Electronegatividad Leland C. Allen

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J . A m. Chem. Soc . 1989, 1 1 1 , 9 0 0 3 - 9 0 1 4 9003

Electronegativity Is the Average One-Electron Energy of the

Valence-Shell Electrons in G round -State Free Atoms

Leland C . Allen

Contribution fr om the Department of Chem istry, Princeton University,Princeton, New Jersey 08544- 1009. Received February 27, 1989

Abstract: It i s argued t hat e lect ronegat ivi ty is the thi rd dimension of the Periodic Table, and th at xSpcc (me, + n t , ) / ( m+ n , for representat ive elements where tp, t S re t he p , s ionization energies an d m , n t he num ber of p , s elect rons. Values

of spectroscopic x are obtained to high accura cy from the Na t ional B ureau of Sta nda rds atomic energy level tables and closelymatc h the widely accepted Paul ing and Al l red Rochow scales. xspcc at ional izes the diagonal separat ion between metalsand non-meta l s i n the Periodic Table, the format ion of noble gas molecules, metal l izat ion of the elements as one descends

groups I-V, and the force defini tion used by Al l red Rochow. Axs t he energy d i f fe rence of a n a v e r ag eelectron in a tom A and in a tom B, i s ab l e t o sys t emat i ze proper t ies of tE v a s t a r ra y2 Kh ow n mater i a ls : i oni c so li ds , cova len tmolecules, metals, minerals, inorganic and organic polymers, semiconductors, etc. Transi t ion-m etal elect ronegat ivity can notbe s imply de termined because of t he na ture of d-orbi tal radial d ist ribut ions and this i s reflected in i t s pauci ty of use among

transi t ion-me tal chemists. Est imates for fi rst t ransi t ion series xSw are ob t a ined and a computa t i ona l method to address t h i sproblem is given. It also proves possible to trans late free atom, ground-s tate xSp into the in si tu m olecular orbi tal defini t ionof average o ne-elect ron energy for orbi tals local ized on an atom ic center. This leads to an improved defini tion of group (orsubst i tu ent) elect ronega t ivi ty, extension and refine ments in the use of elect ronegat ivi ty pertu rbat ion s in qual i tat ive andsemiquant i ta t i ve molecul ar o rb i t a l t heory , and unders t anding of hybr id orb i t a l e l ec t ronegativ i t y order ing ru l es such as sp >

= x -

sp* > sp3.

I . Electronegativity: Connection to Perio dic Table andProperties

I t i s t h e h y p o t h e s i s o f t h i s p a p e r t h a t e l e ct r o n e g a ti v i ty i s ani n t i m a t e p r o p e r t y o f t h e P e r i o d i c T a b l e and t h a t i t s d e f i n i t io n

fol lows from t h i s r e l a ti o n s h ip . T h e c o n t i n u i n g a n d o v e r a r c h i n g

chemical organizing abi l i ty of t he Per iod i c Ta ble s t rongly sugges t s

t h a t an addi t i ona l var i ab l e , beyond the c h a n g e i n Z cross a row

and the c h a n g e i n s h e l l n u m b e r d o w n a c o l u m n , m u s t p l a y a key

role in t h e c h a r a c t e r i z a t i o n o f s o l i d -s t a t e a n d m o l e c u l a r b i n d i n g .

It is most l i ke ly t ha t t h i s new th i rd d imension i s an energy because

t h e S c h r G d i n g e r e q u a t i o n i ts e lf i d e n t i f ie s e n e r g y a s t h e c e n t r a l

p a r a m e t e r for d e s c ri b in g t h e s t r u c t u r e o f m a t t e r . S i n c e t h e Pe-r i o d ic T a b l e i s c o m p r i s e d of r o w s i n w h i c h a subshe l l i ncreases

its o c c u p a n c y i n o n e - e l e c tr o n s t e p s u n t i l c o m p l e t i o n a t a n o b l e

g a s a t o m , and since successive rows simp ly add subshe l l s , t he new

p r o p e r t y m u s t b e t h e e n e r g y o f a subshe l l . From t h e A u f b a u

Pr inc ip le , we know (for t he represent a t i ve e l ement s) t ha t subshe l l s

a r e spec i fi ed by t he numb er of s and p e l ec t rons and t h us i t fo l lows

tha t e l ec t ronegat iv i t y is d e f i n e d on a per-e l ec t ron (or a v e r a g e

o n e - e l e c t r o n e n e r g y ) basis as

mep + nes-- m + n

w h e r e m a n d n a r e t h e n u m b e r o f p a n d s val ence e l ec t rons ,

respec t i ve ly . T h e corresp onding one-e l ec t ron energ i es , tp a n d tS

are t h e m u l t i p l e t - a v e r a g e d t o t a l e n e r g y d i f f e r e n c e s b e t w e e n a

g r o u n d - s ta t e n e u t r a l a n d a s in g ly i o n i ze d a t o m , a n d t h e a t o m i c

energy l eve l da t a requ i red t o determine t h e m is a v a i la b l e a t h i g h

a c c u r a c y from t h e N a t i o n a l Bureau o f S t a n d a r d T a b le s .' xspcci s term ed spect roscopic electronegat ivity and a t h ree-d imensiona l

I ) Moore, C. E. Af om ic Energy Leuels, National Bureau of StandardsCircular 467, Vol. I (1949), I1 (1952), 111 (1958) (reprinted as NS RDS-NB S35, Vol. I, 11, 111). Supplents on selected second-row elements plus Si,NSRDS -NBS 3, Sections 1-1 1 (1965-1985). Na, M g, Al, Si, and P Martin,W. C., et al. J . Phys. Chem. ReJ D ata 1979,8 ,817; 1980, 9, 1; 1981, 10, 153;1983, 12, 323; 1985, 14, 751. Pettersson, J. E. Phys. Scr . 1983, 28, 421(mult iplets of (3s)(3p) ' for S 11). Li, H.; Andrew, K. L. J . Opt. SOC m .1971, 61, 96; 1972,62, 255 (multiplets of ( 4 ~ ) ( 4 p ) ~or As 11). Arcimowicz,B.; Aufmuth, P. . Opt. Soc. Am . 1987,4 B, 1291 (?S2' multiplet of (5s)(5p)'for Sb 11). Bibliography on the Analysis of Optical Atomic Spectra, NBSSpecial Publications 306 (1968), 306-2 (1 969), 306-3 (196 9), 306-4 (1969).Bibliography on Atomic EnergV Levels and Spectra, NBS Special Publications363, 1968-1971, 1971-1975, 1975-1979, 1979-1983. Also valuable for up-dates: Radzig, A. A.; Smirno v, B. M. Reference Data on Atoms, Molecules,and Ions; Spring -Verlag : New York, 1985; Spring er Series in ChemicalPhysics, Vol. 31.

0002-7863/89/1511-9003 01.50/0

Table I. Electronegativities for R epresentative Elements (P aulingUnits)

atom xSwn x p b XAQR' XME xMe

H

LiBeBCN0

FN e

N a

Mg

AISiPSClAr

KCaGaGeAsS eBrKr

R bS r

InSnS bTeIXe

2.300

0.9121.5762.0512.5443.0663.6104.1934.787

0.8691.293

1.6131.9162.2532.5892.8693.242

0.7341.0341.7561.9942.21 12.4242.6852.966

0.7060.963

1.6561.8241.9842.1582.3592.582

2.20

0.981.572.042.553.043.443.98

0.931.31

1.611.902.192.583.16

0.821 oo1.812.012.182.552.96

0.820.95

1.781.962.052.102.66

2.20

0.971.472.012.503.073.504.10

1.011.23

1.471.742.062.442.83

0.911.041.822.022.202.482.74

0.890.99

1.491.721.822.012.21

1.902.603.083.624.00

1.581.872.172.643.05

1.751.992.212.462.75

3.059

1.2821.9871.8282.6713.0833.2154.4384.597

1.2121.630

1.3732.0332.3942.6513.5353.359

1.0321.3031.3431.9492.2562.5093.2362.984

0.9941.214

1.2981.8332.0612.3412.8802.586

Th e scale factor set by xsw average of Ge and As = the combinedaverage of the Allred and Rochow and Pauling values for Ge and As.Thus absolute values in Rydbergs were multiplied by 2.30016.bPa uling , data from ref 6. CAllred and Rochow, ref 7. dBoyd andEdgecombe, values taken directly from Table I of ref 8. CMulliken.data from ref 30. Sam e scale factor convention as used for x , ~ ,husabsolute values in kJ were multiplied by 0.004419.

m a p f o r t h e r e p r e s e n t a ti v e e l e m e n t s t h r o u g h t h e 5 t h r o w i s g i ve n

a s F i g u r e 1. T h e n u m b e r s f r o m t h e N B S t a b l e s g iv e a b s o lu t e

v a l u e s in R y d b e r g s a n d a s i n g le s c a le f a c t o r ( s e t n e a r t h e c e n t e r

0 1 9 8 9 A m e r i c a n C h e m i c a l S o c i e t y



9004 J . A m . Chem. SOC. ,Vol. 1 I I No. 25, 1989 Allen

Figure 1. Electronegativity,xapa= ( m e , + n c , ) / ( m + n , where m, ep,

n , e, are the number and ionization potentials (multiplet averaged) of pand s electrons in the valence shell of representative elements throughthe 5th row. e, and L, were obtained from National Bureau of Standardshigh-resolution atomic energy level tables (ref 1). Cross-hatched atomsare those of the metalloid band.

of the metalloid band ) conve rts them to the Pauli ng scale. It isto be noted especially tha t both xsw and the Periodic Table aredefined by free atoms in their ground states.

xsp versus Rows. If we plot xsw versus row number (Figure2 and Table I ) we may expect a set of nearly parallel curves thatfall off rather rapidly with increasing row number. This resultfollows because atoms are nearly spherical and each left-to-rightisoelectronic step is of increasing subshell radius, therefore de-creasing energy. Each vertical step downward from one curveto another removes one electron therefore yielding successivedecreases in average valence shell energy as one moves from topto bottom in Figure 2. Th e upward shift at the fourth row forp-block elements (and consequent alternation in groups I11 andIV) is a result of incomplete s, p valence shell screening by 3delectrons as one passes through the first transition series. (T hescreening primarily affects the s electrons because of their non-zerocharge density at th e nucleus. The incomplete d screening affectsthe fifth row as well as the fourt h.) Th e metalloid band is alsodesignated in Figure 2 and it is apparent that this region en-compasses just those atoms, and only those atoms, known to be

metalloids.2 This band is approp riately narrow since it representsthe most important diagonal relationship in the Periodic Table,that w hich separates the non-metals from the metals. The diagonalnature of this dividing band implies that it is a consequence ofthe Periodic Table’s third dimension. We show below that ther eis a fixed relationship between xsP, free atom energy levelspacings, and energy band w idths in solids, thereby establishingthe connection between x and th e physical basis for differen-tiating metallic solids a n fi q u id s from non-metals.

