Revised Mulliken Electronegativities II. Applications and Limitations Steven G. Bratschl University of Texas. Austin, TX 78712 In Part I of this series (I), the Mulliken electronegativity scale (2, 3) has been revised and extended to consider 50 elements. I t is shown that an alternate method of assignment of "typical" singly bonded valence states to main groups V-VII of the periodic table allows conversion of Mulliken electronegativities to the well-known Pauling units (4, 5) with unprecedented accuracy and solves a half-centuryold puzzle concerning the physical dimensions of the two scales. In this work, some of the more important applications of Mulliken electronegativities to topics in chemical education are discussed, problems within the Mulliken system are identified (with suggestions for their solution where possible), and recommendations are made for future research. Calculation of Partial ionic Charges on Combined Atoms from Mulllken Eleclronegativities Calculated partial ionic charges on combined atoms are valuable aids in understanding and teaching periodic variations in chemical and physical properties (6-8).A method of partial ionic charge calculation applicable to Mulliken electronegativities has developed from the Iczkowski-Margrave atomic charge-energy function (9),as extended by Hinze, Whitehead, and Jaffe (10). For multiple ionizations, the relative energy E of an atom A is approximately a quadratic function of the charge, 6 ~If. the energy of the neutral atom A in a specified valence state is set a t zero, the energy of A+ is the valence-state ionization potential, IPv,and the energy of A-is thenegative of thevalence-state electron affinity, EAv. This gives
Assuming the energy as defined by eq 1to be a continuous and differentiable function of charge,
Iczkowski and Margrave (9) and Hinze, Whitehead, and Jaffe (10) have equated d E ~ l d with 6 ~ electronegativity, considering the Mulliken definition (2,3) to he a special case of eq 2, with 6~ = 0. Huheey (Sb, 11, 12) has suggested the following notation: IP, + EA, a, = inherent electronegativity = XA(M)= (3) 2
The equalized Mulliken electronegativity a., in a molecule or polyatomic ion may he calculated by (14,15)
where u represents the number of atoms of a particular element in the species formula (e.g., for BFa-, U B = 1and UF = 4) and q represents the net charge on the species (e.g., for BFI-, q = -1). The partial ionic charge on an atom A in the species may he calculated by rewriting eq 2 as
Using BF4- as an example, assigning the valence states B (V = 3) = te and F (V = 1) = 14.3%s (I).
The partial ionic charges have been given to three decimal places in this example merely to minimize round-off error. As discussed in the section concerning bond energy calculation, they are not accurate to this level.
Correlation ol Mulllken a and b Parameters and the Principle of Hard and Sofl Aclds and Bases A highly electronegative atom will usually have both a high valence-state ionization potential and a high valencestate electron affinity. Thus, it is not surprising that "the values of a and b for an element parallel each other" (11,12), or that the ratio bla remains fairly constant for many elements (16). A correlation of the "typical" singly bonded valence states discussed in Part I of this series (1)gives Nalewajski (16) has reported bla = 1.52 f 0.22, from groundstate ionization potentials and electron affinities for several elements. Substitution of eq 7 into eqs 5 and 6 gives
~XA(M) b~= charge coefficient = -= IP,, - EA, d6A The calculation of partial ionic charges from the Mulliken a and b Darameters is accomnlished throueh the ao~lication of ~anderson'sprinciple o i electronegat';vity equalization (fia, 7 o . I J j :"When twoor more atoms unite. their elecrronegativities become adjusted to the same inte&ediate value."
..
'
~~
~
Present address: University of Connecticut-Greater Hartford Campus. 85 Lawler Road. W. Hartford. CT 061 17.
