Electronegativities and hardnesses of open shell atomsJosé L. Gázquez and Elba Ortiz Citation: The Journal of Chemical Physics 81, 2741 (1984); doi: 10.1063/1.447946 View online: http://dx.doi.org/10.1063/1.447946 View Table of Contents: http://scitation.aip.org/content/aip/journal/jcp/81/6?ver=pdfcov Published by the AIP Publishing Articles you may be interested in Development of algebraic techniques for the atomic open-shell MBPT(3) J. Math. Phys. 51, 123512 (2010); 10.1063/1.3520516 Orientation, alignment, and hyperfine effects on dissociation of diatomic molecules to open shell atoms J. Chem. Phys. 84, 3762 (1986); 10.1063/1.450217 A treatment of open shells J. Chem. Phys. 71, 4969 (1979); 10.1063/1.438310 Electronegativity J. Chem. Phys. 43, S124 (1965); 10.1063/1.1701474 Electronegativity Effects of Hybridized Carbon Atoms J. Chem. Phys. 33, 1881 (1960); 10.1063/1.1731536

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Electronegativities and hardnesses of open shell atoms Jose L. Gazquez and Elba Ortiz Departamento de Qu{mjca. Division de Ciencias Basicas e Ingenieria. Universidad Autonoma Metropolitana­Iztapalapa, c.P. 55-534. Mexico. D.P.. 09340. Mexico

(Received 1 March 1984; accepted 11 May 1984)

A Taylor series expansion of the energy of an atomic system around the neutral atom value, which introduces the first and second derivatives of the energy with respect to the number of electrons (electronegativity X, and hardness TJ, respectively) is proposed. The relaxed first derivative and the unrelaxed second derivative of the Xa and hyper-Hartree-Fock methods are used to relate X and TJ with the Lagrange multiplier Ei , and the self-repulsion integral J (i) of the highest occupied atomic orbital for the case of an open shell. A simple model, based on screening effects, is developed to get a better representation of a relaxed second derivative. This model replaces J (11 by !(r- 1

}i and leads to TJ = l(r- 1) i' The use of this relation, together with theXa expression for

electronegativity, X = - Ei , and a simple charge transfer model for electronegativity equalization leads to values of molecular electronegativities which are in very good agreement with the values obtained through the use of atomic or molecular experimental information. The relations here derived only need information obtained from a neutral atom calculation.

I. INTRODUCTION

Recently,l-12 the concept of electronegativity has re­ceived great attention because of its relation with the chemi­cal potential of density functional theory. I Electronegati­vity13 X is the negative of the slope of the ground-state electronic energy (of an atom or a molecule) as a function of the number of electrons, but this is the negative of the chemi­cal potentialp ofthe density functional theory of Hohenberg and Kohn, I thus

X = - p = - (aE laN)v (v is the external potential). (1)

Through this identification, it has been possible to calculate systematically within the same context, atomic ionization potentials, 14 electronegativities, and electron affinities5

•12 by

making use of an approximate energy density functional. In particular, Bartolotti, Gadre, and Parr used the transition state concept within Xa theory to obtain an expression for the electronegativity in terms of the Xa Lagrange multipli­ers. Their results were in good agreement with other elec­tronegativity scales.

On the other hand, Parr and Pearson 10 have recently defined the absolute hardness of an atom or a molecule as

(2)

and using Eqs. (1) and (2), they were able to derive theoreti­cally the principle of hard and soft acids and bases, and pro­vided relations to quantify the amount of charge transferred in the formation of a molecule, in terms of the electronegati­vity and the hardness of the isolated atoms.

The overall situation indicates that electronegativity and hardness will playa central role in an atoms in molecules description within density functional theory. The object of the present work is to analyze these quantities within the Xa and hyper-Hartree-Fock (hyper-HF) methods. 15.16 The starting point considers the Taylor expansion of the energy of an atomic system around the neutral atom value, which introduces the first and second derivatives of the energy with respect to the number of electrons. Next, we examine the

expressions for these derivatives in the Xa and HF methods, relaxation effects are discussed mainly in connection with the second derivative. In Sec. III, we propose a simple model based on screening effects to obtain improved estimates of the second derivative, and finally in Sec. IV we consider some aspects of charge transfer and molecular electronegati­vity.

