Electronegativity and Atomic Charge James L. Reed Clark Atlanta University, Atlanta, GA30314
The electron distribution is fundamental in determining chemical and physical properties of substances. In a very qualitative fashion electronegativity is used throughout the chemistry curriculum a s a n indicator of atomic cbarge--and thus electron distribution. Because electronegativity is a far more fundamental concept than previously thought, it should be continually developed in sophistication throughout the curriculum. However, after electronegativity is introduced in the first semester of the first course (131 little is done to develop the concept further(4-10). For example, electron distributions and bonding are covered in most physical chemistry texts, yet electronegativity,which is essential to these wncepts, is all but neglected. The Value of the Quantitative Approach
Unfortunately, the student's understanding of electronegativity remains very qualitative, and atomic charge discussions are subject to a great deal of hand-waving. Of course detailed charge distributions are available from molecular orbital calculations, but such computations are beyond the capabilities of beginning students. As a result, the use of reasonably detailed atomic charges becomes a matter of exposition in which the instructor simply presents representative atomic charges and then correlates these with properties. I t is far more exciting and effective to make the process one of exploration, rather than exposition In this article, we suggest a methodology by which a student may explore the influence of atomic charges on chemical behavior by determining, in a rather simple manner, very reasonable atomic charges. This discussion begins with a n abbreviated description of the development of the electronegativity function. The full develo~mentmav be found elsewhere (I1). This discuslevel appropriate for the advanced students. sion is at ; The later discussions on atomic charee and reactivitv, however, are very appropriate for the first year and in the intermediate levels.
charge of an atom in a molecule. The method has its basis in basic LCAO-MO theory. The only input required is the Mulliken electronegativities (18-20) and the simple valence bond structure of the molecule or ion. Whereas many other schemes have been proposed for computing atomic charges (21-251, our method uses few new parameters and wnstants, and it is firmly rooted in basic molecular orbital theory. The equations are simple in form, and they can be interpreted in terms of common chemical and physical models. Thus, they are amenable for use in instruction and other applications. The Energy Function
The concept of electronegativity was originally proposed by Pauling (26). Since his orginal proposal, it has been formulated in various ways (18-20,2732). The most extensively used and explored formulation has been that of Iczkowski and Margrave who formulated the electronegativity X , as the potential. X
dE,
-,-ds,
(1)
where q, is the charge on the atbm m, and E is its energy. To a very good approximation, the energy of a n atom is found empirically to be a quadratic function of its charge. 1 Em = amp, + -b q2 + C, 2 "Lm
(2)
and ~n =am+ bmqm
(3)
where a, and b, are the empirical Mulliken constants for a particular valence state of a gaseous atom. They are determined semiempirically h m spectroscopic data. Klopman has given these empirical constants in terms of the standard atomic integrals (3337).
UnderstandingReactivity
In general, the reactivity of a molecule is controlled in one of two ways, and there are two types of reactions that result from this control. These are: frontier-controlled reactions, which are controlled by the outermost electron density charge-controlledreactions, which are controlled hy the net electron density about each atom Frontier-controlled reactions receive a great deal of attention in three areas of study: contributing resonance forms (12); Huckel n molecular orbital theory (13,141; and frontier orbital theory (15,161. When relatively large charge separations occur, a knowledge of atomic or partial charges is commonly useful.
where and are the valence orbitals of the atom. If these are given in terms of the integrals characteristic of the atom m (111, we get -a, = I ,
+ b,NK
and
A Convenient Way to Compute Atomic Charges
In recent articles from our laboratories, the concept of electronegativity bas been examined in the context of the LCAO-MO approximation (11,171. Among the several very useful results was a simple method for computing the
where Q, is the number of valence orbitals on neutral m, andNO, is the number of electrons. Volume 69 Number 10 October 1992
785
In the LCAO-MO approximation, the molecular orbitals yrk are given by
where the summation is over all the atomic orbitals in the basis set, and ckiare constants. Substituting this into the Schrodinger equation yields the following standard expression.
where H,, is derived from the resonance integrals, and S, is derived from the overlap integrals. Thus, after some rather straightforward but tedious manipulations, we arrive at eq 8. It is an expression for the molecular electronic energy in terms of atomic populations and the Mulliken electronegativity constants. Electronegativity and Electronegativity Equalization Differentiation of eq 8 with respect to any arbitrary atomic population nz yields
Because Huckel, CNDO, and similar approximations have no overlap populations, we get the following equation.
