4591
J. Phys. Chem. 1994, 98, 4591-4593
The Hard and Soft Acids and Bases Principle: An Atoms in Molecules Viewpoint Jose L. Ghzquez’ and Francisco Mhdez Departamento de Quimica, Diuisibn de Ciencias Bhsicas e Ingenieria, Uniuersidad Autbnoma Metropolitana-Iztapalapa,A.P. 55-534, Msxico, D.F. 09340, Mexico Received: February I , 1994’
The chemical potential equalization principle is used to define the fukui function of the kth atom in a molecule A with N A eleCtrOnS,&k = q A k ( N A ) - q A k ( N A - I), for electrophilic attack, and& = q A k ( N A + 1) - q A k ( N A ) , for nucleophilic attack, the softness of an atom in a molecule, s i k = s x k , where SA is the global softness, and the hardness of an atom in a molecule = l/& ( q A k is the charge of the kth atom in the molecule). With these definitions it is shown that, in general, the reactive site of a molecule is located at the atom with the largest value of the fukui function (the softest atom), however, the interaction between two chemical species will not necessarily occur through their softest atoms, but rather through those whose fukui functions are approximately equal.
vik
I. Introduction
The hard and soft acids and bases (HSAB) principle has been very useful to explain the behavior of many chemical systems.’-‘ Recently, this principle has been invoked in a local sense in order to explain, in terms of density functional concepts such as the fukui function,5 the response of a chemical system to different kinds of reagents.*q6I3 The extrapolation of the general behavior “soft likes soft” and “hard likes hard”, locally, together with the idea that the larger the values of the fukui function, the greater the reactivity, seems to be a very useful approach to explain the chemical reactivity of a wide variety of systems. Certainly, the determination of the specific sites at which the interaction between two chemical species is going to occur is of fundamental importance to determine the path and the products of a given reaction. The object of the present work is to make use of the chemical potential equalization principle to introduce the concepts of the fukui function, and the hardness and the softness of an atom in a molecule and to show through these quantities that, indeed, greater values of the fukui function imply greater reactivity and that the HSAB principle may be invoked as a criterion to determine the reactive sites of two interacting species.
two quantities on the total number of electrons of the molecule A, NA, is explicitly indicated), N o k is the number of electrons of the isolated atom, and PAk and q~ are the chemical potential and the hardness of the kth atom in the molecule A. In general, these two quantities will be different from the isolated atom values, p,,k and l]ok, because they correspond to an effective chemical potential and an effective hardness. This may be seen better by considering the expression for the chemical potential of molecule A with ( N A - 1) electrons. According to eq 1, C(k(NA
-
= PA, + vAkqAk(NA -
(2)
Because of the chemical potential equalization principle, C ( ~ ( N A ) = MA(NA) and P k ( N A - 1) = ~ A ( N -A l), for all values of k,from is the chemical potential of molecule A with 1to K. Here /LA(NA) N A electrons, while ~ A ( N-A1) is the chemical potential of the same molecule with (NA- 1) electrons. Thus, the combination of eqs 1 and 2 leads to
and
11. The Effective Chemical Potential and the Effective
Hardness of an Atom in a Molecule Consider a molecule A formed by the binding of K atoms, with a total of N A electrons. According to the chemical potential equalization principle, the chemical potential of each atom in the molecule must be equal to the chemical potential of the molecule. This equalization is achieved through charge transfer among the constituent atoms and through the distortion of the electronic density produced by the change in the external potential (u(r)) of each atom due to the presence of all the other atomse5 Now, even though the chemical potential is a function of the number of electrons and the external potential, here we assume that the variation of the chemical potential of an atom in a molecule may be determined accurately through the expression
where Pk is the chemical potential when the number of electrons associated with the kth atom in A is N A k (the dependence of these
(4)
If the charges of all the atoms in A, q A k ( N A ) and q M ( N A - l), and the chemical potentials, c(A(NA)and ~ A ( N A l), are determined first through an independent procedure, for example, through molecular orbital calculations, eqs 3 and 4 indicate that PAk and TAk are, in general, different from the isolated atom values pok and q d because, through this procedure, one is indirectly taking into account the changes in the external potential of all the atoms when they form part of the molecule. The values of pAk and,qAkare unique for the kth atom in A. If the same atom forms part of a different molecule, B, it will have different values of P B k and t)Bk because it will be in a different chemical environment. However,it is assumed that they remain unchanged when the total number of electrons in the molecule changes with a fixed geometry. Now, the quantity ~ A ( N A-) I.(A(NA - 1) is the left finite differences approximation to the derivative ( a p ~ / & ” ) ~ which , is equal to the global hardness3 of the molecule, that is,
Abstract published in Advance ACS Abstracts, March 15, 1994.
