J . Phys. Chem. 1989, 93, 3087-3089
3087
Comparison between the Geometric and Harmonic Mean Electronegativity Equilibration Techniques Mahlon S. Wilson and Shinichi Ichikawa* Department of Chemical and Nuclear Engineering, University of California, Santa Barbara, California 93106 (Received: September 12, 1988)
The geometric and harmonic mean electronegativity equilibration techniques are demonstrated to be functionally identical when applied to the electronegativity systems from which the methods originate. From this we conclude that (1) a constant ratio of the hardness to the electronegativity, 7 , is implicit in Sanderson electronegativities, (2) hardness equilibrium obeys the harmonic mean, and (3) the geometric mean is not universally applicable for electronegativity equilibration.
Introduction
x
The concept of electronegativity has spawned numerous approaches and definitions since Pauling’ formulated the original system over 50 years ago. Many of the prominent electronegativity systems,2-5as well as work functions,6 behave similarly to Pauling’s original scale; that is, the relationships are surprisingly linear. One notable exception is the electronegativities of Sanderson, which correlate well with the other systems but in a decidedly nonlinear manner. Furthermore, Sanderson formulated the geometric mean postulate’ for electronegativity equilibration and he utilizes it successfully in his polar covalent model to accurately predict the energies of many different types of bonds.* This success has been followed by several”3 generalized applications of the geometric mean to electronegativity equilibration. However, the geometric mean is not compatible with all electronegativity systems.I0 Consequently, investigation of the relationship between the geometric and harmonic mean as applied to Sanderson and the other electronegativities is expected to yield insight into the viability of the equilibration techniques. On a general note, we are interested in reconciling the differences between Sanderson and the other electronegativity systems because the former offers a successful technique for the prediction of bond energies and the latter provide a physical foundation for the manifestation of electronegativity.
=
xo + 7Oq
(3)
where the nought superscripts denote the neutral state before the charge transfer occurs. In this expression, 7 satisfies the definition of the “hardness”12 as
= a2E/aq2
(4)
In the original presentationI6 of the hardness, a prefix of is used in the definition. The ratio of the hardness to the electronegativity, y, is found to possess a relatively constant value for the majority of the elements.’1-12,i6 In the formation of simple compounds, the electronegativities of the component atoms are postulated by Sanderson to equilibrate to a single, intermediate value that is characteristic of the molecule or compound.’’ If the charge dependency of the atomic electronegativities is described by eq 3, the electronegativity of the molecule can be found by solving for the common intermediate value xm of the atomic electronegati~ities.~For a diatomic molecule Xm
= XI’
+ 71’41 = x
~ O
+ 72’42
(5)
After solving for q utilizing q1 = -q2, further rearrangement yields
Harmonic Mean Electronegativity Equilibration Density functional theoryi4 yields the following relationships between electronegativity (x),chemical potential (F), and energy (E) X=-F=
-(aE/alv),
(1)
for an atom with N electrons and a nuclear charge of Z. If the energy of an atom exhibits a parabolic charge (9) dependencyi5
If we take advantage of the approximation that y is a constant for all elements, then the above equation reduces to the simple form described by Nalewajski as the harmonic meani1 Xm = 2(
-)
X1°X20
XI0
+ X2O
(7)
The generalized expression of the harmonic mean for polyatomic molecules is
the electronegativity as a function of charge is described by eq 1 as
(1’) Pauling, L. J . Am. Chem. SOC.1932, 54, 3570. (2) Allred, A. L.; Rochow, E. G. J. Inorg. Chem. 1958, 5, 264. (3) Mulliken, R. S. J. Chem. Phys. 1934, 2, 782. (4) Hinze, J.; JaffE, H. H. J . Am. Chem. SOC.1962, 84, 540; J . Phys. Chem. 1963, 67, 1501. ( 5 ) Bartolotti, L. J.; Cadre, S. R.; Parr, R. G. J . Am. Chem. SOC.1980,
Geometric Mean Electronegativity Equilibration The use of the geometric mean’ to describe the equilibrated electronegativity of a molecule
102, 2945.
