Orbital Interactions and Chemical Hardness - American Chemical

1994, 98, 1436-1441. Orbital Interactions and Chemical Hardness. Hiroshi Fujimoto* and Shinichi Satoh. Division of Molecular Engineering, Kyoto Univer...
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J. Phys. Chem. 1994, 98, 1436-1441


Orbital Interactions and Chemical Hardness Hiroshi Fujimoto' and Shinichi Satoh Division of Molecular Engineering, Kyoto University, Kyoto 606, Japan Received: October 5, 1993'

Electron delocalization between molecules in chemical reactions is represented compactly in terms of the pairs of localized interacting orbitals. These interacting orbitals are illustrated to be reproduced approximately by projecting on to the occupied and unoccupied M O subspaces of a n isolated reactant molecule a reference orbital function that has been chosen to specify the local characteristics of a given reaction. These projected reactive orbitals allow us to estimate the potential of a n atom or a functional group for electron donation or electron acceptance in the reactant molecule. An analysis of the potential yields a novel view of orbital interactions that they are governed by local electronegativity and local hardness of a structural unit and also by localizability of the interaction on the structural unit in each of the reagent and reactant parts. The localizability of the interaction is nothing but the Fukui function amplified on the reaction sites. The physical significance of these terms is discussed in relation to the types of chemical bonds to be formed in reactions.

Introduction The classification of "hard and soft acids and bases" (HSAB) proposed by Pearson is one of the useful concepts in chemistry.1-2 It is often connected with the energy levels of the highest occupied (HO) MO and the lowest unoccupied (LU) M O of a m o l e c ~ l e . ~ Parr defined some years ago the "absolute hardness" and "chemical poter1tial".~,5 Chemical hardness is an outcome of the density functional theory6.7 and appears to be a current topic in theoretical chemistry.8-10 The stabilization arising from the interaction of two molecules has been related there to the net electronic charge shifted between them.4~5 This raises, however, a difficulty when it is applied to more general cases of molecular interactions, particularly between two neutral species. The formation of new bonds between the reagent and the reactant and, therefore, stabilization of the interacting system have been shown to be brought about by mutual electron delocalization.11J2 Pearson pointed out then that it is not the difference but the sum of electronic charges transferred between the reagent and the reactant that is really important in chemical interaction^.^^ The principle of maximum hardness says that "at equilibrium, chemical systems are as hard as possible".gJ3 The transition state of a reaction should, on the other hand, have a minimum energy gap between the HOMO and the LUMO or low-lying excited states to facilitate the unimolecular decomposition of the activated complex along the reaction coordinate.14J5 We should note that acid and base are the properties that are related primarily to the power of an atom or a functional group in a molecule to attract or release electrons. The ability of a molecule for electron donation and acceptance should be discussed by defining some local quantities, taking the effects of molecular periphery properly into account. It seems worthwhile then to see how the concept of hard and soft is applied in molecular interactions. We report here that the hardness and electronegativity revive in the orbital interaction scheme when we attempt to look into local characteristics of chemical reactions. Discussion Following Iczkowski and Margrave, the electronegativity defined by16

x is

= -(aE/am where E is the total energy and N is the number of electrons. It is estimated usually by a familiar relation proposed by Mulliken Abstract published in Advance ACS Absrracrs, January 1, 1994.

0022-3654/94/2098- 1436$04.50/0

in terms of the ionization potential and the electron affinity by ( I + A)/2 or in terms of the orbital energies of the HOMO and LUMO by -(€HOMO + eLuMO)/2.l7Meanwhile, Parr and Pearson defined the chemical potential p and the hardness 71 of a molecule by applying the density functional the or^:^^^^