II. TAYLOR EXPANSION

Nonintegral numbers of electrons do not occur in na­ture, however the concept of nonintegral populations of elec­trons on atoms in molecules has been generally accepted as valid and useful in chemistry. Since the amount of charge transferred or accepted by an atom in the formation of a molecule is, in general, small compared to the total number of electrons of the atoms involved, one could propose a Tay­lor expansion of E (Z,q) (the total energy of an ion of atomic number Z with charge q = Z - N) around the neutral atom value E (Z,O). Thus we write

E(Z,q)=E(Z,O)+qaEI +q2 ~~I + .... aq q=O 2! aq q=O

(3)

Such expansion, using only the terms explicitly displayed, has already been considered successfully by Lawes, March, and Yusafl7 to explain the regularity found empirically by Pyper and Grantl8 between successive ionization potentials over a wide area of the periodic table.

Using Eqs. (1) and (2) one can write Eq. (3) in terms of the absolute electronegativity and hardness of the atom as

E (Z,q) = E (Z,O) + qX + q2TJ . (4)

It is interesting to analyze this relation for the cases corresponding to the first ionization potential (I) and the electron affinity (A ). For the first one we have

I=E(Z, + l)-E(Z,O)

=X+TJ, (5)

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2742 J. L. Gazquez and E. Ortiz: Electronegativities and hardnesses of atoms

while for the second one we have

A = E (Z,O) - E (Z, - 1)

=X-1], (6)

the combination of these relations leads immediately to the expression for electronegativity given by Mulliken, 19 name­ly,

X=W+A) (7)

and to the expression for hardness suggested by Parr and Pearson 10

1] = !(I -A), (S) which correspond to the finite difference formulas for the first and second derivatives.

Now, Bartolotti, Gadre, and pa.rr have suggested that electron affinities may be estimated in the Xa theory from the calculated electronegativities and ionization potentials using Eq. (7). On the other hand, Robles and Bartolotti 12 have obtained the hardness from Eq. (S), using the calculated ionization potential and the electron affinity determined through Eq. (7). However, in view ofEqs. (5) and (6), one can see that there is an alternative way for calculating the ioniza­tion potential and the electron affinity, exclusively in terms of properties of the neutral atom, if one could evaluate di­rectly its electronegativity and hardness. This is possible in those theories which allow for the number of particles to be varied continuously, such as Xa and hyper-HF.

A. First and second derivatives in Xa theory

The energy functional in the Xa method for the spin nonpolarized case is given byl5

E [nk,'/Ik] = ~ ni J '/I~(l)( - !Vi)'/Ii (l)dTI

_ZJP(l)dTI +~JJ p(11o(2) dT1dT2

71 2 712

- ! aCx J [p(1)]4/3dTI , (9)

where ni is the occupation number of the orbital '/Ii' a is precisely the parameter a of the Xa theory (and its value may be fixed in different ways), Cx = 3(3/S1T)1/3, and

p(l) = L ni'/l~(l)'/Ii(l). (10) i

The minimization of the total energy with respect to the orbi­tals, subject to the orthonormality constraints, leads to one­electron equations of the form

[ - !Vi + Vow] '/Ii (l) = Ei '/li(l) , (11)

where

Vow = - Z + J p(2) dT2 - aCx pl/3 . (12) 71 712

The variation of the total energy, as given by Eq. (9), with respect to one of the orbital occupations, allowing the orbitals to relax, leads t020

(13)

By taking into account all the orbitals involved in going from the positive to the negative ion, one may combine Eqs. (1) and (13) to express the atomic electronegativities6in terms of the Lagrange multipliers Ei • The easiest case corresponds to those open shell atoms in which a single orbital is involved, as shown in the following diagram:

Ei ____ _ X Ei ____ _ XX Ei ___ _

XX XX XX positive ion neutral atom negative ion

For this case, the electronegativity will be given by5.12

X=-E i • (14)

The other cases (closed shell and some transition metal atoms) involve two orbitals, and although the electronegati­vity may be approximated by the negative of the arithmetic average of the two Lagrange multipliers, 5,6 the calculation of the hardness, especially in connection with the model devel­oped in Sec. III, is not clear to us at the moment, so we will restrict here to the open-shell single orbital case.

