6866
J. Phys. Chem. 1991, 95,6866-6870
Electronegativity: An Atom in a Molecule James L.Reed Department of Chemistry, Clark Atlanta University, Atlanta, Georgia 30314 (Received: December 14, 1990)
The concept of electronegativityas the property of an atom in a molecule is examined in detail by utilizing the LCAO-MO methodology. The result is a series of electronegativity functions which equalized to minimize the molecular energy. The form of these functions yields several significant insights into the nature of electronegativity itself and into the nature of an atom in a molecule. One of the more pragmatic consequences of this study is a simple but powerful method for estimating the atomic or partial charge of an atom in a molecule. The veracity of these results is also discussed.
Introduction
Although Pauling defined electronegativity some 30 years ago,' it has yet to be satisfactorily formulated."11 Probably the most significant recent advance in electronegativity theory has been the proof by Parr and co-workers12that Sanderson's13 original proposal of electronegativity equalization is in fact true for all natural orbitals. Parr showed that for the ground state of a system the chemical potential is equal for all natural orbitals and that this chemical potential could be identified with the IczkowskiMargrave electronegativity formu1ation.l' Reed" showed that the Iczkowski-Margrave electronegativities, which are in fact chemical potentials, do not equalize upon bond formation. It was pointed out that these electronegativities are, as are most electronegativity formulations, isolated atom f~ncti0ns.l~ This being the case,they do not conform to Pauling's original definition which requires the property of an atom in a molecule.' They are thus at best useful approximations to a true electronegativity. In addition to the failure of these electronegativitiesto equalize upon chemical combination,several investigators have noted the seeming paradox that, should such electronegativities equalize, all atoms of the same element must have the same atomic or partial ~ h a r g e . ' ~ This J ~ is contrary to chemical experience and physical evidence. In an attempt to equalize electronegativities and to reconcile the seeming paradox, investigators have attempted to use isolated or prebonded atom electronegativities in various equalization The strategies used have included taking the geometric and harmonic means as well as the more complex p d u r e s used by Reed and Huheey.'"l Furthermore, in attempting to estimate partial charge, generally these schemes fail to consider the nature of particular chemical bonds or how the shared electron density might be partitioned among bonded ( I ) Pauling, L. The Nature o/ the Chemlcal Bond, 3rd ed.; Comell University Reu: Ithaca, NY, 1960, p 91. (2) Ray, N.K.; Samueh, L.; Parr, R. 0 . J . Chem. Phys. 1979,70,3680. (3) Parternak, A. J . Chem. Phys. 1980, 73, 593. (4) Part", A. Chem. Phys. 1977,26,101. ( 5 ) Gody, W. Phys. Rev. 1946.69.604. ( 6 ) Allred, A. L.; Rockow, E. G. J . Inor#. N u l . Chem. 1958, 5, 264. (7) Sandeon, R. T. J. Chem. Ed. 1952.29, 539. (8) Boyd, R. J.; Markur, G. E. J . Chem. Phys. 1981, 75, 5385. (9) Bartolotti, L. J.; Gadre, S. R.; Parr, R. G. J . Am. Chem. Soc. 1980, 102,2945. ( I 0) Klopman, G. J . Chem. Phys. 1965,13, S 124. ( I 1) Iczkowslri, R. P.; Margrave, J. L. J . Am. Chem. Soc. 1%1,83,3547. (12) Parr, R. 0.;Donnelly, R. A.; Levy, M.; Palk, W. E. J . Chem. Phys. 1978. 68. 3801. ( I 3) Sanderaon, R. T. Science 1955, 121, 207. (14) Rwd, J. L. J. Phys. Chem. 3981,85, 148. (IS) Smith, D. W. J . Chem. Ed. 1990,67, 559. (16) Sanderaon, R. T. J . Chem. Ed. 1988,65, 277. (17) Jolly, W. L.; Berry, W. B. J . Am. Chem. Soc. 1973, 95, 5442. (18) Parr, R. G.; Bartolotti, L. J . Am. Chem. Soc. 1982, 104, 3901. (19) Bartsch, S. C. J . Chem. Ed. 185,62, 101. (20) Wilmshurst, J. K. J. Chem. Ed. 1%2,39, 132. (21) Huheey, J. E. J . Phys. Chem. 1965,69, 3284.
