45 7.03 7.52
5.45 8.67 8.49 9.29
6.76 9.32 9.23 9.95
8.51 10.48 10.86 11.48
5.19 8.96 8.86 9.78
7.03 9.85 9.88 10.68
7.85 10.36 10.82 12.1
3.79 8.85 8.87 10.00
6.76 10.14 10.33 11.28
29.0 ((16.1)
20.1 ((17.0)
% errorb a
b
( with RM from eq 6
R(
Ritt
t-Ritt
Ritt 4.74 6.72 6.28 6.77 4.43 6.86 8.77 8.49 8.98 5.63 6.57 9.77 9.72 10.51 5.88 6.56 9.95 9.97 10.87 5.55 5.09 9.68 9.81 10.93 4.30 12.1 ((9.0)
t-Ritt 4.87 6.78 6.35 6.83 4.58 7.10 8.91 8.67 9.15 6.04 7.58 10.26 10.31 11.05 7.24 7.79 10.62 10.78 11.61 7.46 7.18 10.69 11.01 12.02 7.30 6.5 ((6.8)
( with our RM (from eq 7)
Ritt 4.89 6.83 6.40 6.89 4.93 7.19 9.00 8.74 9.22 6.72 7.09 9.85 9.82 10.59 7.55 6.89 10.06 10.10 10.98 7.64 5.61 9.77 9.90 10.99 7.10 9.0 ((7.4)
t-Ritt 5.07 6.92 6.51 6.99 5.10 7.42 9.16 8.95 9.41 7.01 7.90 10.37 10.49 11.23 8.29 7.93 10.72 10.94 11.78 8.54 7.26 10.71 11.10 12.12 8.35 4.7 ((4.8)
Abbreviations used: Ritt, full-Rittner model (i.e. eq 1); t-Ritt, t-Rittner model (i.e. eq 3). The experimental µ values are taken from ref 7. Average value with the standard deviation given in parentheses.
19810 J. Phys. Chem., Vol. 100, No. 51, 1996
Hati et al.
TABLE 3: Names of the Potentials Described by O(r) ) ar-m exp(-brn) for Various Values of m and n m
n
potential name
ref
m 0 0 1 2 -1 0 -7 3
0 1 2 1 1 1 1.5 1 1
Born-Lande Born-Mayer Barshni-Shukla I Hellmann Varshni-Shukla II Varshni-Shukla III Varshni-Shukla IV Wasastjerna this work
a b c d e 6 6 6
Our present work originates from our continued efforts to explore the relevance of polarizability in chemistry.22,34,35 At one time, chemists used to believe that polarizability is a more physical than chemical quantity.36 This notion has changed since 1990 when Nagle37 empirically correlated R with electronegativity, one of the most important concepts in chemistry. In 1994, we have shown that R can also be quantitatively related to hardness,34,38-40 another major concept of the contemporary inorganic chemistry that was introduced by Pearson41 in 1963 in connection with the classification of various Lewis acids and bases into three broad categories: hard, soft, and borderline. The presently accepted working formula for the hardness η of a chemical species is eq 8, where I is the ionization potential and A is the electron affinity of the species under consideration.42
a Born, M.; Lande, A. Verhandl. Deut. Phys. Ges. 1918, 20, 210. Born, M.; Mayer, J. E. Z. Phys. 1932, 75, 1. c Varshni, Y. P.; Shukla, R. C. J. Chem. Phys. 1961, 35, 582. d Hellmann, H. J. Chem. Phys. 1935, 3, 61. e Varshni, Y. P.; Shukla, R. C. ReV. Mod. Phys. 1963, 35, 130. b
η ) (I - A)/2
diatoms by the t-Rittner model. It has also been realized that a simple inverse power or an exponential form of φ(r) is insufficient. Here we have employed a general form of the type φ(r) ) ar-m exp(-brn) and optimized m and n (to integer values) to obtain the best possible results for the alkali halides and hydrides. For given integer values of m and n, the parameters a and b have been determined by using the relations dW/dr ) 0 and d2W/dr2 ) k at the equilibrium bond length, where k is the force constant. In Table 3 it is indicated that most of the commonly used forms of φ(r) are obtained by using appropriate ( values (eq 7) values of m and n. We have found using our RM that the t-Rittner model yields the best agreement with the experimental data7 on various quantities when m ) 3 and n ) 1 (Table 4).30 The force constants and other necessary input data such as rotational constant, vibrational constant, and reduced mass are taken from a compilation by Huber and Herzberg.7 It is found that C calculated by the Slater-Kirkwood approach31 gives results better than those with C obtained from London’s formula.32 Table 4 shows that the final overall reproduction of the various experimental quantities for the alkali halide and hydride diatoms by our approach is satisfactory.33
(8)
