Chemistry Reference
In-Depth Information
e
gross
and |
F|”, it has been shown that
e
gross
can be evaluated from the absolute
modulus of the difference structure amplitudes for the case that the residual density
obeys a Gaussian distribution. Obviously, this number can be compared to the
number obtained from summing over the grid. Equality of these numbers is another,
independent indicator of a Gaussian distribution of the residuals, additional to the
parabolic shape and to the expectation value for
d
f
(0).
For the evaluation of the residual density grids, the software jnk2RDA [
10
]
was used throughout. It accepts residual density grids from XD, MoPro [
11
,
12
],
TONTO [
13
], and BayMEM [
14
]. For BayMEM, the grid format has to be
modified before evaluation. Output data is a Postscript file with an RDA-plot,
i.e., a plot
d
f
(
r
0
)versus
r
0
for a range
D
1.0 e
˚
3
. Additionally, the
values for
d
f
(0),
r
0,min
,
r
0,max
, and their difference
Dr
0
¼ r
0,max
r
0,min
are
given. If the data were in XD format, some additional information from
the master file is automatically extracted, like the unit cell data for calculation
of the volume. The multipole model used, thegridsizeoftheresidualdensity
grid and the XDFOUR section are printed into the Postscript file. From version
1.4alsotheentropy
S
of the absolute values of the residual density distribution is
given [for a definition of
S
see (25)]. This software is available for Unix/Linux
and Windows machines.
þ
1.0 to
3.1
...
to Idealized Distributions
To check the basic assumptions and to have a look at how noise appears in the
RDA-plot, ideal, noise-free data are generated from a multipole model for S(N
t
Bu)
3
up to a resolution of sin
y
/
l
= 1.14
˚
1
. These data have been added to Gaussian
noise with mean value zero according to the following equations:
I
noise
hkl
I
ideal
hkl
I
error
hkl
¼
þ
hk
q
RandomGauss
I
ideal
I
error
hkl
¼
p
1
ð
x
Þ
2
p
s
e
ð
x
m
Þ
2
1
RandomGauss
ð
x
Þ¼
2
s
2
s ¼
1
0
The noise increases with
p
1
. In Fig.
4
, the RDA-plot is shown for four different
values of the noise control parameter
p
1
: 0.000, 0.222, 0.444, 0.888, and in Table
1
the residual density descriptors are given. Additionally, there is a horizontal and a
vertical red bar drawn in Fig.
4
. The horizontal bar indicates the flatness (actually
only, when drawn at
d
f
(
r
0
)
m ¼
Dr
0
¼ r
0,max
r
0,min
. This value depends
explicitly solely on the following four factors:
¼
0),
(1) The experimental resolution
(2) The residual density grid resolution
(3) The density and thermal motion model
(4) The noise in the data