Assuming that p should have the same values for the two reactant molecules A and B in the composite reacting system, the net amount of the electronic charge shifted from B to A in the interaction has been given by5cJ8

where ~~0 and XAO indicate the chemical potential and electronegativity of A in an isolated state, respectively. An important conclusion obtained there is very simple, BO - ~ A O ) > 0, in order to cause an electron shift from B to A. The relation (pea - ~ A O ) > 0 can be derived, however, very easily within the orbital interaction scheme.19 Electronegativity has indicated originally the power of an atom in a molecule to attract electrons to itself.20 In contrast, the electronegativity given by the ionization potential (or -€HOMO) and the electron affinity (or -CLUMO) applies to the whole system which may be an atom, a molecule, an ion, or a radical. The same is also true for the absolute hardness defined by eq 3. The properties of each atom can hardly be recognized in a large molecular species in this approach. Chemical interactions are local by nature, reactivities being determined primarily by the local power of an atom or a functional group in a molecule. There is another important factor, Le., orbital phase, in discussing reactivities and selectivities in reactions.21 The following example illustrates a possibility of discussing the hardness and electronegativity within interference of the wavefunctions or more conventionally in the orbital interaction scheme. Multifariousnessand Similarityin ChemicalInteractions. There are two aspects of chemical reactions that are essential but have not been invoked explicitly in most of the existing reactivity theories.22 First, a variety of compounds, different in size and structure, tend to undergo the same type of reaction, if they have a common functional group. Second, a compound undergoes a variety of reactions against different types of reagents. The 0 1994 American Chemical Society

Orbital Interactions and Chemical Hardness

The Journal of Physical Chemistry, Vol. 98, No. 5, 1994 1437 ............

,. ,......






W a


Figure 2. Projected unoccupied reactive orbital of ethyl chloride for

bimolecular nucleophilic substitution reaction (a) and for @-elimination reaction (b).

of methyl chloride. An orbital localized on the C r H bond is obtained in the case of @-elimination reaction. e This simple example shows us an ability of molecules in interactions. Different molecules provide similar orbitals for a ,...... given type of reaction. A variety of compounds that are different /..... :“.,: iii : * in size and structure but have a common functional group can thus undergo the same type of reaction. On the other hand, a molecule provides different interactingorbitalsfor different types of reactions. Thus, a compound undergoes a variety of reactions. Such a trend is general and becomes clearer when the reactant molecules are allowed to deform along the intrinsic reaction coordinate.1 lb,27 Projected Reactive Orbitals and Hardness. The analysis of the Figure 1. Unoccupied orbital of ethyl chloride participating in electron wave function for the composite interacting system has revealed delocalization with an attacking hydride ion. The position of the anion an important consequence of electron delocalization that it is indicated by a dot. specifies the active structural units of the reactant molecules for a given reaction. Now, let us examine the physical significance similarity and multifariousness in chemical reactions are related of the interactingorbitals. Projection of a certain referenceorbital obviously to local characteristics of chemical interactions. function, 6,, on to the occupied and/or unoccupied MO spaces Electron delocalization between two molecular species is of a molecule, large in size and complicated in structure, gives represented by a combination of the occupied MO’s of one part rise to theoccupied orbital, &(fir), and/or the unoccupied orbital, and the unoccupied MO’s of the other ~art.1~923 The number of dun, (&), that are closest to the function 6,. They should show the MO’s relevant for the interaction increases as the interacting the largest amplitude in the region spanned by 6,. species become larger. Then, by carrying out simultaneous Figure 2 shows the projected unoccupied orbitals of ethyl transformations of canonical MO’s within each of the two chloride. The reference orbital was chosen very simple, being 1s fragment species, we have demonstrated in our previous papers A 0 of the @-hydrogen for the elimination reaction and the that electron delocalization is represented concisely in terms of a-carbon components in the LUMO of methyl chloride for the a few pairs of localized interacting orbitals, by evaluating the substitution reaction. The interacting orbitals that have been contributions of all the occupied and/or unoccupied M O ’ S . ~ ~ obtained in Figure 1 by the analysis of electron delocalization in Figure 1 shows the orbital of an ethyl chloride molecule which the composite interacting system of an ethyl chloride molecule interacts with an attacking hydride ion. Here, to lift the influence and a hydride ion may practically be replaced by these projected of molecular distortion on the orbital shape, we fixed the structure orbitals. Incidentally, for morecomplicated cases, we may choose of ethyl chloride molecule. The STO-6G minimal basis set was 6, in such a manner as to give the HOMO and/or the LUMO utilized to illustrate simply the trend.25~” Electron delocalization when it is projected on to the smallest reactant molecule. The represented originally by the combinations of the hydride 1s A 0 application of the same reference function to larger reactant and the unoccupied MO’s of ethyl chloride is now presented by molecules should lead to the occupied or unoccupied projected the unoccupied orbital of ethyl chloride shown in Figure 1 and orbitals that are closest to the HOMO or the LUMO of the the hydride 1s AO. The latter orbital is not shown here, since smallest one in the amplitude distribution and orbital phase. it remains unchanged in the orbital transformation within the Reactivities of molecules are discussed in experimental chemfragment at the present level of calculation. istry very often on the basis of the quantities of molecules in an At an early stage of the interaction represented by the models isolated states2* The orbital 4, (6,) and the orbital & , (6,) are (i) and (ii) in Figure 1, electron delocalization takes place from defined for an isolated reactant molecule for the given reference the hydride 1s orbital to the @-hydrogen of ethyl chloride, the function 6129 @-hydrogenbeing located closer to the attacking anion. As the anion comes closer to the a-carbon as illustrated by the model (iii), the LUMO of ethyl chloride participates more significantly in the interaction. The hybridization of the LUMO with the other unoccupied M O s gives rise to the orbital that has a larger M M amplitudeon the carbon center. At a late stage of the interaction (iv), the orbital is localized almost completely on the C,-Cl bond. The orbital is antibonding with respect to the C,-C1 bond, indicating that the bond will be loosened upon electron acceptance where dr,, and dj,r are the coefficients of the occupied MOs, from the attacking hydride. Note that the orbital of ethyl chloride 4i (i = 1, 2, ..., m), and the unoccupied MO’s, $j 0’ = m + 1, m a t the stage (iv) bears a close resemblance rather to the LUMO