0022-3654/94/2098-459 1$04.50/0
0 1994 American Chemical Society
4592 The Journal of Physical Chemistry, Vol. 98, No. 17, 1994
Ghzquez and MCndez
On the other hand, ( q A k ( N A ) - q A k ( N A - 1)) iS the left finite differences approximation to the derivative (aq/aIV)v,which is the condensed fukui function for electrophilic attack? that is, and .&k
= (aqAk/aNA);
qAdNA)
- qAk(NA -
(6)
Thus, substituting eqs 5 and 6 in eq 4 one finds that
Since the global softness is the inverse of the global hardnes~,~ one can express eq 7 in the form
where the changes in the external potential of A and B have been neglected, and the charge transfer from one species to the other one is such that /LAB = p~ + '/2?)A ANA = p~ 4- '/*qe A.~'B. Since ANA + Ah$ = 0, then
wheresib = 1/?&, is the softness of the kth atom in the molecule. Performing the summation over k one finds that K
K
and
It is interesting to note that, in the present context, not only the global quantities s i and q A are inverses of each other but also the local quantities S i k and q i k are inverses of each other. This situation implies that
where we have made use of the inverse relationship between SA and q ~ The . change in the grand potential of each species is given by"
K
in contrast with the usual definition of the local hardness and the local softness, or other definitions of the hardness and softness of an atom in a molecule.14J5 It is also important to mention that the additivity rule for the softness of the atoms in the molecule (eq 9) does not contradict the arithmetic mean law,16 because in the latter case one makes use of the isolated atom values of the softness, while in the former one makes use of the effective softnesses of the atoms in the molecule, and these two values, as we have already seen, are not necessarily equal to each other. One may carry out a similar derivation for the difference between the chemical potential of molecule A and its corresponding anion. In this case the condensed fukui function is expressed in terms of the right finite differences approximation to the derivative (dq/dN),, that is,
= (aq/an?:
E
qAk(NA
+ l) -
qAk(NA)
(l l )
and the global hardness is expressed in terms of the right finite differences approximations to the derivative ( a p A / a N A ) , , that is
In this case the properties expressed in eqs 7-10 remain the same, except that thesuperscript "-I must be replaced by the superscript This situation simply indicates that for each atom in the molecule there are two different reactivity indexes. For electrophilic attack one needs to use the left finite differences approximations, while for nucleophilic attack one needs to use the right finite differences approximations.
"+".
111. The Hard and Soft Acids and Bases Principle
Let us consider now the interaction energy between two chemical species A and B. For simplicity we will eliminate the "+" and "-" superscripts, however, it is implicitly assumed that one of the species will act as a nucleophile, while the other one will act as an electrophile. Thus, from a global point of view, one has that
and
where eq 15 has been used, and AQA + AQB = MA + AEB. In order to prove the HSAB principle, Chattaraj, Lee, and Parr" have shown that for a given chemical potential difference, pg - PA, and a given SB, the minimization of AQA with respect to SA leads, precisely, to SA = SB, while the minimization of AQB with respect to SB at fixed ( p -~ PA) and SA leads, precisely, to SB = SA. Under these conditions (An,),, = (Ail&,. If one considers the internal structure of A and B, each atom k in A must fulfill the condition AB = PA + V A k m A k , and each atom I in B must also fulfill the condition AB = FB + V B l A N B I . All atoms kin A start from p~ and arrive at AB through charge transfer. The same situation occurs with all the atoms I in B, which start from p~gand arrive at NAB. Since the value AB is determined from the global interaction between A and B and it is given by eq 15, one finds that
and
While the energy change and the grand potential change for k in A and for 1 in B is given by MAk
1 2 2 = z s A k ( p A B - PA) 1
mB/= ~sB/(P:B
-
(22)
(23)
The Journal of Physical Chemistry, Vol. 98, No. 17, 1994 4593
The Hard and Soft Acids and Bases Principle
and
One can see that because of the additivity rule expressed in eq 9 the summation of h E A k and AQAk over all atoms k in A and the summation of A& and AQBIover all atoms 1 in B leads to the global expressions, eqs 16 and 18 and 17 and 19, respectively, and therefore z f = l h Q A k z k l h n ~=~z f = i M A k + z k l A J ! ? ~ p Equations 20 and 21 indicate that the larger the value of the fukui function the greater the charge transfer. Since it seems reasonable to assume that the reaction site will be the one where the charge transfer is greatest, one may conclude that, indeed, the larger the fukui function, the greater the reactivity.18 Following Chattaraj, Lee, and Parr, one may also consider the minimization of AQAk with respect to SA at fixed ( l e - PA), SB, and fAk. From eq 24 one finds that the change of the grand potential of the kth atom becomes a minimum when SA = SB. Similarly, the minimization of AQB/ with respect to SB at fixed ( p ~ - p ~ ) , ~ ~ fBlleads , a n d tosg=sA. Attheminimum, (AQAk)min = - ( ' / 8 ) h B - PA)2sAfAk and (AQB/)min - ( ' / d b B - PA)2sAfB/. Thus, the global HSAB principle implies that the grand potential of all the atoms in A and of all the atoms in B becomes a minimum when both species have an approximately equal global softness. Now, if the interaction between A and B occurs through the kth atom of A and the lth atom of B, one may assume that the most favorable situation correspond to
+
(AoAk)min
(AoB/)&
(26)
fAk =fB/
and since sA= SB, then, using eq 4, one has that the interaction sites may be characterized through the condition %
'BI
IV. Final Remarks It should be mentioned that eq 28, as a result of the equality between SA and SB and the equality between fAk and fB/, is a particular case of a general expression in which SA may or may not be equal to SB and fAk may or may not be equal to fer, but (SAfAk) = (safer), and therefore s a = SBI. This more general statement could have been obtained directly from eqs 15-19 by assuming that when A and B interact through the kth and the lth atoms, respectively, one should replace SA and SB by SAk and S B ~ .In this case, the equilibrium chemical potential AB is determined by the softnesses of the atoms that participate directly in the interaction, and the minimization of the grand potential leads to the conditions SAk = SBI and (AQAk)fin = (AQB/)mint even if SA # SB and fAk # fsl. Thus, through this procedure one is led to a less restrictive verison of the local HSAB principle. However, we believe that it is necessary to carry out an extensive study of different systems, in order to test if the most favorable interaction occurs when s,, = SB andfAk =fB/ or when SAk = SBI,independent of the individual values. Nevertheless, the analysis presented in this work provides support to make use of the local HSAB principle to study the behavior of the reactive sites of molecules. Acknowledgment. We would like to thank A. Martinez for many valuable discussions. This work has been aided by a research grant from the Consejo Nacional de Ciencia y Tecnologla.