Now, if the second derivative of the energy with respect to one of the orbital occupations is evaluated allowing the orbitals to relax, the final expression will contain complicat­ed terms that arise from the derivative of the effective poten­tial, Eq. (12). The numerical evaluation of these terms would imply a great computing effort, and probably would not al­low for a simple interpretation of hardness in terms of some atomic properties. However, for unrelaxed orbitals, the final expression is simply given by

(a2~) =J(i)+K(i), (15) ani nj'Fni

where

J(i) = J I '/Ii(lWI'/Ii(2W dTI dT2 712

(16)

and

K(') = - 1 aC J I '/Ii(lWI'/Ii(lW dT . I 3 x [p(1)]2/3 1

(17)

Thus, for the open shell single orbital case, the combination ofEq. (2) with Eq. (15) will give for the hardness

1]unr = HJ(i) + K(i)] , (IS)

where the SUbscript "unr" stands for unrelaxed. If Eqs. (14) and (IS) are substituted into Eq. (5), one

obtains for the first ionization potential

1= -Ei + HJ(i) +K(i)] . (19)

At this point it is important to call attention to Koopmans theorem within theXa theory. Gopinathan21 considered the calculation of the first ionization potential using only the neutral atom orbitals and found 1= - Ei + Y (i). The ab­sence of the term !K (i) that appears in Eq. (19) comes from the expansion proposed by Gopinathan, since the terms re­tained in the demonstration amount to the assumption that the exchange term in Eq. (9) has a linear dependence on the occupation numbers. Therefore, if such linear dependence were assumed from the beginning, there would be no contri­bution from such term to the second derivative, and Eq. (19)

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J. L. GAzquez and E. Ortiz: Electronegativities and hardnesses of atoms 2743

would reduce to the result found by Gopinathan. Thus, we see that the difference between both approaches is not a mat­ter dealing with relaxation effects, but rather a consequence of the assumption of a linear dependence on n; of the ex­change term, and we may conclude that the approach pre­sented here based on the truncated Taylor series expansion, and the determination of a relaxed first derivative and an unrelaxed second derivative for a particular functional is equivalent to a Koopmans theorem type of approach22 to determine the first ionization potential. This conclusion will be more evident in the hyper-HF case.

Now, if Eqs. (14) and (18) are substituted into Eq. (6), one obtains for the electron affinity

A= -E;-HJ(i)+K(i)]

and combining Eqs. (19) and (20) one finds that

I-A =J(i) +K(i).

(20)

(21)

If K (n were very small compared withJ (I), we could neglect it in Eq. (21) and therefore

I - A 'ZJ (I) (22)

is just the Pariser formula23 for self-repulsion, which is so important in semiempirical theories of electronic structure of molecules. 24 It is interesting to note that although Eq. (22) has been derived starting from the Xa energy functional, it does not contain the Lagrange multipliers (which differ sig­nificantly from the HF Lagrange multipliers), it is given in terms of a self-repulsion integral which will give very similar results if evaluated with Xa or HF orbitals.

B. First and second derivatives In hyper-HF theory

The hyper-HF method was introduced by Slaterl5 as an extension of the usual HF method, in order to be able to vary continuously the number of particles. The basic idea consists of setting up a formula for the energy of the atom, averaged over all multiplet states found in the configuration. This for­mula iSl6

Eav = L [n;l(i) + !n;(n; - 1)J(i)] ;

+ L n;njK (i,j) , (23) pairs ;,j,''''' j

where J (i) is the self-repulsion integral given by Eq. (16), ni is again the occupation number of the ith orbital, I (i) is a one­electron integral, and K (i,}1 is a two-electron integral; both are defined in Ref. 16 and their explicit formula is not impor­tant in this work.

The minimization of this average energy with respect to the orbitals, subject to the orthonormality constraints, leads to one-electron equations that contain the Lagrange multi­pliers, which are given by

E; = I (i) + (ni -1)J(i)+ LnjK(i,JI. j#i

(24)

It is important to mention that while the hyper-HF method reduces to the HF method for the closed shell case, they differ for open shell atoms. However, the numerical differences are in most cases very small, for the canonical orbitals, as may be seen by comparing the results obtained by

Mann25 (hyper-HF) with the results obtained by Froese­Fisher26 (HF). Thus, although formally we are dealing with the hyper-HF energy functional, the numerical results would be very similar to those that would be obtained through the HF energy functional, but while the analysis through the hyper-HF functional is relatively simple, it would be very complicated3 with the HF functional because it does not allow for the number of particles to be varied continuously.