Fujimoto and Satoh

1438 The Journal of Physical Chemistry, Vol. 98, No. 5, 1994

+ 2, ...,M), in the linear expansion of 6, in terms of the canonical

MO's. The local electron-donating potential and the electronaccepting capacitance of the structural unit specified by 6, are given then by29

(7) u


where ei and ej signify the energy of MOs 4iand 6) The quantities defined by eqs 7 and 8 for an isolated reactant molecule do not apply to a whole molecule, but give different values for different 61's.

The reference function is now written as

s(r) = S f ( r ) (14) The definition of local hardness V(r) is somewhat different. The integration of the product of V(r) and f(r) over the space gives rise to the global hardness 7. The Fukui functions governing an electrophilic attack f -(r) and a nucleophilic attack f +(r) have been defined respectively by



= af#J,(6,)

and against electron acceptance, respectively. The orbital 9,(6,) and $ ~ , , ~ ~ can ( 6 ~ be ) closer in energy to 6,, when both the inactive fraction of the orbitals and ~(6,)are made smaller. Considering that eq 13 is formally similar to eq 3 and the argument that chemical systems are as hard as possible at e q u i l i b r i ~ m , ~ J 3 - ~ ~ we may call ~(6,)the hardness of a structural unit in the molecule and 2(1 - a2)~(6,)and 2a27(6,) the basic hardness and acidic hardness, respectively. Parr and collaborators defined the local softness s(r) in terms of the global softness S and the Fukui functionf(r):SbJO

+ (1 - U 2 ) q J u n , ( 6 , )

(0 Ia2 I1)


The electron occupancy of the reference orbital function, b,, is 2a2. That is, the reference orbital can donate the electron population up to 2a2 and accept the electron population up to 2( 1 - a2) in the molecule on which it is projected. The electrondonating or -accepting ability of 6, is estimated by m


