Chemistry Reference
In-Depth Information
obtained from two-component spin-orbit ZORA (PbCl
2
), four-component Dirac
(Bi
2
H
4
), scalar-relativistic ZORA [(CH
3
)
2
SAuCl], and nonrelativistic calculations
(all model systems) are shown in this Fig.
7
. In the case of PbCl
2
, the major
changes in the Fukui function when using a relativistic Hamiltonian are observed
directly at the Pb atom, where the relativistic effects are expected to be most
pronounced. The nonrelativistic results predict the Pb atom to be the preferred
position for an electrophilic attack, whereas the spin-orbit ZORA calculations
indicated the opposite, i.e., an attack at one of the two chlorine atoms. For the
case of Bi
2
H
4
, the four-component Dirac calculations predict a high reactivity at
both bismuth atoms on the opposite side of the hydrogen atoms. Regarding the
nonrelativistic results, the reactivity on the whole isosurface is significantly
lowered. Only in the case of the gold complex, the scalar-relativistic result
exhibits only minor changes at the gold and the chlorine atoms when comparing
to the nonrelativistic calculations.
6.2 The Quantum Theory of Atoms in Molecules
Bader's atoms in molecules (AIM) theory [
1
,
138
] is an interpretative theory which
divides the electron density in a molecule into atomic basins {
O
i
} such that it is
possible to define properties
G
(
O
i
) of an atom
i
in a molecule, which can then be
determined as integrals over the property density
r
G
(
r
) of the corresponding basin:
ð
d
3
rr
G
ðrÞ:
G OðÞ¼
(62)
O
These “atomic” properties are then additive and sum up to molecular properties
if all basins are taken into account. In the limiting case of a single atom, the AIM
property must be equal to the corresponding property of the free, unbound atom.
The individual atoms in a molecule are in this context separated from each other by
the interatomic surfaces (IAS), which are usually called zero-flux surfaces. At each
point of the IAS, the normal vector
) is orthogonal to the gradient of the electron
density, which is expressed through the so-called zero-flux boundary condition
n
(
r
nðÞrr ðÞ¼
0
:
(63)
AIM was first formulated by Bader [
1
] as a nonrelativistic theory and then later
studied by Cioslowski and Karwowski [
139
] within the relativistic regime. In the
nonrelativistic case, the partitioning of the molecule is uniquely defined in terms of
open quantum systems (atoms) through the zero-flux boundary condition, which
follows from the properties of the Lagrangian density. Due to a certain arbitrariness
in the expression for the relativistic Lagrangian density, the partitioning of the
atoms is no longer uniquely defined in four-component theory [
139
].