In eq. 8, N = Z ( U )= the number of atoms in the species formula. Nalewajski (16) has derived eq 8 for neutral species (q = O), calling it the "Harmonic Mean Law." Equations totally analogous to eqs 8 and 9, but using Pauling electronegativities (4, 5) and setting bla = 1, have been suggested recently by the author (15). Pearson's principle of hard and soft acids and bases (1720) has served as a qualitative guide for the prediction and Volume 65 Number 3 March 1988
223
rationalization of many chemical reactions: "Hard acids prefer hard bases and soft acids prefer soft bases." Classification of chemical species as hard or soft has been made on the basis of such properties as charge, size, electron configuration, and closeness in spacing of energy levels. Over the years, much confusion has developed over the definitions and underlying causes of hard-soft acid-base phenomena (8c, 20,21). In particular, hardness has often been confused with inherent acid or base strength (21). Evans and Huheey (22-24) have observed that typical hard chemical species tend to have large Mulliken b parameters (eq 4). More recently, Parr and Pearson (25) have suggested that hardness be formally defined in this way, interpreting the Mulliken b parameter as the resistance of the electronegativity of a chemical species to change in the number ~~- of electrons. Parr and Pearson have tabulated hardness parameters for several atoms, molecules, and ions of chemical interest: for the sake of internal consistencv and because of the frequent lack of valence-state promotion energies, thev have used eround-state ionization ~otentialsand electn,; affinities t6roughout their work. ~bmparisonbetween s ~ e r i e with s different charges - should urohahly he made with caution. The Mulliken b parameter for an atom A in a specified valence state is rigorously the energy change for the process ~
~
~~~
~
~
2A(vs)
-
At(vs) + A-(vs)
(10)
Thus, according toParr and Pearson (25),the hardness of an atom A may he defined as its resistance to disproportionation into A+ and A-. Evans and Huheey (22-24) have found i t unnecessary to consider hardness or softness explicitly in the evaluation of the energy of homolytic bond cleavage and have suggested that, for such processes, "hard-hard" interaction could he viewed as "small-small" interaction. In general, small atoms tend to be harder than large atoms, although exceptions are known (8c, 17-21). Onemay therefore expect arough inverse correlation between atomic size and the Mulliken b parameter (26). For the "typical" singly bonded valence states discussed in Part I of this series (1). 0.92 0.30 b= (11) r
*
where r is the nonpolar covalent atomic radius in nanometers (76). Substitution of eq 11into eqs 5 and 6 gives
Rav. -. Samuels. and Parr (26) . . have derived ea 12 for neutral species ( q = 0). The Mulliken atomic b ~arameter,a quantitative measure of hardness (25), is roughly relateddirectly to atomic electronegativity (eq 7) and inversely to atomic size (eq 11). Interestingly, thedatain Part I of this series (I) indicate that atoms that show positive deviations from eq 7 tend to show negative deviations form eq 11, and vice versa. Therefore, a combination of eqs 7 and 11should lead to a partial cancellation of errors and allow the estimation of the Mulliken b parameter with increased accuracy. The geometric mean of eqs 7 and 11is as illustrated in the figure. The plotted points deviate systematically from a linear relationship, perhaps indicating that eqs 7 and 11should not he combined with equal weights. Still, a general correspondence between b and (ah)'" is 224
Journal of Chemical Education
Conelationof Mulliken atomic a and b parameters and atomic size.
obvious. Thus, b, (alr)1'2 or simply a h could be used as an index of relative atomic hardness. I t is noteworthy that the Mulliken b parameters for pure s and pure pvalence states (I) often deviate widely from eqs 7, 11, and 14. This supports the view that most pure s and p electronegativities have little direct bearing on chemistry, being merely contributing terms to actual s-p hybrid electronegativities (1). Calculation of Mulllken Group Electronegatlvltles The concept of electronegativity evolved largely from the desire of organic chemists to understand reaction mechanismsin terms of the inductive effects of different functional groups (27). Hinze, Whitehead, and Jaffe (10) have calculated Mulliken group electronegativities a c by an iterative method, while Huheey (11,12) has obtained both a c and bc by plots of a., against net group charge 6c. Explicit equations for a c and bc can be derived from the work of Reed (14):
In eqs 15 and 16, u represents the number of atoms of a particular element in the group (e.g., for CH3, uc = 1and U H = 3). Using CHz as an example, assigning the valence states C(V = 4) = te and H(V = 1)= ~ ( 1 ) .
Mulliken electronegativities for 35 groups are presented in the table. These values often differ significantly from the extensive tabulations given by Huheey (11,12), because dif-
Mvlliken Group Electronegativitles (eV)
CHI C2Hr CsHs CHO COCHs COOH CFs CClo CN SiHs SiFa NNN NH. NO NOn NF2 NCO NCS pH2 OH OCI OClO OCiO? 0CiO3 OBrO? OIO* ON0 0NOz OCHs OCsHs OCHO OCOCH. OCN SH SCN
and are probably superior because they take into account the dependence of electronegativity on hybridization. The Mulliken b parameters for groups are generally much lower than the Mulliken atomic b parameters ( I ) , decreasing with increasine" Nc. -. the number of atoms in the erouD (86. 11,12). In agreement with eq 7, the 35 groups in the table give
-
te, s te, s 1,. S lr. 5 , h tr, tr, te, s tr, tr. ', s te. '
bc = (1.20 + 0.17)a,lNc
. te. di, dl te. s
tr, di, h
-. .. S
tr. v tr, te
tr, di, tr tr, di, tr S
..,. ', ...,te, . .. . 5
16.7% s, ' 20% r. ' 20% s.
