Theor Chem Acc (2015) 134:117
1 3
The most widely used approach is Bader’s quantum theory of atoms in molecules (QTAIM), which depicts AIM
as nonoverlapping [ 3 , 7 – 10 ]. Other popular models, such
as those of Stewart [ 4 , 11 ] and Hirshfeld [ 5 ], depict AIM
as overlapping spherical fuzzy electron densities. Both the
advantages and disadvantages of these methods have been
previously discussed [ 1 ].
This paper builds upon the preliminary fi ndings by Warburton et al. [ 1 ], which suggest that employing molecular
radial density to partition molecules into atomic and bonding contributions may provide a more intuitive scheme to
defi ne atoms and bonds. The ABIM model represents AIM
as atoms with overlapping electron densities and bonds in
molecules (BIM) as the region where this overlap occurs.
Radial density provides a good starting point for an
ABIM theory because the atomic radial density, r 2 ρ(r), of a
free atom in space has shell structure and contains a realistic
number of electrons in each shell. The connection between
the number and position of the critical points in the atomic
radial density distribution function and the atomic shell
structure has been well established [ 12 , 13 ]. In 1976, Politzer
and Parr [ 12 ] showed that the minimum in the atomic radial
density provides a well-defi ned and physically meaningful
separation of electron density into a core and valence region
for the second period atoms. Boyd [ 13 ] then established
that analogous minimum surfaces are also found for heavier
atoms and further showed that the fi rst 54 neutral atoms in
their ground state have the expected shell-occupation numbers. Therefore, partitioning the electron density of atoms
based on the minimum in radial density is quite effective. By
extension, we postulate that a molecular radial density function may effectively partition the molecular electron density
in an intuitive and physically meaningful way. However,
as Smith pointed out, extension of atomic radial density to
molecular systems is not straightforward [ 14 ].
Recently, the molecular radial density, ρ rad (r i ) , at a
grid point, r i , was defi ned as the sum of the radial densities of all molecular atoms at that grid point as given by
Equation 1 [ 1 ]. The weight, W A (r i ) , is a partition function
that represents the fraction of density owned by a particular atom in a molecule at each grid point in space at a distance r iA from atom A . The weight allows us to calculate
the electron density distribution associated with each AIM
(W A (r i ) ρ(r i ) = ρ A (r i )) . In particular, the Becke weight
is employed in this paper, which defi nes the ownership
of each point in the molecular space based on r iA and the
atomic Bragg–Slater radii [ 15 ].
(1)
ρ rad (r i ) =
N A
A
r
2
iA W A (r i ) ρ(r i )
=
N A
A
r
2
iA ρ A (r i ) 0 ≤ W A (r i ) ≤ 1
It is necessary to point out that any weighting scheme that
partitions the molecular space into atomic contributions
can be used, and the Becke weight is just one of many
possible choices. Other possibilities include Hilbert space
decomposition (e.g., Mulliken) and 3D space partitioning
(e.g., QTAIM [ 3 ]). The Becke weight was originally chosen to partition the molecular space because of its simplicity and ease of implementation. One additional benefi t of
the Becke weight is that it does not employ spherical averaging, and thus, distortion of atoms upon bonding can be
investigated.
The molecular radial density along the internuclear
axis has been studied previously [ 1 ]. This paper looks
at the radial density in 2D planes of various molecules
which reveal topologically interesting features residing
off the internuclear axis. We also present the terminology
employed to classify the radial density topology of atoms
and molecules. Most importantly, properties of ABIM will
be calculated and discussed.
2 Methodology
All the calculations were performed with the program
MUNgauss [ 16 ]. The molecular electron and radial densities were calculated at HF/6-311++G(d,p)// HF/6311++G(d,p), specifi cally using RHF for closed shell
and ROHF for open-shell molecules. All of the visual aids,
including contour, relief, and gradient plots, were created
using the Mathematica version 9.0 graphing package. The
gradient vector fi eld plots were calculated in Mathematica
from input radial density data.
Throughout this paper, length is given in bohr, radial
density ( ρ rad ) is given in e/bohr, the gradient of radial density ( ∇ρ rad ) is given in e/bohr
2
, and electron density is in
e/bohr
3
. The terms ρ rad , ∇ρ rad , and ∇ 2 ρ rad are all calculated
analytically using MUNgauss. Please see supporting information for the derivation of ∇ρ rad and ∇ 2 ρ rad .
For both atoms and bonds, the shape ( S
o in bohr ) and
volume ( V o in bohr
3
) were calculated using Eqs. 2 – 5 . In
Eq. 2 , S o is calculated at a specifi ed origin: the nuclear
position for AIM and the radial bond critical point for
BIM. S o can be diagonalized as in Eq. 3 , which results in
the diagonal matrix shown in Eq. 4 . The eigenvalues correspond to principal axes of the electronic second moment.
The shape of the atom or bond is defi ned as ( S o
x , S o
y , S o
z ),
where S o
i =
i 2 o . Using the eigenvalues of shape, we
can calculate the volume of an atom using Eq. 5 . All of the
investigated properties (shape, volume, dipole, number of
electrons, r and r 2 ), are calculated through numerical
integration using the standard SG-1 grids [ 17 ]. In particular, r and r 2 can be calculated for atoms using Eqs. 6
and 7 , as well as for bonds using Eqs. 8 and 9 .