where 6 = 6~(i!A(~
where A(i) is the atom about which orbital i is centered. Thus, after some manipulation, we get (11)
The term
from eq 13 is equal to the negative of the atomic electronegativity. Partitioning the Overlap to Compute the Electronegativity where the n,'s are the atomic populations that derive from the constants expressed by chi.The summations are over all the atoms. In addition to the atomic terms, several interatomic terms now appear. The nonbonded interatomic interactions are grouped together in HES.Also, the integral P,. has been introduced in which
and A A
N A O A O
In approximate theories that deal with overlap populations, the overlap population must be partitioned among the atoms. The Mulliken analysis, which is used almost universally, partitions the overlap density equally among the participating atoms. Clearly, this does not reflect what actually occurs. We have proposed instead to partition the overlap density inversely as the electron-electron interaction energy, which is given by the Mulliken b, constant (11).Thus, to an excellent degree of approximation the electronegativity is given by the following equation.
MOAOAO
CCpmn=CCCCCpkdno=CCCpk@ where the i and j atomic orbitals are given in terms of orbitals 1 and o on atoms m and n, respectively. A precise definition is not given for the P,, term, but in the simplest cases it describes the occupation of a localized chemical bond. The r, term is defined below.
in which
The similarity to eq 3 is obvious. Because the Mulliken function gives the electronegativity-or, more correctly, the chemical potential (39, 40)'--of an isolated gaseous atom, eq 15 is for an atom in a molecule. Thus, it is faithful to Pauling's original definition. As earlier, a, is the potential that attracts electron density to the nucleus, and b, is the magnitude of the electron-electron repulsion, which has also been called the hardness (37).The r, term, which is also a potential, is composed of r,. terms. They give the attractions of electron density toward the interatomic regions, and thus away from the nuclei. Minimizing the Energy to Compute the Net Charge
N O 0
and 'There is still wntroversy over which of these is actually the electronegativity and which is the chemical potenial (38,39). 786
Journal of Chemical Education
In an earlier report, Reed (11)showed that equalization of the Mulliken electronegativities fails to minimize the electronic energy of a molecule, as required by the Sanderson Principle (40).Equation 13, however, shows that according to the Sanderson Principle, equalization of the modified atomic electronegativities minimizes the electronic energy. Thus, for any pair of atoms m and n,
Table 1. Atomic Charges for the Bicarbonate Ion and a Sample Calculation for the Hydroxyl Oxygen
Then solving for q , and summing q. over all atoms, we get the net charge 2.
Molecule
am
bm Hybridization
lrm 2
9m
Solving for qmyields
where X* is a global electronegativity characteristic of all the atoms in the molecule or ion.
where these molecular constants, introduced earlier by Reed (17).are given below. Atomic electronegativities were taken from ref 19.
The m'olecular constants a*, r*, and b* have the same meaning as the corresponding atomic constants, except that they are characteristic global values for the molecule. Thus, we have a simple equation for the determination of the atomic charge of an atom in a molecule: eq 17. Atomic Charges Many schemes have been proposed for computing atomic charges, but few of these have been adopted for instruction (22-25) for many reasons. For the first time a n approximate electronegativity function (eq 15)has been proposed that arises naturally from the LCAO-MO expressions. This meets Pauling's criterion of a n atom in a molecule, and it can be used for instruction in electronegativity. It also permits facile computation of atomic charges. These computations may be further simplified if the WolfbergHelmholz (42)' approximation is made for the resonance terms.
Examples
A sample calculation for the bicarbonate ion is given in Table 1. The values for the Mulliken electronegativities are taken from the tabulations of Hinze and Jaffe (19-20).This Darticular example is informative for several reasons. hi bicarbonate ion contains three different types of oxygen of which one carries a formal charge of -1. Although this formal charge is not explicitly included in the computation, its affect is clearly evident in the resulting oxygen atomic charges. The effect of formal charge is even more evident for azide ion.