atoms. Both of these considerations are fundamental to an attempt to estimate partial charge and are intimately tied to a concept of an atom in a molecule. Nonetheless, these schemes have been successful in estimating useful partial charges. However, the electronegativity equalization and averaging procedures are difficult to justify on a theoretical basis as are the extensive parametrizations. Furthermore, they provide little insight into what might constitute an electronegativity function which meets the criteria of both Pauling' and S a n d e r ~ n . ' ~ The concept of electronegativityand electronegativityequalization have been profitabl scrutinized from the viewpoint of density functional theory.%' In the paragraphs to follow we will attempt to provide insight into what might constitute an adequate electronegativity function. With few exceptions, detailed examinations of the chemical and physical properties of molecules are dominated by LCAO-MO theory. It thus seems appropriate that electronegativity and electronegativity equalization be examined in the context of LCAO-MO theory. We will use an approach similar to that used by Klopman for atoms.22
r
Calculations Ekch.ollic bergks. Our point of departure is the LCAO-MO approximation for the molecular orbitals with a basis set consisting of the valence atomic orbitals
the Slater determinant for the N-electron system, and the expectation value for the molecular energy. This yields the familiar expression for the electronic energy
where
HI], Jll, and K,]are the resonance, Coulomb, and exchange integrals, respectively. The summations are over the occupied molecular orbitals, N, and atomic orbitals, AO. The term p is the through-space nonbonded interactions, and S = SA oA where A(i) is the atom about which orbital i is centered. b e g e continuing with the development of eq 2, it will be necessary to develop a few simplifying relationships. (22) Klopman, G. J . Am. Chem. Soc. 1964,86, 1463.
OO22-3654/91/2095-6866$02.50/0Q 1991 American Chemical Society
The Journal of Physical Chemistry, Vol. 95, No. 18, 1991 6867
Electronegativity: An Atom in a Molecule At& Populatiolls. The overlap populatiod5@for a molecule is given by N AOAO CkFkPlj
C k I#
I
Realizing that where m # n the interaction is a though-space electrostatic interaction and is contained in P, and letting N
N A O A O
=
x
xckmFkdmlno k m l n o
nm4 = CCkm4Qm2
(3)
k
we get
which can be reduced to terms over atoms (4) The total number of electrons is given by m
m
m
Energies and Atomic Population Optimization. Substituting these results into eq 2 yields
m# n
where Nmois the number of electrons on neutral m, 9,,, the partial charge on each atom, and nm8$mn =
N O 0 x ~ C C k m ~ k d m l n o k I# o
(6)
Thus
A
A
m
m
E = Cnm21m + XC(1 - l/Qm)(Jm - Km)nm4A A
A A
1/2C x(nm2Hmn/smn) + Y4C C@mnHmn/Sm)+ p (16) m# n m# n It can be shown that the Iczkowski-Margrave electronegativity parameters are related to these terms in the following manner: -a, = I , + bmNmo b, = (1 - l/Qm)(Jm- K,) (17) and A
m
m
m
m+
(7)
n
rm
= (l/Qm)
A
C (Hmn/Smn) = (l/Qm) C
n(#m)
n( # m)
rmn
(18)
A
Letting nmnSmn
= fmnnd'nm + f m A J m n
(8)
E = C(-amnm2- b,,,Nmonm2+ v2bmnm4 - v2rmnm2)+ m
A A
fi$CCPm~mn + p (19) m+ n
wheref,, and f,, partition the overlap population between atoms n and m. qm
A
-
Nmo Nm
Nm'
- nm2 - n WCm )f
d d m n
Resonance Terms. Let Pkf] (ckdI(1) + ckfl](2)lck&l(1) + ck#](l))
(9)
(10)
This is an expression for th energy as a function of the atomic populations, nm2. The optimal values of nm2are those for which the energy of the system is minimized, and this occurs when the following conditions are met for all x and when the molecule is in its equilibrium geometry:
where the atomic orbitals i a n d j are on different atoms. Rearranging, multiplying by HI,, and summing yield
where summations over atomic orbitals have been reduced to summations of atoms as described above. Corr md cwkmbT m The requirement of electronegativity equalization is satisfied when the valence orbitals on each atom are equivalent.