(6) Gordy, W.; Thomas, W. J. 0. J . Chem. Phys. 1956, 24, 439. (7) Sanderson, R. T. Science 1955, 121, 207. (8) Sanderson, R. T. Polar Covalence; Academic: New York, 1983. (9) Huheey, J. E. J . Phys. Chem. 1965, 69,3284. (10) Parr, R. G . ; Bartolotti, L. J. J . Am. Chem. SOC.1982, 104, 3801. ( I 1) Nalewajski, R. F. J . Phys. Chem. 1985, 89,283. (12) Yang, W.; Lee, C.; Ghosh, S. K. J . Phys. Chem. 1985, 89, 5412. (13) Datta, D. J. Phys. Chem. 1986, 90,4216. (14) Parr, R. G.;Donnely, R. A,; Levy, M.; Palke, W. E. J. Chem. Phys. 1978, 68,3801. ( 1 5 ) Iczkowski, R. P.; Margrave, .I. L. J. Am. Chem. SOC.1961,83, 3547.
0022-3654/89/2093-3087$01.50/0
is an integral component of Sanderson’s polar covalent model8 for the prediction of bond energies. Considering the success of this model, the geometric mean expression can be considered a proven description of the equilibrated electronegativity (as far as the polar covalent model is concerned). However, Parr and (16) Parr, R. G.; Pearson, R. G. J . Am. Chem. SOC.1983, 105, 7512. (17) Sanderson, R. T. Science 1951, 114, 670.
0 1989 American Chemical Society
Wilson and Ichikawa
3088 The Journal of Physical Chemistry, Vol. 93, No. 8, 1989 14 L
c
5
I INITIAL ELECTRONEGATIVITY (ATOM 2) = 12
1210:
6 -
INITIAL ELECTRONEGATIVITY (ATOM 1)
Figure 1. Equilibrated Mulliken-JaffE diatom electronegativitiesfrom
the harmonic mean (lines) and from the geometric mean applied to a correlation with Sanderson electronegativities(squares). Bartolotti’O indicate that an electronegativity system that can appropriately utilize the geometric mean for equilibration should vary exponentially with charge, in contrast to Sanderson electronegativities which vary linearly by definition (eq 14). Consequently, the Sanderson electronegativity system does not satisfy the geometric mean application criteria.
Geometric vs Harmonic Mean If we consider the equilibrated intermediate electronegativity of a diatomic molecule calculated from either the geometric or harmonic mean of the initial electronegativities, the two solutions diverge considerably from one another as the difference between the initial electronegativities increases5 due to the dissimilar responses of the two functions. Clearly, both techniques cannot be appropriate for the prediction of equilibrated electronegativities if applied to the same electronegativity system. In a similar vein, the different characteristics of Sanderson’s and the linearly related suggest that the geometric family of electronegativity ~ystemsl-~ mean, which is demonstrably successful in the former case, is not directly applicable in the latter. We illustrate the latter point by considering the linearly related set of electronegativities that are akin to the density functional theory definition and Sanderson’s electronegativities, which exhibit a nonlinear correlation with the above set. If Mulliken-JaffE4 electronegativities, x, are arbitrarily chosen from the above set, simple correlations of Sandersonis electronegativities, S, of the form S = a + bf(x) are
S = -2.60
Hardness Equilibration Yang et a1.12 propose that the molecular softness ( l / q ) is the average of the atomic softnesses, whereas other^^^^^^ suggest that it is the sum of atomic softnesses. In addition, the geometric meani3 has also been applied to hardness equilibration. If y is a constant, then vi = yxi, and the introduction of this identity into eq 8 yields an expression for the equilibrated hardness from the harmonic mean
0:
2
hibiting a linear interrelation (assuming y is constant).