Atomic Charge and Reactivity (21)
Thus, the Mulliken electronegativity constants are the only data required for the computations. For each atom the valence state is taken to be that in which all valence orbitals are equivalent. The number of valence orbitals is equal to the number of electron pairs (o,z, and nonbonding) about the atom. The r, term is evaluated using eq 21 to evaluate rm.for each bond (oand n) to atom m. 'For simplicity a single value of k has been used for eq 21. Improved results are realized when different values are used for different bond orders, which is the usual practice.
It would be useful to repeat two fundamental principles that help to interpret the influence of atomic charge on chemical reactivity. The atomic charges in a ground-state molecule represent the optimal electron distribution. Thus, any transfer of charge among the atoms is, by itself, an endothermic pmcess, usually of considerable enera. The transfer of charge causes an approximately quadratic increase in energy. Thus, less energy is required to kansfer several small increments of charge, than to transfer the same amount as a single increment. Volume 69 Number 10 October 1992
787
Table 2. Various Hydrides with the Computed Hydrogen Atomic Charges
Species
9H
Species
Table 3. Various Hydroxyl-ContainingSpecies and the Computed Hydrogen and Hydroxyl Atomic Charges 4H
Species
LiH
4.234
NH3
0.112
NaOH
BeHz
4.085
Hz0
0.263
Mg(OH)z
HF
0.636
AI(OHh
CHn
0.012
Species
Si(0H)q Ionization is the most directly observed charge-transfer process. Consider the energies for the following ionizations, which are formally oxidations of nitrogen.
OP(OH)3 OzS(0H)z H30+
Molecules with a hydrogen charge less than 0.263 will he undissaciated in aqueous media. Moleruler wth a hydrogen charge between 0.263 and 0.426 *,ill hr partially disaocmted. 'hlolrrulrs wth a hvdroaen chareerrester thm 0426 will he - completely dissociated.Parallel arguments can be made for the dissociations that form hydroxyl ion. Although formally, the molecular charge resides on the nitrogen, i t is really distributed over all atoms in small increments. Thus, the ionization energy is much lower for amines than for nitrogen itself. Furthermore, even though the methyl group is more electronegative than hydrogen (171, the ionization energy of methylamine is lower than that of ammonia because the charge is distributed over a larger number of atoms. This is the principle behind the inductive stabilization of charge. This series and the examples to follow are meant for illustration only. The strength of eqs 15 and 18 lies not in the exposition, but in the ease with which students can use them to make similar analyses, hopefully arriving a t similar conclusions. Binary Hydrides A number of representative binary hydrides may be found in Table 2 with their hydrogep atomic charges. The transition in properties, from basic to acidic hydrides across a period, is evident from the examples. The group 1 and 17 hydrides carry the largest charges. This is consistent with the chemistry of these substances, being characteristic of the hydride ion and of the proton, even though i t is evident from the charges that none of these substances contains ions. Because these hydrides already carry large charges, less energy is required to form ions or ionlike intermediates. By the same reasoning the hydrogens of the hydrides in the middle of the period are nearly electroneutral. For such compounds, considerable energy is required for any ion-producing process. Thus, these compounds participate in processes producing uncharged radicals. Aqueous Chemistry The observations made thus far are very straightforward and can be readily deduced from qualitative considerations of electronegativity. the hydrogens in the hydroniumion carry a 0.426 charge. I n water, on the other hand, the hydrogen carries only a 0.263 charge, and about two molecules in a billion are dissociated. Thus, in a semiquantitative sense, one might expect the following to be true. 788
Journal of Chemical Education
Third-PeriodHydroxides
