Theor Chem Acc (2015) 134:117
1 3
graph in red. The core region of each nucleus is easily
identifi ed in this cross section as the symmetrical, volcanoshaped cones that surround the nuclei. The core is separated from the distorted valence by a minimum in radial
density. Thus, the boundary between the core and valence
shells still exists for the atoms in the molecule.
Unlike the homonuclear diatomic F 2 , which has a symmetrical distribution of radial electron density, the heteronuclear diatomic ClF has an asymmetric distribution as
predicted due to the unequal sharing of electron density.
When Cl (at y = 0 bohr) and F (at y = 3.050 bohr) combine to form ClF, the single core shell in the F atom and
both core shells in the Cl atom remain mostly unchanged
in the bonded molecule as shown in Fig. 2 b. This fi gure
also shows that the RBCP is skewed toward the fl uorine
nucleus. This feature makes sense because F is more electronegative than chlorine, and Cl is fairly polarizable. In
Fig. 2 d, the cross section of the ClF molecule depicts the
two core shells for Cl and the single core shell for F.
As shown in Fig. 2 , F 2 and ClF show very interesting
and intuitive features with respect to the core, bonding, and
the nonbonding regions. In the core region, we see that the
core shells are transferable from an atom to a molecular
atom. In the bonding region, the distorted valence shells
overlap to form a bond with a single-bond critical point.
There are also topological features, such as the peaks off
the internuclear axis that require further investigation.
3.1.2.2 Second period hydrides Hydrogen atoms in molecules can often be problematic to model and detect due
to their relatively low electron density. However, although
small, hydrogen can undergo very diverse chemistry
depending on whether it is anionic, cationic, or neutral.
Molecular radial electron density provides an aesthetic way
to model hydrogen containing compounds and qualitatively
determine the nature of the hydride in these molecules.
The molecular radial density of hydrogen containing
compounds is more aesthetically pleasing to model and
visualize than electron density. When plotting molecular
electron density of hydrogen containing compounds, one
must readjust the scale and truncate the electron density of
heavier atoms in order to identify the electron density peak
of a hydrogen constituent. On the other hand, the molecular
radial density function of both hydrogen and heavy atoms
can be viewed on the same scale. This is a consequence of
the balance between the electron density and the squared
distance from the nucleus that allows the valence region to
become emphasized and the nuclear region to become less
overwhelming in comparison with plotting solely the electron density function. As a result, radial density is generally
shown on a scale between 0 and 2 e/a o .
Hydrogen AIM can be easily identifi ed in Figs. 3 and
4 , which show the radial density ( ρ rad ) and the gradient
paths of radial density ( ∇ρ rad ), respectively, for the second
period hydrides. Since hydrogen does not have any core
shells, we can easily identify the hydrogen nuclei as points
of zero radial density that are surrounded by a single distorted valence shell as shown in Fig. 3 . This leads to the
depressions (‘dimples’) seen in the radial density graphs. In
Fig. 4 , there exists a distinct boundary path that completely
encircles each hydrogen nucleus that results in hydrogen
having a very similar and easily identifi able shape. Larger
versions of the plots found in Figs. 3 and 4 can be found in
Figure S1–S16 of the supporting information.
The nature of hydrogen in the second row hydrides is
chemically diverse. Crossing the second period from LiH
to HF, hydrogen shifts from being anionic in LiH to more
covalently bonded in CH 4 to cationic in HF. Ultimately, we
would expect the molecular radial density to give insight
into the type of bonding (ionic/covalent) and the nature
of the hydrogen (cationic/anionic/neutral) in these types
of molecules. One promising method for identifying bond
type is by analyzing the curvature of the RBCP. In the
∇ρ rad plots, the boundary path at the RBCP is concave for
LiH and BeH 2 , fairly fl at for BH 3 , CH 4 , and NH 3 , and convex for H 2 O and HF, all with respect to the hydrogen. The
observed trend is concave curvature for anionic hydrogen,
fl at curvature for covalently bonded hydrogen, and convex
curvature for cationic hydrogen.
3.2 Topology
3.2.1 Terminology
The radial density of atoms and molecules is rich in topological features such as critical points, critical rings, and
critical spheres. These critical architectures can be identifi ed and characterized using the gradient and diagonalized
Hessian of radial density, respectively. Each critical feature
includes points where the gradient of radial density vanishes (∇ρ rad = 0) . The classifi cation (point, ring, sphere)
and magnitude (maximum, minimum, or saddle) of a critical feature is found by evaluating the diagonalized Hessian
of radial density at these points to yield three characteristic
eigenvalues that are invariant to coordinate rotation.
The set of eigenvalues { 1 , 2 , 3 } is referred to as
a spectrum. For consistency, the following convention
is employed to order the eigenvalues: 1 ≤ 2 ≤ 3 . The
rank and the signature are crucial for classifying critical architecture. The rank ( r ) is the number of nonzero
eigenvalues. A rank of one, two, or three represents a
critical sphere, ring, and point, respectively. The signature ( s ) is the sum of the signs of the three eigenvalues.