Expanding the Coulombic interactions and factoring the summation over o yield
where x is any arbitrary atom and pm and p are taken to be very weak functions of n,. This expression may be rewritten in terms of N, and Nn to yield
Condition 20 is met if the terms in the parentheses are equal for each atom. Thus for any pair of atoms m and n (dE/dN,),T = (dE/dNn)NT (22) and if
-
(dnm2/dNm)c (dn,2/dNn)fl
-
1
(23)
then (dE/dnm2)fl = (dE/dn,Z)p
(24)
Letting where the last term arises because the summations over m and I and over n and o are identical. Now letting
A
bn/bm = C
A fmynm,+Smy/y(5n{nynn,+Sny
v(*m)
(25)
and substituting eq 9 into eq 24 yields 0
and similarly for the summation over I where Q, and Qn are the number of orbitals on n and m, we get
am
+ Y2rm+ bm9, = an + V2rn+ bAn
(26)
Solving for 9n and summing over all atoms, we get A
Z = (a,/b*)
+ 1/2(rm/b*) + ( b m q m / b * ) - E(an/bn) + A
CY2(rn/bn) (27)
Reed
6868 The Journal of Physical Chemistry, Vol. 95, No. 18, 1991
where Z is the total charge and A
(b*)-’ = E ( 1 /bn) n
This may be solved for q,,, to obtain qm = (a* V2r* b*Z - a,,, - f/zr,,,)/b,
+
+
(28)
(29)
where A
A
a* = E(a,,/bn)b* n
r* = X(r,,/b,,)b*
(30)
+ Y2r* + b*Z
(31)
(dE/dnz)c = a*
n
and (dE/dn,,,2)Nr = a,,, = -I,,,
+ y2rm+ b,q,
- (1 - l/Q,,,)(J,,, - K,,,)N,,, +
Discussion The intent of this study has been to examine the concept of electronegativity and the principle of electronegativity equalization in the context of the LCAO-MO methodology. As part of this process we have also developed a simple method for estimating the partial charge of atoms in molecules and molecular ions. The guiding principle in this work has been to arrive at expressions which are intuitively manageable, but are at the same time as rigorous as possible given this constraint. Derivation. For atoms the chemical potential which may be identified with electronegativityis that for an atom in a molecular environment.’J2 Furthermore, such an atom is in its ground state and thus must have equal orbital electronegativities.” For the Iczkowski-Margrave and other isolated atom electronegativities the latter requirement is satisfied, if on each atom all of the valence orbitals are equivalent. It has been suggested that for Iczkowski-Margrave electronegativities the valence state be taken to approximate an atom in a molecule.’’ Not only has an atom in a molecule been variously formulated, the concept itself is far from unambigu~us.’~J” In addition, it should be noted, however, that although equivalent valence orbitals satisfy the equal orbital electronegativity requirement for isolated atoms in their ground states, that is not generally true for atoms in a molecular environment. This notwithstanding, the use of equivalent valence orbitals has permitted significant simplifications in eqs 12 and 14 and has been used as an approximation here. No attempt has been made to rigorously define the integrals p,,,,,. In the simplest diatomic cases they would describe reasonably well the occupancy of a localized chemical bond. Thus at constant N7,p,,,,, is rather insensitive to changes in the atomic populations and should also be small or zero for atoms not bonded to each other. In addition, factoring n, in eq 11 is the result of using an average resonance term, eq 18. The LCAO-MO methodology most naturally partitions electron density into atomic and overlap population^.^^^' Because both electronegativity and partial charge are predicated on the concept of an atom in a molecule, the overlap population must be partitioned among the atoms. In eq 8 the partitioning coefficients f,, are introduced. In the Mulliken population analysis25the coefficients,fmn, are all set q u a l to 0.5, but this partitioning is far from unique.23J” One of the most obvious partitionings occurs when bond distances are partitioned into the covalent radii. Parr and -workers2 have shown that the covalent radii are inversely related to b,,,. Since at least qualitatively the covalent radii are an indication of the partitionings of the overlap region, it is (23) Bader, R. F. Acc. Chcm. Res. 1975, 8, 34. (24) Gusc, M. P. J . Chcm. fhys. 1981, 75, 828. (25) Mulliken, R. S. J . Chcm. fhys. 1955, 23, 1832. (26) Mulliken, R. S. J . Chcm. fhys. 1955, 23, 1841. (27) Politzer, P.; Weinstein, H. J . Chcm. fhys. 1979, 71, 4218.
