Chemistry Reference
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and the gray lines represent the grid (Fig.
3a
). The short black lines in Fig.
3b
indicate those gridlines, which are intersected by the residual density value zero. For
the fractal dimension determination of the residual density value zero
d
f
(
r
0
¼
0)
only those short black lines contribute. For a three-dimensional grid, the borders
between adjacent grid points are no longer lines but surfaces.
Equations (8) and (9) still depend on
e
, which is a bit of a drawback when one wants
to have an absolute scale. However, for comparisons of different density or thermal
motion models fitted to the same experimental data, this is already sufficient.
The dependence on
e
raises the question: is there a natural choice for the
residual density grid? On the one hand, the grid should be fine enough to show
all relevant details, however, it also needs to be in balance with the experimental
resolution.
2.1.6 Resolution in Real and in Reciprocal Space
When diffraction data are collected up to a given resolution and the collected data are
Fourier-transformed to restore the original function, the high frequency components
above the resolution threshold are missing, one has effectively applied a low-pass
filter. Therefore, the restored function shows less detail, it is smoother. Accordingly,
if this restored function is sampled in real space, there is a minimum threshold for the
spacing, which, when used, will enable to exactly reproduce the Fourier coefficients.
If the spacing is made finer, only computing power and memory is wasted.
For example, an audio CD is sampled with a rate of approximately 42 kHz. This
is sufficient to restore all frequencies in the music up to 21 kHz, which is assumed to
be satisfactory for most people.
The connection between sampling frequency and the maximum frequency
reconstructible, the bandwidth limit, is given by the Nyquist sampling theorem
[
9
]. This theorem imposes restrictions on the choice of
n
x
,
n
y
,and
n
z
in (8) and (9).
We are now going to have a look what the Nyquist sampling theorem implies for
the RDA.
Suppose the data were measured completely up to a resolution now given in the
maximum indices
h
,
k
, and
l
. From the Nyquist sampling theorem, we can immedi-
ately tell the spatial resolution to be chosen for the residual density grid.
n
x
¼
2
h
jj
max
n
y
¼
2
k
jj
max
:
(10)
n
z
¼
2
l
jj
max
2.1.7 Expectation Value for d
f
(0)
If we take (10) as a convention, we can put it into (8) and (9) for the calculation of
an expectation value for
d
f
(
r
0
¼
0). How many times can the residual density value
zero be assumed in a residual density grid?