The series of third-period hydroxides may be found in Table 3 along with their hydroxyl ( q +~qo) and hydrogen charges. The M-O-H functionality is common to all of the examples. In contrast to trends in organic chemistry, similar structures among these hydroxides may lead to vastly different chemistries. I n sodium hydroxide the hydrogen is nearly electroneutral, and thus does not have Bmnsted-acid properties. The hydroxyl group, on the other hand, carries a very large negative charge, -0.656, compared to the 4 . 2 6 3 for hydroxyl in water. We would expect it to be completely dissociated in water, and it is. Consistent with its atomic charges, the aluminum hydroxide behaves a s a weak B r ~ n s t e dacid and weak Arrhenius base. The hyroxide of the next element is called orthosilic acid with a hydrogen charge somewhat greater than 0.263. I t is significantly dissociated in water, as expected. The remainine comoounds have electroneutral or oositive hydroxyl groups i d do not exhibit Arrhenius base behavior. For these. the hvdropen charpes are cornoarable or greater than those in the h&oniu& ion, and the hydrogens are completely dissociated in water. Phosphoric acid is a relatively strong acid. The dissociation of the first hydrogen is virtually complete in aqueous solution. In contrast, the second hydrogen is only weakly dissociated, and the third is hardly dissociated a t all. The following charges are consistent with this behavior 'A charge of 0.435 for the hydrogens in OP(OH)3 A charge of 0.362 for the hydrogens in O,P(OH);
(This is hetween the charge for hydrogen in water and the charge for hydrogen in hydronium ion.) . A charge of 0.260 for the hydrogen in 0 3 ~ ( 0(This ~)2 is close to the charge for hydrogen in water.) Various Hydroxyl-ContainingSpecies
Sulfuric acid is a strong acid ( q =~0.531) for which several derivatives may be prepared. Fluorosulfuric acid is a stronger acid (q, = 0.6851, and the protonated derivative, OSF(OH12+,is even stronger ( q =~ 0.714). I t is the active
0
PROTON AFFiNITYleVl
Figure 1. Plot ofthe chlorine atomicchargeversus the freeenergy for the reduction to chloride for several chlorine-containing species: (1) Cl, : (2)HOCI; (3)HCIO, : (4)CIO; (5)C10;. species in superacid. Methylsulfonic acid, on the other hand, is weaker than sulfuric acid (q, = 0.434). When magnesium hydroxide and most group 2 hydmxides are treated with water, they tend to form insoluble oxides. Its hydroxyl charge is -0.369, suggesting that magnesium hydroxide should be appreciably dissociated in water. However, the MgOHi ion that forms upon dissociation has a hydrogen charge of 0.559. I t is expected to be completely dissociated, thus forming the metal oxide. Arguments such as these need not be limited to aqueous chemistry. For example, one can use the autopmtolysis of ammonia and its hydrogen charges to explore known and anticipated liquid ammonia chemistry. Other Possibilities Chemical Reactivities
There is a vast variety of chemical and physical phenomena that can be readilv exdored bv students using atomic charges. Both the rea&iviiy towapds substitutionand the site of substitution depend on a combination of bond euergy and charge effects. Thus, formation of neutral~adicals is favored by small charge separations, and larger charge separations favor ionic intermediates. For example, nucleophilic substitution at the C-0 bond of methanol is very disfavored, but substitution at the C-0 bond in methvl tosvlate is facile. This is due to the increasedchargesepa~ationin the tosylareand the inductive stabilization of the tosvlate ion bv the delocalization of the negative charge over the ion. q c = ~ 0.200 in methanol
versus q c ~ = 0.386 in
methyl tasylate
Manv other behaviors can be subiected to interoretations cased on considerations of atokic charge. ~he'effectiveness of various Lewis acid and Lewis base catalvsts are due to charge effects. Although the oxidizing and reducing abilities of substances depend on many factors, the charges carried by specific atoms tend to correlate well with redox reactivity To illustrate, the chlorine atomic charge is plotted against the free energy for the reduction to chloride ion3for several 3The free energies were computed from the standard reduction potentials (433.