Of the electron pair provided by 4, of a reactant, the fraction of 2a2 is shown above to be donated via the active region of the molecule and, of the electron holes provided by & ,, the fraction of 2(1 - a2) is utilized by the active region of the molecule to make bonds with the assumed reagent. By replacing the HOMO in eq 15 by 4, and the LUMO in eq 16 by I#J~,,, and by replacing the A 0 of the reaction site in p(r) by 6,, one sees that the active density of our projected reactive orbital gives nothing but the value of the Fukui function of the structural unit. Then, we may write

f-(s,) The function 6, is given by a certain atomic (hybrid) orbital or by a combination of a few atomic orbitals, representing the orbital of the reactant molecule that takes part in bond formation with the assumed reagent in the given type of reaction. Thus, -A(&) shows the power of an atom or a functional group in a molecule to attract electrons. This is in line with the organic chemist'sview of electronegativity. If 6, happens to be represented only by the occupied MOs, Le., 6, =, @ , the reaction center acts solely as a basein thereaction, whereas thereaction center behaves solely as an acid when 6, is coincident with &noc. In most cases, 6, is given by a combination of 4, and & ., That is, a reactant molecule serves partly as an electron donor and partly as an electron acceptor in reactions. Then, the active part of &(6,) is (12, i.e., ( 6 , ~ # ~ ~ ( 6 ,=) )a, and that of ),6(& is 1 - a2. The remaining part, 1 - a2 in & and a2 in #J,,, is not involved directly in the interaction. Thus, we are able to measure by a2and 1 - a2the efficiency of a reactant molecule in utilizing the atomor the functionalgroup for the interaction with a reagent. Now, we may write

in which

The quantities given by eqs 11 and 12 represent, respectively, how apart and &,oc are located in energy from the orbital 6, that is most suitable for the bond formation in the given reaction. That is, they stand for a kind of resistance of an atom or a functional unit of the reactant molecule against electron donation C#J~


2 2 / 2 = a2


The Fukui function represents the softness of the reaction sites. Having 0 If(6,) I1, eqs 11 and 12 may be rewritten as31 Xw(6r)

= X(6r) - (1 -f-(6r)l

X 29(6,)

for electrophilic attack (1 9) Aunw(6r)

= X(6r) + 11 -J+(6r)I x 2q(6r) for nucleophilic attack (20)

The reaction sites should have a stronger electron-donating ability or a larger electron-accepting capacitance when f has a larger density on the sites to make 1 - f smaller. Soft acids and base are polarizable, whereas hard acids and bases are not in their original definition.' The orbitals &(Sr) and &,oc(6r) are delocalized over a bond or over a few atoms, the former being bonding and the latter being antibonding between the reaction centers and the adjacent atoms. These occupied and unoccupied orbitals mix with each other to interact with a reagent by giving the orbital 6,. Perturbation theory tells us that polarizability of a bond is inversely proportional to the energy gap between the occupied and unoccupied orbitals.32-34 The hardness defined by eq 13 is regarded as providing a scale of polarizability of the reaction sites. One sees then that the three most important factors in interactions, the local power of an atom or a functional group in a reactant molecule to attract (or to donate) electrons (local electronegativity, -A), polarizability of the bonds around the reaction centers (local hardness, v), and the efficiency of the reactant molecule in utilizing the reaction site or the functional group (localizability, are combined altogether concisely in &(ar) and XUn,(6,). The localizability is

Orbital Interactions and Chemical Hardness

The Journal of Physical Chemistry, Vol. 98, No. 5, 1994 1439

TABLE 1: Local Electron-Accepting Capacitance and Acidic Hardness of Boron Compounds BH3' B(CH3)ib BF3" 9.13 L4,) (ev) 6.32 8.03 6.32 4.51 -1.18 V 6 r ) (ev) 11.36 13.05 da,)(ev) 1

1 -a2

0.845 3.52 1.33 0.963

local acidic hardness (eV) 0.00 LUMO energy (eV) 6.32 1 LUMO amplitude on B D3h symmetry. b C3" symmetry.