'. tr. h, te. s di, dl
-, S ', di, dl
8.21 10.94
5.71 3.88
3.10
.
quat ti on 15.
.-
.
ZPv = oA + b,12
(18)
EAv = a, - bA12
(19)
2.7118
(*melinking atom 13wrinen first. .-
(17)
In addition to beinga measure of hardness (25), the Mulliken b parameters is an inverse measure of charge capacity, the ability to donate or withdraw charge as a function of electronegativity change. Groups are effective reservoirs of charge capacity, increasingly so with increasing Nc. Charge capacity considerations explain interesting chemical differences between atoms and arouns. such as the abilitv of groups ro funct~onboth as hette; charge donors and acceptors than atoms of wnilar inherent elecrroneaativit\ (8h,Rd, 28). Equations 15 and 16 do not take group structure into account and predict that isomeric groups have the same electronegativity, provided that all valence states remain the same. This is in agreement with Sanderson's principle of electronegativity equalization (6a, 70, 13) hut has caused some concern (11.12). If Sanderson's orincinle is valid. and there is theoretick sipport that it is (i9-31j, then strueturallv denendent inductive effects. so imnortant in organic chemistry, should probably not be interpreted as polar effects (charge effects operating through bonds), but as field effects (charge-distance effects operating directly through space) (28). Equations 3 and 4 may be combined to give
18.
'.
&
...
14 Equation 16.
I*Conmrted from &as described in ref 1 10 Ref 28. 18 Pmbabiy lm low (see text).
ferent valence states have usually been assigned to singly bonded atoms of elements in periodic groups V-VII (I). Partial ionic charge calculations with Mullikeu group electronegativities are performed identically to those with Mulliken atomic electronegativities, that is, via eqs 5 and 6. Using CH3F as an example, assigning the valence state F(V = 1) = 14.3% s (I).
The value obtained for a,, (8.61) is the same as that obtained fromeq 5 with atomic electronegativities, and also gives 6c = +0.040 and 6~ = +0.112. Mullikengroup a parameters may be converted to Pauling units (4, 5) as described in Part I of this series (I). The author (28) has recently calculated group electronegativities directly from thermochemically derived Pauling values (5). Both results are listed in the final two columns of the table. The agreement between the two sets of values is generally good except for groups containing multiply bonded atoms; in such cases the converted Mulliken a parameters are higher
Eauations B a n d 19 are strictlvvalid onlvfor atoms. hut it is of &erest to test their applicibility to gr"oups. ~nfo;tunatelv. eas 18 and 19 ~ r e d i cthe t ualence-state ionization poten(ial a i d electron affinity, which are generally unavailable for groups, so comparison with experiment is difficult. (For ex&pie,the isolated methyl gro"p does not maintain tetrahedral HCH bond angles through the sequence CH3-, CH3. CH3+.)Nevertheless, Reed U4) has calculated Mulliken a and b parameters for a number of groups and stable molecules via eas 15 and 16 and has then attempted to oredict ionization potentials via eq 18. Reed has f&nd that eq 18 predicts (valence-state) ionization potentials for nonbonding electrons within f 1 0 % of the experimental (groundstate) values. For bondine electrons.. the .re dictions of ea 18 are 20-30% too low, whilefor antibonding electrons theyare 30-70% too high. On the other hand, eq 19 predicts group and molecular (valence-state) electron affinities that are two to four times as large as the experimental (ground-state) values (8e), indicating that Reed's occasionally good ionization potential results (14) may be somewhat fortuitous. The questionable integrity of the Mulliken b parameter in bonded atoms (discussed in the following section) may lead to a propagation of errors which is more obvious in the group electron affinity because it is smaller than the group ionization potential. Calculation of Bond Energies from Mulliken Eleclronegativities
The calculation of the stahilitv of unknown chemical snecies is at the very heart of practical theoretical chemistry and is a valuable teaching aid (6b, 7c). Ferreira (32) and Evans and Huheey (22-24) have utilized Mulliken electronegativities in the calculation of "bond energies". the decrease in total energy of binary molecular com~oundsrelative to isoVolume 65
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225
lated atoms in their valence states. They have ascribed the energy decrease to three contributing terms: 1. The covalent energy,Ec, a lowering in energy of a molecule due to overlap of bonding orbitals. Ec has been calculated by the aritbmetic mean (32) or the geometric mean (22-24) of the nonpolar
tnm wrthdrswal by other. more elcrtrunegative aroma 181 groups !I, bd, dg, 34). In mder to accommodnrr the nohle gases. i1 has brrn suggestrd that the word arrrnrr in Psulmg's definition be replaced by the phrase attract or hold (34).