The larger the value of η, the harder the chemical species. We have demonstrated that for an open shell atomic species η is directly proportional to (1/R)1/3. However, this proportionality is lost in closed shell atomic species; in such cases, η increases linearly with (1/R)1/3.49 Equation 8 defines the global hardness of an isolated chemical species;42,44 moreover, it is not applicable experimentally to the free monovalent monoatomic anions since the A values of such anions are not accessible in practice.22 The point is that eq 8 cannot be used as such to characterize the hardness of an ion in a molecule.44,45 From our earlier works34,43 ( 1/3 it follows that (1/RM ) can be a measure of the gas-phase ( values (Table 1) hardness of an ion in a molecule. Our RM give rise to the following hardness orders (in gas phase) for the various closed shell monovalent ions studied here: Li+ > Na+ > K+ > Rb+ > Cs+ and F- ≈ H- > Cl- > Br- > I-. For + is alkali metal cations, the trend observed in terms of RM essentially the same as that produced by free ion R (i.e. R+ in Table 1). On the other hand, this is not the situation with the anions. According to the R- values in Table 1, H- is the softest free anion of the five anions considered here; but in an ionic
TABLE 4: Experimental and Calculated (by t-Rittner Model) Values of Binding Energy and Various Spectroscopic Constants for Alkali Halides and Hydridesa 104Re
W AB
a
b
calc
LiF 26.124 0.742 180.2 LiCl 68.709 0.742 144.5 LiBr 93.016 0.719 137.0 LiI 134.051 0.683 126.1 LiH 0.733 0.166 144.7 NaF 88.435 0.914 152.4 NaCl 205.627 0.869 126.6 NaBr 240.050 0.816 120.8 NaI 315.086 0.765 112.6 NaH 11.908 0.241 130.1 KF 171.963 0.882 138.5 KCl 444.779 0.870 114.4 KBr 533.388 0.831 109.1 KI 707.910 0.788 101.7 KH 26.412 0.324 115.5 RbF 228.201 0.878 134.4 RbCl 620.977 0.878 110.7 RbBr 753.871 0.843 105.5 RbI 1010.263 0.804 98.4 RbH 35.493 0.349 111.6 CsF 276.922 0.845 132.7 CsCl 821.186 0.865 107.8 CsBr 1027.144 0.837 102.7 CsI 1428.615 0.806 95.6 CsH 49.022 0.366 107.8 5.1 ((3.3) errorb
expt
calc
107γe
ωexe expt
184.1 206.817 202.9 153.3 92.381 80.096 147.4 66.571 56.44 138.7 49.788 40.90 165.1 2400.31 2132.0 153.9 44.972 45.586 132.6 17.944 16.248 127.7 10.434 9.409 120.3 7.596 6.477 150.3 1463.61 1353.0 139.2 24.102 23.350 118.0 8.746 7.899 113.6 4.614 4.048 106.1 3.044 2.678 127.3 898.69 810.0 133.6 16.011 15.277 113.6 5.194 4.536 109.0 2.066 1.860 101.9 1.241 1.095 119.6 774.44 720.0 130.5 13.102 11.756 112.3 3.839 3.376 108.6 1.389 1.240 101.1 0.775 0.683 115.4 665.45 579.0 11.5 ((4.9)
calc 8.401 5.285 4.292 3.620 25.810 3.564 2.020 1.400 1.132 19.727 2.451 1.333 0.848 0.662 15.230 1.931 0.961 0.527 0.380 14.222 1.709 0.825 0.420 0.290 13.183 8.3 ((6.2)
expt
calc
1010βe expt
calc
expt
7.929 1422.332 -1141.135 -1240.0 4.5 508.024 396.6 -209.234 -190.0 3.5 323.010 244.0 -108.744 3.3 218.691 153.0 -55.410 23.20 42859.26 7500.0 -178630.5 -160000.0 3.4 186.175 233.5 -42.911 2.0 56.429 51.4 -5.324 -8.3 1.5 26.800 24.5 -2.063 -5.0 1.0 17.732 14.3 -0.913 -0.5 19.72 23045.29 -80795.779 -30000.0 2.4 75.446 69.63 -9.352 -2.0 1.3 20.817 16.3 -0.284 -0.83 0.80 8.916 7.7 -0.045 -0.02 0.574 5.051 3.88 0.053 0.04 14.3 11249.73 -26644.46 1.9 42.223 33.0 -3.178 -3.7 0.92 10.295 7.0 0.150 0.23 0.463 2.916 2.14 0.048 0.335 1.487 1.18 0.038 0.053 14.21 9082.413 3000.0 -19230.05 -13000.0 1.61 30.157 -1.587 3.1 0.731 6.429 0.221 0.38 0.374 1.630 0.052 0.064 0.250 0.755 0.029 0.023 12.90 7094.533 -12719.137 62.4 ((114.3) 70.0 ((86.2)
For the meanings of the symbols, see text. Units used: a and b, a.u.; W, kcal/mol; other quantities, cm-1. Sources of experimental data: W, ref 3; other quantities, ref 7. b Average % error; standard deviations are given in parentheses. a
Applications of Rittner’s Model to Alkali Halides ( value, it becomes almost as molecule, as revealed by the RM hard as F-. Not many scales for hardness of an ion in a molecule are at present available. Elsewhere we have tried to develop one by thermochemical means45-47 in the spirit of Pauling’s thermochemical scale for electronegativity.48,49 According to our thermochemical scale for the gas-phase hardness of an ion in a molecule,43,45 the pertinent trends are Li+ > Na+ > K+ > Rb+ ≈ Cs+ and H- > F- > Cl- > Br- > I-. Thus ( here are the two gas-phase hardness series generated by RM more or less in line with those expected chemically. In conclusion, here we have tried to estimate the polarizability ( ). Our of a monovalent ion in an ionic diatomic molecule (RM ( RM values are transferable from one ionic molecule to another one. With these, we have re-examined the applicability of Rittner’s model to alkali halide and hydride diatoms critically. In the process, a new phenomenological form of the crucial repulsive part of the t-Rittner model has emerged. We have ( values can be used to rank the also pointed out that the RM various monovalent monoatomic ions in terms of their gas-phase chemical hardness in a molecule.
