Chemistry Reference
In-Depth Information
where the density at a point r within an atom
is seen as determined by the sum
of two contributions: the basin average of the potential at r exerted by the
Laplacian of the density (the basin average of the local source LS), plus the flux
through the surface of
O
of the electric field density at r due to the electron density
on the surface boundary,
r
(r
s
).
4
For an isolated molecule or a molecular complex,
O
O
may be clearly taken as the whole space and (6) becomes identical to (1), as the
electron density vanishes at infinity. Formula given in (6) should be applied to
those systems which have finite boundaries, like a (unit) cell in a crystal or any of
its composing atoms or group of atoms. It should also be used for large molecular
systems, or for convenient and well-defined part of them. One could so replace the
integration of the LS over all (potentially infinite) atoms of the system, with just
one basin's and one surface's integration. For instance, the effect of the environ-
ment on a cluster of molecules within a liquid or crystal might be conveniently
investigated, through the SF approach, using the surface boundary of the cluster
in (6). The surface integral can then be envisaged as a sum of contributions
from each of the interatomic surfaces
S
(
O
0
) composing the total surface
S
(
O
,
O
)
bounding
.
As shown in [
1
], (6) may be obtained in several ways: through the use of
the equation of motion for the generator r
O
r
j j
1
for a proper open system
O
(a QTAIM basin), or by solving Poisson's equation for a potential given by the
density
r
(r), or also in a purely mathematical manner, using the Green's theorem
Z
A
ð
I
2
v
2
u
dr
0
¼
u
r
v
r
Þ
d
S
ð
u
r
v
v
r
u
Þ;
(7)
S
A
with A being an arbitrary basin and
S
A
its enclosing surface. By making use of
the well-known [
5
] identity,
r
j jÞ
1
2
r
0
Þ
r
ð
r
¼ð
4
pdð
r
and by setting
r
j j
1
and
v
r
0
Þ
u
¼
r
¼ rð
, one easily gets (6)
provided
the basin A fulfills the
QTAIM zero-flux recipe
½
rrð
r
Þ
n
ð
r
Þ¼
0
8
r
2
S
A
, to get rid of the surface term
u
r
v
.
2.2 Local and Integrated Forms of the Source Function:
Definitions and Use
In Sect.
2.1
, the local form of the source function, (8)
r
j jÞ
1
r
0
Þ¼ð
2
r
0
Þ;
LS
ð
r
;
4
p
r
r
rð
(8)
4
In fact,
)
1
r
(r
s
)
3
r
(
j
r - r
s
j
¼
(r
r
s
)/
j
r
r
s
j
r
(r
s
)
¼ e
(r, r
s
), with
e
(r, r
s
) being the electric
field density at r due to the electron density at r
s
.