0.604 10.31 9.13 0.960

enolate ion hard



thiocyanate ion

1 2 9 eV ...................... Q



5.87e~ N






one wishes to estimate how the softness or hardness of each center is modified by substituent groups and by the change in peripheral structures. Incidentally, it might appear to be strange a t first sight to argue the local electronegativity and local hardness of a structural unit in a molecule in the light of the "electronegativity equalization principle".'* Chemical potential should be constant everywhere in a system at equilibrium. In reacting systems, however, electron delocalization takes place between the reaction sites or between some functional groups of the reagent and reactant to generate new chemical bonds.11J2 It is the role of polarization interaction to redistribute electrons between the structural unit and the remaining part within each of the reagent and reactant molecules torelax thesystemor toequalize thechemical potential everywhere in each species. The polarization interaction is important in shifting the ionic or radical center within a molecule in reactions, but its energy is much smaller than the delocalization stabilization in usual cases.32bJ7 Chemical Reactivities. In the interaction between an electron acceptor A and an electron donor B, we obtain

hard soft Figure 3. A comparison of local hardness of different centers in enolate ion and in thiocyanate ion.

connected directly with the Fukui function amplified in the frontier of interaction to indicate to what extent the reaction sites are made free from the remaining part of the molecule to interact with the reagent. Table 1 presents thelocal hardness q and theelectron-accepting capacitance A,, of BH3, B(CH3)3, and BF3. The 2pa A 0 of the boron atom has been taken as 6,. The 2p A 0 of the boron atom is shown to have a lower energy in B(CHJ3 than in BH3, as indicated by the smaller A value. Donation of electronic charge from the methyl groups to the B 2p A 0 gives the A 0 a probability of being found in the occupied M O space. The electron-accepting capacitance of the boron center is weaker due to the hardness term. The electrophilicity of the 2p A 0 of the boron center has been reduced significantly in BF3. In experiment, BH3 is classified as a soft acid and BF3 as a hard acid, while BR3 is located on the boundary.3s One may notice here that BH3 does not give 6 , for the boron 2 p r A 0 as 6, and, hence, the hardness is not defined. A similar situation happens also in the absolute hardness by Parr and Pearson in the case of p r ~ t o n . ~AJ ~proton does not have the HOMO level. Furthermore, the HOMO and the LUMO of BH3 belong to different symmetries and, therefore, it seems to have little physical or chemical significance to define 1/2(€LUMO - €HOMO) as the absolute hardness of the molecule. Figure 3 compares the hardness of the carbon center with that of the oxygen center in an enolate ion and the hardness of the nitrogen center with that of the sulfur center in a thiocyanate ion. The 2 p r or 3pa A 0 of each center has been taken as the reference and projected on to the occupied M O space of each species in STO-6G M O calculations. The result of calculation is clear, being in agreement with what has been deduced from experiments as d e m ~ n s t r a t e d .The ~ ~ present treatment will be useful when

where 6,and 6, are defined for A and for B, respectively, to specify the orbitals that are responsible for the formation of new bonds between the reaction sites of these species. The subscripts A and B indicate that the quantities are defined for the reactants A and B in an isolated state. It has been shown in eqs 11 and 12 that A is located a t a point between X, and A,., Thus, in usual donor-acceptor interactions, we have AB > AA in accordance with the conclusion derived from the density functional treatment; (pBO- pAo)> 0 in eq 4. However, the second term on the righthand side of eq 21 is positive. The existence of the hardness term leads us to the familiar orbital interaction scheme (Scheme 1) in which electrons delocalize from the occupied orbital of a lower orbital energy to an unoccupied orbital of a higher orbital energy. The magnitude of the first term depends principally on the type of reaction in question. In the interaction between an electron donor and an electron acceptor, electrons are delocalized more easily from the former to the latter as the first term becomes larger in the absolute value. In contrast, the first term will not be of significance when the reaction centers are like atoms in like molecules. Now, the chemical bonds formed in reactions are classified into three types in terms of the first and second terms. An ionic bond is formed when the second term is large and the first term is small in the absolute value. This corresponds to the interaction between a hard acid and a hard base and, therefore, to the charge-controlled reaction^.^ Next is the case in which the second term is small, but the first term is large in negative value. Electron delocalization takes place predominantly from B to A and, hence, the bonds to be formed are of the coordinate nature. The other is the case in which both the first term (in the absolute value) and the second term are small. Electron delocalization takes place from B to A and also from A to B in a similar magnitude. The bonds to be formed are of covalent nature and it is no more evident to specify which one is the acid and which one is the base. The latter two cases may correspond altogether