Noble Gas Atom xsw The noble gas atoms are logically thetop curve in Figure 2 and xsp characterizes the known chemistryof these elements. Thus, e.g. N e (Figur e 2) has a higher xspqEhanany open-shell ato m an d therefore holds its electrons too tightly

(2) Rochow, E. G. he Metolloids; D. C. Heath Co., 1966.

4 -

Xe

I

Te METALLOIDBAND

....snI n

Sr

Rb

th

Figure 2. xaps rom experimental atomic energy level data versus rows.Family of curves for groups I-VI11 of the representative elements.Pauling units (scale set by equating xspaaverage of Ge and As to com-bined average of Allred Rochow and Pauling scale Ge and As values).Metalloid band has Si as the lower limit and As as the upper limit.

to permit chemical bonding. On th e other hand, the differencein+ alues between Xe and F or 0 easily rationalizes the knownoxides and fluorides of Xe, a nd the relatively large Axsp betweenKr and F accounts for the existence of KrF2 . But Axsp betweenXe and C1 and between Xe and N are sufficiently smaller to

suggest that no binary molecules free of a stabilizing environment(produced, e.g., by crystallization) ar e likely to be found.3a W ear e still left with the tantalizin g question of whether A rF 2 canbe realized (Axspec cross Ar F is only 8% smaller than across XeO)and with the outside possibility of a krypton oxide. No ble gasatoms form the hinge of the Periodic Table because their elec-tronegativity is two-sided: they have the values shown for theirrole of holding electrons, but all have a xSp of zero for attractingelectrons.3b

Comparison with Pauling and Allred Rochow Scales.Pauling’s electronegativity scale was first published in 1 9324 andmany others have been proposed since then. However, an extensivesearch of the literature (textbooks, journal articles, and reviewpapers 5) show tha t values from only two, Pauling’s scale (asup-dated by Allred in 1961)6 and those from Allred Rochow’s

force definition,’ have been frequently and systematically employedby chemists and physicists to guide them in answering practicalproblems in chemical bonding. Figur e 3 comp ares values fromthese two with x s w , and it is immediately apparent th at xsw isreproducing th e pattern established by the Pauling and AllredRochow scales. In fact, xsw seems to adjudicate them. It maybe that x+ for F, and perhaps 0 ar e 1-2% too high, this pos-sibility arising from their extremely high density thereby p roducing

(3) (a) XeCI, is obtainable in a xenon matrix but is too unstable to bechemically characterized. The non-binary compounds, Xe(CF,)* and FXeN-(S02F), , have strong electron-withdrawing groups attached to C or N . (b)Allen, L. C.; Huheey, J. E. J . Inorg. Nucl. Chem. 1980, 42, 1523.

(4) Pauling, L. J . Am. Chem. SOC. 932, 54, 3570.5 ) Allen, L. C., to be submitted to Chem. Rev.

( 6 ) Allred, A . L. J . Inorg. Nucl. Chem. 1961, 17, 215.7) Allred, A. L.; Rochow, E. G. J . Inorg. Nucl. Chem. 1958, 5 , 264.



Electronegativity of the Valence-Shell Electrons

I

Te

Sb

3

In

Sr

Rb

th

Figure 3. xSpe,, xperimental values (solid lines), compared to Pau lingscale (dashed lines) and to Allred Rochow scale (dotted lines).

differentially high electron-electron correlation energy correctionswhen they bind to form molecules or solids. Similar ly, grou p Ixspcmay be 5 4 oo low because of the differentially high chargetransfer binding that may be expected when they bond into solidsor molecules.

Th e Allred Rochow force definition,' xA= o.359zA/rA2+0.744, where ZA s the Slater rule determined effective nuclearcharge and r A s the Pauling covalent radius, has appealed con-ceptually to many chemists. It is thus satisfying tha t a set of linear

relationships exist between xspecnd the force on he outermostelectrons at their radial m axima (shown in part VI below).

Comparison with Boyd Edgecombe's Sca le. Very recently,Boyd a nd Edgecombe* have determined electronegativities fromcomputed electron density distributions for a number of repre-sentative element hydrides, X H. Atomic radii were determinedby the point of minimum c harge density along X-H and elec-tronegativity was assumed to be a direct function of the char gedensity a t the minimum, the number of valence electrons, and th eX-H separation and an inverse function of the atomic radii. Thisappears to be a plausible and promising approach and comparisonbetween their values and xspecor p-block elements is given inFigure 4 (and Table 19 . The Boyd-Edgecombe definition is verydifferent from that of either Allred Rochow or Pauling andthus the striking agreement obtained in Figure 4 is encouraging.

Axswand Keteiaar's Triangle. Pauling's well-known procedure

for equating the bond energy above that expected fro m a perfectsharing distribution to a function of xA- xB designated as th eionic cha racte r of has a clear conceptual relationship to

(8) Boyd, R. J.; Edgecombe, K. E. J. Am. Ch em. S OC.1988 110.4182.(9) Problems with this definition for the six low-electronegativity elements

of groups I and 11, seemingly from charge -transfer effects, eads to values veryfar from those of Allred Rochow and Pauling.

(10) Pauling, L. The Nature of the Chemical Bond, 3rd ed.; CornellUniversity Press, 1960; Chapter 3.

(1 1) In the original construction of his scale' and in further res earchsumm arized in The Nature oft he Chemical Bond,IoPauling has dire cted hisquantitative efforts toward establish ingelectronegativity values for free atoms.However, i n his qualitative introductory paragraph, he associates electro-negativity with the power of an atom in a molecule to attra ct electrons toitself . We identify *... in a molecule ..." with the all embracing molecularproperties ordering capability of the Periodic Table.

J . Am. Chem. SOC.Vol. 1 1 1 . No. 25, 1989 9005

F

4-

x o

N3-

C

02-

I -

0-

Figure 4. xSp, experimental values (solid lines), compared to 15 pblockelements from Boyd Edgecomb e values (dashed lines) computed fromelectron density distributions.

xtpec x : ~ . Thus we identify xtP - x:pec Axsp as the ioniccharacter or bond polarity of bond AB. Axs and the PeriodicTable of G ~ overn the three types of bond slycov alent, metallic,and ionic) traditionally employedI3 to characterize th e chemicaland physical properties of materials in the temperature andpressure range where metals and ionic compounds occur as solidsa n d 1 i q ~ i d s . l ~his is demonstrated most easily by visualizingKetelaar's triangleI5 whose vertices are labeled covalent, metallic,

and ionic. Covalent and metallic bonds have long been recognizedas originating from the same basic quantum mechanical maximumoverlap-exchange forces16 an d therefo re along this side of thetriangle we ar e simply moving right to left in the P eriodic Table.For the elements themselves, the change from diatomics (F2,02N2) o metals (Li, Na) is ruled by xaY see the description ofthe non-metal/metal transition given in part I11 below). Forheteroatomic bonds, e.g., H F compared to H2and F2, it is readilya parent that Axswdeterm ines bond polarity. Thu s, xrp and

needed to construct the usual molecular orbital energy level di-agr am commonly employed to explain the bonding in HF . Thedifference in these average one-electron energies measures the shiftin molecular charge distribution giving rise to the HF bond po-larity.

xsF re the average energies of th e one-electron atom ic orbitals

(12) The fo urth type of bonding, London dispersion forces and molecularmultipole interactions, characterizes the cohesion in molecular liquids andsolids, including inert gas liquids and solids. The magn itudes of many suchinteractions are also governed by Axspcf,but the Periodic Table plays a less

direct role in organizing them than it does for the other three.(13) Spice, J. E. Chemical Binding nd Structure; Pergam on Press: New

York, 1964. See], F . Atomic Structure and Chemical Bonding Methuen:Londo n, 1963. Ketelaa r, J. A. A. Chemical Constitution; Elsevier: NewYork, 1958. Companion, A. L. Chemical Bonding McGraw-Hill: New York,1964. Pimental, G. C.; Spratley, R. D. Chemical Bonding Clarijied ThroughQuantum Mechanics; Holden-Day: San Francisco, 1969.

(14) At high tempe ratures, of course, matter is solely in the form of atomsand covalently bound molecules.

(15) Ketelaar, J. A. A. Chemical C onstitution; Elsevier: New York, 1958;Chapter I.

(16) Slater, J. C. Introduction to Chemical Physics; McGraw-Hill: NewYork, 1939; Chapter 22.



9006 J . Am. Chem. Soc.. Vol. 111. No. 25, 1989 Allen

Table 11. Stru ctur al Classification of Ionic-Covalent Compo unds

A. Classification of Representative Element Fluorides in Their Group Oxidation State s

I I1 111 IV V VI VI1

K FR b F InF3 SnF4 SbFs

ionic polymeric molecular covalent

B. A List of Simple Oxides of the Represe ntative Elements

I I1 I11 IV V VI VI1 VI11

L i 2 0 B e 0 B2°3 co2 N O , N20, N203 0 2 F 0 2 , 0 4 F 2

N a 2 0 MgO Si02 P4°6r p4°10 s o 2 , so3 Cl20, c102

K 2 0

co N2°4r N2°S F202

c12 7

C a O Ga203 G e 0 2 A 5406 S e 0 2 , S e 0 3 B r 2 0 , B r 0 2A s - 0 .