The following scale-independent definition of atomic or group electronegativity addresses all these issues, yet recovalent homonuclear hond energies. Evans and Huheey (22-24) have also attemoted to account far oossible re~ulsionbetween mains simple enough for beginning chemistry students: eleclone pairs of electrons on adjacent bonded atoms. Ec becomes tronegatiuity = the ability of a n a t o m orgroup to attract or less negative with increasing charge transfer, due to decreasing hold electrons to itself when combinine with other atoms or overlap of handing orbitals. groups. The word combining is a dynamic term (which is not 2. The Madelung (ionic or electrostatic) energy, EM,a lowering in synonymous with combined), and is compatible with Sanenergy of a molecule due to attraction of opposite charges.EMhas dersou's principle. been calculated by a classical eoulombic model. EM becomes more negative with increasing charge transfer and decreasing Conclusion internuelear distance. A common misconception among chemistry teachers is 3. The elertnmcgativity rnerm, E x , a lowering in rnrrg). of n molethe belief that the lack of accurate electron affinities severecule due 10 chargr rmnsfer irmn a les, ro a muw elertronegarivr atom 1Y.10,. E x hsr hemralculated by zim~~lrnne~~u.rapplir~1iun ly limits the application of the Mulliken electronegativity of eq 1to all atoms in the molecule. Ex becomes more negative scale to topics in chemical education. However, Hinze and with increasing charge transfer until electronegativity equalizaJaffe (35) have pointed out that this has never been a valid tion is reached; thereafter it becomes less negative and may excuse for avoiding the Mulliken scale because the uncereventually become positive. tainty in the electron affinity is always small compared to the electronegativity. Moreover, the accumulation of accuFerreira (32) and Evans and Huheey (22-24) have varied rate, experimental electron affinities for many elements (36) these terms with partial ionic charge to minimize the total has nullified this areument. molecular energy. Their calculated hond energies agree reaIn the author's oGnion, future research into the Mulliken sonably well with experiment. However, because E, and EM electroneeativitv should be directed toward two work in opposition to the total energy and because EMis " svstem main objectives: always more negative than the Ec, it replaces (6c, 7d, 8f, 2224, 32), the total energy minimization procedure leads to 1. The evaluation of reliable Mulliken electronegativities for the charge transfer that is significantly larger than predicted by transition elements. eqs 5 and 6. 2. The development of an extensive method of bond-energy calculaFerreira (32) and Evans Huheey (22-24) have speculated tion, applicable to a wide variety of compounds, and including a that the discrepancy in partial ionic charge could indicate quantitative relationship between the Mulliken b parameter that Sanderson's principle of electronegativity equalization (dXn(M)ldb~) and internuclear distance. (6a, 7a, 13). although useful, is only approximately true. It is I~elievedthat two major theoretiraldiffirultiesremain Ferreira (32) has made the provocative comment that Sanassociated with the Mulliken system: derson's principle is valid only if there are no interactions between the atoms, that is, if the "bonded" atoms are sepa1. The need for a generalized method of valence state assignment rated hy an infinite distance! (1).This is a very serious problem. However, Sanderson's principle has received support 2. The need for a better understanding of the importanceof outer dthrough the quantum-mechanical arguments of density orbital participation in chemical bonding (I). functional theory (29, 301, and Politzer and Weinstein (31) As long as the concept of atoms in molecules remains have shown that it can he derived without specifying any useful. it is likelv that the electroneeativitv conceDt will be particular theoretical model. Therefore, i t appears that Ferretainkd in