which means that
'Ak
these cases the interaction is not favored by the charge-transfer process between the kth atom in A and the lth atom in B. Equation 29 also indicates that the greater the values of fAk and fB/, the greater the stabilization energy, in agreement with the idea that the largest values of the fukui function are, in general, associated with the most reactive sites. However, if one assumes that eq 21 may characterize the reaction site, then the interaction between A and B will not necessarily occur through the softer atoms but through those whose fukui functions are approximately equal. This statement is in agreement with experimental facts.*
(28)
That is, the interaction between A and B is favored when it occurs through those atoms whose softnesses are approximately equal: the local HSAB principle. An argument in favor of eq 26 is that not only both atoms will be equally satisfied but also, because of eq 27, the summation of eqs 20 and 21 would be equal to zero, which means that the charge removed from one of them will be exactly equal to the charge gained by the other one. In addition, one can see that if the interaction energy is assumed to be dominated by the summation of the change in the energy of the atoms located at the reaction site, then, when fAk = &, from eqs 22 and 23 one finds that
References and Notes (1) Pearson, R.G.Hard andSofi Acids and Eases;Dowden, Hutchinson and Ross: Stroudsville: PA, 1973. (2) Klopman, G.,Ed. Chemical Reactiuiry and Reaction Paths; Wiley: New York, i974. (3) Parr, R. G.;Pearson, R. G. J. Am. Chem. SOC.1983,105, 7512. (4) Nalewajski, R. F. J . Am. Chem. Soc. 1984,106,944. (5) Parr, R. G.;Yang, W. Density-functional theory of atoms and molecules; Oxford: New York, 1989. (6) Yang, W.;Mortier, W. J. J . Am. Chem. Soc. 1986,108, 5708. (7) Lee,C.;Yang, W.;Parr,R.G.J.Mol.Struct.(THEOCHEM) 1988, 163, 305. (8) Langenaeker, W.;De Decker, M.; Geerlings, P. J . Mol. Struct. (THEOCHEM) 1990,207, 115. (9) Mbndez, F.;Galvh, M. In Density FunctionaIMethods in Chemistry; Labanowski, J. K., Andzelm, J. W., Eds.; Springer-Verlag: New York, 1991; p 387. (10) Langenaeker, W.;Demel, K.; Geerlings, P. J. Mol. Srruct. (THEOCHEM) 1991, 234, 329. (1 1) Mhdez, F.; Galvan, M.; Garritz, A.; Vela, A.; GBzquez, J. L. J. Mol. Struct. (THEOCHEM) 1992,277,81. (12) Galvan, M.; Dal Pino, A.; Joannopoulos, J. D. Phys. Rev. Lerr. 1993,
-.(13) Dal Pino, A.; Galvh, M.; Arias, T. A.; Joannopoulos, J. D. J . Chem.
70. 21. -~
This expression indicates that the change in the energy will be negative (greater stabilization) independently of the values of PA, PB, SA, and SB. However, when fAk and fB/ are very different, then, depending on the values of PA, PB,SA, and SB, the summation of eqs 22 and 23 may lead to positive values, indicating that in
Phys. 1993,98, 1606. (14) Mortier, W. J.; Ghosh, S.K.; Shankar, S. J. Am. Chem. Soc. 1986, 108. 4315. (15) Baekelandt, B. G.; Mortier, W. J.; Schoonheydt, R.A. In Chemical Hardness;Sen,K. D., Ed.;StructureandBonding 80;Springer-Verlag: Berlin, 1993;p 187. (16) Yang, W.;Lee, C.; Ghosh, S. K. J . Phys. Chem. 1985,89, 5412. (17) Chattaraj, P.K.; Lee,H.; Parr, R. G. J. Am. Chem. Soc. 1991,113, 1855. (18) Pearson, R.G.J . Chem. Educ. 1987.64,561.