1440 The Journal of Physical Chemistry, Vol. 98, No. 5, 1994

Fujimoto and Satoh

to the frontier-controlled reactions in the earlier clas~ification.~ One may notice here another case in which both the first and second terms are large in absolutevalues. This type of interaction is governed not by the hardness but mainly by the electronegativities of reaction centers and may not be included in the usual HSAB theory. If eq 21 happens to give a negative value, it signifies that an electron will be transferred from B to A before bond formation takes place.38 The hardness estimates to what extent the reactive unoccupied orbital of the electron-acceptor part is elevated from AA and to what extent the reactive occupied orbital of the electron-donor part is lowered from AB. Equation 21 takes not only the local hardness but also the local electronegativity of the reaction sites into account, together with the efficiency of the donor and of the acceptor in utilizing their reactive units for the interaction. Thus, X, and A,, will serve by themselves or in combination as useful scales of reactivities, by defining the type of bonds to be formed and examining the local specificity of m o l e c ~ l e s . 3 ~ - ~ The stabilization energy associated with electron delocalization from B to A may be given in a second-order perturbation form by

Chemical hardness and softness have been discussed so far within the framework of density functional theory. The present study has demonstrated, however, that the concept of hardness, though not in its strictly mathematical form but in the form of finite difference approximation, can also be applied in the orbital interaction scheme when we focus our attention on local characteristics of interactions. The HOMO and the LUMO are the simplest form of our reactive orbitals and, accordingly, the argument presented above coincides with the discussion by Parr and Pearson at that e11d.~J3 Examination of local properties of molecules becomes more and more important, as we are concerned with more realistic problems. Our reactive measure makes it possible to compare or predict single-centered and multicentered reactivities of atoms and functional groups in large and complicated chemical systems by MO calculations, keeping alive the original concept of HSAB and the concept of phase matching of interacting orbitals.

m j + A


2fA+(6r)f~(6s)h(6r,6s)2 Xunoc:A(8r) - 'OC:B(~S)


in which h(6,, 6,) is a quantity in theunit of energy, being regarded approximately to be constant for a given type of reaction. Note here that the Fukui functions appear both in the numerator and in the denominator. The numerator will be large, while the denominator will be small, when both A and B are soft reactants. A similar relation is derived also for electron delocalization from A to B. Finally, Politzer and collaborators proposed a few years ago a reactivity scale for the one-center reactions called average local ionization p~tential.~l The scale is obtained formally by replacing di,? in eq 7 by 4i2and ci by Jeil. It includes the contribution of all theoccupied MO's, but thelocal natureof chemical interactions is not taken into account. For the multicentered reactions in which a certain symmetry is conserved, e.g., cycloadditions,2l we can utilize two reference orbital functions, one for the symmetric orbital interactions and one for the antisymmetric orbital interactions. The relative rates of the Diels-Alder reactions of substituted acetylenes, for example, have been correlated successfully with X, and A,, derived by taking the in-phase and out-ofphase combinations of the p r AO's of two reaction centers.42 The hardness may be defined for each set of symmetric and antisymmetric orbitals in this case. It is practical, however, to utilize X, and A,, as a simple measure of estimating nucleophilicity and electrophilicity of a structural unit by means of M O calculations. Conclusion Electron delocalization is the major driving force of chemical reactions. An analysis of electron delocalization has disclosed an attracting behavior of molecules in reactions that they recombine their M O s to give the orbitals that are most suitable for the formation of new bonds and the breaking of old bonds. These interacting orbitals have been illustrated to possess the interesting physical or mathematical properties that they are reproduced approximately by projecting the frontier M O of the smallest molecule or its principal component on to the occupied and unoccupied M O subspaces of larger reactant molecules. This is related obviously with the fact that chemical interactions are local by nature: reactants different in size and structure undergo the same type of reaction, while a compound undergoes a variety of reactions, as observed in experiments. We can thus clarify the factors that should govern the electron-donating ability and electron-accepting capacitance of an atom and a functional group in molecules.

