15.1 Introduction
The analysis of the electron charge density is one of the most often applied and
standard approaches to describe the system investigated; molecule, ion, greater
cluster or even crystal [1–3]. This is why the Quantum Theory of Atoms in
Molecules (QTAIM) seems to be a useful tool for such analysis since it concerns
directly the electron charge density distribution of the system analyzed [4, 5]. One
of ideas of the QTAIM approach is the partitioning of 3D electron charge density
space into fragments attributed to atoms (atomic basins); the properties of those
fragments are often transferable from one system to another one. In such a way it is
possible to consider volumes of atoms or to calculate charges integrated over those
volumes. This is important that the mentioned here theoretically calculated volumes
and charges may have experimental equivalents since it is possible to perform the
crystal structure determination to have the experimental electron charge density
distribution in crystal and further apply the QTAIM approach [1–3].
However that is not all; the detailed properties of the electron density, ρ(r), of the
system considered may be analyzed [5–7]. The critical points (CPs) denoted by the
coordinates r C , are those where the gradient of the electron density, ∇ρ(r C ), vanishes
and they correspond to minima, maxima or saddle points of the electron density.
The CP is labeled by giving the duo of values (ω,σ), where ω is the rank of CP
while σ is its signature. There are the following critical points; (3,−3)—the local
maximum often named as attractor is attributed to the position of atom, (3,−1)—the
saddle point which often is called the bond critical point (BCP), (3, +1)—the saddle
point which is called the ring critical point (RCP) and (3, +3)—the local minimum,
i.e. the cage critical point (CCP). The physical interpretation of critical points
mentioned here is very well known and it is discussed in numerous monographs and
review articles. This is important to announce here that the positions of (3,−3)
critical points (attractors) are attributed to the positions of atoms. However there is
an excellent agreement between the positions of non-hydrogen attractors and the
corresponding nuclei (at least the differences are much smaller than the experimental or theoretical errors) but there is noticeable disagreement between the
positions of hydrogen atom attractors (local maxima of the electron density) and
their nuclei [8]. This is later discussed in this chapter.
There is another important term useful to describe the distribution of the electron
density—the bond path which links pair of attractors [9, 10]. The two gradient paths
which originate at the bond critical point and terminate at each of the two attractors
define the bond path [11]. In other words the bond path (BP) is a line of the
maximum electron density linking the nuclei (more precisely attractors) of two
atoms. The bond critical point is that one at the bond path where the electron
density attains the minimum value. There are numerous studies on properties and
physical meaning of the bond path. It was pointed out that every bond path is
accompanied by a virial path [12]. The latter one is a line linking the same nuclei as
those connected by the bond path. The virial path is characterized by the maximally
negative potential electron energy density thus it is maximally stabilizing with
400
S.J. Grabowski
The analysis of the electron charge density is one of the most often applied and
standard approaches to describe the system investigated; molecule, ion, greater
cluster or even crystal [1–3]. This is why the Quantum Theory of Atoms in
Molecules (QTAIM) seems to be a useful tool for such analysis since it concerns
directly the electron charge density distribution of the system analyzed [4, 5]. One
of ideas of the QTAIM approach is the partitioning of 3D electron charge density
space into fragments attributed to atoms (atomic basins); the properties of those
fragments are often transferable from one system to another one. In such a way it is
possible to consider volumes of atoms or to calculate charges integrated over those
volumes. This is important that the mentioned here theoretically calculated volumes
and charges may have experimental equivalents since it is possible to perform the
crystal structure determination to have the experimental electron charge density
distribution in crystal and further apply the QTAIM approach [1–3].
However that is not all; the detailed properties of the electron density, ρ(r), of the
system considered may be analyzed [5–7]. The critical points (CPs) denoted by the
coordinates r C , are those where the gradient of the electron density, ∇ρ(r C ), vanishes
and they correspond to minima, maxima or saddle points of the electron density.
The CP is labeled by giving the duo of values (ω,σ), where ω is the rank of CP
while σ is its signature. There are the following critical points; (3,−3)—the local
maximum often named as attractor is attributed to the position of atom, (3,−1)—the
saddle point which often is called the bond critical point (BCP), (3, +1)—the saddle
point which is called the ring critical point (RCP) and (3, +3)—the local minimum,
i.e. the cage critical point (CCP). The physical interpretation of critical points
mentioned here is very well known and it is discussed in numerous monographs and
review articles. This is important to announce here that the positions of (3,−3)
critical points (attractors) are attributed to the positions of atoms. However there is
an excellent agreement between the positions of non-hydrogen attractors and the
corresponding nuclei (at least the differences are much smaller than the experimental or theoretical errors) but there is noticeable disagreement between the
positions of hydrogen atom attractors (local maxima of the electron density) and
their nuclei [8]. This is later discussed in this chapter.
There is another important term useful to describe the distribution of the electron
density—the bond path which links pair of attractors [9, 10]. The two gradient paths
which originate at the bond critical point and terminate at each of the two attractors
define the bond path [11]. In other words the bond path (BP) is a line of the
maximum electron density linking the nuclei (more precisely attractors) of two
atoms. The bond critical point is that one at the bond path where the electron
density attains the minimum value. There are numerous studies on properties and
physical meaning of the bond path. It was pointed out that every bond path is
accompanied by a virial path [12]. The latter one is a line linking the same nuclei as
those connected by the bond path. The virial path is characterized by the maximally
negative potential electron energy density thus it is maximally stabilizing with
400
S.J. Grabowski