R b2 O S r O I n 20 3 S n 0 2 S biO ;, S b 4 0 6 T e 0 2 , T e 0 3 I2O4, I4O9 XeO ,SnO Sb205 I2O5 X e 0 4

C. Stru ctura l Classification of Representative Element Oxides

I I1 111 IV V VI VI1 VI11

Li Be I B I C N 0 F~ ~~

P I S c1C a S e B r

R b S r S b T e Ig 1F

AsN aK

Xe

ionic polymeric I molecular covalent

An important subcategory along the covalent-metallic side isthe semiconductors. Thes e ar e th e covalent solids of metalloidband elements (e.g., Si, Ge) or binary compounds with a metalloid(e.g., GaAs, InSb, S ic ) or binaries straddling this band (e.g., Alp,G ap ). As discussed previously, the metalloid band is defined bya specific range of xsp values and thu s xsp and Axs, classifythe bonding in these materials.

Along the ionic-covalent leg of Ketelaar's tri angle decreasingAxsp determines the properties of typical binary compounds asillustrated by the species shown in Table I I . I 7 J 8 On the left sideof Table IIA are the representative element fluorides that adoptclassic ionic structu res (e.g. LiF, MgF,) and on the right a col-lection of ten well-known covalently bound individual molecules.

Between these limits is the interesting diagonal-shaped region,delineated quite accurately by Axspcc,the structure of whosecomp ound s ar e polymeric solids (e.g., in AlF,, GaF ,, a nd InF,the metals are coordinated octahedrally with shared vertices; BeF2has a silica-like structu re). Tab le IIB lists simple oxides of therepresentative elements and Table IIC classifies them structura lly.Again we find well-recognized ionic compounds on the left (e.g.,MgO , sodium chloride structure; N azO , antifluorite; AlzOs, co-rundum and its closely related C- M z0 3 onic structur e for GazO,and In203both of which have approximately octahedral coor-dination around M3+and approximately tetrahedral arou nd 02-and S nO z with an ionic 6:3 rutile structure) and on the rightdiscrete covalently bound molecules (e.g., C 0 2 ,SOz, NO, O4F2,F 0 2 , XeO,). In between are polymeric materia ls, (e.g., BzO3, a3-connected, silicate-like network; SiOz and G e0 2 he a- qua rtz

structure; P406and P40,,,, metas table molecules transfo rmableinto two-dimensional or three-dimensional networks that ar e alsoformed by the arsenic and antimony oxides; Se02,3-connectedinfinite chains; TeO z layer and three-dimensional networks).Tables IIA and IIC with their diagonal-shaped polymeric regions

(17) Puddephatt, R. J.; Monaghan, P. K. The Periodic Table of the Ele-ments, 2nd ed.; Clarendon Press: Oxford, 1986. Wells, A. F. SiructuralInorganic Chemistry, 4th ed.; Clarend on Press: Oxford , 1975.

(18) An excellent quantum me chanica l discussion of the relationship be-tween ionic and covalent bonding is given in: Slater, J . C. Quantum Theoryof Molecules and Solids, Volume 2, Symmetr y and Energy Bands in Crystals;McGraw-Hill: 1965; Chapte r 4. J . Chem. Phys. 1964, 41, 3199. Slater'sanalysis provides the basis for the ability of Axsw to characterize the changein bond polarity as bonding changes from ionic to covalent, thus playing a rolefor the ionic-covalent leg of Ketelaar's triangle similar to that which his workcited in ref 16 provided for the metal-cov alent leg (see par t 111).

16-2 - ,

3 8 -X

I IIFigure 5. xSF experim ental values, versus groups for t he firs t five rowsof representative elements. Pauling units. Values are given in Table I.

defined by xspcc re typical of a large amount of data that canbe classified in this fashion.17 Similarly, along th e metallic-to-ionicside Axsp characterizes the sequential change from pure metalsto ionic crystals (e.g., Li, Li,Sb, Li3As, Li3P, Li3N, Li 20 , LiF:Li,Sb has the intermetallic Fe3A1 structure derived from cubicclose packing; Li3As has complex alloy-like phases; Li p forms aswell as Li,P and the lithium phosphorus bond is intermediate,neither metallic or ionic, while L i3N, Li2 0, and LiF a re ionicspecies progressing to the extreme).

xsp versus Groups. Figure 5 plots xsp versus group num berand the pattern of interrelationships displayed is surprisinglydifferent from that of Figure 2, even though the data ar e the samein both maps. For each row, xSp is rising nearly linearily withZ , but the incomplete d-screening in the transition series, dif-




ferences in screening between s and p electrons, an d th e succes-sively lower slopes due to increasing radii lead to an intricatelyorder ed set of values for groups 111-V of the 3rd, 4th, an d 5throws. This forcefully brings out the require ment for the high-accuracy values provided by xspc nd speaks against the impressionof many chemists th at only order-of-magnitude electronegativityestimates are needed. Contemporary research on semiconduc-tor-electrolyte and semiconductor-metal junctions is one exampleof solid-state physics and chemistry where accurate electroneg-ativity values for groups 111-VI of the 3r d, 4th, an d 5t h rows will

prove u~ e fu 1 .l ~ or some time the magnitude of 5th row elec-tronegativities has been in dispute and this is readily app arent intheir differing magnitudes on the Pauling and Allred Rochowscales (Table I ) . Another long-standing uncertainty has been thevalue for CI relative to N, many chem ists favoring the Paulingscale assignment of chlorine greater tha n nitrogen. Table I showstha t the reverse is true and the difference is comparable to thatbetween Ge and As or C1 an d Br. On e consequence of the lowerelectronegativity of CI compared to N is the structural dissim ilaritybetween FN, and ClN3.20

As we descend any of the group I, 11, 111, IV, or V lines inFigure 5, x s decreases and characterizes the increasing me-tall ization.2PbcThe, p, and d levels are getting closer togetherand there is a corresponding loss of bond directionality. Gro upIV is a classical example: starting with diamond, th e tetrahedrallycoordinated, covalently bonded, insulator (or with the highlydirectional bonding in graphite layers), the next three elements,Si, Ge, gray Sn , are tetrahedrally co ordinated semiconductors withsuccessively decrea sing energy gaps. The final two ar e metalswith increasing conductivities: white Sn, distorted from tetrahedr alcoordination by compression along the c axis giving it approxi-mately six nearest neighbors and Pb, face-centered cubic with 12nearest neighbors.

H bonds, A-H.-B, ar e anoth er well-known form of bondingwhose dominant characterizing parameter is electronegativitydifference, A x = x A- xH.22Less well-known, but equally im-portan t, ar e Alcock’s secondary bonds,z3a type of non-hydrogen,hydrogen bond, A-Y.-B. For example,23 the solid ClF2 +SbF 6-contains a linear bonding arrangem ent, F-Cl-F, where r(F-C1)= 1.52 A and r(CI--F) = 2.38 A. As in hydrogen bonds, A-Y-aBis typically linear, B has an electron donor lone pair, and 4Y-B)

is significantly larger than r(A-Y) but shor ter than the sum ofvan der Waals radii. A great variety of such bonds are found ininorganic solids and Axspec x - x3 c controls r(Y-B), bondstreng th, and othe r properties, as it does in the case of hydrogenbonds.z4 In summary , the description of bonding patterns in thisand the previous section suggests that x S w , because it can bedetermined accurately and is the third dimension of the PeriodicTable , may replace many specialized and ad hoc explanations ofsolid-state and molecular properties, thereby fulfilling the roleof Occam’s razor.

Trans ition-Metal Electronegativity. An important conclusionis that transition element electronegativities cannot be simplydetermined du e to the nature of d-orbital radial distributions. Thisis not surprising: the literature of transition-metal chemistryroutinely employs antibonding d-orbital occupancy, oxidation state,and formal charge on ligands as characterizing p arameters but

electroneg ativity is rarely mentioned. In the represen tative ele-ments the radii of the outermost s and p electrons are approxi-mately the same while in the transition elements the s orbital radiiare 21 /z o 31/z imes greater than the d orbital radii. Nevertheless,it is well-known that d-electrons contribute significantly to thebonding orbitals in both pure metals and complexes with non-

(19) Sculfort, J.-L.; Gautron, J. J . Chem. Phys. 1984 80 3767 and ref-erence therein.

(20) Allen, L. C.; Peters, N. J. S., in preparation.(21) Adams, D. M. Inorganic Solids; John Wiley Co.: New York,

1974.

(22) Allen, L. C. J . A m . Chem. Soc. 1975, 97, 6921; Proc. Nat l . Acad.

(23) Alcock, N. W. Ado. Inorg. Radiochem. 1972, 15, 2.(24) Desmeules, P. J. ; Nuchte rn, J.; Allen, L. C., to be published.

Sci . U.S.A. 1975, 72, 4701.