Acknowledgment. This work was supported by the Ministry of Education, Science, and Culture of the Japanese Government through a grant-in-aid for scientific research. References and Notes (1) (a) Pearson, R. G. J . Am. Chem. SOC.1963,85,3533. (b) Pearson, R. G. J . Am. Chem. SOC.1985, 107, 6801. (2) Pearson, R. G. Hardandsoft Acids andBases; Doweden, Hutchinson and Ross: Strousbourg, PA, 1973. ( 3 ) Klopman, G . Chemical Reactivity and Reaction Paths; Wiley-Interscience: New York, 1974; DD 55-165. (4) Parr, R. G.; Donnelly, R. A.; Levy, M.; Palke, W. E. J . Chem. Phys. 1978, 68, 3801. (5) (a) Parr, R. G.; Bartolotti, L.J. J . Am. Chem. SOC.1982,104,3801. (b) Parr, R. G.; Yang, W. J . Am. Chem. SOC.1984,106,4049. (c) Parr, R. G.; Pearson, R. G. J . Am. Chem. SOC.1983,105, 7512. (6) (a) Hohenberg, P.; Kohn, W. Phys. Rev. 1964,136, B864. (b) Kohn, W.; Sham, L. J. Phys. Rev. 1965, 140, A1133. (7) Parr, R. G.; Yang, W. Density Functional Theory of Atoms and Molecules; Oxford Press: New York, 1989. (8) Sen, K. D. Structure and Bonding 80, Chemical Hardness; Springer-Verlag: Berlin, 1993. (9) Pearson, R. G. Acc. Chem. Res. 1993, 26, 250. (10) Parr, R. G.; Zhou, Z. Acc. Chem. Res. 1993, 26, 256. (11) (a) Fukui, K.; Fujimoto, H. Bull. Chem. SOC.Jpn. 1968, 41, 1989. (b) Fukui, K.; Fujimoto, H. Bull. Chem. SOC.Jpn. 1969, 42, 3399. (12) Fujimoto, H.; Inagaki, S.; Fukui, K. J . Am. Chem. Soc. 1976,98, 2670. (13) Pearson, R. G. J . Chem. Educ. 1987,64, 561. (14) Bader, R. F. W. Can. J . Chem. 1962,40, 1164. (15) Pearson, R. G. Symmetry Rules f o r Chemical Reactions; Wiley-Interscience: New York, 1976. (16) Iczkowski, R. P.; Magrave, J. L.J . Am. Chem. SOC.1961,83,3547. (17) Mulliken, R. S. J . Chem. Phys. 1934, 2, 782. (18) Sanderson, R. T. Science 1955, 121, 207. (19) Adopting the definitions of chemical potential and hardness by Parr and Pearson in terms of M u s , we obtain CLUMO:. - fHOMGB = (pa0 - p ~ o ) (TA+ 1 8 ) . Theinteractionbetweenanacid Aanda baseB may becharacterized by a stronger electron delocalization from B to A than that from A to B. Perturbation theory tells us that electrondelocalization is inversely proportional to the energy gap. Thus, we get (ZLUMOB - C H 0 M o . d - (fLuMoa - f H o M o 8 ) = 2(PB0 - p*O) > 0. (20) Pauling, L. The Nature of the Chemical Bond; Cornel1 University Press: Ithaca, NY, 1960. (21) Woodward, R. 8.; Hoffmann, R. The Conservation of Orbital Symmetry; Academic: New York, 1970. (22) Fukui, K. Theory of OrientationandStereoselection;Springer: Berlin, 1975 and references cited therein. (23) Mulliken, R. S.; Person, W. B. Molecular Complexes; Wiley: New York, 1969. (24) (a) Fukui, K.; Koga, N.; Fujimoto, H. J . Am. Chem. Soc. 1981,103, 196. (b) Fujimoto, H.; Koga, N.; Hataue, I. J . Phys. Chem. 1984,88, 3539. (25) As for the applications of the analysis to more complicated systems including transition metals, see, (a) Fujimoto, H.; Yamasaki, T.; Mizutani, H.;Koga,N.J. Am. Chem.Soc. 1985,107,6157. (b) Fujimoto,H.;Yamasaki, T. J . Am. Chem. SOC.1986, 108, 578. (26) Frisch, M. J.; Head-Gordon, M.; Trucks, G. W.; Foresman, J. B.; Schlegel, H. B.; Raghavachari, K.; Robb, M.A.; Binkley, J. S.;Gonzalez, C.; Defrees, D. J.; Fox, D. J.; Whiteside, R. A.; Seeger, R.; Melius, C. F.; Baker, J.; Martin, R. L.; Kahn, L. R.; Stewart, J. J. P.; Topiol, S.; Pople, J. A. GAUSSIAN 90; Gaussian Inc.: Pittsburgh, PA, 1990. (27) Fukui, K. J . Phys. Chem. 1970, 74, 4161.