+ 1 . 9 8 ~ l / ~S, = -1.73 + 2.26 In x
(loa, lob)
By using the geometric mean (eq 9) to equilibrate the Sanderson electronegativities, we have the following relationship for the equilibrated Mulliken-JaffE electronegativities Thus, we now have a second means, in addition to the y = constant harmonic mean equilibration expression (eq 7), to calculate the equilibrated Mulliken-JaffE electronegativities (x,). For the experimental range of initial electronegativities xI and xz of about 2-12 eV, Figure 1 demonstrates that the two methods yield molecular electronegativities that are virtually identical. The first correlation is used in this figure (eq loa), but the logarithmic expression is equally satisfactory for the demonstration. Additionally, the comparison is relatively insensitive to the values of the coefficients a and 6 in the correlations. Thus, to all appearances, the geometric mean applied to Sanderson electronegativities is achieving the same result as the harmonic mean applied to the group of electronegativities ex(18) Sanderson, R. T. J . Am. Chem. SOC.1983, 105, 2259
This expression supports the average of atomic softnesses for molecular softness of Yang et al. In molecules with initial atomic hardnesses that are somewhat alike, the harmonic and geometric mean techniques yield comparable equilibrated values. The differences between the two methods increase in higher polarity molecules.
Charge Formation An interesting manifestation of the electronegativity is the amount of charge formed on each atom in the equilibration of the molecule. Although we have shown that the equilibrated electronegativities are very much similar when using the two different systems, the charge predictions are not in as complete agreement. Unfortunately, direct measurement of the charge is not possible although it can be inferred from dipole strengths, etc.; therefore, it is not possible to experimentally corroborate the charge formation predictions when utilizing the various electronegativity systems. Consequently, we can only compare the charge formation predicted by the techniques. The utilization of a constant value of y in Huheey’s9 approach for calculating charge formation in diatomic molecules results in a charge dependency on y per X2O - XI0 41=-=-72 + v i o
1 y
xzo - xio X2O + xio
(13)
In Sanderson’s polar covalent model the charge formed on the atoms in a diatomic molecule is calculated by using8
s, - SI
41=-=
1.57S,i/2
S21/2- s I 1.57
(14)
Once again a relationship of the form S = a + b f ( x ) is used to substitute for the S’s in eq 14 in order to calculate the charge formed as a function of initial Mulliken-JaffE electronegativities. In this latter case, no assumptions (such as a y value) are necessary. A comparison of Huheey’s (eq 13) and Sanderson’s (eq 14) methods for calculation of the charge formation as a function of initial Mulliken-JafE electronegativities is shown in Figure 2. In this comparison, a value of unity is used for y in eq 13. The agreement between the two is not as good as that for the equilibrated electronegativities (Figure l), but the results are similar for the larger values. The disagreement is not unexpected, because both Mulliken-JaffE and Sanderson electronegativities are linearly dependent upon charge (eq 3 and 14) in their respective electronegativity systems, and the systems themselves are not linearly related (eq 10a and lob).
Discussion One often wonders how Sanderson’s polar covalent model can be as successful as it is considering the charge dependency of the electronegativities is described by only a single parameter, S, whereas energy correlations require at least two parameters (e.g., x and 7 in eq 3). A possible answer is that the variation of (19) Ray, N. K.; Samuels, L. S.;Parr, R. G. J . Chem. Phys. 1979, 70, 3680. (20) Reed, J L. J Phys. Chem. 1981, 85, 148
1.5
1.0
-
0.5
-
0.0
-
-0.5
-1.0 -1.5
-
fact proportionally constant. However, the experimental data from which the values for Mulliken style electronegativities are derived reflect the “global” energies (ionization potential and electron affinity), which incorporate energy components that are not manifested on the local scale. Chief among the extraneous energies is the work involved in transferring the manipulated electrons to and from infinity, which may add substantial and variable contributions to the ionization and affinity energies. The size and electron density of an atom (upon which Sanderson electronegativities are based) may be more appropriate indicators of the local electronic environment. In partial confirmation of this discussion, the deviation of the Mulliken-Jaff5 electronegativity values for the first-row elements from those predicted by the simple correlation (eq loa) with the Sanderson electronegativities varies roughly in proportion to the deviation of the y values about the norm.
; INITIAL ELECTRONEGATIVITY (ATOM 2) = 2 m
m
.
.
4
o 12 ;
.. . m
I
I
I
I
I
I
Figure 2. Charge formation in a diatomic molecule as predicted by the techniques of Huheey (lines) and Sanderson (squares). See text.
