Computational Chemistry

and from Eq.7.34

¼

1

2

@^2 E

@N^2



v

¼

1

2

@m

@N



v

¼"

1

2

@w

@N



v

(7.38)

The one-half factor is [ 150 ] to bringinto line with Eq.7.32, where this factor
arises naturally in applying the three-point approximation and the definitions ofI
andAto the rigorous Gibbs equation (Eq.7.30) for electronic chemical potential.
Electronegativity has also been expressed in terms of orbital energies, by takingI
as the negative of the HOMO energy andAas the negative of the LUMO energy
[ 150 ]. This gives

¼

ðELUMO"EHOMOÞ

2

(7.39)

Like the analogous expression for electronegativity (Eq.7.33), this requires only
a “one-pot” calculation, of the HOMO and LUMO. Much of what was said about
Eq.7.33applies to Eq.7.39. Table 7.12 gives values ofcalculated analogously to
thewvalues discussed above. The HOMO/LUMO hardness values are in even
worse agreement with theI/Aones than are the HOMO/LUMO electronegativity
values with theI/Avalues (the zero values for the HOMO/LUMO-calculatedof
the radicals values arise from taking the half-occupied orbital [semioccupied MO,
SOMO] as both HOMO and LUMO). The orbital view of hardness as the HOMO/
LUMO gap is discussed by Pearson, who also reviews the principle of maximum
hardness, according to which in a chemical reaction hardness and the HOMO/
LUMO gap tend to increase, potential energy surface relative minima represent
species of relativemaximumhardness, and transition states are species of relative
minimumhardness [ 150 ]. In papers by Tore-Labbe ́and Nguyen these general ideas
about hardness are expounded [ 152 ] and the reciprocal concept of softness is used
(with the Fukui function) to rationalize some cycloaddition reactions [ 153 ].
The Fukui function (the frontier function) was defined by Parr and Yang [ 144 ] as

fðrÞ¼

dm

dvðrÞ



N

¼

@rðrÞ

@N



v

(7.40)

This says thatf(r) is the functional derivative (Section7.2.3.2,The Kohn–Sham
equations) of the chemical potential with respect to the external potential (i.e. the
potential caused by the nuclear framework), at constant electron number; and that it
is also the derivative of the electron density with respect to electron number at
constant external potential. The second equality showsf(r) to be the sensitivity of
r(r) to a change inN, at constant geometry. A change in electron density should be
primarily electron withdrawal from or addition to the HOMO or LUMO, the
frontier orbitals of Fukui [ 154 ] (hence the name bestowed on the function by Parr
and Yang). Sincer(r) varies from point to point in a molecule, so does the Fukui

7.3 Applications of Density Functional Theory 503