Theor Chem Acc (2015) 134:117
1 3
number of critical spheres observed is related to its period.
The number of SX is equivalent to its atomic period, while
the number of SM equals one less than the period. The
boundary traced out by the minimum critical spheres marks
the boundary between adjacent shells for an atom. It can be
readily inferred from both the spherical symmetry of atoms
and Fig. 1 that the radial density topology for a H, F, and
Cl atom have 1, 2, and 3 SXs, as well as 0, 1, and 2 SMs,
respectively.
We emphasize that the trend between the atomic period
and the number of observed SXs and SMs is limited to the
fi rst three periods [ 14 ]. In a study by Smith et. al. of the
radial density of neutral atoms from hydrogen through uranium, the radial density of atoms that were heavier than the
third period exhibited fewer SXs than the actual number of
shells [ 18 ].
3.2.3 Molecular topology
The radial density topology of molecules is more complex
than atoms. The topology of the simple F 2 diatomic has
numerous critical points and critical rings as depicted in Fig. 5
and described in Table 1 . A PM occurs at each F nucleus.
Two more PM arise adjacent each F nucleus along the internuclear axis which marks the boundary between the core and
valence regions. Also along the internuclear axis, residing
in the middle of the two nuclei, is the maximum bond point
(PX) depicted in red. This shows a buildup of radial density
at the center of the bond. A saddle minimum point (PSM) is
found in the nonbonding region for each F atom on the internuclear axis, which is shown as a green point. More interesting features also arise off the internuclear axis including two
maximum rings (RX) and two saddle rings (RS).
Table 1 Name, acronym,
sign of eigenvalue, rank, and
signature for each critical
architecture in ABIM
Architecture
Name
Acronym
1
2
3
r
s
Point
Max point
PX
−
−
−
3
−
Saddle max point
PSX
−
−
+
3
−1
Saddle min point
PSM
−
+
+
3
+1
Min point
PM
+
+
+
3
+3
Ring
Max ring
RX
−
−
0
2
−2
Saddle ring
RS
−
0
+
2
0
Min ring
RM
0
+
+
2
+2
Sphere
Max sphere
SX
−
0
0
1
−
Min sphere
SM
0
0
+
1
+
Trivial case
Infi nite point
IP
0
0
0
0
0
Fig. 5 The contour ( left ) and
gradient vector fi eld ( right ) of
ρ rad (r) for F 2 shows numerous critical points (PX = red ,
PSM = green , PM = blue )
and critical rings (RX = red ,
RS = green )
64
Reprinted from the journal
1 3
number of critical spheres observed is related to its period.
The number of SX is equivalent to its atomic period, while
the number of SM equals one less than the period. The
boundary traced out by the minimum critical spheres marks
the boundary between adjacent shells for an atom. It can be
readily inferred from both the spherical symmetry of atoms
and Fig. 1 that the radial density topology for a H, F, and
Cl atom have 1, 2, and 3 SXs, as well as 0, 1, and 2 SMs,
respectively.
We emphasize that the trend between the atomic period
and the number of observed SXs and SMs is limited to the
fi rst three periods [ 14 ]. In a study by Smith et. al. of the
radial density of neutral atoms from hydrogen through uranium, the radial density of atoms that were heavier than the
third period exhibited fewer SXs than the actual number of
shells [ 18 ].
3.2.3 Molecular topology
The radial density topology of molecules is more complex
than atoms. The topology of the simple F 2 diatomic has
numerous critical points and critical rings as depicted in Fig. 5
and described in Table 1 . A PM occurs at each F nucleus.
Two more PM arise adjacent each F nucleus along the internuclear axis which marks the boundary between the core and
valence regions. Also along the internuclear axis, residing
in the middle of the two nuclei, is the maximum bond point
(PX) depicted in red. This shows a buildup of radial density
at the center of the bond. A saddle minimum point (PSM) is
found in the nonbonding region for each F atom on the internuclear axis, which is shown as a green point. More interesting features also arise off the internuclear axis including two
maximum rings (RX) and two saddle rings (RS).
Table 1 Name, acronym,
sign of eigenvalue, rank, and
signature for each critical
architecture in ABIM
Architecture
Name
Acronym
1
2
3
r
s
Point
Max point
PX
−
−
−
3
−
Saddle max point
PSX
−
−
+
3
−1
Saddle min point
PSM
−
+
+
3
+1
Min point
PM
+
+
+
3
+3
Ring
Max ring
RX
−
−
0
2
−2
Saddle ring
RS
−
0
+
2
0
Min ring
RM
0
+
+
2
+2
Sphere
Max sphere
SX
−
0
0
1
−
Min sphere
SM
0
0
+
1
+
Trivial case
Infi nite point
IP
0
0
0
0
0
Fig. 5 The contour ( left ) and
gradient vector fi eld ( right ) of
ρ rad (r) for F 2 shows numerous critical points (PX = red ,
PSM = green , PM = blue )
and critical rings (RX = red ,
RS = green )
64
Reprinted from the journal