Acknowledgment. Thanks are due to Prof. D. Mukherjee of the Department of Physical Chemistry for extending the computational facilities. We wish to thank the reviewers for their constructive criticism. References and Notes (1) Fajans, K. Z. Phys. Chem. 1934, B24, 103. (2) Ruffa, A. R. Phys. ReV. 1963, 130, 1412. (3) Kumar, M.; Shanker, J. J. Chem. Phys. 1992, 96, 5289. (4) Coker, H. J. Phys. Chem. 1976, 80, 2078. (5) Rittner, E. S. J. Chem. Phys. 1951, 19, 1030. (6) Varshni, Y. P.; Shukla, R. C. J. Mol. Spectrosc. 1965, 16, 63. (7) Huber, K. P.; Herzberg, G. Molecular Spectra and Molecular Structure IV. Constants of Diatomic Molecules; Van Nostrand Reinhold Co.: New York, 1979. (8) Redington, R. L. J. Phys. Chem. 1970, 74, 181. (9) Brumer, P.; Karplus, M. J. Chem. Phys. 1973, 58, 3903. (10) Shanker, J.; Agrawal, H. B.; Agrawal, G. G. J. Chem. Phys. 1980, 73, 4056. Kumar, M.; Kaur, A. J.; Shanker, J. J. Chem. Phys. 1986, 84, 5735. (11) DeWijn, H. W. J. Chem. Phys. 1966, 44, 810. (12) See also the references given in ref 3. (13) Fajans, K. Struct. Bonding (Berlin) 1967, 3, 88. (14) Pauling, L. Proc. R. Soc. A 1927, 114, 181. (15) Kucharski, S. A.; Lee, Y. S.; Purvis, G. D.; Bartlett, R. J. Phys. ReV. A 1984, 29, 1619. (16) Bagdanovich, P.; Vaitickunas, P. SoV. Phys. Collect. 1985, 25, 15. (17) Kutzner, M.; Kelly, H. P.; Larson, D. J.; Altun, Z. Phys. ReV. A 1988, 38, 5107. (18) Fowler, P. W.; Sadlej, A. J. Mol. Phys. 1991, 73, 43. (19) Das, A. K.; Ray, D.; Mukherjee, P. K. Theor. Chim. Acta 1992, 82, 223. (20) Harbola, M. K. Int. J. Quantum Chem. 1994, 51, 201. (21) Sources of the polarizability data: alkali metal cations, ref 22; H-, ref 16; F-, ref 15; Cl-, ref 17; Br-, ref 23; I-, ref 20. (22) Hati, S.; Datta, D. J. Phys. Chem. 1996, 100, 4828. (23) Fowler, P. W.; Tole, P. J. Chem. Soc., Dalton Trans. 1990, 86, 1019. (24) Since the R values of the anions are quite sensitive to the nature of the method of calculation, it is necessary to appraise the quality of these values used here. The R of H- used by us is actually an experimental one.16 For F- and Cl-, as the polarizabilities (Table 1) were obtained by high-level correlated calculations,15,17 these seem to be reasonably accurate.
