Chemistry Reference
In-Depth Information
Use of the local expression of the virial theorem [
6
]
1
4
r
2
rð
r
Þ¼
2G
ð
r
Þþ
V
ð
r
Þ;
(4)
enables one to express the local source in terms of the positively defined kinetic
energy density, G(r), and of the electronic potential energy or virial density, V(r),
and to so introduce [
1
] a further interesting interpretation of the local source
1
p
2G
ð
r
Þþ
V
ð
r
Þ
r
0
Þ¼
r
0
Þþ
r
0
Þ:
LS
ð
r
;
¼
LG
ð
r
;
LV
ð
r
;
(5)
r
j j
r
Equation (5) discloses that LS is related to the failure to locally satisfy the virial
relationship between twice the integrated kinetic energy and the virial field den-
sities. It also shows that LS may be seen [
13
] as given by the sum of a kinetic
energy, LG, and of an electron potential energy, LV, local source contribution.
3
Molecular regions, where the electron density is concentrated (
2
r
(r
0
)
0) and
where the potential energy dominates over the kinetic energy, act as a positive
source
for the electron density at a point r, whereas regions of charge depletion,
(
r
<
2
r
(r
0
)
0), and of dominant kinetic energy act as a negative source, a
sink
, for
the same point. The effectiveness of the electron density at r
0
to be source or sink for
the electron density at another point r is then related to the magnitude of its charge
concentration or depletion at r
0
, weighted by the inverse of the distance of these two
points.
A given atomic source function value, S(r;
r
>
), will always be the result of the
sum of local positive and negative contributions and can thus be either globally
positive or negative. As
r
(r) is positive everywhere, S(r;
O
) will also be positive at
any r for an isolated atom, since its own basin is the only one determining the
density. For an atom in a polyatomic system, the local sources are usually found to
beat the local sinks in determining the electron density at its intervening bcps, but
the opposite may also hold true in specific circumstances. A typical example is the
source from the hydrogen atom involved in standard hydrogen bonds to the electron
density at the hydrogen bcp (see Sect.
3.3
).
This section ends by introducing a second powerful formula for the density at a
point in terms of external sources and by showing how both (1) and this formula
may easily be derived from a single mathematical theorem. The alternative expres-
sion for
r
(r) is given by [
1
]
O
Z
I
2
rð
r
0
Þ
r
d
r
0
þ
j
1
rð
rð
r
Þ¼ð
14
p=
dS
ð
r
S
Þr
j
r
r
S
r
S
Þ
;
(6)
r
j j
r
O
S
O
3
LS could also be expressed in terms of local contributions related to the total electronic energy
density H(r)[
6
] and to the kinetic energy density G(r)since¼
2
r
r¼
2G(r)
þ
V(r)
¼
G(r)
þ
H(r).