Orbital Interactions and Chemical Hardness (28) Hammett, L. P. Physical Organic Chemistry; McGraw-Hill: New York, 1940; pp 184-199. (29) Fujimoto. H.; Mizutani, Y.; Iwase, K. J . Phys. Chem. 1986,90,2768. (30) (a) Yang, W.; Parr, R. G. Proc. Natl. Acad. Sci. U.S.A. 1985,82, 6723. (b) Ghosh. S. K.; Berkowitz, M. J . Chem. Phys. 1985,83, 2976. (c) Berkowitz, M.; Ghosh, S. K.; Parr, R. G. J . Am. Chem. Soc. 1985,107,681 1. (d) Berkowitz, M.; Parr, R. G. J . Chem. Phys. 1988,88, 2554. (3 1) The Fukui function is a measure of softness. Since 0 S / 1, it seems more reasonable to take 1 - f as a measure of hardness than to take I/f. A soft acid (or base) may be regarded as the one which has a small hardness as an acid (or as a base). The change in hardness should be rather gentle and linear. (32) (a) Fujimoto, H.; Hoffmann, R. J . Phys. Chem. 1974,78,1874. (b) Fujimoto. H.; Inagaki, S . J . Am. Chem. SOC.1977, 99, 7424. (33) Coulson, C. A.; Longuet-Higgins, H. C. Proc. R . SOC.London, A 1947, 192, 16. (34) Libit, L.; Hoffmann, R. J . Am. Chem. SOC.1974, 96, 1370. (35) Huheey, J. E.; Keiter, E. A.; Keiter, R. L. Inorganic Chemistry,

The Journal of Physical Chemistry, Vol. 98, No. 5, 1994 1441 Principles of Structure and Reactivity; Harper: New York, 1993; pp 344355. (36) (a) House, H. 0.; Auerbach, R. A.; Gall, M.; Pcet, N. P. J. Org. Chem. 1973,38,514. (b) Mitchell, P.C.H.; Williams, R. J. P.J. Chem.Soc. 1960, 1912. (c) Jergensen, C. K. Inorg. Chem. 1964, 3, 1201. (37) Fukui, K.; Fujimoto, H.; Yamabe, S . J . Phys. Chem. 1972,76,232. (38) See,for example, Eberson,L. Electkw Transfer Reactions in Organic Chemistry; Springer: Berlin, 1987. (39) Fujimoto, H.; Hatakeyama, K.; Kawasaki, S.;Oishi, Y. J. Chem. Soc., Perkin Trans. 2 1991, 989. (40) Reactions involving radical speciescan be treated in a similar manner by applying the unratricted open-shell scheme. (41) (a) Sjoberg, P.;Murray, J. S.; Brinck, T.; Politzer, P.Can. J . Chem. 1990, 68, 1440. (b) Murray, J. S.; Brinck, T.; Politzer, P. J . Mol. Srruct. (THEOCHEM 1992, 255, 271. . (42) Fujimoto, H.; Endo, J.; Mizutani, Y.; Fukui, K. J . Mol. Struct. (THEOCHEM) 1988, 163, 249.