Now, the variation of the total energy, as given by Eq. (23), with respect to one of the orbital occupations, allowing the orbitals to relax, leads t027

(25)

where Eq. (24) has been used. Therefore, for the open shell single orbital case, the electronegativity will be given by

x = - E; - Vii) . (26)

Such definition of eIectronegativity was proposed originally by Mulliken,28 who demonstrated that it is simply his defini­tion applied to the HF atom. Moffit29 also proposed the same definition, and later Peters30 gave some numerical evidence. More recently, Donnelly has derived it using arguments based on an energy functional of the first order density ma­trix for the HF case. 3

As for the Xa method, if the second derivative is ob­tained allowing the orbital to relax, one would obtain com­plicated expressions. Therefore, here we will consider only the case for unrelaxed orbitals, which gives

(a2E) =J(i) (27)

an; nJ""n,

and the hardness will be given by

1Junr = Y (i) . (28)

This relation is almost identical to the one obtained in the X a

method [Eq. (18)] except for the term K (i). This results for X and 1J reflect some of the differences

and similarities between the Xa and hyper-HF methods. That is, the Lagrange multipliers are quite different [notice the electronegativity expressions, Eqs. (14) and (26)], but the integrals evaluated with either orbitals are quite similar [no­tice the hardness expressions, Eqs. (18) and (28)].

If Eqs. (26) and (28) are substituted into Eq. (5), one obtains for the first ionization potential

1= -Ei , (29)

which is simply a verification of Koopmans theorem within HF theory. However, it is important to recall that this rela­tion has been obtained through the use of a relaxed first deri­vative and an unrelaxed second derivative as was discussed earlier for the Xu method. Thus, in order to go beyond Koopmans approximation, it is necessary to include relaxa­tion effects in the second derivative.

Now, if Eqs. (26) and (28) are substituted into Eq. (6), one obtains for the electron affinity

A = - E; -J(i) (30)

and combining Eqs. (29) and (30) one finds that

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2744 J. L. Gazquez and E. Ortiz: Electronegativities and hardnesses of atoms

TABLE I. Comparison between Xa and hyper-HF methods for several atoms (atomic units)."

Atom J(I) _~F [ - ~i + !J (I) (. ! - ~, +! [J(/) + K(I)llx.

hyper-HF Xa

Li 0.234 0.232 0.196 0.211 0.192 B 0.403 0.411 0.310 0.331 0.316 C 0.512 0.527 0.407 0.453 0.437 N 0.619 0.640 0.509 0.576 0.559 0 0.724 0.750 0.616 0.704 0.686 F 0.829 0.859 0.730 0.834 0.816 Cl 0.474 0.495 0.506 0.546 0.536 Br 0.412 0.430 0.457 0.481 0.473 I 0.346 0.362 0.403 0.419 0.412

• ~ll values correspond to the highest occupied orbital. The values for hyper-HF were calculated from the data 10 Ref. 25. The Xa values were obtained from self-consistent field calculations using the Herman-Skillman program with the a parameters of Ref. 38.

I -A =J(l} , (31)

which is equal to Eq. (22), however it has been derived start­ing from the hyper-HF energy functional.

Although here we have made emphasis on the calcula­tion of the first ionization potential through the knowledge of the electronegativity and the hardness ofthe atom, it may be noted that within the hyper-HF context, one could make use of the transition state concept to calculate ionization en­ergies. In fact, this has already been done by Brandi, Matos, and Ferreira3

! and the results thus obtained compare better with the experimental values than those obtained through Koopmans approximation, and are very similar to those ob­tained by the transition operator method.32

It is important to note that Eqs. (26) and (31), that have been derived through the definitions provided by Eqs. (1) and (2), are in agreement with the results of Gopinathan and Whitehead,9 who have recently discussed in great detail the dependence of the total energy on occupation numbers. Through the hyper-HF functional, they have derived Eqs. (26) and (31) by considering the appropriate energy differ­ences and using only the neutral atom orbitals.

C. Results

The relations presented here for the Xa and hyper-HF functionals show remarkable similarities between the two

TABLE II. Electronegativities and hardnesses for several atoms (eV)."

Atom

Li B C N o F Cl Br I

hyper-HFb

2.15 2.95 4.11 5.43 6.91 8.59 7.32 6.83 6.26

• See reference of Table I. b Equation (26). <Equation (14).

X

Xa < hyper-HF'

2.58 3.18 3.41 5.48 5.14 6.97 6.98 8.42 8.94 9.85

11.01 11.28 8.12 6.45 7.25 5.61 6.46 4.71

d Equation (28). "Equation 118).