reasonable and has proven convenient here to let these partitionhgs be inversely proportional to b,,,, eq 25. They are thus inversely related to the Coulomb exchange integral terms, eq 17. This would seem to be an improvement over the Mulliken partitioning, since it partitions the electron density in favor of the atom having the smaller Coulomb exchange energy. Population Optimizatioo. One of the results of this derivation is a rather simple expression for energy as a function of atomic populations which uses the Iczkowski-Margrave electronegativity constants, eq 19. The optimal atomic populations are obtained by differentiation of this energy expression with respect to the atomic populations. This yields a set of atomic electronegativity-like functions, (dE/dn,)@, which are the chemical potentials of the atomic populations. Furthermore, a minimum-energy state is realized when the electronegativitiesof all the atoms equalize. Of particular interest is the relationship of these atomic electronegativities to the isolated atom electronegativity functions of Iczkowski and Margrave. (33) X m = a m + bmqm Two expressions have been generated for the chemical potentials of the atomic populations. Equation 31 yields the characteristic potential for a molecular species and is dependent on its charge. Equation 32 yields the chemical potential of an individual atomic population and depends on the partial charge of the atom. To the extent that the derivatives in eq 23 approximate unity, these chemical potentials equal the atomic electronegativities. This is usually a reasonable approximation, but for Huckel, CNDO, and other approximate LCAO-MO theories where the overlap populations are neglected, the derivatives do equal unity and eqs 31 and 32 yield the atomic electronegativities. Politzer and Weinsteinz7have extended the conclusions of Sanderson” and Parr2 by showing not only that the electronegativities (chemical potentials) of the atoms and orbitals are equal but also that the chemical potentials of all arbitrary portions of the molecule are equal. This being the case, the chemical potentials of the atomic populations must be equal to the atomic and orbital electronegativities. Electronegativity Function. As was our original objective, the form of the electronegativity functions is intuitively manageable and particularly informative. The electronegativity function consists of the Iczkowski-Margrave function, eq 33, plus the term, r,,,,which is a potential that attracts electron density away from the atomic cores and into the interatomic regions. Analogously, the molecular function, eq 32, is the same as that derived by Reed plus the r* term. The atomic parameter, r,,,, is composed of orbital contributions, eq 18. The form of these suggests that for an atomic orbital participating in bonding the electron density moves away from the atomic core and to the periphery of the orbital and that this is not the case for lone-pair electrons. This same idea has been proposed by Huzinaga28 who has described atoms in a molecule as deformed or polarized. The interatomic densities and their relationship to electronegativity have been discussed in the bond-charge model of density functional theory.’ Similar conclusions to those arise quite naturally in this treatment. Since the rmnterms are not generally equal for all valence orbitals on the same atom, equivalent valence orbitals, which are required by electronegativity equalization for free atoms, do not satisfy the equal orbital electronegativity requirement for atoms in a molecule. This supports our earlier observation concerning a choice of valence state. The Iczkowski-Margrave electronegativities themselves might be expected to equalize when the r,,,,,’s are zero or when they are all equal. The former has been discussed previo~s1y.l~The rmnterms are equal in the case of diatomic molecules and for polyatomic molecules where all of the atoms and their bonding are identical. The case of diatomic molecules has appeared in several textbooks without noting that this is a special case.29*30 (28) Huzinago, S.; Sakai, Y.; Miyoshi, E.; Nareta, S. J . Chcm. fhys. 1990,
93, 3319.