J. Phys. Chem., Vol. 100, No. 51, 1996 19811 However, such correlated calculations being computationally very difficult for the two large anions Br- and I-, these, to our knowledge, have not yet been reported. Still in the case of Br-, though the method of calculation is coupled Hartree-Fock (CHF), a fairly large basis set has been employed,23 and it appears that the resulting R value is near its CHF limit. So far, to our knowledge, there are only two reports on the estimation of R for I-;20,25 in the recent one,20 the method is based on density functional theory, and in the older one,25 CHF calculations have been performed numerically. In any case, both the methods have yielded almost the same R value for I-. It is possible that inclusion of correlation might affect the R values of Brand I- to some extent. However, the effect of correlation on the polarizability cannot be predicted beforehand. While in the case of Fcorrelation increases the value of R (relative to that obtained from SCF or other uncorrelated calculations),15 R for Li- decreases26 with the incorporation of correlation; on the other hand, the available theoretical evidence shows that for Cl- R estimated by correlated methods17 (Table 1) is comparable to the CHF limit27 (4.675 Å3). In a later part of the text, we shall discuss the effects of variation in the R values of Br- and I- by (10% on our calculations. (25) Schmidt, P. C.; Weiss, A.; Das, T. P. Phys. ReV. B 1979, 19, 5525. (26) Canuto, S.; Duch, W.; Geertsen, J.; Muller-Plathe, F.; Oddershede, J.; Scuseria, G. E. Chem. Phys. Lett. 1988, 147, 435. Archibong, E. F.; Thakkar, A. J. Chem. Phys. Lett. 1990, 173, 579. (27) McEachran, R. P.; Stauffer, A. D.; Greita, S. J. Phys. B 1979, 12, 3119. (28) Miller, T. M.; Bederson, B. AdV. At. Mol. Phys. 1977, 13, 1; 1988, 25, 37. (29) Huheey, J. E. Inorganic Chemistry: Principles of Structure and ReactiVity, 3rd ed.; Harper and Row: New York, 1983; p A-90. (30) Varshni-Shukla II potential [see Table 3; m ) 2 and n ) 1 in φ(r)] gives the best possible agreement with the experimental data for all the quantities except βe. The average % errors produced by this potential are as follows: W, 4.1 ( 2.4; Re, 9.4 ( 5.5; ωexe, 6.5 ( 4.7; γe, 30.5 ( 70.4; βe, 110.8 ( 164.1. The rms error for the five experimental quantities for this potential is 51.7%, and that for our potential is 42.5%. (31) Slater, J. C.; Kirkwood, J. G. Phys. ReV. 1931, 37, 682. (32) London, F. Z. Phys. 1930, 63, 245. (33) In a footnote earlier,24 we mentioned that it is possible that the best R values for Br- and I- might differ somewhat from those used here (Table 1). However, we have checked that a variation of (10% in the R values of Br- and I- leads to some minor variation ((3%) only in the calculated βe and has marginal effects on all other calculated quantities (i.e. µ, W, Re, ωexe, and γe). The point is that a change in the R values of Br- and I- within (10% does not affect any of our conclusions drawn here. (34) Hati, S.; Datta, D. J. Phys. Chem. 1994, 98, 10451. (35) Hati, S.; Datta, D. J. Phys. Chem. 1995, 99, 10742. (36) Jorgensen, C. K. Struct. Bonding (Berlin) 1967, 3, 106. (37) Nagle, J. K. J. Am. Chem. Soc. 1990, 112, 4741. (38) Prior to our work,34 the existence of a correlation between polarizability and hardness (or softness, the reciprocal of hardness, which some authors39 have described as “charge capacity”) was felt by several workers at the qualitative level.37,39,40 (39) Politzer, P.; Huheey, J. E.; Murray, J. S.; Grodzicki, M. J. Mol. Struct. (Theochem) 1992, 259, 99. (40) Sen, K. D.; Bohm, M. C.; Schmidt, P. C. Struct. Bonding (Berlin) 1987, 66, 99. (41) Pearson, R. G. J. Am. Chem. Soc. 1963, 85, 3533; Science 1966, 151, 172; Coord. Chem. ReV. 1990, 100, 403. (42) Parr, R. G.; Pearson, R. G. J. Am. Chem. Soc. 1983, 105, 7512. (43) Hati, S.; Datta, D. Proc. Indian Acad. Sci. (Chem. Sci.) 1996, 108, 143. (44) Pearson, R. G. J. Am. Chem. Soc. 1988, 110, 7684. (45) Datta, D.; Singh, S. N. J. Chem. Soc., Dalton Trans. 1991, 1541. (46) Datta, D. J. Chem. Soc., Dalton Trans. 1992, 1855. (47) Hati, S.; Datta, D. J. Chem. Soc., Dalton Trans. 1994, 2177. (48) Pauling, L. J. Am. Chem. Soc. 1932, 54, 3570. The Nature of the Chemical Bond, 3rd ed.; Oxford and IBH Publishing Co.: New Delhi, 1967; Chapter 3. (49) Datta, D.; Singh, S. N. J. Phys. Chem. 1990, 94, 2187; 1991, 95, 10214.
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