TJunr

Xn e

2.63 5.18 6.73 8.24 9.73

11.20 6.46 5.62 4.74

methods. In fact, they would give very similar results if: (i) the orbitals obtained by the two methods were almost identi­cal and (ii) the Lagrange multipliers were related by2!,22

~F = i:a - V (Ita (32)

[through Eq. (32) one may derive the hyper-HF relations starting from the Xa relations, except for the term K (i), or vice versa]. Both conditions are only approximately fulfilled in practice and therefore, the quantitative results differ slightly. This may be seen in Table I, where we compare the Xa and HF values of J(i) and the relation between the La­grange multipliers.22 In principle, the integrals for J (i) and

K (i) depend on the quantum number m for a given I, how­ever, we have adopted the convention m = 0 in every case, as is usual inXa theory.

In Table II, we have reported the electronegativities and hardnesses of several atoms that correspond to the open shell single orbital case discussed here. The electronegativi­ties for the Xa method correspond to the values reported by Bartolotti, Gadre, and Parr.5 We can see that both methods exhibit known trends in general. However, in view ofEq. (8), it seems that hyper-HF electronegativities have rather small values and that the unrelaxed hardnesses for both Xa and hyper-HF have rather large values.

III. RELAXATION EFFECTS IN THE SECOND DERIVATIVE

The results obtained in Sec. II, indicate that relaxation effects in the second derivative are very important to deter­mine appropriate values of atomic hardnesses, which will be very useful to calculate atomic and molecular properties.

A direct approach to calculate relaxed second deriva­tives would require the determination of the energy of the system for several values of q (or n j ). Thus, 1] could be ob­tained through the numerical differentiation of the resultant E vs q data. However, such procedure would imply a great computing effort and would not allow for a simple physical interpretation. Here, we propose a simple model to include relaxation effects. The development will be framed within the Xa theory .

The starting point is to replace in Eq. (11) the Coulomb and exchange interactions of the effective potential [Eq. (12)]

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J. L. GlU:quez and E. Ortiz: Electronegativities and hardnesses of atoms 2745

by an average screening constant S, whose value will be a function of q. Then Eq. (11) becomes

[ _ J.- V2 _ Z - S(q)] 1//. = £.1//. (33) 2 r 1 1 I

and therefore

£;(q) = _ ~ [Z -~(qW I

(34)

where V; is the principal quantum number of the ith orbital. It follows from Eq. (34) that the difference between the

neutral atom eigenvalue £; (0) and the eigenvalue correspond­ing to some charge q is given by

£;(q) _ £;(0) = ~ [Z -StOW ~ [Z -S(qW (35) I

Assuming that the difference betweenS (q) andS (0) is directly proportional to the charge q, i.e.,

S(q) -S(O) = kq, (36)

where k is a constant, and substituting this relation in Eq. (35), one obtains that

£;(q) _ £;(0) = [Z - S(O)] kq _ k2q2 . (37) o 20

Now, at this point we recognize that for a hydrogenic orbital, the average value of r- I is given by

(r- I ); = Z -:(0) . I

(38)

Therefore, we can write Eq. (37) in the form

-I k 2q2 £;(q) = £;(0) + (r );kq - 2v7 .

I

(39)

On the other hand to fix the value of the constant k, one may recall that the effective potential [Eq. (12)] behaves as - Z Ir when r-+O and as - (q + l)1r when r----+-oo (this long-range behavior is not obeyed by the Xa potential which behaves as - qlr for r large, because the exchange potential decays

exponentially, however here we will consider the correct one). In terms of the simple screened potential ofEq. (33) this means that S (q) goes to zero when r-+O, and goes to Z - q - 1 when r----+- 00. Therefore, one could take the arith­metic average, S (q)::::: (Z - q - 1 )12, which would lead to

S(q) -S(O) = -!q.

This relation shows that the assumption ofEq. (36) is reason­able, and that k::::: - !. Inserting this value in Eq. (39) one finally obtains that

£;(q):::::£;(O) - Hr-I);q, (40)

where we have dropped the last term ofEq. (39) because it is much smaller than the other two.

