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be reproduced by Gauss-type basis functions. The picture-change-affected DKH
(n,0) curves exhibit large errors in the close vicinity of the nucleus and are located
above the Dirac curve, whereas the picture-change-corrected DKH(
n
,
n
) contact
densities match the Dirac contact densities [as would the picture-change-corrected
DKH(
n
,
n
) and Dirac curves], even for the second-order DKH transformation of the
property operator.
6 Electron Density in Conceptual Theories
6.1 Conceptual Density Functional Theory
Many principles of chemical reactivity such as electronegativity, chemical hard-
ness/softness and frontier orbital theory had been proposed mostly as rather phe-
nomenological descriptive concepts some decades ago. Many pioneering studies
discussed for instance the Mulliken electronegativity [
115
-
118
], the hard and soft
acid and base (HSAB) principle [
119
-
123
], and frontier molecular orbital concepts
[
124
-
129
]. These concepts were then later unified and rigorously defined in the
framework of conceptual DFT. Conceptual DFT was defined by Parr and coworkers
[
130
], who recognized that the electronegativity can be defined as the negative of
the derivative of the energy with respect to the number of electrons for a constant
external potential. A detailed description of the history of conceptual DFT and
various reactivity descriptors can be found in [
131
]. In principle, conceptual DFT
relies on a Taylor series expansion of the total-energy density functional [
132
] with
respect to perturbations in the external potential
v
(
r
) and the number of electrons
N
:
Ev
½
ðÞþD
v
ðÞ;
N
þD
N
N
D
ð
dE
dv
2
E
@
E
1
2
D
@
2
ðÞ
d
3
r
¼
Ev
½
ðÞ;
N
D
N
þ
v
þ
ðÞ
N
@
N
ðÞ
@
N
2
v ðÞ
v ðÞ
ðð
N
ð
d@
d
2
E
1
2
E
2
d
3
r
d
3
r
0
þD
d
3
r
þ
N
D
ð
v
ðÞ
Þ
D
v
ðÞ
þ:
dv
ðÞdv
rðÞ
dv
ðÞ@
N
(59)
The external potential and the number of electrons are both functionals of the
electron density. The derivatives of the energy with respect to either the external
potential or the number of electrons or both of them can be identified with several
chemical descriptors, shown in Table
1
, that are related to chemical reactivity.
The mixed derivative which is first order in the external potential and in the number
of electrons is called the Fukui function
f
(
). It can be interpreted in two different
ways. The definition as a derivative of the electron density implies that it represents
the change in the electron density
r
(
r
when the number of particles
N
changes. However, the Fukui function characterizes also the response of the chemical
r
) at the point
r