(29) Jolly, W . L. Inorganic Chemistry; McGraw-Hill: New York, NY, 1976; p 52.
Electronegativity: An Atom in a Molecule
The Journal of Physical Chemistry, Vol. 95, No. 18, 1991 6869
The chemical potential (or electronegativity), eq 31, is a chemical potential characteristic of a molecule (and its atoms and orbitals) and is subject to the constraint that the total number o.2 0.1 of electrons remain constant. The chemical potential (or electronegativity) of a molecule, on the other hand, is not subject to this constraint, a fact that is not always a p p r e ~ i a t e d . ~ . ’ ~ . ~ ~ Therefore, these results do not provide for the determination of molecular or group electronegativities, although the derivation of the latter is a rather straightforward extension of what is presented here. This notwithstanding, as a,, r,,,, and b,,, are atomic -0.4 parameters, a*, P , and b* are molecular parameters with similar -0.5 meanings. By analogy to the relationship of a, and b, to the ionization energy of atoms, the relationship of a* and b* to the -0.6 0 molecular ionization energy has been demonstrated in a previous Nltrogca 1. Electron Bindlng Energy ( e V ) comm~nication.’~ The electronegativity functions as we have derived them are emperical functions. However, because the Figure 1. Plot of the nitrogen Is binding energies versus the partial emperical constants are related to the fundamental atomic intecharge computed via eq 29 for a series of nitrogen-containing anions. grals, eq 17, they can be rendered in a nonemperical form also,
t
/ I
I
eq 32. In addition to relating the newly derived molecular electronegativity constants a*, r*, and b* to the molecular ionization energy, several investigators have found emperical correlations between various electronegativitiesand other molecular properties. but perhaps the most intriguing is the This began with Pa~ling,)~ correlation of the weighted harmonic mean electronegativity with the superconductivecritical temperature of inorganic materials.” In addition, Benson and c o - w o r k e r ~have ~ ~ *extensively ~ studied the correlation of a new electronegativity, the unshielded core potential, with the heats of formation of homologous families of compounds. All of these correlations have been emperical. It is most likely that they will find their basis in the more fundamental relationships developed here. Putw or Atomic Charge. The role of the electron distribution in determining the chemical and physical properties of a molecule cannot be overstated. Whereas a detailed description of the electron distribution is difficult to obtain and interpret, partial or atomic charges have proven to be just as informative in many instances. Equation 29 provides a simple method for determining the partial charge of atoms in molecules and ions. It is based on the LCAO-MO model and the electronegativity equalization which arises naturally upon energy minimization. Furthermore, it is a simple equation requiring a minimum of data and only a knowledge of the simple bond structure of the molecule. Whereas atomic information is contained in the Iczkowski-Margrave constants (a, and b,), bonding and structural information is contained in the constants rmn. Finally, the form of eqs 29 and 30 renders the results easily interpretable. For example, one of the results of the inductive effect is that charge is distributed over several atoms, which reduces the Coulombic repulsion. In eq 28 this is reflected clearly in that b* is smaller than any b, in the molecule or ion. Because both the concepts and the formulations of an atom in a molecule vary widely, the absolute value of the partial charge, which either explicitly or implicity depends on this concept, also varies widely. Whereas partial charges determined via various methods are rarely in quantitative agreement, the qualitative agreement is usually quite r e a ~ o n a b l e . ~ Charges ~ ’ ~ ~ determined by diffmnt methods usually correlate reasonably well, even though the absolute values may differ enormously. To illustrate the lack (30) Huhssy, J. E. Inorganic Chemfsrry, 3rd 4.;Harper and Row: Cambridge, MA, 1983, p 159. (31) Huheey, J. E. J . Org. Chem. 1962, 31, 2365. (32) Hendrickson, D. N.; Hollander, J. M.; Jolly, W. L. Inorg. Chem. 1969,8, 2642. (33) Pauling, L. J . Am. Chem. Soc. 1947,69, 542. (34) Ichikawa, S. J . Phys. Chem. 1989,93, 7302. (35) Luo, Y.; Benmn, S. W. J. Phys. Chem. 1988, 92, 5255. (36) Luo, Y.; Benmn, S. W. J . Phys. Chem. 1989, 93,4644. (37) Jolly, W. L.; Perry, W. B. J . Am. Chem. Soe. 1973, 95, 5442. (38) Politzer, P.; Harris, R. R. J . Am. Chem. Soc. 1970, 92, 6451. (39) Huheey, J. E.; Watts, J. C. Inorg. Chem. 1972, 10, 1553. (40) Hinze, J.; Jaffe, J. H. J . Am. Chem. Soc. 1%2,84, 540. (41) Hinze, J.; Jaffe, J. H. J . Am. Chem. Soc. 1963, 85, 148.
