QUANTUM COMPUTATIONAL STUDIES ON PORPHYCENE - Theoretical and Computational Research in the 21st Century - page 165

f ( r ) = [δμ/δv( r )] N (16)

Assuming that the total energy E as a function of N and functional of v ( r )
is an exact differential, the Maxwell relations between derivatives may
be applied to write the Fukui function or the frontier function as f ( r ) =
[∂ρ( r )]/∂ N ] v .
A very important condition can be found for the Fukui function is that
it must be normalized to 1.

∫ f ( r )d r = 1 (17)

Owing to the discontinuity of the chemical potential at integer N , the de-
rivative of μ= (d E /d N ) will be different if taken from the right or the left-
hand side. One has μ when the derivative is taken from above ( N + δ,
δ→0), μ when the derivative is taken from below ( N − δ, δ→0), and for the
cases where is not a net charge exchange a good approximation is to use
the average μ = ½(μ + μ). Therefore, one has three different functions f ( r )
= [∂ρ( r )/∂ N ], where the subscript α = ± indicate whether the derivative is
evaluated at ( N ± δ, δ→0), and f ( r ), which is the average of the other two.
Now, to answer the question “If a reagent ( R ) approaches to substrate ( S ),
what direction will be preferred?” one can assume that in the usual cases
the preferred direction is the one for which the initial dμ for the species S is
a maximum. The first term from the RHS of the Eq. (17) involved only the
global quantity and at the large distant is ordinary less direction sensitive
than second. Parr et al. [26, 28], therefore, opinioned that “ the preferred
direction is the one with largest f(r) at the direction side ” and gave us the
firm prediction -
Governing electrophilic attack f − ( r ) = [∂ ρ ( r )/∂ N ] − v (18)

Governing nucleophilic attack f + ( r ) = [∂ ρ ( r )/∂ N ] + v

(19)

Governing neutral attack f 0 ( r ) = [∂ ρ ( r )/∂ N ] 0 v

(20)

These three cases have μ s >μ r , μ s <μ r and μ s ~μ r .
Computation of the local reactivity indices like the Fukui function in
the real molecular space is a very demanding task but diffi cult to evaluate
without further approximation. A “frozen core” approximation now gives
d ρ = d ρ value in each case and therefore governs electrophilic attack,

f − ( r ) ≈ ρ (HOMO) ( r ) (21)