Chemistry Reference
In-Depth Information
Clearly, the integration on the whole space in (1) may be apportioned among the
pieces of any conceivable mutually exclusive partitioning of space or in terms of
one of the many proposed fuzzy boundary partitioning schemes [
10
]; however,
since the QTAIM basins have the unique property of being proper open quantum
systems and have also been amply demonstrated to be the atoms or group of atoms
of “chemistry” [
6
], only such a specific partitioning may ensure an unbiased and
quantum-mechanically rigorous association of S(r;
O
) with the atoms or group of
atoms of a system.
A closer inspection to (1) and to the definition of the local source LS reveals
[
1
] that
r
(r) is given by an expression that resembles that for the electrostatic
potential at r due to the electron distribution at all other points of the space,
with
ð
4
pÞ
1
2
rð
r
0
Þ
replacing
r
(r
0
) in the numerator of the integrand. Indeed,
both the electrostatic potential
V
elec
and
r
(r) are a solution [
11
]
r
Z
4
p
r
r
j jÞ
1
ð
r
0
Þ
dr
0
;
'ð
r
Þ¼
ð
q
(3)
2
of the Poisson's equation
r
'ð
r
Þ¼
q
ð
r
Þ
, with
'
being, respectively,
V
elec
or
r
,
q
2
being, respectively,
r
or
r
r
, and exploiting the definition of
V
elec
in terms of
1
The electron density
r
(r) may thus
be seen as the potential generated by its Laplacian distribution [
1
], in agreement
with the physical interpretation given earlier as of
2
V
elec
ð
Poisson's equation,
r
r
Þ¼þ
4
prð
r
Þ:
2
r
(r
0
) causing or determining
r
r
(r).
It has been recently claimed that [
12
] one should focus more on the physical
interpretation of the SF, namely that the Laplacian distribution
determines
the
electron density at any point in space, rather than on the “formal” mathematical
interpretation of (2) that a basin
contributes
to the density. Although we believe not
be responsible of any serious misconception in this regard, it seems yet worth
spending few words to further clarify the point. Since the Laplacian distribution
determines r
at any point in space, integration of such a distribution within an
atomic basin, weighted by the influence function, just singles out the distinct
contribution
from the basin to the
determination
of
r
.
2
It is in this sense that, within
the SF approach, an atom gives its contribution to
r
(r), and not clearly in terms of
the direct contributions from its basis functions to the total density, even admitting
one could always define them (which is not the case, for instance, for electron
densities given numerically on a grid). An interpretation of the SF (and of its
integrated form, see
infra
) in terms of a standard population analysis would also
be totally at odd with the exhaustive and mutually exclusive partitioning of the
space adopted within the SF approach.
1
The right-hand member has a positive sign since the electron density
r
(r) is taken as a positive
quantity, despite the electron is negatively charged.
2
Incidentally, one should note that a uniform distribution has no sources, since
2
r
would vanish
everywhere in this case and the only source for the density at point r is the point itself.
r