Nitrogen 1s Electron Blndlng Energy (.V)
Figure 2. Plot of the nitrogen Is binding energies versus the partial charge computed via eq 29 for a series of nitrogen-containing cations (0) and neutrals (0).
of quantitative agreement, several examples of the partial charges that have been reported for the hydrogens of methane, a presumably simple case, are given here: 0.018,’2 0.007,” 0.116,’3 0.124,” 0.015,” -0).0107,u0.0108,’5 and 0.125.46 Our method for computing the methane hydrogen partial charge yields 0.012. Although a detailed discussion of the quality and limitations of this method for computing partial charges is appropriate, discussion here will be limited to a few illustrative examples. Hinze and Jaffe4s4I have evaluated the electronegativity constants a, and b, for a variety of valence states for most elements. Selection of the appropriate valence state is based on the requirement that each atom have equivalent valence orbitals and, in the case of multiple bonding, equivalent bond orbitals are f0rmed.~4~’At first sight it would appear that both the resonance and overlap integrals must be evaluated in order to determine the r,,,,,’s needed to compute partial charges. However, considerable simplification may be effected, if the Wolfsberg-HelmholzQ ,. Both approximation is made for the resonance integrals, H ionization energy and electronegativity have been used in the Wolfsberg-Helmholz approximation for the resonance integral. Both can be expressed in terms of the Iczlowski-Margrave electronegativity constants to yield rmn
rmn = t/zk(am + 4 = k[L/(a, + an) + f/2(bm + bn)I
(34)
(42) Dawn-Micovic, L.; Jcremic, D.; Allinger, N. L. J. Am. Chem. Soc. 1983, 105, 1716. (43) Chrilian, L. E.; Francl, M. M. J. Compur. Chem. 1987, 8, 894. (44) Salahub, D. R.; Sandorfy, C. Theor. Chim. Acta 1971, 20, 227. (45) Diner, S.; Malrieu, J. P.; Claverie, P. Theor. Chim. Acru 1969, 13, 1. (46) Sichel, J. M.; Whitehead, M. A. Theor. Chfm. Acta 1966, 5, 35. (47) Linnett, J. W. The Electronic Srrucrure ojMolecules; Spobwood Ballantyne Co.: London, 1966; p 37. (48) Wolfsberg, M.; Helmholz, L. J. Chem. Phys. 1952, 20, 837.
6870 The Journal of Physical Chemistry, Vol. 95, No. 18, 1991
Reed
TABLE I: Carbon and Nitrogen Is Binding Energies .ad Computed
Partid Chrrges
compound anions NaN03 NaNOZ NaAONNQ) NadONN02) Na(NNN) Na(NNN) Na&Oz
binding energy,’ eV
partial chargeb
407.4 404.1 403.9 400.9 403.7 399.3 401.3 399.0 398.3 398.5 407.2 398.5
0.260 -0.043 0.050 -0.347 -0.100 -0.450 -0.265 -0.600 -0.511 -0.385 0.260 -0.478
405.3 405.1 403.7 402.2 400.0 398.4 398.0 398.8 401.8 399.4 294.6 296.1 299.7 303.1
0.093 0.084 -0.086 -0.059 -0.229 -0.277 -0,276 -0.337 -0.120 -0.281 0.186 0.253 0.386 0.501
402.5 407.2 402.1 400.2
0.021 0.260 0.030 -0.086
KCN
KOCH NaAPODW3 NH4NO3 KSCN neutrals HOC~H~BOZ C6HSN02 C5Hllo80
(CHhNO (CONH2)Z C6HsCN C5H5N
NH3
NH3SO3
C6H5CONHz
-CH3CH202CCF3 CH~CH~OZCCF~ C H ,C HzOZCF, CH3CH2OzCCF3 cations hbH6S04 NHZNO3 NH,OH*HCI E5H&HCI
ONitrogen binding energies were taken from ref 37, and the carbon binding energies, from ref 23. bPartialcharges were computed by using q s 29 and 43 (& = 1.75). The electronegativitieswere taken from ref 41, and the recommended values for fluorine were taken from ref 14.
