Chemistry Reference
In-Depth Information
2.1 Mathematical Derivation and Physical Interpretations
More than 10 years ago, Bader and Gatti showed [
1
] that the electron density at any
point r within a system may be regarded as consisting of contributions from a local
source, LS(r, r
0
), which operates at all other points of the space:
Z
LS
r
0
Þ
dr
0
:
rð
r
Þ¼
ð
r
;
(1)
r
j jÞ
1
r
0
Þ¼ð
2
rð
r
0
Þ
The local source, given by LS
ð
r
;
4
p
r
r
, where
r
j jÞ
1
is a Green's function or an influence function [
5
], representing
the effectiveness of how the cause, the Laplacian of the density at r
0
,
ð
4
p
r
2
r
(r
0
), gives
rise to the effect, the electron density at r,
r
(r). If the local source is integrated over
an atomic basin or a group of atomic basins defined as in the quantum theory of
atoms in molecules (QTAIM) [
6
], that is over those regions of space
r
O
bounded by
surfaces
S
which are never crossed by
rr
(r) vectors
½
rrð
r
Þ
n
ð
r
Þ¼
0
8
r
2
S
O
;
then the electron density at r may be equated to a sum of contributions S(r;
O
), each
of which is termed as the
source function
from the atom or group of atoms
to
r
(r).
When compared to
r
(r), it represents a measure of how this atom or group of atoms
O
O
contributes to determine the density at r, relative to the contributions from other
atoms or group of atoms in the system. One may visualize the electron density at a
point within a given basin
as determined by an internal SF self-contribution and
by a sum of SF contributions from the remaining atoms or groups of atoms within a
molecule:
O
X
; O
0
Þ:
rð
r
Þ¼
S
ð
r
; OÞþ
S
ð
r
(2)
O
0
6¼O
Decomposition afforded by (2) enables one to view the properties of the density
from a new perspective and anticipates the SF as a tool able to provide chemical
insight. Let us consider, for instance, the well-known and highly debated concept of
bond path (BP) [
7
,
8
] in Bader's theory. Topologically, a BP is associated with the
only two atoms it connects [
6
], but its shape and the values of the electron density at
any point along the path, including the so-called bond critical point (bcp), clearly
depend, though to different extents, on the whole set of physical interactions present
in a system and accounted for by its Hamiltonian operator. Using the SF, such an
apparent inconsistency automatically fades away since, according to (2), the elec-
tron density at the bcp, taken as the most representative density point for the two
bonded atoms [
6
], is determined not only from the contributions of these two atoms
but also, in principle, from those of all the remaining atoms in the system, so
bringing to the fore interesting “nonlocal” roles into the bonding. One immediately
envisages that the more covalently will be two atoms bonded to each other, the
higher will be their relative contributions to the density value at their intervening
bcp [
9
].
