Parr and Yang defined the Fukui function, f(r), as the functional derivative of the
chemical potential with respect to the external potential at constant number of
electrons [5, 6]
f ðrÞ ¼
dl
dvðrÞ
N
ð8:4Þ
Note that because of Eq. 8.3, the Fukui function is really the second derivative of
the energy with respect to the electron number and the external potential. Assuming
that the energy is an exact differential, the crossed second order derivatives are
equal, and
f ðrÞ ¼
@qðrÞ
@N
v
ð8:5Þ
As for the chemical potential, the Fukui function has different values if the
derivative is taken from the right or the left. It can be demonstrated that, in the
canonical ensemble at zero temperature, the Fukui function is given by
f
þ
ðrÞ ¼ q N þ 1 ðrÞ À q N ðrÞ
ð 8:6Þ
When the derivative is taken by the right, and
f
À
ðrÞ ¼ q N ðrÞ À q NÀ1 ðrÞ
ð 8:7Þ
when the derivative is taken by the left.
Equations 8.5–8.7 are not valid if the ground state or its vertical anions are
degenerate states. In such cases, the functional derivative of Eq. 8.4 is ill-defined
and, a proper perturbation theory for degenerate states must be used to define Fukui
function analogues [25–27]. The Fukui function, approximated using the Eq. 8.6,
represents the molecular reactivity as an electron acceptor; whereas the Fukui
function, approximated using the Eq. 8.7, plays the same role in the reactivity of
electron donors. In a frozen orbital approximation Eqs. 8.6 and 8.7 are just the
square of the frontier orbitals, LUMO and HOMO, respectively. In this way, they
recover all the frontier orbital reactivity developed by Fukui [5, 28], making their
chemical interpretation clear. The regions of the molecule where the Fukui function
has a big value are the most prone to accept or donate electrons.
Hence, both Fukui functions are useful reactivity descriptors especially for
orbital controlled reactions. Considering the Eqs. 8.6 and 8.7, it is easy to see that
the Fukui function integrates to one:
Z
f ðrÞdr ¼ 1
ð8:8Þ
8 Topological Analysis of the Fukui Function
229
chemical potential with respect to the external potential at constant number of
electrons [5, 6]
f ðrÞ ¼
dl
dvðrÞ
N
ð8:4Þ
Note that because of Eq. 8.3, the Fukui function is really the second derivative of
the energy with respect to the electron number and the external potential. Assuming
that the energy is an exact differential, the crossed second order derivatives are
equal, and
f ðrÞ ¼
@qðrÞ
@N
v
ð8:5Þ
As for the chemical potential, the Fukui function has different values if the
derivative is taken from the right or the left. It can be demonstrated that, in the
canonical ensemble at zero temperature, the Fukui function is given by
f
þ
ðrÞ ¼ q N þ 1 ðrÞ À q N ðrÞ
ð 8:6Þ
When the derivative is taken by the right, and
f
À
ðrÞ ¼ q N ðrÞ À q NÀ1 ðrÞ
ð 8:7Þ
when the derivative is taken by the left.
Equations 8.5–8.7 are not valid if the ground state or its vertical anions are
degenerate states. In such cases, the functional derivative of Eq. 8.4 is ill-defined
and, a proper perturbation theory for degenerate states must be used to define Fukui
function analogues [25–27]. The Fukui function, approximated using the Eq. 8.6,
represents the molecular reactivity as an electron acceptor; whereas the Fukui
function, approximated using the Eq. 8.7, plays the same role in the reactivity of
electron donors. In a frozen orbital approximation Eqs. 8.6 and 8.7 are just the
square of the frontier orbitals, LUMO and HOMO, respectively. In this way, they
recover all the frontier orbital reactivity developed by Fukui [5, 28], making their
chemical interpretation clear. The regions of the molecule where the Fukui function
has a big value are the most prone to accept or donate electrons.
Hence, both Fukui functions are useful reactivity descriptors especially for
orbital controlled reactions. Considering the Eqs. 8.6 and 8.7, it is easy to see that
the Fukui function integrates to one:
Z
f ðrÞdr ¼ 1
ð8:8Þ
8 Topological Analysis of the Fukui Function
229