J. Phys. Chem. zyxwvu 1994, zyxwvuts 98, zyxwvu 4591-4593 4591 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 zyx 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 for nucleophilic attack, the softness of an atom in a molecule, sik zyxwvut = sxk, where SA is the global softness, and the hardness of an atom in a molecule vik = l/& (qAk 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. A with NA eleCtrOnS,&k = qAk(NA) - qAk(NA - I), for electrophilic attack, and& = qAk(NA + 1) - qAk(NA), 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. 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 NA 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 NAk (the dependence of these Abstract published in Advance ACS Abstracts, March 15, 1994. 0022-3654/94/2098-459 1 $04.50/0 two quantities on the total number of electrons of the molecule A, NA, is explicitly indicated), Nok 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, zy p,,k and l]ok, because they correspond to an effective chemicalpotential and an effective hardness. This may be seen better by considering the expression for the chemical potential of molecule A with (NA - 1) electrons. According to eq 1, C(k(NA - = PA, + vAkqAk(NA - (2) Because of the chemical potential equalization principle, C(~(NA) = MA(NA) and Pk(NA - 1) = ~A(NA - l), for all values of k, from 1to K. Here /LA(NA) is the chemical potential of molecule A with NA electrons, while ~A(NA - 1) is the chemical potential of the same molecule with (NA - 1) electrons. Thus, the combination of eqs 1 and 2 leads to and (4) If the charges of all the atoms in A, qAk(NA) and qM(NA - l), and the chemical potentials, c(A(NA) and ~A(NA - 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 qd 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,qAk are unique for the kth atom in A. If the same atom forms part of a different molecule, B, it will have different values of PBk 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(NA) - I.(A(NA - 1) is the left finite differences approximation to the derivative (ap~/&”)~, which is equal to the global hardness3 of the molecule, that is, 0 1994 American Chemical Society