J . Am. Chem. Soc., Vol 111, No. 25, 1989 9007

metals but their degr ee of participation is difficult to assign. Th enumber of participating d-electrons decreases across a row becausethe ratio of the outer s radial maxima to the d radial maximabecomes significantly larger and because the d orbital energysharply decreases (becomes more core-like). As described in thesection below, we have devised a quantum mechanical compu-tational method to quantify the bonding contribution and theeffective number of d-electrons for particular molecules and s olids,but we can also make an approximate estimate for the firsttransition series by using well-known information from inorganic

chemistry. Thu s we use our xspccormula for s- and d-electronsplus the following assumptions. (i) Because xsFc s the thirddimension of the Periodic Table, the highest value for a transitionelement must be equal to or less than the lowest value in themetalloid band (Si, 1.916). (ii) The first half of the d-block istreated like the first half of the p-block because the observedmaximum oxidation statesz5 suggest that all of the s- and d-electrons can fully particip ate in chemical bonding. However,except for Ru and Os, the elements in the second half do not realizemaximum oxidation states equal to the number of outer s- andd-electrons in the f ree atomsz5 this is a result of the decrease inthe nu mbe r of effective d-electrons caus ed by the reasons givenabove). For the first transition series we interpret the sequentialmaximum oxidation state decrease in Fe, Co, and Ni by one foreach step to th e rightz5 as successive lose of on e d-electron andassume this pattern for Cu and Zn . (iii) Th e 4s orbital is assumeddoubly occupied because the 4s is widely split on molecule for-mation and the bonding MO s dominated by this orbital will alwaysbe doubly occupied. (iv) The one-electron energies used in theXSP= equation are taken from Herm an Skillman (see part Vbelow for references and comparison w ith experiment). xSw values(reference to Au(V) = 1.90 to satisfy (i)) are the following: Sc,1.15; Ti, 1.28; V, 1.42; Cr, 1.57; Mn, 1.74; Fe, 1.79; Co, 1.82;Ni, 1.80; Cu, 1.74; Zn, 1.60 (if the recently observed Cu(IV) isused instead of Cu(III), it raises x by a few percent). Thesevalues closely parallel the experimental numbers from the Paulingscale definition obtained by Allred6 (with the single exception ofMn whose low-value Allred attributes to a crystal field stabilizationeffect). Th e trends in xsw are those expected from the traditionaltransition element groupings in the Periodic Table: a smooth risewith 2, ike that of the representative elements, for Sc through

Mn; quite similar, high values for Fe, Co, and Ni; Cu by itselfand slightly lower than th e Fe, Co, N i triad; Zn very much lowerwith a value close to that of ALZ6

In SituElectronegativity. We have emphasized throughout thispaper tha t electronegativity is a free-atom, ground-state q uantityand that it derives its meaning as an extension of the PeriodicTable. By the same argumen t, the Periodic Table itself achievesits miraculous ability to organize chemical phenomena to a con-siderable extent because xsps s its third dimension. Thus , thePeriodic Table an d xSw together comprise the small set of rulesand num bers tha t help rationalize th e observed properties of the10 million known compounds, but they stand a par t from th e vastlycomplicated bonding details represented in these many moleculesand solids. There fore, an in situ electronegativity defined for a

(25) Huheey, J. E. Inorganic Chemistry, 3rd ed.; Harpe r Row: New

York, 1983; 569-572. Purcell, K. F.; Kotz, J. C. Inorganic Chemistry; W.B. Sanders, 1977; pp 530-531.

(26) Because x , ~s a free atom quantity with a single defining numberfor each element, it cannot depend on oxidation state. While it is certainlytrue that atoms have different electronegativities for different oxidation states,e.g. F,CH versus CH, (this can be readily measured by the difference inhydrogen bond energy realized between F3C-H an d H,C-H as proton dono r),search of the literatu re5 hows that represen tative element chemists generallyhave not found a need for assuming a strong dependence. Repres entativeelements of groups I and I1 have such low xaF that they are always in theirgroup number oxidation state. Polar covalent bonds formed by p-block ele-ments carry atomic charges that are seldom close to their oxidation numbertherefore suggesting small changes in electronegativity with change in mo-lecular environment. On he other hand, the closely spaced levels found intransition m etal containing molecules and solids has given rise to the beliefthat transition metal electronegativity should be strongly dependent on oxi-dation state. As in the case of main group elements, some dependence iscertainly to be expected, but at present the strength of this dependence isunknown.



9 8 J . Am . Ch em . Soc., Vol. 1 1 1 No. 25, 1989 Allen

solid or molecule can only produc e numbers of very specific interestfor the particular compo unds studied and not t he generality ofthe free-atom values.26 W e also recall tha t, as Pauling pointedout in his original paper: it is th e difference in electronegativityacross a bond AB that is of prime importance and that this dif-ference is to be a measu re of ionic character of AB and to notinclude its covalent component. In spite of these considerations,it is a useful and straightforward task to set up a projectionoperator a nd apply it t o a molecular orb ital wave function to yieldin situ average one-electron energies for atoms in molecules. Th edifference of these numbers for a t o m A an d B, when appropriatelyreferenced to homonuclear bonds AA and BB, is then t he desiredin situ electronegativity difference and we have designated it theBond Polarity Index, BPIAB. (The mathematical formulae aregiven in part VI1 below.) Within the context of free-atom xsw,th e role of BPI,, is 2-fold: (1) It makes connection betweenfree-atom A x s w and th e detailed determination of bond polarityin molecules a nd solids from a b initio electronic wave functionsas well as the incorporation of electronegativity concepts intoqualitative and semiquantitative molecular o rbital theory.27 ( 2 )Ab initio calculations on a wide variety of transition-metal com-plexes can hopefully lead to atomic electronegativities for thetransition elements including their dependence on oxidation state.%A set of preliminary calculations are quite encouraging in this

regard.28 It is also a simple ma tter to define an Fractional Po-larity** from the Bond Polarity Index and this is the equivalentof Pauling’s dipole moment referenced “percent ionic character”.Results from the sam e set of preliminary calculations are similarlysupportive.

11. Other Definitions of Electronegativity

We have already noted the unique standing of the Pauling an dAllred Rochow scales as being those which xsw was obligatedto match. In this section we discuss central features of theMulliken definition because it is also unique in providing theprincipal conceptual alternative to xsw (even though varioustabulations of values based on Mulliken concepts have not enjoyedwidespread acceptance among practicing chemists and physicists).We address only those aspects that may directly challenge x sas the energy dimension of the Periodic Table. In a soon o &comple ted review a r t i ~ l e , ~e attempt to treat the other p arts ofthe vast and complex history of electronegativity.

T he sim ple st fo rm of th e M ~ l l i k e n ~ ~efinition is given as

where I is the lowest first ionization potential of a ground -statefree atom and A is its correspon ding electron affinity. A principalattrac tion of this definition (as well as the second form discussedbelow) among theoretical chemists and physicists has been tha tit has the units of energy per electron. W e regard this as importantsupport also for hpc, hich is expressed in the same units. Figure6 plots x M versus rows for this definition with use of the latestexperimental data30 (referenced in the same way to a point in themetalloid band, as employed in Figure 2; values given in TableI ) . Comparing Figure 6 to Figures 2 and 3 shows a very roughsimilarity, but many incorrect features are readily apparent. Th e

most obvious are the following: (a) Th e metalloid band is muchtoo wide and contains the metals Be and Sn n addition to the

X M = ( I + A ) / 2

(27 ) T he Pauling concept of relating the elect ronegativit y difference forbond AB to the excess energy above the mean of the AA and BB bondenergies’O remains as the best basis for obtaining experimental estimates offree atom values from molecular and solid-state data. The Bond PolarityIndex then forms the link between free atoms and (referenced) in situ atom.It is obvious, of course, that the comparison between free atom Ax and in situAx can only be made on a relatiue basis because the average one-electronenergy of a free atom is very different than an average one-elect ron atomicenergy in situ due to molecular bonding energy. Controversies and practicalproblems with the Pauling scale are discussed in ref 5.

(28) Allen, L. C.; Egolt, D. A.; Knight, E. T.; Liang, C., to be submitted.The principal feature of EI,,and B P I A ~s their ability to assign a local energv(albeit, one-electron energy) to an atom or to the di fference between bondedatoms.

(29) Mulliken, R. S. J . Chem. Phys. 1934, 2 , 782.(30) Reference 1 for ionization potentials. Hotop, H.; Lineberger, W. C.

J . Phys. Chem. ReJ Duta 1985, 24, 731, for electron affinities.

Pd 3rd qth

I

Xe

Te METALLOID‘” BANDSb

S?i

In

Sr

Rb

5th

Figure 6. xM = ( I + A ) / 2 , Mulliken definition of electronegativity,versus rows. I and A are experimental ground state first ionizationpotentials and electron affinities, respectively (values from ref 30).

Pauling units xM values and referencing given in Table I). Metalloidband set to include known metalloids E, i, Ge, As, Sb, Te).

metalloids. (b) Hydrogen has a higher value than carbon. (c)There is no alternation in x M magnitudes between 3rd an d 4throw elements (Si-Ge, Al-Ga). As noted previously, alternationis due to incomplete screening of 4s electrons by the 3ds of thefirst transition series and it is a n experimentally well-establishedcriterion for judging electronegativity scale^.^'+^^ (d) The group

VI1 halogens are too high and cross with the noble ga s atoms.(e) Groups I1 and I11 cross, with th e former too high and the lattertoo low. Th e origins of these problems are largely the result oftwo basic flaws in the Mulliken definition. Th e first is lack ofs electron representation. The s electrons certainly enter intorepresentative element chemical bonding, chan ge their relativecontributio ns across a row, and must obviously influence bondpolarity. Th e second is that Z and A are treated sym metricallyin the x M equation. Z may be identified as the energy of a n electronin the outer shell of a neutral atom an d A the corresponding energyin the negative ion, but in most neutral molecules the atomiccharges are much nearer to 0 than to - 1 .