Now, differentiating the energy in Eq. (3) with respect to q, we have that

_aE---,(Z-...:.,=q) = _aE_1 + q _~_E f aq aq q=O aq2 q=o'

(41)

which may be rewritten in the form

~EI £;(q) = £;(0) - q-aq2 q=O

(42)

TABLE III. Comparison between the relaxed second derivative (,-1) i and

J (I) + K (I) for the Xa method (atomic units). a

Atom WElai)!q_ob ~(,-I)i J(I) +K(I)

Li 0.173 0.178 0.193 F 0.564 0.642 0.823 Na 0.165 0.159 0.187 CI 0.367 0.375 0.475 Br 0.326 0.314 0.413

• See reference in Table I. bThe second derivative has been calculated numerically from the E vs q data.

Comparing this result with Eq. (40), we see that the second derivative may be expressed as

~~ I :::::! (r- I); (43)

aq q=O

In order to test this relation, we have calculated the second derivative numerically. The results are reported in Table III, together with the values of 1I2(r- I

); and of J(i) + K(I)[recail Eq. (15)]. We can see that 1I2(r- I

); pro­vides a much better representation of the relaxed second de­rivative than J (i) + K (i). Therefore, we may conclude that a better approximation to the hardness of an atom is

11 = !(r- I); (44)

In Table IV, one can see that the results obtained through this relation are in good agreement with other esti­mates. IO

,12 It is important to mention that the values in the first two columns l2 require two independent calculations, one for the transition state corresponding to the ionization process (half-occupancy in the highest occupied orbital), and another one for the electronegativity, which for the open­shell single orbital case corresponds to the neutral atom.

TABLE IV. Hardnesses of several atoms (eV).

Atom

Li B C o F Na AI Si P S CI K Br Rb I

A

4.03 4.33 5.50 6.14 7.37 3.58 2.83 3.55 5.93 4.08 4.79 2.84 4.28 2.61 3.70

Other estimates"

B C

3.06 2.38 4.39 6.01 5.49 5.00 6.42 6.08 7.52 7.01 2.91 2.30 2.94 2.77 3.61 3.38 5.42 4.86 4.28 4.12 4.91 4.70 2.35 1.92 4.40 4.24 2.21 1.85 3.81 3.70

Thisworkb

2.42 4.14 5.32 7.61 8.73 2.16 2.64 3.31 3.93 4.53 5.11 1.74 4.28 1.60 3.54

a The values in columns A and B were taken from Ref. 12. and were obtained from spin-polarized calculations with the Xa (A ). and the local-spin-den­sity (B ) (Ref. 39) functionals. The values in column C were taken from Ref. 10. and were obtained through the formula 1] = (1/2) (1 - A ). using the experimental first ionization potentials and electron affinities.

bEquation (44) with the Xa method. See reference in Table I.

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2746 J. L. Gazquez and E. Ortiz: Electronegativities and hardnesses of atoms

TABLE V. First ionization potentials (/) and electron affinities (A ) for several atoms determined through theXa

method (eV).··b

Atom / A

TSc This workd Exp 2X _Ie Thisworkf Exp

Li 5.03 4.99 5.39 0.12 0.16 0.62 B 7.92 7.54 8.30 - 1.12 -0.73 0.28 C 10.63 10.46 11.26 -0.37 -0.18 1.27 0 16.24 16.55 13.61 1.60 1.33 1.46 F 19.19 19.74 17.42 2.81 2.28 3.40 Na 4.70 4.49 5.14 -0.05 0.17 0.55 AI 5.22 4.89 5.98 -0.72 -0.38 0.44 Si 7.18 6.91 8.15 0.01 0.30 1.39 P 9.16 8.92 10.48 0.86 1.07 0.75 S 11.21 11.05 10.36 1.82 2.00 2.08 CI 13.32 13.22 13.01 2.90 3.01 3.62 K 3.89 3.66 4.34 -0.05 0.19 0.50 Br 11.86 11.53 11.84 2.62 2.97 3.36 Rb 3.65 3.39 4.18 -0.08 0.20 0.49 I 10.42 10.00 10.45 2.48 2.93 3.06

• See reference in Table I. bExperimental values taken from Ref. 34. C Values calculated through the transition state appropriate for the first ionization. d Equation (45) of the text. e Values taken from Ref. 5. X and I are calculated separately by the Xa method using the appropriate transition state in each case.

fEquation (46) of the text.

Then 7J is given by I-X. On the other hand, Eq. (44) only requires neutral atom information. However, a more direct comparison with columns A and B would require to extend the present approach to the spin-polarized case.