1
2 ATOMIC
3
1
5
6
CHARGE
Figure 2. Plot of the proton affinity versus a' + (li2)b' for several nitrogen bases (44):( 1 ) N2 ;(2)NF3;(3)NCI3; (4) NH3; (5)NH2 (CH3); (6) NH(CH3)z; (7)N(CHd> chlorine-containing species (see Fig. 1).There are many other examples. Electronegativityas a Fundamental Property
That eledronegativity can be so diversly formulated suggests that it is indeed a fundamental property. In fact, Allen has gone as far as to suggest that it is the third dimension of the periodic table (10). Thus, in addition to atomic charges, a myriad of additional useful information may be gained through understanding electronegativity. It has been shown that Reed's molecular electronegativity constants a* and b* yield surprisingly accurate estimations of molecular ionization energies (17).Furthermore, it ie'well-known that the ionization energies of a homologous series of bases correlate well with their proton affinities. The correlation of a* + l b * 2
with the proton affinity of a series of amines is shown in Figure 2. Agreement with Other Methods
Although we have presented atomic charges that are both chemicallv reasonable and self-consistent. there mav be concern about how well these agree with those det& mined bv other methods.'l'hisauestion has beenaddressed elsewheie (11). Unfortunately, the reported atomic charges may differby more than one order of magnitude even for determinations made for very simple molecules using very sophisticated techniques. In some cases determinations may not even agree in sign. The constant k of the Wolfsberg-Helmholtz approximation (eq 21) is an empirical parameter. It could be adjusted to yield atomic charges that agree with more authorative values, should such values come into existence. Conclusion The concept of eledmnegativity, which dates back to the classification of substances by Berzelius, is still embroiled in controversy today-even to the point of serious challenges to Pauling's orginal definition. Regardless of the final disposition, should there truly be such a quantity as electronegativity,the following statements can be made. Volume 69 Number 10 October 1992
789
Eleetronegativity would dominate the determination of the distribution of charee within a molecule. Eleetronegativity w&ld be the property of an atom. .As a corollary of the preceding statement, electronegativity would also be the property of an atom within a molecule. We would suggest that electronegativity does indeed exist. arising naturally i n the LCAO-MO theory (11,17), a s seen by eqs 15 a n d i g . Because the quantities comprising eqs 15 and 19 are so fundamental, they are dominant in many other atomic and molecular properties. We would further suggest that, because of this, it has been possible to formulate electronegativity sucessfully using vastly different approaches and properties. Summary I n the preceding paragraphs we have examined a new electroneeativitv formulation that is faithful to P a d i n & original leiinition. We have demonstrated a method kr the simple computation of atomic charges, and we have provided a few examples from among many possibilities to show how students can use atomic charges and electronegativity to explore chemical behavior. Acknowledgment The author wishes to thank Dr. Henry C. McBay for his verv h e l ~ f u discussions l in the ~ r e ~ a r a t i oofnthis manuscript.
-
5. Jolly, L. J. ThoPriineplrs o f l m ~ ~ a nChemistry; k m a w - H i m : NewYork, 1976; pp 4254. 6. MoeUer,T.InoqankChemistry4MadrmI1ffddti0:Wiley:NewYork, 1982;pp ...i u e 7. Greenrumd, N. N.; E m h a w , A . Chpmishy ofLkEkmenls: Pergarnor., 198%p 30. 8. Bmmkrg, P Physkal Chemdry; Ally" and Bacon: B o h , 198%p 613. 9. A t b s , P. W. Physlml Chemistry; W H. Freeman:San fiancisco. 1982; p 4.58 10. Allen, L. C. J.Amr. Chem. Sac. 1989,111,90&9024. 11. Reed, J. L.J Phys Cham. 1991,95,68664871. W.B. Saundaa:Philadelphia. 1979; 12. Temay,A.L.C~ntempomryOrgonkCh~mistry: pp 5 M 4 . W. B. Sauoders:Philadelphis, 1979; 13. Temsy,A.L.Co~empor~ryOognnirChpmistry; pp 54b561. 14. Purcell, K F;Kotz. J. C. I n o r p i e Chemistry; W B. Saundem:Philsdephia, 1971; pp 88L896. 15. Temay,A.L.ConlampwyOlgankCbmisfry;W. B. Saunders:Philadelphis, 1979: pp 551-561. R. G. Symmetry R& for C h i m l Roocfionn:W~ley:New Ymk. 1976; pp 16. Pe-n, 17. 18. 19. 20. 21. 22.
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W ,&(I; p 91. Dordy, W.Phya Re". 1946.69.101. Allred, A. L.; Raekow, E. G. J. I n o g Nvcl Chpm. 1%55,5,26&268. Sandereon,R. T. J Chem Edvc. 1962.29.539544. Smith, D. W J Chem Ed-. 1990,67,559413. Allen, L. C. J. Am. Chom. Soc 1969.111.90034014. Boyd, R.: Edgemmbe, K E . J . Am. Chem Soc lW, 110.41824186. P m , R G.:Peamn,R.G.J A m . Chem.Soc 1983,105,75127514. Nalewejski, R. F J. Am. Chem Soe 19&1.106,94&949. PoliUeyP. J. Chem. Phys. lM7,86,1072-1076. Peermn.R.G J. C h . Educ. 1987.64.561.
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