someuform.The Mulliken-system offers-a simple, reira's (32) comment about Sanderson's principle should elegant definition of absolute electroneeativitv that has nrobahlv he directed instead toward the Iczkowski-Marwithstood meticulous theoretical scrutiny-lf the-theoretical grave atomic charge-energy function (91, eq 1.It is suggested difficulties stated above can be resolved, it is predicted that that the Mulliken b narameter ( s u ~ ~ o s e d al vmeasure of the Mulliken system of electronegativity is the one that will dXA(M)/d6~) decreasks from its isoi&ed-atom value of IPv endure after all the others have been deposited into the - EAv (eq 4) as an atom approaches other atoms, due to the archives of chemical history. interaction of electrical fields. The possibility of a distancedependent Mulliken b parameter preserves Sanderson's Literature Cited principle but also accounts for the results of Ferreira (32) 1. Brabeh, S. G. J . ChmL Educ., in press. and Evans and Huheey (22-241, because charge transfer 2. Mu1iiken.R. S. J. ChanPhya. 1934,2,782. increases as b decreases (eq 6). 3. Mulliken,R. S. J. ChemPhys. 1935.3.573. 4. Pau1ing.L. J.Am. Chem.Sac. 1932, S4.3570. 5. A1lred.A. L. J . Inorg. NucLChrrn. 1961.17.215. 6. Sanderaan,R. T. Inorganic Chemistry: Reinhold: New York,1967:la) pS1,lbl pp 121A Scale-Independent Definition of Atomic or Group 138, 14 pp 124-130, Id) p 305. Electronegativity 7. Sanderaan, R. T. Polor Coudenee: Aesdemie: New York, 1983:la) pp 37-39, lb) p 41, Pauling's original definition of electronegativity, "The nower of an atom in a molecule to attract electrons to itself' (33), needs to be revised to accommodate our modern views 9. Ierkowski, R. P.:Margrave, J. L. J . Am. Chom. Soc. 1961,83,3647. of the electronegativity concept: 10. Hinze,J.; Whitehead,M. A.:snd Jsffe,H. H.J . A m , Chsm.Soc. 1963.85,148. n. Huheey, J. E. J. Phys Chem 1965.69.3284, 1. The word pourer implies a specific physical dimension, energy1 12. Huhely, J. E. J.Phys. Chem 1966.70.2086. 13. Sanderson, R. T. Science 1951.114.670. time. A better word is ability. 14. Reed.J. L. J.Phvr. Cham. 1981.85.148. 2. The word atom should be replaced by the phrase atom or group. 15. ~~st&h,S.G. J . C ~ ~ ~ . E 1984,61,588. ~ U C . 16. Nalewajski. R. F. J. Phys Chem. 1985.89,2831. 3. Pauling's definition implies that different bonded atoms in a 17. Pearsun, R. G. J.Arn.Cham.Soc. 1963,85. 3533. single chemical species have different electronegativities. Thus, 16. Pearson, R. G. J. Chzm. Edur. 1968.45.581. 19. Peanon, R. G. J. Chem. Educ 1968.45.643. it violates Sanderson's principle of electronegativityequalization 20. Pearson, R. G. Hard and Soft Acids ond Boaer: Dowden, Hutchinson and Ross: (6a, 70,13). Stroudshurg, PA, 1973. 4. Pauling's definition excludes the noble gases, whose very high 21. ~ e n r e nW , . B. he ~ e w i a~ c i d - ~ o sConcepts: e A" Oueruialu: Wiiey: New York. 1980: pp 253-303. electronegativitiesindicate only their relative resistance to elec-
-
S~
~
226
Journal of Chemical Education
22. 23. 24. 25. 26. 27. 28. 29.
Eusna, R. 8.; H u h w , J. E. J. lnorg. Nurl. Cham. 1970,32.373. Huheey, J.E.:Evans, R. S. J I n o r g . NurLChrm. 1970.32.383. Even8.R. 8.: Huheev.J. E. J. I w r s Nuel Chem. l970.32.777. ~ ~ ; J.~rn.?hem.Suc, 1983, i05.7512. Parr, k.G . ; ~ e a m R.G. Ray,N. K.:Samueln,L.;Pan,R.G. J.Chem. Phyl. 1979,70,3680. Pritehard, H. 0.; Skinner, H. A. Chsm. Re". 1956. 55.745. Brat*ch.S. G. J. ChemEdue. 1985,62,101. Parr, R. G.: Donnelly, R. A: Low, M.; Palke, W. E. J. Chem. Phys. L918,6B, 3801
30. 31. 32. 33.
Donnelly, R . A . ; P r r R . G. J. Chem Phyr. 1918,69,4431. Politrer, P.; Weinsbin, H. J.Chern.Phy% 1979. 71,4218. Feneira. R. J . Phys. Chem. 1964.68, 2240. Pauling, L. TheNorvre oflhs Chm&d Band: CornellUniversity: Ithscs, NY, 1939:p
"".
cs
34. Allen, L. C.;Huheey, J. E. J. Inorg. Nucl Chem. 1980,42,1523. 35. Hinze,J.;Jaffe,H.H. J. Phys. Chem. 1963,67,1501. 36. Hoiop. H.:Lineberger, W. C. J.Phyr. Cham. ReJData 1985.14.731.
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