Chemistry Reference
In-Depth Information
Many readers will hopefully agree that the shape is clearly reminiscent of
a Gaussian: there is only one maximum, the distribution is symmetric, and there is
the typical change in curvature. Only in the periphery, the symmetry is not main-
tained strongly and there are small gaps, but this may as well be so just by accident.
The logarithm of the frequency distribution is shown in Fig.
2
.Intherange
close to the center, it resembles a parabola, however, in the periphery the devi-
ations from the parabolic shape are quite distinct and they do not give the
impression of random statistical fluctuations but of being systematically too
large. This is indeed the case.
While Figs.
1
and
2
demonstrate the importance of the logarithmic scale, there
are still two important steps missing to convert Fig.
2
into an RDA plot. The
difference in the RDA is
what
is counted and
how
it is counted. Let us start with
the latter. Whereas in Fig.
1
and
2
it was counted how many grid values are within a
given small range, e.g., between 0.10 and 0.11 e
˚
3
, i.e., the data were binned, in
the RDA it is counted which residual density values are between neighboring
residual density grid points, i.e., the data are not binned.
For example, when at adjacent grid points, say in
x
-direction, the residual density
values are
0.0976 e
˚
3
then in the frequency table the values for all
integer multiples of 0.01 e
˚
3
between these limits are incremented by one. That is
for the residual density values
0.1682 and
0.16,
0.15,
0.14,
0.13,
0.12,
0.11,
0.10 e
˚
3
. This is done for each of the three independent directions. So far for
the “how”, now for the “what”.
While in the histogram approach small grid volumes are counted (i.e., grid points
with attached volume, with a residual density value within certain limits), in the
RDA the faces of grid points containing certain residual density values (such as all
positive and negative integer multiples of 0.01 e
˚
3
) are counted. For an example,
see Fig.
3
. The blue line represents the residual density value zero as an example
a
b
0.4
0.4
0.2
0.2
0
0
-0.2
- 0.2
- 0.4
- 0.4
-0.4
- 0.2
0
0.2
0.4
- 0.4
- 0.2
0
0.2
0.4
Fig. 3 (a) Residual density value zero (
blue curved line
) and residual density grid (
straight gray
lines
) of an arbitrary example structure. (b) Evaluation of a residual density plane. The short black
lines contribute to
d
f
(
r
0
¼
0)