A second form of the Mulliken definition, “valence sta te x,”,identifies a specified ato mic hybridization and co mputes I , andA , from ground-state I and A plus promotion energies to theatomic excited state designated. Data from the NB S atomic

energy level tables’ a re used to ob tain promotion energies of thes and p electrons into excited state s with orbital occupancies of0, 1, and 2 corresponding to the desired valence state. Thisdefinition clearly avoids the problem of omitting s electrons.Bratsch ha s given an up-to-date accoun t of this scheme33and ina recent complete tabulation of values Bergman n and Hinz e” list124 possible hybridizations and corresponding Pauling-scaleelectronegativities for the sulfur atom alone. This method seeksto address a specific atom in a specific molecule and t he chiefdifficulty, of course, is that there is no a priori way to mak e such

(31) Sanderson, R. T. J. A m . Chem. SOC. 952, 74,4192.(32) Allred, A. L.; Rochow, E. G . J . Inorg. Chem. 1958, 5 , 269.(33) Bratsch, S. G. J . Chem. Ed., 1988, 65, 34, 223.(34) Bergmann, D.; Hinze, J. Electronegativity and Charge Distribut ion.

In Structure and Bonding;Sen,K. D., Jmgensen, C. K., Eds.; Springer-Verlag:

New York, 1987; Vol. 66.



Electronegativity of the V alence-Shell Electrons

+- T<12s l~12p l~>

--r(2si2~013- 4s- - - - - _ _ - - -@?1 Go

BORON OXYGEN 6 ONE-ELECTRON ENERGY

V i r ) r- V ( r ) r- B C N S z

Figure 7. Left side: Schematic of effective radial potentials for boronand oxygen with energy values for cp, cs, and xSpc esignated. Right side:One-e lectron energies, cp, ts and xsm, versus atomic number,Z,or B,C, N , and 0.

valence state assignments. As Bratsch c0n cludes,3~ This is a veryserious problem .

111. Non-Metal to Metal Transition

It is the purpose of this section to explain why a single pa-rameter, the magnitude of xsp, is adequate to characterize thetransition from non me tal to metal for the elements in the tem-perature and pressure range for which metals in their typicalstructures (face-centered cubic (fcc), body-centered cubic (bcc),hexagonal close-packed (hc p)) occur. Th e basis for this expla-

nation is to be found in an elegant description of interatomicinteractions given by Slater some time agoI6 and by a thoroughtreatm ent of elementary elec tronic structure m odels for solids givenby W a ~ g h . ~ ~n the right, high x , ide of the Periodic Tableonly diatom ic covalent bonds occur N2,02, z). The averageenergy of an electron (x,,) is very high and s and p atom ic energylevels e,, tp are widely separated: there is no other bonding optionand exchange energy locks them into homonuclear covalent bonds.As shown in Figu re 7 , large xsp and large energy level spacingare directly related. Like the H atom itself, the effective potentialseen by an electron in an atom is funnel-shaped and this form ofpotential has the property that lower energy is correlated withlarger spacing between levels. Thu s high xsPc mplies widelyspaced levels, both for occupied and unoccupied levels, andtherefore also large HOM O-LU MO energy gaps. As we progressto the left along a given row of the Periodic Ta ble the energy level

spacing decreases as xspec ecreases. Now if a given ato m issurrounded by others, energy bands will appear and the width ofa band is inversely proportional to the energy of the level fromwhich it arises. Thu s small x,, corresponds to small energy levelspacing and broad, frequently overlapping, energy bands. Smallspacing and broad bands gu arantee electron deficiency along withthe electrical conductivity and ductility associated with the metallicsta te. If high x atoms are clustered together they will formdiatomic molecules showing saturation of valence rather thanmetals because t he barrier between coalescing atoms will be high,the levels widely separated, and the bands narrow.

IV. Experimental Determination of xsPBecause Pauling's original assignment of 4.0 for fluorine's

electronegativity was arbitra ry and because of the long-sustaineduncertainty as to its appropriate u nits, it has been difficult to obtain

physical and chemical insight as to what magnitude and rangeof values to expect. This quand ary is mediate d by definingelectronegativity as th e average valence shell ionization potentialand by th e experimen tal finding tha t its values extend fro m ap-proximately 4 to 25 eV , just the spread of ionization potentialsand energies one expects to find identified with electronic phe-nomena in chemistry an d solid-state physics. It is also useful torecognize that on the Pauling scale a chang e of 1 in the first placeafter the decimal point corresponds to approximately 1 4 kcal/mol.

tp and E,. In his 1955 Physical Review paper, and more ex-tensively in his books on atomic structure, Slater defined one-electron energies and how they are to be determined from the

35). Waugh, J. L. T. The Constitution of Inorganic Compounds; Wiley-Interscience: New York, 972.

J . Am . Ch em .Soc. Vol, I 1 1 No. 25,I989 9009

SINGLE CONFIGURATION, LS COUPLED, MULTIPLET AVERAGED, ONE ELECTRON ENERGIES

a n 011

2P-,

(36) Slater J. C. Quantum Theory of Atomic Structure; McGraw-Hill:New York, 1960;Vol. I and 11. Slater, J . C. Phys. Reu. 1955, 98, 1039. In

Table I of the Phys . Reu. article and Table 8-2, page 206 of Vol. I, Slater hasgiven cp and e, values for H to Sr and this served as a valuable check on thenumbers we report in Table 111. W e have extended t he number of atomsincluded, tabulated additional significant figures,corrected three or four errors,and used the u p g r ad ed spectra available since Slater 's tabulation.

(37) Hansen, J. E.; Persson, W . J . Opt . SOC. m . 1974, 64, 696.(38) Persson, W.; Pettersson, S.-G. Phys. Scr. 1984, 29, 308. It should

be noted here that E,,, the spherically averaged energies from which wedetermine cp and cI are independent of the coupling scheme39 LS, LK, jK,or j j ) .



9010 J . Am . Chem.SOC. ol. 1 1 1 , No. 25, 1989 Allen

Table 111. Experimental and Computed xlpc Values (Rydbergs)

Hartree- Hartree-atom epo 6 exptl“ Fockb Fock-Slatef

H 1 oooo 1 oooo 1 oooo

LiBeBCN0FN e

N a

gAISiPS

CIA r

KC aG aG eAs

S eBrK r

R bS rIn

SnS bT eIXe

0.60980.78380.9687

1.16461.37091.5870

0.43930.57160.70950.8537

1.00461.1627

0.43590.55440.6738

0.79510.91771.0453

0.41 80.51550.61780.71960.82150.9235

0.39630.68521.03231.42821.8784

2.37962.95263.5628

0.37780.56200.83201.09421.3848

1.66891.85422.1491

0.31900.44930.92701.17961.3921

1.57101.79142.0222

0.30700.41860.87381.07021.23011.37501.53521.7196

0.3963

0.68520.89151.10601.3326

1.56961.82282.0810

0.37780.56200.7010.83290.9796

1.12541.24731.4093

0.31900.44930.76330.86700.961

1.05371.16731.2895

0.30700.41860.71980.79280.86270.93811.02541.1226

0.3926

0.61860.86621.13901.4374

1.67221.94142.2408

0.36420.50600.66460.83681.02701.1696

1.33681.5252

0.29480.39100.70500.84060.9920

1.09561.22021.3626

0.27560.35680.62800.74140.86700.94681.04501.1580

0.4039

0.60120.77920.97491.1851

1.40861.64581.8953

0.37770.50510.61570.73890.8719

1.01411.16551.3257

0.30860.39870.67910.76270.8597

0.96611.08011.2003

0.29050.36820.61180.67810.75610.84170.93281.0277

“ x lW = (me, + nc , ) / m + n ) ; m , n = number of p and s valenceelectrons, data from ref 1. bData from ref 41. cData from ref 44.

using the radial Coulom b and exchan ge integrals as fitting pa-rameters. E,, v alues fo r Br I1 ( 4 ~ ) ~ ( 4 p ) ~nd ( 4 ~ ) ( 4 p ) ~ere

obtained from those ofKr 111

and SrV

and checked by predictionof Rb IV. Th e assumption of linearity is only in error by 0.05%and is close to the limits of uncertainty in H ansen and P ersson’sE,, values. There are two unobserved multiplets in (4 ~ ) ( 4 p ) ~fS e I1 and th e same extrapolation procedure was used on the E,,values from Persson and Pettersson3* for Kr IV, R b V , and SrVI. For Te I1 ( 5 ~ ) ( 5 p ) ~o unambiguously assigned spectra existsand thus cSSwas extrapolated from the other 5th row one-electronenergies. This was not difficult because the e,, versus Z curvesare smooth an d t SS losely parallels cis.

Configuration Interaction Corrections. Instantaneous elec-tron-electron correlatio n effects must be addressed and thisp ro blem i s w ell t re at ed in th e bo ok by C ~ w a n . ~ ~e illustratet hi s for th e ( 4 ~ ) ( 4 p ) ~onfiguration of G e I1 using t he extendedconfiguration interaction study of Andrew, Cowan, and G iac-c he tt ie 40 Bo th ( 4 s ) h s an d ( 4 ~ ) ~ n donfigurations can mix with

( 4 ~ ) ( 4 p ) ~nd those invest igated were the (4 ~ ) ~ 4 d ,4 ~ ) ~ 5 s ,nd( 4 ~ ) ~ 5 d .its to the spectral lines from these configurations, usingtheir radial interaction integrals as disposable parameters, weremade and E,, determined. It was found tha t 4 s 2 5 s contributednegligibly and that a very accurate fit could be achieved with onlythe ( 4 ~ ) ~ 4 dnd ( 4 ~ ) ~ 5 donfigurations yielding E,, = 69 784 cm-’compared to E,, = 66 866 cm-’ that we obta in directly from th eNB S data without configuration interaction. Thus correlationcorrections change t4 from 1.1530 to 1.1796 Ry and x from0.8537 to 0.8670 Ry, a 1.5% increase. Because of the close analogybetween the spectra in the 4th a nd 5th rows we have computed

39)Cowan, R. D. heory of Atomic Structure and Spectra; University

(40) Andrew, K. .; Cowan, R. D.; Giacchetti, A. J. Opt. SOC.A m . 1967,of California Press, 1981.

57, 15.

