THE TIME EVOLUTION OF THE HARD AND SOFT ACIDS AND BASES THEORY - Theoretical and Computational Research in the 21st Century - page 39

several sites of a molecule determines which atom will form the coordi-
nate covalent bonds. This leads to the local HSAB principle.

1.2.16 LOCAL HARDNESS AND SOFTNESS

The hardness is not restricted to be constant everywhere, unlike the chemi-
cal potential. Therefore, local hardness can be assigned. Although Pearson
[9] made first contribution to the concept in early 1963, the notion of local
hardness was first introduced by Berkowitz et al. [32].
Berkowitz et al. [32] have derived the expression for local softness that
reveals its relation to its reciprocal property, local hardness.
The idea is to defi ne the appropriate two-variable kernels for hardness
and softness, and then to generate local hardness and local softness from
the corresponding kernel equations.
Here, we consider a ground state, or change of one ground state to
another for which ρ( r ) determines all properties; it determines μ and v ( r ).
Starting from the Hohenberg-Kohn theorem, the modifi ed potential can be
written as follows:

u ( r ) = v ( r ) − μ = δF[ρ]/δρ( r ) (58)

where u ( r ) is a functional of ρ( r ), and the functional derivatives looks thus:

−2η( r , r ) = δu[ r ]/δρ( r ') = δu[ r ']/δρ( r ) (59)

where η( r , r ) is the hardness kernel.
The calculation of the local hardness is diffi cult; therefore, the local
softness defi ned by Yang and Parr [60(b)] is as follows.
As u ( r ) is a functional of ρ( r ), the reverse i.e., ρ( r ) is a functional of
u ( r ) and its functional derivative also exists.
The softness kernel is defi ned as follows:

-s( r , r ') = δρ [ r ]/δu( r ') = δρ [ r ']/δu( r ) (60)

Now, using Eqs. (59) and (60), we can identify the local hardness and
softness terms as follows:

s ( r ) = ∫ s ( r , r ′)dr′ (61)