The sign is +1 for a positive eigenvalue, −1 for a negative eigenvalue, or zero for a vanishing eigenvalue. The
signature relative to the rank determines whether the
61
Reprinted from the journal
1 3
graph in red. The core region of each nucleus is easily
identifi ed in this cross section as the symmetrical, volcanoshaped cones that surround the nuclei. The core is separated from the distorted valence by a minimum in radial
density. Thus, the boundary between the core and valence
shells still exists for the atoms in the molecule.
Unlike the homonuclear diatomic F 2 , which has a symmetrical distribution of radial electron density, the heteronuclear diatomic ClF has an asymmetric distribution as
predicted due to the unequal sharing of electron density.
When Cl (at y = 0 bohr) and F (at y = 3.050 bohr) combine to form ClF, the single core shell in the F atom and
both core shells in the Cl atom remain mostly unchanged
in the bonded molecule as shown in Fig. 2 b. This fi gure
also shows that the RBCP is skewed toward the fl uorine
nucleus. This feature makes sense because F is more electronegative than chlorine, and Cl is fairly polarizable. In
Fig. 2 d, the cross section of the ClF molecule depicts the
two core shells for Cl and the single core shell for F.
As shown in Fig. 2 , F 2 and ClF show very interesting
and intuitive features with respect to the core, bonding, and
the nonbonding regions. In the core region, we see that the
core shells are transferable from an atom to a molecular
atom. In the bonding region, the distorted valence shells
overlap to form a bond with a single-bond critical point.
There are also topological features, such as the peaks off
the internuclear axis that require further investigation.
3.1.2.2 Second period hydrides Hydrogen atoms in molecules can often be problematic to model and detect due
to their relatively low electron density. However, although
small, hydrogen can undergo very diverse chemistry
depending on whether it is anionic, cationic, or neutral.
Molecular radial electron density provides an aesthetic way
to model hydrogen containing compounds and qualitatively
determine the nature of the hydride in these molecules.
The molecular radial density of hydrogen containing
compounds is more aesthetically pleasing to model and
visualize than electron density. When plotting molecular
electron density of hydrogen containing compounds, one
must readjust the scale and truncate the electron density of
heavier atoms in order to identify the electron density peak
of a hydrogen constituent. On the other hand, the molecular
radial density function of both hydrogen and heavy atoms
can be viewed on the same scale. This is a consequence of
the balance between the electron density and the squared
distance from the nucleus that allows the valence region to
become emphasized and the nuclear region to become less
overwhelming in comparison with plotting solely the electron density function. As a result, radial density is generally
shown on a scale between 0 and 2 e/a o .
Hydrogen AIM can be easily identifi ed in Figs. 3 and
4 , which show the radial density ( ρ rad ) and the gradient
paths of radial density ( ∇ρ rad ), respectively, for the second
period hydrides. Since hydrogen does not have any core
shells, we can easily identify the hydrogen nuclei as points
of zero radial density that are surrounded by a single distorted valence shell as shown in Fig. 3 . This leads to the
depressions (‘dimples’) seen in the radial density graphs. In
Fig. 4 , there exists a distinct boundary path that completely
encircles each hydrogen nucleus that results in hydrogen
having a very similar and easily identifi able shape. Larger
versions of the plots found in Figs. 3 and 4 can be found in
Figure S1–S16 of the supporting information.
The nature of hydrogen in the second row hydrides is
chemically diverse. Crossing the second period from LiH
to HF, hydrogen shifts from being anionic in LiH to more
covalently bonded in CH 4 to cationic in HF. Ultimately, we
would expect the molecular radial density to give insight
into the type of bonding (ionic/covalent) and the nature
of the hydrogen (cationic/anionic/neutral) in these types
of molecules. One promising method for identifying bond
type is by analyzing the curvature of the RBCP. In the
∇ρ rad plots, the boundary path at the RBCP is concave for
LiH and BeH 2 , fairly fl at for BH 3 , CH 4 , and NH 3 , and convex for H 2 O and HF, all with respect to the hydrogen. The
observed trend is concave curvature for anionic hydrogen,
fl at curvature for covalently bonded hydrogen, and convex
curvature for cationic hydrogen.
3.2 Topology
3.2.1 Terminology
The radial density of atoms and molecules is rich in topological features such as critical points, critical rings, and
critical spheres. These critical architectures can be identifi ed and characterized using the gradient and diagonalized
Hessian of radial density, respectively. Each critical feature
includes points where the gradient of radial density vanishes (∇ρ rad = 0) . The classifi cation (point, ring, sphere)
and magnitude (maximum, minimum, or saddle) of a critical feature is found by evaluating the diagonalized Hessian
of radial density at these points to yield three characteristic
eigenvalues that are invariant to coordinate rotation.
The set of eigenvalues { 1 , 2 , 3 } is referred to as
a spectrum. For consistency, the following convention
is employed to order the eigenvalues: 1 ≤ 2 ≤ 3 . The
rank and the signature are crucial for classifying critical architecture. The rank ( r ) is the number of nonzero
eigenvalues. A rank of one, two, or three represents a
critical sphere, ring, and point, respectively. The signature ( s ) is the sum of the signs of the three eigenvalues.
The sign is +1 for a positive eigenvalue, −1 for a negative eigenvalue, or zero for a vanishing eigenvalue. The
signature relative to the rank determines whether the
61
Reprinted from the journal