0

X

N

3

C

H

2-

Be

LI-

O- i n d I

3rd

Figure 9 . x experimental values (solid lines), compared to Hartree-Fock xspc(Sgihed lines). Data from ref 41.

the corresponding correlation correction for Sn I1 ( 5 ~ ) ( 5 p ) ~asedon the E,, ratios found for Ge. Andrew, Cowan, and GiacchettiMchose to study the Ge I1 ( 4 ~ ) ( 4 p ) ~ase because of its rather largedeviation from LS coupling and the expected strong configurationmixing. In surveying configuration intera ction studies in the

literature we found none with greater deviations than th e G e case,therefore we have not mad e furth er correlation corrections forthe entrees in Table 111and believe them to be within the 1.5%error range found for Ge. In addition to the analysis of exper-imental spectra given here, direct quantum-mechanical calculationsof the atomic wave functions can also be employed to obtain cp

and tgand, a s described below, these provide a close check on th evalues listed in Table 111.

V. Computational Determination of xspeThe most obvious and straightforward way to obtain tp and e,

values is from canonical Hartree-F ock solutions. Resu lts fromthe Clem enti Roetti Hartree-Fock tabulations4’ are given asFigure 9 (and Table 111). Because of Koopmans’ theorem42 andthe immediate interpretability of the H a r t r e F o c k equations, onevery much hopes that using these t and 4 in our formula will yield

values close to experimental xsps. kortunately, Figure 9 does showtha t all of the significant features of the p attern a re preserved,but there remains a considerable spreading out of the computedvalues that is undesirable. It arises from two sources: (1) theClementi Roetti wave functions are determined for ground-statemultiplets and therefore no multiplet averaging has been carriedout, and (2) correlation corrections.

A second approach atte mpts to eliminate both of these sourcesof error but still retain the simplicity of interpretation and easeof compu tation embodied in the canonical Hartree-Fock method.Figure 10 (and Table 111) gives the results of u sing tp and ts valuesin our formula that were calculated by the H artree-Fock-Slater

41) lementi, E.; oetti, C. Atomic Data and Nuclear Data T ables 1974,

42)Koopmans, T.C. Physica 1939, 1 , 104.1 4 , 177; . Chem. Phys . 1974, 60 3342.




X

3 -

2-

I

J . A m . Chem. Soc.. Vol. 1 l l No. 25, 1989 9011

BORON

r -

OXYGENV( r r -

-- N g

Figure 10. x S w , experimental values (solid lines), compared to Har-tree-Fock-Slater xlpc (dashed lines). Data from ref 44.

method employing the Slater statistical exchange potential.43 T hedata.to construct this f igure were obtained from the book byHerm an Skillman who numerically compu ted these wavefunctions, their one-electron energies, and their effective potentialsfor all atom s in the Periodic Table.44 Coulo mb and exchangeeffective potentials were also averaged over angles thereby yieldingmultiplet-averaged solutions, and it is obvious from Figure 10 thatfor our purposes these approximations ar e very good indeed. Th eHartree-Fock-Slater scheme is computation ally simpler than thestraigh t Hartree-Fock method itself and it was originally intro-

d uc ed fo r th is p urp ose . S ub se qu en t r e s e a r ~ h ~ ~ . ~as showed thatthe statistical exchange approximation employed partially com-pensates antiparallel spin pair correlation effects, thereby pro-ducing good estimates for exp erimental o ne-electron energies-justthe ingredients we need for constructing xsw. Thus, the goodagreement between experimental xsw and xsw computed fromHerm an Skillman 's tabulation of the Hartree-Fock-Slaterone-electron energies provides an independent test of xsw values(Tabl e It is also the justification for use of Herman

(43) Slater, J . C. Phys. Reu. 1951, 81, 385.(44) Herman, F.; Skillman, S. tomic Structure Calculations; Prentice-

Hall: New York, 1963. The Hartree-Fock-Slater scheme employed in thesecalculations is only defined for two or more electrons an d therefore no valuefor the hydrogen atom is obtained.

(45) Slater, J. C. Phys. Reu. 1968,165,658. The remarkable agreement

between experiment and Hartree-Fo ck-Slater one-electron energies for atomsthroughout the Periodic Ta ble was already known to Herm an and Skillmanand is impressively demonstrated by Figures 5 and 6 in their book.u

(46) McNaughton, D. J.; Smith, V.H. Int . J . Quuntum Chem. 1970, IIIS775.

(47) We have ignored relativistic effects because none of the calculationswe have used for comparison, ref 7,8,41, and 44, have employed relativisticwave functions. However, we can estimate the relativistic mass-velocity andDarwin corrections for representative atoms by perturbation theory using themethods and numerical results given by Herma n and Sk illmanu (energies inRydbergs):

X*ps 0 S sc Te Po

nonrelativistic 1.4086 1.0141 0.9661 0.8417 0.8002relativistic 1.4118 1.0230 1.0084 0.9313 1.0341percent error 0.22 0.86 4.19 9.63 22.6

Thus t he relativistic correction is less than 10% for the atoms we have con-sidered, but it is quite large for the sixth row.

X - Large3.61){-Lorge

Figure 11. Schematic of effective radial potentials showing xsw andforce = - A V / A r for boron and oxygen. The diagra m illustrates the

qualitative correlation between xspcand the force at a n atomic radius.

50

2 0 2 4 2 8 3 2 3 6 4 6 4 4/ c ' - , 8 0

x for P rw

2

4 0

- 6 0

- 4 0

- 2 0

I

- I I

I 6 18 2 0 22 24 26 28 30

Figure 12. Relationship between xrw (Pauling units) and force (-A V / A r , Ry/A) at atomic radi i equal to the outer radial maxima of thevalence orbitals (A). Data from ref 44.

Skillman 's tabulated effective potentials to determin e the rela-tionship between xSp and force described below.

VI. Relationship between xSF and Force at an AtomicRadius

As noted previou sly, the p ractical utility, ease of obtaining valuesfor all atoms, and the heuristic app eal of a force definition havemad e the Allred Rochow scale7 an important reference pointin all discussions of electronegativity. Because of the intrin sicproperties of funnel-shaped effective potentials it is easy to makea qu alitative connection between xsP and force = - A V / A r , asshown in the Figure 11 schematic. For typical atoms, B with amedium value of xsw,and 0 with a large value of xspec,t is clear



9012 J . Am . Ch em. SOC.,Vol. 1 1 1 , No. 25, 1989 Allen

Table IV. Relationship between xsga and Force at the Outer RadialMaximum

atom rM b AVIA F X.-d

LiBeBCN0F

N eN a

M gA1SiPSCIAr

KCaG aG eAsS eBr

KrR bS rIn

SnS bT eIX e

1.6311.0090.7620.6340.5240.4640 .410

0.3541.7321.2891.2551.1000.8930.8140.7400 .669

2.1741.7501.3211.1191.0140.91 1

0.857

0.8032.3261.9481.4621.2901.2001.112

1.0251.019

0.2121.0221.9764.5488.111

11.37317.423

25.5360.1880.61 1

0.8451.4532.8053.8895.3197.489

0.1140.3710.8471.5402.2423.0394.147

5.2800.1040.3140.7641.2441.5972.269

3.0943.283

1.0471.5582.0202.5273.0723.6514.266

4.9120.9791.3091.5961.9152.2602.6283.0213.436

0 800

1.0331.7601.9772.2282.5042.799

3.1110.7530.9541.5861.7581.9602.1822.4182.664

All data from Herman and Ski l lman, Atomic Structure Culcula-rions, ref 44. 'rM in A . < A V / A r in Ry/A. dPaul ing units. xsPesv-erage of Ge and As matched to combined average of Allred and Ro-chow and Pauling Ge and As, thus absolute values in Rydbergs aremultiplied by 2.59184.

that the slope, A V / A r , a t a n r value near to where xs w crosses

the potential curve has a medium magnitude for B and a largemagnitude for 0. Beyond this qualitative relation we mu st bemore specific as to the definition of the radius. Allred Rochowuse the Pauling covalent radii which are based on self-consistentdivision of experimentally determined internuclear separationsfor many covalently bound molecules and solids. For the qua n-tum-mechanical wave functions and effective potentials from whichwe compute AV/Ar , the most physically reasonable and generallyemployed radius is the outermost radial maximum of the atomicorbitals. Th e crossing point between xsw and t he effective po-tential schematics of Figure l l , however, represents th e classicalturning point and corresponds to very high quantum numbers.Our ground-state orbitals therefore have outer radial maximumsignificantly sm aller than these crossing points.

The graphs relating force and xsw, shown in Figure 12 forp-block elements were calculated numerically by interpolating anddifferencing in the Herm an Skillman atomic wave function andeffective potential tables described above (values given in TableIV) . Th e interesting a nd significant featu re of these graphs isthe almost linear relationship between force and xsFc or eachp-block row and that the lines for the 3rd, 4th, and 5th rows areclose together.49 These numerical facts plus the qualitativerelationship expressed by the schematic of Figure 11 enrich theconceptual and intuitive chemical utility of el ec tr on eg at i~ it y. ~~

(48) Pauling, L. J . Am. Chem. Soc. 1947.69, 542.(49) The lines for the s-block elements of groups I and I1 show the same

relationship to one another as those of the pblo ck (Figure 12). but the s-blocklines have a smaller slope than those of the pb loc k. Therefor e there is nocontinuous single linear relationship between xsps nd any row.