Now in terms ofEqs. (14) and (44), the first ionization potential and the electron affinity are given by

(45)

and A Xa = - €; - 1(,-1); . (46)

The results obtained through these equations are re­ported in Table V. It may be seen that there is a very good agreement between the first ionization potential calculated by the transition state method5

,14 (which includes relaxation effects up to the second derivative), and the one calculated with Eq. (45). The results for the electron affinity are less accurate with respect to the experimental values, but very similar to those of Bartolotti, Gadre, and Parr which re­quire two independent calculations. Nevertheless, the values are good to about 1 eV, and the trends predicted are quite reasonable. On the other hand, the results of Robles and Bartolottp2 for I andA, especially those obtained by includ­ing correlation effects, are in general better than ours. This indicates that the spin polarization plays an important role and should be included in the present formalism.

The definition of hardness given by Eq. (44) could also be applied to the hyper-HF method, However, in view ofEq. (25), the results will depend strongly on the difference between 1/U (i) and 1/4(,-1); and the values thus obtained are, in general, less satisfactory than the ones obtained through the Xa method.

Now, it is interesting to note that subtracting Eqs. (45) and (46) one obtains that

(47)

In Fig. 1, we have plotted the experimental values of I - A vs the Xa values of (,-1);. It may be seen that there is, in gen­eral, a good agreement between these two quantities. Thus, Eq. (47) provides an alternative way to Eq. (22), to evaluate the important quantity (I - A ). One immediate advantage is that while J (i) is a two-electron integral, (,-1); is a one­electron integral.

(I-A>exp

16

• 12

12

FIG. 1. Plot of the difference between the experimental first ionization po­tential and electron affinity vs one-half of the expectation value r- 1 of the highest occupied atomic orbital, corresponding to an Xa calculation. Fif­teen dots correspond to the atoms of Table V, straight 1iney = x for a com­parison. All quantities in eV.

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J. L. G4zquez and E. Ortiz: Electronegativities and hardnesses of atoms 2747

IV. CHARGE TRANSFER AND MOLECULAR ELECTRONEGATIVITY

The amount of charge transferred in the process of for-mation of a molecule AB in its ground state from atoms A and B in their ground states, may be estimated from the electronegativity equalization principle33 (which corre-sponds to a chemical potential equalization), if one knows the electronegativities of the atoms involved as a function of the charge. That is, the charge transferred is such that at the end, atoms A and B are in states having electronegativities XA andXB such thatXA = XB = XAB'

In the present case, the atomic electronegativity as a function of charge is given by [from Eq. (4)]

XA =X;" +2q1];", (48)

where X;.. and 1];" represent the electronegativity and hard-ness of the neutral atom in its ground state. Thus, after the formation of the molecule, the electronegativity equalization principle demands that

(49)

Since q A = q = - qB' one finds that

X; -X;" q= ,

2(1];" + 1]~) (50)

which is the same as the result obtained by Parr and Pear­son. 10 The molecular electronegativity will be given by

X;"1]~+X~1];" XA8 = 0 0

1]A + 1]8 (51)

which differs from the geometric mean principle for elec­tronegativity equalization.33,34

Now, substituting Eqs. (14) and (44) in Eq. (51), one obtains the molecular electronegativity within the Xa meth­odas

XA8 = (52)

where E;", E~, (r- I > A' and (r- I >8 are the eigenvalues and the r- I expectation values of the highest occupied orbitals of the neutral atoms A and B, respectively.

In Table VI, we can see that the results obtained through Eq. (52) are in good agreement with other estimates. It is important to mention that the values in the second col­umn4 were obtained through an electronegativity equaliza­tion, using the atomic energy expression E (N) = a IN + a~ 2, so the final expression is quite similar to our Eq. (52), however the constants at and a2 , which are propor­tional to the electronegativity and hardness, were fixed through the use of the experimental first ionization potential and electron affinity of the atoms involved in the molecule; the values in the third column4 make use of the knowledge of the experimental equilibrium bond length of the diatomic molecules AA, BB, and AB, and the values in the fourth column3s were obtained from the geometric mean principle and the atomic electronegativities of Mulliken which also require knowledge of the experimental first ionization po­tential and electron affinity of the atoms. Thus, the agree­ment with Eq. (52), which only uses information obtained

TABLE VI. Electronegativities of several AB systems (eV).

