electron density (AIM) and the Electron Localization Function (ELF) [3, 4, 35, 45].
The Fukui function, being a scalar field in <
3 itself, can also be analyzed topologically. In this case, the critical points correspond to maxima, minima and saddle
points. They can be located by the analysis of its gradient fields. The maxima are
called attractors, which many times have a physical interpretation. For instance, the
Fukui function, like the electron density, has a cusp condition at the nuclei positions. Therefore, it will always have an attractor at the atomic positions. However, it
can also have attractors in other positions. We will see that this characteristic has an
interesting chemical interpretation. It is useful to define the f-localization domains
as the volumes enclosed by the isosurface f ðrÞ ¼ f involving all the points for
which f ðrÞ ! f . They are called reducible when they contain more than one
attractor and, irreducible when they contain only one attractor. Each attractor is
characterized by its basin, which is the set of points lying on the trajectories ending
in this attractor. The basins are irreducible domains (they do not overlap) and the set
of all basins fills the complete space. Hence, the whole space is partitioned into
basins of attractors, and any observable physical property can be defined in those
regions. For instance, for a basin X k , one can calculate the number of electrons
contained in this basin as:
N k ¼
Z
X k
qðrÞdr
ð8:17Þ
The overall sum of the N K naturally results in the total number of electrons. One
can also define the condensed Fukui function as its integration over each basin as:
f
Æ
k ¼
Z
X
Æ
k
f
Æ
ðrÞdr
ð8:18Þ
The interpretation remains the same. Note that the basins of the Fukui function
f
þ and f
À will be different.
8.4 Some Selected Applications
In this section we present some representative examples to show how the topological analysis of the Fukui function works. Accordingly with the literature [44,
46, 47] the results are commonly reported in two ways: the first involves a
3D-representation of f r
ð Þ where one isosurface is selected to plot in a way that can
represent all the Fukui basins. Accompanying these isosurfaces are the values of the
Fukui function condensed in the corresponding basins. The second way makes it
simple to compare with other methods used to condense the Fukui function, due to
the fact that all atomic contributions are reported as a single value over each atom
k. The geometrical optimization and electronic structure calculation have been
232
P. Fuentealba et al.
The Fukui function, being a scalar field in <
3 itself, can also be analyzed topologically. In this case, the critical points correspond to maxima, minima and saddle
points. They can be located by the analysis of its gradient fields. The maxima are
called attractors, which many times have a physical interpretation. For instance, the
Fukui function, like the electron density, has a cusp condition at the nuclei positions. Therefore, it will always have an attractor at the atomic positions. However, it
can also have attractors in other positions. We will see that this characteristic has an
interesting chemical interpretation. It is useful to define the f-localization domains
as the volumes enclosed by the isosurface f ðrÞ ¼ f involving all the points for
which f ðrÞ ! f . They are called reducible when they contain more than one
attractor and, irreducible when they contain only one attractor. Each attractor is
characterized by its basin, which is the set of points lying on the trajectories ending
in this attractor. The basins are irreducible domains (they do not overlap) and the set
of all basins fills the complete space. Hence, the whole space is partitioned into
basins of attractors, and any observable physical property can be defined in those
regions. For instance, for a basin X k , one can calculate the number of electrons
contained in this basin as:
N k ¼
Z
X k
qðrÞdr
ð8:17Þ
The overall sum of the N K naturally results in the total number of electrons. One
can also define the condensed Fukui function as its integration over each basin as:
f
Æ
k ¼
Z
X
Æ
k
f
Æ
ðrÞdr
ð8:18Þ
The interpretation remains the same. Note that the basins of the Fukui function
f
þ and f
À will be different.
8.4 Some Selected Applications
In this section we present some representative examples to show how the topological analysis of the Fukui function works. Accordingly with the literature [44,
46, 47] the results are commonly reported in two ways: the first involves a
3D-representation of f r
ð Þ where one isosurface is selected to plot in a way that can
represent all the Fukui basins. Accompanying these isosurfaces are the values of the
Fukui function condensed in the corresponding basins. The second way makes it
simple to compare with other methods used to condense the Fukui function, due to
the fact that all atomic contributions are reported as a single value over each atom
k. The geometrical optimization and electronic structure calculation have been
232
P. Fuentealba et al.