function. Parr and Yang argue that a large value off(r) at a site favors reactivity at
that site, but to apply the concept to specific reactions they define three Fukui
functions (“condensed Fukui functions” [ 105 ]):
f$ðrÞ¼@rðrÞ
@N$
v$¼þ;"; 0 (7.41)The three functionsf+,fk", andfk^0 refer to an electrophile, a nucleophile, and a
radical. They are the sensitivity, to a small change in the number of electrons, of the
electron density in the LUMO, in the HOMO, and in a kind of average HOMO/
LUMO half-occupied orbital. Practical implementations of these condensed Fukui
functions are the “condensed-to-atom” forms of Yang and Mortier [ 155 ]:
fkþ¼qkðNþ 1 Þ"qkðNÞ for atom k as an electrophile
fk"¼qkðNÞ"qkðN" 1 Þ for atom k as a nucleophilefk^0 ¼1
2
½qkðNþ 1 Þ"qkðN" 1 Þ for atom k as a radical(7.42)
Hereqk(N) is the electron population (not the charge) on atom k, etc. (see
below). Note thatfk^0 is just the average offkþandfk". The condensed Fukui functions
measure the sensitivity to a small change in the number of electrons of the electron
densityat atom kin the LUMO (fkþ), in the HOMO (fk"), and in a kind of
intermediate orbital (fk^0 ); they provide an indication of the reactivity of atom k as
an electrophile (reactivity toward nucleophiles), as a nucleophile (reactivity toward
electrophiles), and as a free radical (reactivity toward radicals).
The easiest way to see how these formulas can be used is to give an example. Let’s
calculatefk"for the anion SCN". We’ll calculatefS",fC", andfN", to get an idea of the
nucleophilic power of the S, C and N atoms in this molecule. We need the electron
populationqon each atom or, what gives us the same information, the charge on each
atom: for an atom in a molecule, electron population¼atomic number"charge. To
see this, note that if an atom had no electron population, its charge would equal its
atomic number. For each electron added to the atom, the charge decreases by one. So
charge¼atomic number"electron population. We perform the calculations for the
N-electron species (SCN") and the (N"1)-electron species (SCN.). If we were
interested in the nucleophilic power of the atoms in a neutral molecule M, then to get
fk"we would calculate the electron populations or charges on the atoms in M and in
M+, and for the electrophilic power of the atoms in neutral M, to getfkþwe calculate
the electron populations or charges on the atoms in M and M". The calculations are
performed for the two species at the same geometry. In introducing the condensed
Fukui function Yang and Mortier [ 155 ] used for each pair of species a single
“standard” (presumably essentially average) geometry, with accepted, reasonable
bond lengths and angles, and other workers do not specify whether they use for,
say, M and M+, the neutral or the cation geometry. We will adopt the convention that
for a calculation on M (¼þ,"or.), both geometries will be those of M*, the
504 7 Density Functional Calculations