Thus one resorts to contour maps or isosurface plots to represent them.
Unfortunately, they show only a part of the information contained in the function,
since they depend on the contour (or isosurface) value choice one decides to plot. In
order to have a more unambiguous way to analyze a three-dimensional (or higher
dimension) function, one could use the framework of the topological analysis. In
theoretical chemistry, this has already been done in the pioneer works of Bader,
which originated the Quantum Theory of Atoms in Molecules (QTAIM) [1]. Later
this topological analysis was applied to interpret the Electron Localization Function
[2–4], and lately it has been applied to the study of the Fukui function [5–7], which
is namely the object of this chapter.
We will start with the presentation of the Fukui function in the framework of the
Density Functional Reactivity Theory and its chemical interpretation, [8–14] followed by a brief account of the different ways to analyze it and ending with its
topological analysis. Finally, several applications of this analysis will be shown,
and some open problems will be discussed.
8.2 Fukui Function
Density Functional Theory is based on the existence of a functional of the electron
density, qðrÞ, which gives the ground state energy:
E½q ¼ F½q þ
Z
vðrÞqðrÞdr
ð8:1Þ
where F½q is the universal functional of Hohenberg and Kohn [15], and vðrÞ is the
external potential. Besides, there is a variational principle, which yields the following Euler-Lagrange equation:
l ¼
dF½q
dqðrÞ
þ vðrÞ
ð 8:2Þ
where l is the chemical potential present in the equation as the Lagrange parameter.
It can be demonstrated that the chemical potential is the derivative of the energy
with respect to the electron number N [16]
l ¼
@E
@N
v
ð8:3Þ
The derivative is well defined only for open systems and presents a discontinuity
at integer number of electrons [17–24]. Therefore, the derivative at integer number
of electrons has different values whether it is taken either by the right or the left.
228
P. Fuentealba et al.
Unfortunately, they show only a part of the information contained in the function,
since they depend on the contour (or isosurface) value choice one decides to plot. In
order to have a more unambiguous way to analyze a three-dimensional (or higher
dimension) function, one could use the framework of the topological analysis. In
theoretical chemistry, this has already been done in the pioneer works of Bader,
which originated the Quantum Theory of Atoms in Molecules (QTAIM) [1]. Later
this topological analysis was applied to interpret the Electron Localization Function
[2–4], and lately it has been applied to the study of the Fukui function [5–7], which
is namely the object of this chapter.
We will start with the presentation of the Fukui function in the framework of the
Density Functional Reactivity Theory and its chemical interpretation, [8–14] followed by a brief account of the different ways to analyze it and ending with its
topological analysis. Finally, several applications of this analysis will be shown,
and some open problems will be discussed.
8.2 Fukui Function
Density Functional Theory is based on the existence of a functional of the electron
density, qðrÞ, which gives the ground state energy:
E½q ¼ F½q þ
Z
vðrÞqðrÞdr
ð8:1Þ
where F½q is the universal functional of Hohenberg and Kohn [15], and vðrÞ is the
external potential. Besides, there is a variational principle, which yields the following Euler-Lagrange equation:
l ¼
dF½q
dqðrÞ
þ vðrÞ
ð 8:2Þ
where l is the chemical potential present in the equation as the Lagrange parameter.
It can be demonstrated that the chemical potential is the derivative of the energy
with respect to the electron number N [16]
l ¼
@E
@N
v
ð8:3Þ
The derivative is well defined only for open systems and presents a discontinuity
at integer number of electrons [17–24]. Therefore, the derivative at integer number
of electrons has different values whether it is taken either by the right or the left.
228
P. Fuentealba et al.