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Table 2 Residual density descriptors for identical models at different experimental resolution
Cutoff resolution
d
f
(0)
e
gross
r
0,min
r
0,max
Dr
0
[
˚
1
]
[e
˚
3
]
[e
˚
3
]
[e
˚
3
]
[e]
1.14 (no cutoff)
2.7366
8.39
0.32
0.37
0.69
1.00
2.6989
7.25
0.23
0.32
0.55
0.80
2.6543
6.27
0.15
0.21
0.36
Table 3 Residual density descriptors including entropy
S
and percentage of features POF for
experimental IAM and MM refinements and for a perfect MM refinement on synthetic data with
comparable total error
e
gross
and therefore comparable
wR
1
|
w=1
values as in the experimental MM
refinement
d
f
(
r
0
)
e
gross
r
0,min
r
0,max
Dr
0
S
POF
[e
˚
3
]
[e
˚
3
]
[e
˚
3
]
[e]
[%]
IAM (exp.)
2.6826
11.55
0.73
0.59
1.32
69,564
5.3
MM (exp.)
2.7076
7.21
0.24
0.31
0.55
30,574
4.4
MM (theo.)
2.7786
7.36
0.10
0.11
0.21
35,489
1.9
solely by noise. The procedure of adding noise to the structure factors was
described in Sect. 3.1, the noise control parameter was set to
p
1
¼
0.222, because
this noise level generates approximately as many gross residual electrons (7.36) as
exist in the experimental data after the MM (7.21). This provides a check of how the
RDA-plot would appear in the case when all errors come from noise only at a
constant experimental
R
-value (Tables
2
and
3
).
The maximum values in
h
,
k
, and
l
were 21, 21, and 24, which leads to
n
x
¼
42,
from
Equation (14) is calculated to be 2.83198. This leads to percentage features of 5.3%
(IAM), 4.4% (MM, exp.), and 1.9% (MM, theo.). For RDA-plots, see Fig.
6
.
The MM reduced the features by only approximately 0.9%. This is in accordance
with the small reduction in
e
gross
from 11.55 e (IAM) to 7.21 e (MM) and with the
final distribution of
d
f
(0), which is not parabolic. The little progress is due to a small
disorder of the whole molecule. The corresponding occupation factor was, how-
ever, smaller than 1% and could not be refined (see also Fig.
7
and text in Sect. 3.4,
where the same experimental data was used). The POF of the ideal model, however,
is with 1.9% also higher than naively expected. A similar behavior was found in the
previous section for
e
gross
, which did not reduce to zero despite using ideal data with
zero noise. The same effect might take place here. The ideal expectation value is
also calculated under the assumption of 100% completeness of the data, which is
only approximately given.
From the previous discussion, it can be seen that the POF-values indicate that a
reduction is still feasible, at least in principle.
For a comparison with other measures of features, the entropy
S
of the modulus
of the residual density distribution was calculated according to
n
y
¼
42,
n
z
¼
48 for the residual density grid. The expectation value
d
f
ð
0
Þ
X
S
¼
r
jj
ð r
j
:
ln
(25)
r
0