Molecule Other estimates" Thisworke

Ab B" Cd

HF 8.79 11.02 8.62 9.30 HCI 7.88 8.54 7.69 8.06 HBr 7.49 8.01 7.37 7.53 HI 6.97 7.27 6.95 6.98 BrF 8.74 9.19 8.89 8.49 CIF 9.22 9.65 9.29 9.18 ICI 7.51 7.43 7.49 7.14 BrCl 8.00 6.91 7.94 7.64 LiH 4.17 4.17 4.63 3.99 NaH 4.07 3.72 4.50 3.69 KH 3.57 3.30 4.16 3.15 SO 6.78 8.59 8.45 7.42 seO 7.18 8.26 8.22 6.94 TeO 6.27 7.85 7.94 6.44 PN 6.55 7.51 7.59 5.74 AsN 6.31 7.35 7.50 5.46 SbN 5.72 6.62 6.94 5.14

"These values were taken from Ref. 4. bCharge transfer-electronegativity equalization, using experimental first

ionization potential and electron affinity, see Ref. 4. C Based on a simple bond charge model, using experimental equilibrium

bond lengths, see Ref. 4. d Based on the geometric approximation, using Mulliken's electronegativity

determined from the experimental first ionization potential and electron affinity.

e Equation (52) of the text using the Xa values. See reference in Table I.

from a neutral atom calculation with theXa method is high­ly satisfactory.

The results obtained through the electronegativity equalization for polyatomic molecules are reported in Table VII. The values in the second column4 were obtained by the same procedure described for the second column of Table VI, while the values of the fourth column make use of the molecular first ionization potential and electron affinity. Again the agreement is quite good.

TABLE VII. Electronegativities of several polyatomic molecules (eV).

Molecule Other estimates"

NHl CF2

NF2 CO2

N02

H 20 CS2

SOl CH3

BF3 SF3

PCl, COS SFs

Ab

7.30 8.75 9.50 7.05 7.55 7.35 6.25 7.00 6.90 8.25 8.95 7.85 6.60 9.45

"These values were taken from Ref. 4. b See reference b in Table VI. C Based on molecular (I + A )12.

B"

6.10 7.20 7.50 6.60 6.45 7.80 5.55 6.70 5.50 9.10 7.80 5.40 5.80 9.30

Thisworkd

7.63 8.36 9.39 7.35 8.21 8.27 6.11 7.83 7.13 7.87 9.26 7.17 6.63 9.76

dObtained through an extension of Eq. (52) for polyatomic molecules and using the Xa values. See reference in Table I.

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2748 J. L. Gl\zquez and E. Ortiz: Electronegativities and hardnesses of atoms

V. SUMMARY AND CONCLUSIONS

The main results shown in this work are: (i) The Taylor series expansion of the energy around the

neutral system keeping terms up to second order seems to give, in view of the results ofSecs. III and IV, a good descrip­tion of E (q). This in turn supports a parabolic nature of the curve near the neutral system.

(ii) The differences and similarities between the Xa and hyper-HF first and second derivatives have been established. Since for the neutral atom system, hyper-HF and HF theor­ies give very similar results, one can see through the results presented in Sec. II the differences and similarities between Xa and HF for calculating electronegativities and hard­nesses.

(iii) It has been shown that in order to go beyond the Koopmans approximation it is necessary to include relaxa­tion effects in the second derivative. A simple model based on screening effects has been used to obtain a better repre­sentation of a relaxed second derivative in terms of a neutral atom quantity. That is, the replacement of J(i) by !(r-I)i leads to better agreement with the results that are obtained when the relaxation effects are included in the calculations. We have found that if one uses Eq. (39) in connection with the transition state method, the ionization energies of the inner-shell orbitals are described much better than through the use of Koopmans approximation.36 Thus, it seems that the simple screening model is very useful.

(iv) The results obtained through the electronegativity equalization are very satisfactory, however, we feel that a more extensive study should be done in order to establish its usefulness. Besides, the values could be improved through the use of the correct valence states of the atoms. Here we have only used ground state information.

We are at present studying different applications of electronegativity and hardness concepts within the context of the Xa method and through exchange functionals which include correlation. 37.39

ACKNOWLEDGMENTS

We would like to gratefully acknowledge J. Robles for providing us a preprint of his work and the Departamento de C6mputo ofUniversidad Autonoma Metropolitana-Iztapa­lapa for their assistance. Weare also grateful for discussions with M. Galvan and A. Vela.

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