58
Reprinted from the journal
1 3
The most widely used approach is Bader’s quantum theory of atoms in molecules (QTAIM), which depicts AIM
as nonoverlapping [ 3 , 7 – 10 ]. Other popular models, such
as those of Stewart [ 4 , 11 ] and Hirshfeld [ 5 ], depict AIM
as overlapping spherical fuzzy electron densities. Both the
advantages and disadvantages of these methods have been
previously discussed [ 1 ].
This paper builds upon the preliminary fi ndings by Warburton et al. [ 1 ], which suggest that employing molecular
radial density to partition molecules into atomic and bonding contributions may provide a more intuitive scheme to
defi ne atoms and bonds. The ABIM model represents AIM
as atoms with overlapping electron densities and bonds in
molecules (BIM) as the region where this overlap occurs.
Radial density provides a good starting point for an
ABIM theory because the atomic radial density, r 2 ρ(r), of a
free atom in space has shell structure and contains a realistic
number of electrons in each shell. The connection between
the number and position of the critical points in the atomic
radial density distribution function and the atomic shell
structure has been well established [ 12 , 13 ]. In 1976, Politzer
and Parr [ 12 ] showed that the minimum in the atomic radial
density provides a well-defi ned and physically meaningful
separation of electron density into a core and valence region
for the second period atoms. Boyd [ 13 ] then established
that analogous minimum surfaces are also found for heavier
atoms and further showed that the fi rst 54 neutral atoms in
their ground state have the expected shell-occupation numbers. Therefore, partitioning the electron density of atoms
based on the minimum in radial density is quite effective. By
extension, we postulate that a molecular radial density function may effectively partition the molecular electron density
in an intuitive and physically meaningful way. However,
as Smith pointed out, extension of atomic radial density to
molecular systems is not straightforward [ 14 ].
Recently, the molecular radial density, ρ rad (r i ) , at a
grid point, r i , was defi ned as the sum of the radial densities of all molecular atoms at that grid point as given by
Equation 1 [ 1 ]. The weight, W A (r i ) , is a partition function
that represents the fraction of density owned by a particular atom in a molecule at each grid point in space at a distance r iA from atom A . The weight allows us to calculate
the electron density distribution associated with each AIM
(W A (r i ) ρ(r i ) = ρ A (r i )) . In particular, the Becke weight
is employed in this paper, which defi nes the ownership
of each point in the molecular space based on r iA and the
atomic Bragg–Slater radii [ 15 ].
(1)
ρ rad (r i ) =
N A
A
r
2
iA W A (r i ) ρ(r i )
=
N A
A
r
2
iA ρ A (r i ) 0 ≤ W A (r i ) ≤ 1
It is necessary to point out that any weighting scheme that
partitions the molecular space into atomic contributions
can be used, and the Becke weight is just one of many
possible choices. Other possibilities include Hilbert space
decomposition (e.g., Mulliken) and 3D space partitioning
(e.g., QTAIM [ 3 ]). The Becke weight was originally chosen to partition the molecular space because of its simplicity and ease of implementation. One additional benefi t of
the Becke weight is that it does not employ spherical averaging, and thus, distortion of atoms upon bonding can be
investigated.
The molecular radial density along the internuclear
axis has been studied previously [ 1 ]. This paper looks
at the radial density in 2D planes of various molecules
which reveal topologically interesting features residing
off the internuclear axis. We also present the terminology
employed to classify the radial density topology of atoms
and molecules. Most importantly, properties of ABIM will
be calculated and discussed.
2 Methodology
All the calculations were performed with the program
MUNgauss [ 16 ]. The molecular electron and radial densities were calculated at HF/6-311++G(d,p)// HF/6311++G(d,p), specifi cally using RHF for closed shell
and ROHF for open-shell molecules. All of the visual aids,
including contour, relief, and gradient plots, were created
using the Mathematica version 9.0 graphing package. The
gradient vector fi eld plots were calculated in Mathematica
from input radial density data.
Throughout this paper, length is given in bohr, radial
density ( ρ rad ) is given in e/bohr, the gradient of radial density ( ∇ρ rad ) is given in e/bohr
2
, and electron density is in
e/bohr
3
. The terms ρ rad , ∇ρ rad , and ∇ 2 ρ rad are all calculated
analytically using MUNgauss. Please see supporting information for the derivation of ∇ρ rad and ∇ 2 ρ rad .
For both atoms and bonds, the shape ( S
o in bohr ) and
volume ( V o in bohr
3
) were calculated using Eqs. 2 – 5 . In
Eq. 2 , S o is calculated at a specifi ed origin: the nuclear
position for AIM and the radial bond critical point for
BIM. S o can be diagonalized as in Eq. 3 , which results in
the diagonal matrix shown in Eq. 4 . The eigenvalues correspond to principal axes of the electronic second moment.
The shape of the atom or bond is defi ned as ( S o
x , S o
y , S o
z ),
where S o
i =
i 2 o . Using the eigenvalues of shape, we
can calculate the volume of an atom using Eq. 5 . All of the
investigated properties (shape, volume, dipole, number of
electrons, r and r 2 ), are calculated through numerical
integration using the standard SG-1 grids [ 17 ]. In particular, r and r 2 can be calculated for atoms using Eqs. 6
and 7 , as well as for bonds using Eqs. 8 and 9 .
58
Reprinted from the journal