Sanderson electronegativities with charge (hardness) is in some manner constantly proportional to the initial electronegativities. In other words, a constant value of y is implicit in the Sanderson electronegativity values, as supported by Figure 1. However, experimental results11*1z-16 yield values of y that vary considerably and by an amount that would be severely detrimental to the performance of the polar covalent model, if the magnitude of the variations were manifested directly. One explanation that comes to mind to explain these discrepancies is to suggest that on an atomic (“local”) scale the electronegativity and hardness are in
We conclude from the agreement observed in Figure 1 that the concept of the hardness as being constantly proportional to the electronegativity (y is constant) is implicit in the Sanderson electronegativities. Also, in accordance with Parr and Bartolotti,’o we find that the geometric mean electronegativity and hardness equilibration postulates are not appropriate for utilization in electronegativity systems described by density functional theory (eq I ) . The identification of the harmonic mean as an appropriate vehicle for electronegativity equilibration leads to the application of the harmonic mean to hardness equilibration. These conclusions stem from our demonstration that applying the geometric mean to Sanderson electronegativities is functionally equivalent to applying the harmonic mean to the latter group of electronegativities.
Direct Determination of the Limiting High-pressure Rate Constants of the System FSOB F2S20, over the Temperature Range 293-381 K 4- FSO, C. J. Cobos, A. E. Croce de Cobos, H. Hippler,+ and E. Castellano* Instituto de Inuestigaciones Fisicoqurmicas Tedricas y Aplicadas (INIFTA),x C. C. 16, Sucursal 4, (1900) La Plata. Argentina (Received: October 19, 1987; In Final Form: June 7, 1988)
The laser flash photolysis/absorption technique has been used to measure directly the approach to equilibrium in the system FSO, + FSO, == F2S206over the temperature range 293-381 K and the pressure range 250-760 Torr of He and CF4. The rate constants have been found to be independent of pressure, being k,,,, = 4.64 X T/300)(0~72*0~25) cm3 molecule-’ s-’ between 293 and 381 K, and k, = 2.09 X 1014exp[-(11090 f 290)/Tl s-I between 321 and 381 K. The pressure dependence of the equilibrium constants derived here gives a value for of 22.1 f 0.7 kcal mol-I and for of 38.2 f 1.0 cal mol-l K-l. The errors that are quoted correspond to two standard deviations. The low k,,,- obtained can be described by the canonical version of the statistical adiabatic channel model with a standard value of 0.46 for the parameter a/&
Introduction The recombination of radicals and the reverse simple bond dissociation reactions are fairly well-understood processes. Many experimental data of high-pressure rate constants have been recently evaluated in terms of the canonical version of the statistical adiabatic channel model (SACM).’qZ Most of these calculations have been performed for simple bond fission reactions of strongly bound compounds. The comparison of this theory with recombination4issociation experiments in the case of the fission of weak bonds, still stronger than those of van der Waals molecules, appears to be of great importance. The interest arises because the stat Permanent address: Institut fur Physikalische Chemie der Universitat Gottingen, Gottingen, West Germany. t Facultad de Ciencias Exactas, Universid,ad Nacional de La Plata.
0022-3654/89/2093-3089.$01.50/0
tistical theories. which are based on a rapid and complete randomization of the internal energy, might not apply for van der Waals molecules. In this case the reaction could compete with intramolecular dynamics. Recently, some studies on the recomNOz NO2 N204,5 bination reactions NO NOz N203,3,4 and NO2 NO3 N2OS6have been reported. For these com-
+
-+ -
+
-
(1) Troe, J. J . Chem. Phys. 1981, 75, 226. (2) Cobos, C. J.; Troe, J. J . Chem. Phys. 1985, 83, 1010. (3) Smith, I. W. M.; Yarwood, G. Chem. Phys. Lett. 1986, 130, 24. (4) Smith, I. W. M.; Y a r w d , G. Faraday Discuss. Chem. SOC.1987.84,
205.
(5) Borrell, P.; Cobos, C. J.; Croce de Cobos, A. E.; Hippler, H.; Luther, K.; Ravishankara, A. R.; Troe, J. Ber. Bunsen-Ges. Phys. Chem. 1985.89, 337. Borrell, P.; Cobos, C. J.; Luther, K. J . Phys. Chem. 1988, 92, 4377. ( 6 ) Croce de C o b s , A. E.; Hippler, H.; Troe, J. J . Phys. Chem. 1984.88, 5083.
0 1989 American Chemical Society
