J. Phys. Chem. 1995, 99, 6424-6431
6424
Development of Local Hardness Related Reactivity Indices: Their Application in a Study of the SE at Monosubstituted Benzenes within the HSAB Context W. Langenaeker, F. De ProftJ and P. Geerlings" Eenheid Algemene Chemie (ALGC), Fakulteit Wetenschappen, Vrije Universiteit Brussel (VUB), Pleinlaan 2, I050 Brussels, Belgium Received: August IO, 1994; In Final Form: January 30, 1 9 9 9
The local hardness, f j n ( ~ ) a, local quantity describing reactivity following Pearson's hard and soft acids and bases (HSAB) principle, is analyzed in detail. These considerations lead to the introduction of two possible working equations for the local hardness and a new quantity, the hardness density, hA@, better fitting the picture of a counterpart of the local softness. On the basis of the study of intra- and intermolecular reactivity sequences for an electrophilic attack on monosubstituted benzenes PhX (X = 0-, N H 2 , OH, F, CN, NOz, CHO, NH3+, and CHCH2) a better insight into the physical meaning of these density functional theory related quantities, and the molecular electrostatic potential, is acquired. These new insights in tum lead to the introduction of a general form of a new global reactivity index combining the local softness and the newly defined hardness density.
Introduction Density functional theory (DFT)' has been the subject of increasing interest in the past decade under the impetus of R. G. Parr and co-workers.' The theory finds its origin in two famous theorems by Hohenberg and Kohn2 promoting the electron density as the fundamental quantity of a many-electron system, bearing in it all information on atomic and molecular ground-state properties. A series of quantities, which are readily used when considering chemical reactivity, appear in a most natural way within the framework of this theory. For example, a new theoretical basis is found for the use of the frontier molecular orbitals (FMO),being the highest occupied and lowest unoccupied molecular orbitals (HOMO and LUMO), as reactivity indices. This concept was introduced by Fukui in the FMO t h e ~ r y ,a~simplified .~ and widely used form of the perturbation approach to molecular orbital t h e ~ r y .These ~ FMOs, which contain information about the site selectivity of a reaction, can be seen as an approximation to the Fukui functions introduced in DFT. Another quantity appearing in a natural way within density functional theory was identified with the hardness of a system. The concepts of hardness and softness were introduced by Pearsod when discussing acid-base reactions of the type A -t :B
-
A:B
with A being a Lewis acid (electron pair acceptor) and B being a Lewis base (electron pair donor). According to this HSAB (hard and soft acids and bases) principle, hard acids prefer to react with hard bases, and soft acids prefer to react with soft bases. The knowledge of the hardness and softness of a system can therefore help to explain and predict the chemical behavior of Lewis acids and bases. The global softness and global hardness can easily be calculated from the ionization potential (E)and electron affinity (EA) of the system considered (atoms, molecules, groups), quantities for which experimental data are available or which can be calculated in a uniform, nonempirical Their local
* Author to whom correspondence should be sent. t Aspirant N.F.W.O.
@Abstractpublished in Advance ACS Absrracfs, March 15, 1995.
0022-365419512099-6424$09.0010
counterparts, which seem especially interesting when considering site-selective reactivity within a substrate, cannot be obtained as easily. In previous studies the Fukui f u n ~ t i o n ,a~ local softness related quantity, and the local softness* itself, two quantities for which the calculation schemes are well-known, have been investigated for a variety of series of system^.^-'^ In this work some theoretical considerations about the local hardness,15 q n ( ~ )a, quantity for which no routinely applicable calculation scheme is known, are presented. These considerations lead to two possible working equations for the local hardness and the introduction of a new quantity, the hardness density, h,(L), better fitting the picture of a counterpart of the local softness. All of these quantities are investigated, together with the molecular electrostatic potential,16 for a series of systems, the monosubstituted benzenes, which was already the subject of a study of the Fukui function.1° The nine substituents, 0-, N H 2 , OH, F, CN, N02, CHO, NH3+, and CHCH2, present in this series, show different inductive and mesomeric (resonance) effects, which determine both the relative reactivity within a given system (the intramolecular reactivity sequence) and the relative reactivity of different systems in a series (the intermolecular reactivity sequence). Theory
(1) Basic Framework. Within the framework of density functional theory the absolute chemical hardness 7, a global quantity, is defined as17
where p is the electronic chemical potential identified'* as the negative of the electronegativity as defined by Iczkowski and Margrave.lg N is the number of electrons, and Y(:) is the external potential. The quantity 7, known from Pearson's HSAB prin~iple,~ measures the resistance of the chemical potential to a change in the number of electrons, or the influence of changing the number of electrons on the electronegativity. In a finite difference approximation, supposing a quadratic E = E(N) relation, the working equation for the global hardness is given by17 0 1995 American Chemical Society
Local Hardness Related Reactivity Indices
J. Phys. Chem., Vol. 99, No. 17, 1995 6425
+
This quantity has gained recent interest within DFT (see, for example, the contribution of Gazquez in ref 20 and recent papers, mostly by the same author, on the change of global hardness with the The global softness S,* defined as the inverse of the global hardness 7, (3) can be approximated as S=-
1
IE - EA
for the systems with NO 1, NO, and NO - 1 electrons, respectively. (2) Local Hardness. The search for a local counterpart of hardness @([) starts, in analogy with the local softness (eq 3, from writing @(c) as
as deduced from eq 1.l 5 An explicit form of the local hardness @(:) can be obtained starting from the Euler equation resulting from the application of the variation principle to the energy functional:'
(4)
The local softness S(L) can be defined as
with F&(:)] containing the kinetic energy and electronelectron interaction energy. When multiplying eq 14 by a composite function A ( Q ( ~ ) , ' which ~ integrates to N ,
demanding that
J&?(_rN
dL = N
(15)
one obtains after integration of both sides Starting from eq 5, s(iJ can also be written as (7) or, using (3),
where f(r) is the Fukui f ~ n c t i o n . ~The last part of eq 8 indicates that local softness measures the sensitivity of the chemical potential of a system to a local external perturbationz4 and therefore is expected to contain information about the relative reactivity of different sites in a molecule. Considering eq 8, it is obvious that the local softness S(L) contains the same information as the Fukui function Kc)plus additional information about the total molecular softness, and therefore about the global reactivity with respect to a reaction partner with a given hardness (or softness), as stated in the HSAB principle. It is therefore advisable to use&) only in studies of intramolecular reactivity sequences (Le. relative site reactivity in a molecule) and s(c) in studies of intermolecular reactivity sequences. As one expects e(c)to be a discontinuous function of N , two and left types of local softnesses, containing the right V(r)) (f(c)) derivative of e([)with respect to N at the N value considered (NO),are of interest:
Taking the functional derivative with respect to external potential Y,leads to
e, at constant
yielding
Using eq 14, we obtain
This leads to an additional constraint for the composite function
S+(L)
=f