It should be noted, however, that there is no basic quantum-mechanical theorem tha t requires a line ar relationship betweenaverage one-electron energies an d forces at o uter radial maximafor a given row in a given block and none implying that a singleformula, like tha t of Allred Rochow, should be able to fit allof the representative ele men ts5'

VII. Bond Polarity Indexm

A paper of this title is to be subm itted soon,28 n d we give hereonly the resulting qu antum-m echanic al formulae and definitions

that bear on Axsw and a single specific example. We employa single determinant molecular orbital wave function (it is readilygeneralized to a configuration interaction wave function) andassume an L CA O expansion, qi = x C & j 4 & ,nd that +i satisfiesthe Roothaan equations in canonical form. Th e index i designatesthe N MOs, ni their occupancy, and ci their one-electron energies.The indices j , k designate the M AOs in the basis set. W e areinterested in the average energy of atom A, with M(A) AOs, ina molecule or solid with M AOs The projection of 4i on MOi is . fC j i4 j+ i 7 and the energy of this A 0 n the molecule is thesum over all MOs of niqJCji4j+i 7. If this quantity, in turn, issummed over the AOs of atom A we get the in situ energy of atomA. Th e average energy (per electron) of this atom , called theenergy index, EIA, is then given by the formula

where SI s the atomic orbital ov erlap integral, S+j+ d ~ . ~ *heBond Polarity Index represents in situ Axsp and the covalentbonding component must be subtracted out, thus

where EIArefs EIA for the corresponding homonuclear bond AAand sub tracts out its pure covalent energy component. For ex-ample, for the C-F bond in H$-F we compute EIc and EIF, butalso EIcRf n H3C-CH3and EIFd in F-F. With use of the 6-31Gbasis set and ato mic units, EIc = -0.743, EIf = -0.881, EIcre'=-0.665, EIFrrf -0.983, and B PIcF = -0.180 auaZ8W e can convertthese into a Fractional Polarity by dividing BPI,, by th e energ yrequired to ta ke an electron from A to B, and this number is theenergy equivalent of Pauling's dipole moment de termined percent

ionic character.l0 For the H3C-F example ( r C F= 1.383 A) thefractional polarity is 47%.

(50) This qualitative association is enhanced by the work of Boyd andMarkus ( J . Chem. Phys. (1981) 77, 5385), who employed a special set oflarge-valued radii (e.g., C, 1.54 A; N, 1.36; 0, 1.27) and assumed XA

zA/rAz(l - .fv ( r ) dr) where the radial charge density D(r) is obtained fromthe Clementi and Roetti Hartree-Fock atomic orbitals. Their x A show thesame general pattern as the Mulliken definition values of Figure 6 (but groupsI1 and 111 do not overlap), and p erhaps this is not surprising because as r-m the charge density is known to fall off as exp(1,r). Unfortun ately, use ofsmaller values of ?A, such as those used by Allred and Rochow or the outerradial maxima, do not lead to useful x (R. J. Boyd, private communication).This appears to be he result of assuming the force to be inversely proportionalto a radius squared thus imposing the concept of a classical radius onto aquantum mechanical charge distribution.

(51) The fact that the Allred and Rochow formula fits the 2nd row valueswithout re-referencing as would be suggested by the considerably differe nt

slope and range of A V / A r versus xcpc or this row compared to the 3rd, 4th,and 5 th appears to result from a fortuitous choice of covalent radii. Allredand Rochow chose the larger of the two sets of radii offered by Pauling in his1947 tabulation while the radial maxima parallel the smaller set and, in fact,are significantly smaller still.

(52) EIAgives insight into the atomic binding process and the dependenceof electronegativity on oxidation state. When on e representative element bindsto anothe r the average valence shell energy of the atom with lower electro-negativity will move to lower energy and vice versa, i.e. the transfer of chargeduring binding leads to a c ompacting of atomic energ y levels but does not alterthe identity of the atoms. For example, free atom E IA a t 6-31G' comparedto EIA n (in parenthesis, atomic units) are as follows: F, -0.953 (FH,

-0.661); B, -0.426 (BH3, -0.568); Be, -0.302 (BeH2, -0.465); Li, -0.196(LiH, -0.300). The energy level compacting is in the direction expected andthe change in oxidation states does not change the ordering of the electro-negativities thereby rationalizing the observation of representative elementchemists tha t dependence of electronegativity on oxidation state is generallynot required.

-0.892);0 , 0.823 (OHz, -0.772); N , -0.720 (NHI, -0,694); C, -0.567 (CH,,




Group or Substituent Electronegativity. This is defined as thedifference in electronegativity across a bond A B where A is areference atom (or molecular fr agment) and B is the group ofinterest (e.g., -CH3, -CHO, -CCH, -CHCH2, -CH2CH3,X H ,O H , -C 02 H , NH2, -OCH3 , -SiH3, etc.). There are severalexcellent, up-to-date reviews of this important physical organicchemistry t o p i ~ ~ ~ v ~ ~nd it is apparent that there is considerablyless agreement on values for the 25-35 most studied groups thanthere is for the approximately equal number of individual atoms(Allred Rochow and Pauling scales). Calculation of group

electronegativities is a direct application of the Bond PolarityIndex, BPIAB, and results to-da te app ear prom ising.28 A principalpoint for our purpose here is the possibility of making closeconnection between th e success achieved by xspccn reproducingthe Allred Rochow and Pauling scales and the expectation thatits in situ tran slation into BPIAB can similarly provide t he basisfor acc urate an d well-defined values for this physically an d syn-thetically useful quantity .

Electronegativity Perturbations on M olecular Orbital EnergyLevel Diagrams. Much of the conceptual understanding of organicand inorganic chemistry during the last quarter century has beenachieved through the use of qualitative and semiquantitativemolecular orbital theory, and the effect of electronegativityperturbations on molecular orbital energy levels has been aprominent theme.5s5q Are the assumptions concerning the natureand definition of electronegativity throughout this development

compatible with those we have put forward here? The formulagiven for EIA above demo nstrates an aff irmativ e answer an d wealso show that the new definition can lead to an extension andrefinement of the existing qualitative and semiquantitativetreatments.

In the majority of applications a single type of A 0 is used(typically a 2pn ) and a n increase in electronegativity of a specifiedatom has been modeled by lowering the atomic Coulomb integralfor that atom. Changes in the MOs are then determined by first-and second-order perturbation theory.55 With only one type ofAO, average and individual orbital energies are identical, and whenthe sam e approximations with respect to overlaps are m ade in theEIA expression and in the perturbation theory, the resultantelectronegativity perturbations of the MOs will be the same andthus within the conceptual framework put forth in this paper.

Starting with single A 0 homonuclear reference systems, Albright,Burdett, and Whangbo give three illuminating examples55 T

orbitals for A, - B and benzene - yridine, H3- H’H, andHHH’) and show how electronegativity perturbation results fromthese simple, special cases have wide ramifications in the quali-tative understanding of subs tituent effects in organic structureand reactivity. In the examples noted here the actual magnitudeof the electronegativity perturbation is left unassigned as onlyqualitative trends were desired, but if needed, its magnitude couldbe determined directly by the EIA formula.

For m any representative element inorganic systems (e.g., AB4,AB5, AB6) Gim arc has constructed qualitative molecular orbita lenergy level diagrams and invoked electronegativity perturbationsto successfully rationalize trends in their geometrical and spec-troscopic proper tie^.^^ Both s and p A Os ar e required to construct

(53) Prog. Phys. Org. Chem.; Taft, R. W., Ed.; Wiley-Interscience: NewYork, 1987; Vol. 16. See particularly: Taft, R. W.; Topsom, R. D. TheNature and Analysis of Substituent Electronic Effects and Topsom, R. D.Some Theoretical Studies of Electronic Substituent Effects in OrganicChemistry.

(54) Isaacs, N . S.Physical Organic Chemistry; Longmans ScienceTechnology, 1987.

5 5 ) Albright, T. A.; Burdett, J. K.; Whangbo, M.-H. Orbital Interactionrin Chemistry; Wiley-Interscience: New York , 1985; Cha pter 6, pp 80-86.

(56) Gimarc, B. M. Molecular Structur e and Bonding, T he QualitativeMolecular Orbital Approach; Academic Press: New York, 1979; Chapte rs1, 3, 4, 5 , and 7. For the AB4 case see pp 62 and 63 and: Gimarc, B. M.;Khan, S. A. J . Am. Ch em. S OC. 978, 100 2340.

(57) Fleming, I. Frontier Orbitals and Organic Chemical Reactions;Wiley-Interscience: New York, 197 6; Chapters 2, 3, and 4.

(58) Jorgensen, W. L.; Salem, L. The Organic Chemist’s Book of Orbitals;Academic Press: New York, 1973; Chap ter 1.20.

(59) Streitwieser, A., Jr. Molecular Orbital T heory or Organic Chemists;Wiley: New York, 1961; Chap ter 5 .

J . Am . Chem. SOC. ,Vol. 1 1 1 , No. 25, 1989 9013

suitable MOs for these molecules and a useful example is tetra-hedral AB4 subject to a perturbation that significantly lowers theelectronegativity of A (e.g., CI, to I4) . When A lowers itselectronegativity two chang es occur in its orbitals: their centerof gravity moves to higher energy and the s-p separa tion getssmaller. In Td AB4 for 8 valence electrons the LUMO andLU M O+ l levels are 2a,, 2 t, and 2 tl, 2al , respectively,s6 thereversal largely brought ab ou t by the reduction in s-p separation .When molecular orbitals ar e constructed directly from the freeatom AO s their relationship to electronegativity is just x s but

when MOs are to be made out of the orbitals of moEularfragments a nd an a tomic or grou p electronegativity perturbationthen invoked, the most straightforward approach is to constructthe M Os for the systems with and without perturbation and thencalculate B PIA B o determin e, a posteriori, the electronegativitydifference caused by the perturbation.