Thus one need know only the electronegativityconstants and the bond structure in order to obtain partial charges. Partial charge is not an experimentally measurable quantity and thus must be inferred from other molecular pr0perties.~2”-~~ Jolly in one of the pioneering investigations studied the correlation of partial charge with core electron binding energy. He examinined the relationship between the partial charge on nitrogen determined by both extended Huckel and CNDO methods and its Is cote binding energy. Only moderate correlations were found in either case. The nitrogen partial charges (via eq 29) and Is binding energies reported by Jolly” may be found in Table I and are plotted in Figures 1 and 2. The correlation of both anions and neutrals is quite good and is an improvement over the Huckel and CNDO results. Thus if the degree of correlation of computed partial charges with binding energies is indicative of the veracity of a procedure, eq 29 would appear to be a significant improvement over several much more complex procedures. In addition, it would also speak to the validity of the derivation and the electronegativity functions. To further illustrate the strength of the method, two examples have been provided. The azide ion is a prime candidate for the paradox described in the introduction since it contains only one element with all of the atoms having the same hybridization. It has two energetically accessible resonance forms -0.45
N-N-N
4.10
-0.45
4.28
4.10
N B N - N
-0.62
298
296
29h
Carbo.
300
302
/Oh
1 1 E l m t r o m ~ l n l l n gSnargy ( e v )
F i i 3. Plot of the carbon 1s binding energies versus the partial charge computed via q 29 for the carbons in ethyl trifluoroacetate.
with two types of nitrogens in the first and three in the second. With no explicit consideration of formal charge this procedure yields the two different types of nitrogens in the first form and three in the second. Furthermore, the partial charges are consistent with the formal charges assigned to the atoms.
H-C-
,,0.25
Cp.25
..
..
C-
-0.35
0
o-c-c-
F4.62
I
F4.62
A second example is ethyl trifluoroacetate,which contains four different types of carbons. In Figure 3 is plotted the partial charges computed via eq 31 and the 1s binding energies reported by Jolly.‘z The correlation is excellent. One of the deficiencies in eq 31 is indicated in this example by the fact that all of the hydrogens are assigned the same partial charge. There is ample reason to expect that the methylene hydrogens should be more positive than the methyl hydrogens. Again the intuitive form of these equations indicates clearly that this arises, because we have approximated the resonance terms as constants which are independent of the atomic populations.
Summary We have presented for the first time a truly workable electronegativity function for an atom in a molecule. Furthermore, it develops out of the LCAO-MO description and, although intuitive in its presentation, contains few serious approximations. In this treatment electronegativity equalization arises very naturally as a consequence of “khy energy. The analytical form of the resulting functions also provides some useful insight into the nature of an atom in a molecule and provides useful insight into the strengths and limitationsof isolated-atom electronegativity functions. Furthermore, these results suggest, as did the identification of electronegativity with chemical potential, that electronegativity is a much more fundamental property than it was thought to be. One of the more pragmatic outcomes of this study has been a simple method for the computation of the partial charge of atoms in molecules. This method is theoretically based, and the results are intuitively interpretable. The use of this method requires only knowledge of the atomic electronegativities(Iczkowski-Margrave type) and, in the simplest applications, only the localized bonding (valence bond) structure of the molecule.
Acknowledgment. I gratefully acknowledge the support of the National Aeronautics and Space Administration (Grant NAG 3 1080) and Clark Atlanta University for their support of this work.
