Chemistry Reference
In-Depth Information
While
e
gross
is always a positive number,
e
net
may be positive or negative or zero.
From the definitions follows
1
2
e
net
e
gross
e
gross
:
(7)
When one of the equal signs holds, all errors in the volume under consideration
stem from missing or excess electrons.
2.1.5 Features in the Residual Density and d
f
(
r
0
)
First, it is explained how the fractal dimension is defined and then how it is
implemented.
For a given manifold to calculate its fractal dimension, one divides the manifold
in cuboids (or in quadrangles) of characteristic length
e
. Then the number
N
of
cuboids is counted, which contain the part of the structure of interest. In this
application, this structure would be a residual density iso-value
r
0
. The fractal
dimension of this residual density iso-value is then defined as:
ð
N
ðeÞÞ
log
d
f
ðr
0
;
eÞ¼
(8)
log
ð
1
=eÞ
with
3
q
3
1
=e ¼
ð
n
x
1
Þð
n
y
1
Þð
n
z
1
Þ
(9)
and
n
x
,
n
y
,
n
z
being the number of residual density grid values in
x
,
y
, and
z
direction.
The determination of
N
(
e
) is important. The residual density grid mesh is
assumed to be so fine that all values between two adjacent residual density grid
points are taken only once.
Although this concept can be used to quantify features more precisely, as will be
shown later, the most important information is just drawn from a graphical repre-
sentation of the fractal dimension for all residual density values: If the shape is a
parabola, the residuals are distributed according to a Gaussian. The residuals are not
systematic then and there is no structural information contained in it. On the other
hand, if there are shoulders, if the distribution is asymmetric or not a parabola, this
clearly indicates that the residual density contains information for improvement of
the refinement.
For the importance of the logarithmic scale, see Figs.
1
and
2
. These refer to the
same data, however, presented in different ways. A residual density grid was
calculated for the multipole refinement of experimental high-resolution data of
S(N
t
Bu)
3
[
8
]. In Fig.
1
, the frequency of residual density values has been counted