Electronegativityof Hybridized Orbitals. In part I1 of this paperwe have noted that the Mulliken-derived concept of hybrid orbitalelectronegativity suffers from a general inability to make a priorihybridization assignments, yet for certain cases the ordering ofelectronegativity magnitudes according to hybridization is socommon in organic chemistrym that it should be reflected in xSwand EIA. The most important example is the orbital orderingrule (highest to lowest electronegativity) sp > sp2 > sp3. Thequalitative tru th of this ordering is as appare nt in free atom xspfor in EIA as it is in any of the electronegativity definitions that

have been specifically formula ted in terms of hybrid orbitals: thegreater the percent s character the larger the value of xspssimplybecause s electrons have a lower energy than p electrons.61

Electronegativity and A tomic C harge. The definition of xSpas a free atom property precludes any direct relationship with in

situ atomic charge just as the Periodic Table is unable to ch ar-acterize details of the bonding between atoms. On the other hand,chemists have always assumed th e existence of a strong correlationbetween x and in situ atomic charg e. Th is belief is fully justifiedby computational results using the in situ index BPIAB.%However,in making the connection with BPIAB, the long-standing andincompletely resolved problems with quantum-mechanical chargedefinitions themselves need to be addressed.62 Thus, it is sig-nificant that in almost all chemical applications it is bondpolarity,not atomic charge, that is the property of ultimate concern, and

it may turn out that atomic charge will be much less needed inthe fu ture . In addition , since BPIAB is an energy , it follows fro mperturbation theory that it is determined to higher order than thecharge, and therefore less sensitive to basis set changes than mostquantum-mechanical charge definition^.^^

VIII. Experimental Measurement of xAEveryone recognizes that even thougrthe properties of free

atom s are governed by the laws of physics and therefore directlyobtainable from physical measurements, their particular arr ayin rows and columns which constitutes the Periodic Ta ble is not

and EIA

(60) Lowry, T. H.; Richardson, K. S . Mechanism and Theory in OrganicChemistry, 3rd ed.; Har per Row: New York, 1987; Carey, F. A. OrganicChemistry; McGraw-Hill: New York, 1987; Ternay, A. L., Jr. ContemporaryOrganic Chemistry, 2nd ed.; W. B. Saunders, 1979; March, J. AdvancedOrganic Chemistry; McGraw-Hill: New York, 1977.

(61) The set of molecules usually given to illustrate this ordering rule is

H W H , HzC=CHz,and H3C-CH3,and it is certainly true that the acidityof the u orbita l bound hydrogens follows the order given. But it is notnecessary that an in situ electronegativity for the carbon ato m itself followsthis same order because of the difference in multiple bonding for the threecases (in fact, EIc for these molecules shows the opposite order).

(62) A quite different way of relating atomic charge to electronegativity,and one that is as compatible as possible with f ree atom electronegativities,is the Lewis-Langmuir definition applicable to Lewis Dot structures. Foratom A in bond AB this scheme uses the ratio xA/(xA + x e ) to interpolatebetween formal charge (covalent extreme) and oxidation number (ionic ex-treme) (Allen, L. C. J . Am. Ch em. S OC. ,n press).

(63) Since BPIAB epresents the difference in energy between an averageelectron in atom A and in atom B, why do all the electrons not transfer to themost electronegative atom? (1) BPI AB s a one-electron energy difference and,as is well-known in Hartree-Fock theor y, the sum of the one-electron energiesis on the order of one-half the total energy, the rest being electron-electronrepulsion (Coulo mb and exchange terms). (2) Th e Pauli principle restrictspenetration of one atom by another just as it accounts for the impen etrabilityof matter in general.



9014 J . Am . Chem. SOC. ,Vol. 1 1 1 . No. 25, 1989

measurable, Le., the Periodic Table is a chemical pattern recog-nition scheme that belongs to the set of concepts immediately abovephysics in the complexity hierarchy of science, often termedbroken s m m e t r ~ . ~ ~ , ~ ~hus th e corresponding arra y of x t

Periodic Fable gives trends for the chemical property, bond po-larity, a quantity not obtainable from direct physical measurement,even though values for individual elements (and for Ax canbe determined to high accuracy from atomic spectroscopy. Th eusefulness of this categorization is that it removes the frequent

misconception of electronegativity as an inherently fuzzyqu an tit y of somewh at question able significanc e. EIA and BPI,,,chemical indices appropriate for specific molecular a nd solid-stateenvironments, are similarly outside the domain of direct physicalmeasurementa2*

IX. Summary

1, Electronegativity, xs w, is the average one-electron energyof valence shell electrons in ground -state free atoms and may beidentified as the third or energy dimension of the Periodic Table.This definition, xs, = (m ep+ n c , ) / ( m+ n ,where m , ep, n, andc s are the number and ionization energies of the p and s electronsof the representative elements, leads to precision values obtainablefrom high-resolution atomic spectroscopy.

2. Th e electrone gativity of transition-m etal elem ents is likewisethe av erage valence shell energy, (med+ n e , ) / ( m+ n , but it is

often difficult to assign m , the number of d electrons, and thisis reflected in the infrequent usage of this quantity among tran -sition-metal chemists. A simple bonding assumption for the firsttransition series permits approximate estimates.

3. The variation in electronegativity (and increase in groupsI-V metallizatio n) down a column in the Periodic Table and itsincrease across a row, the existence of the diagonal metalloid bandseparating metals from nonmetals, and the chemistry of the noblegas molecules can be deduced from the xsFcdefinition.

4. Th e progressive changes in bonding from covalent to ionic,from the alkali metals through the metal/non-metal transitionto covalent molecules, an d from metallic to ionic bonds are allcharacterizable by changes in x,, and Axs, xfw - xfFC crossbond AB. Many of the changes from one bond type to anotherare separated by regions where polymeric materials predominateand xsFcdelineates the boundaries of these regions.

5. Th e principal properties of other types of bonds, e.g. hy-drogen bonds, A-H.-B, an d Alcock 's seco ndar y bond s, A-Y-B,(which occur widely in ino rganic solids for x , ~ x : ~ ) a r e a lsocharacterizable by xs and Axspec.

6. An analysis of tr e literatur e on chemical usage of electro-negativity since Pauling put fo rth his scale in 19 32 shows tha tonly his and the force definition scale of Allred Rochow havebeen repeatedly employed by practicing chemists, therefore thefinding that xsw reproduces the pattern of these two establishedscales is of first importance. Moreover, the close mat ch achievedenables xspeco referee differences between them , thus showingthat xN> xa and resolving the troublesome discrepancies in 5throw values. Th e tight fit realized between xs, and a very recentreport of p-block electronegativities based on computed electronicchar ge distributions also support its validity. In addition to these

comparisons for representative elements, estimates for firsttransition series x s w parallel those of Allred's Pauling scale ex-perimental values.

(and Ax, ) that we have identif ied as the third dimension of t g

(64) Anderson, P. W. Science 1972, 177, 393.( 6 5 ) Primas, H. hemistry, Quantum Mechanics and Reductionism;

Springer-Verlag: New York, 983.

Allen

7 . Hartree-Fock atomic calculations of x are found toreproduce the principal features of the m ul ti pE averaged ex-perimenta l values. Th e spherically averaged and antiparallelspin-correlated xSp values obtained from Hartree-Fock-Slatercalculations prove to be even closer to the experim ental num bers,thereby acting as an internal check for representative element xspCcand providing useful data for the transition-metal estimates.

8. The chemical bond characterizing capabilities summarizedabove for free atom, ground sta te xspsshow it plus the usual twodimensional Periodic Table capable of organizing overall trends

in chemical structure and reactivity, but do not themselves providenumbers for specific bonds in specific molecules or solids. To getspecific numbe rs for particular molecules or solids it is necessaryto set up the electronic wave function s for the molecule or solidin question and calculate expectation values corresponding to theaverage energy of an atom A or the difference in energy betweenan average electron in atom A and an average electron in anadjacent at om B. Formula e for these in situ atom A and bondAB expectation values have been derived and called the EnergyIndex, EIA, and th e Bond Polarity Index, BPIM, respectively (theyare treated more fully in a forthcoming article).

9. EIA and BPI,, are chemical indices tha t quantify theelectronegativity concepts that have been used in qualitative andsemiquantitative molecular orbital theory during th e last quart ercentury. T hree areas 0: applications are particularly significant:(a) group (substituent) electronegativity-determination of thisimportant and long-debated physical organic chemistry parametercan be carried out as a direct application of BPIAB with A asreference atom or fragment and B as the group of interest; (b)electronegativity perturbations of molecular orbital energy leveldiagrams-A change in electroneg ativity of an atom A in the bondAB from xA o xK implies a change in the average energy 6f theone-electron levels of A to that of A' and is measured by thechange in bond polarity BPI,, - PIA,,; ( c) electro neg ativit yof hybrid orbitals. In most cases of interest to orga nic chemists,e.g., xsp> x s 2 x s , the ordering rules can be deduced imm e-diately from th e definition of free atom xsp without elaborateformulation or calculation . In general, th e electronegativity ofspecific hybrids in specific molecular environments can be com-puted by BPI,,.

Note Added in Proof. We inadvertently omitted helium in ourfigures and tables. xsp for He from the NBS Tables' is 1.8074Rydbergsfor F is also -4.2 and N e at -4.9 is the highest value in t gPeriodic Table. These values predict that HeF, will not be boundand this conclusion is verified by results from extensively correlateda b initio valence bond wave fun ctions (Allen, L. C.; Erdahl, R.M.; Whitten, J. L. J . Am . Ch em . S OC. 965,87, 3769. Allen, L.C.; Lesk, A. M.; Erdahl, R . M . Zbid. 1966 88, 615). The latterreference also includes the repulsive potential energy curve forthe linear symm etric stretch of Ne F2 and this curve rises muchmore steeply than that for He F2 n accordance with the size andxsFcof Ne compared to He. I t may also be recalled that thetextbook repulsive interaction ex ample of He H show s a relativelyflat repulsive curve both theoretically a nd experimentally (Taylor,H . S ;Harris, F. E. Mol. Phys. 1963, 7 , 287) .

Acknowledgment. The autho r thanks his graduate students,Eugene T. Knight and Congxin Liang, for the many stim ulatingdiscussions, suggestions, and critical comments that have greatlyaided this research. H e also acknowledges the num erous usefulsuggestio ns given by Professor Joel F. Liebman and an anonymousreferee.

4.2 in Pauling units (scale factor in Table I ) . x s

Electronegatividad Leland C. Allen

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