Chemistry Reference
In-Depth Information
On the other hand, also the second moment can be calculated from the p.d.f.,
1
DE
¼
2
d
r
0
;
rjj
p
ðr
0
Þ rjj
(19)
1
which yields in the special case of (17)
DE
¼ s
2
rjj
:
(20)
From (18) and (20) follows [
1
]:
r
2
p
DE
rjhi¼
rjj
:
(21)
There is another way to calculate the mean value of the modulus squared of a
function whose Fourier series is known by means of Parseval's theorem:
DE
X
hkl
D
2
r
jj
¼
F
;
j
j
(22)
which leads, when substituted into (15) and (21), to the handy formula:
s
2
p
X
hkl
jD
e
gross
V
UC
¼
2
F
j
:
(23)
Equation (23) gives the connection between the numbers of gross residual
electrons (i.e., the total error including noise) in the whole unit cell and the structure
factor differences. Equation (23) is valid in the case of a Gaussian p.d.f. of the
residual density. Please note that this measure is identical to
wR
1
|
w=
1
[i.e.,
wR
1
,
where the weights have been chosen to be unit weights, see (2)], if the factor of
proportion is chosen accordingly. Unit weights are in accordance with the expan-
sion of a function in an orthogonal and normalized basis set like in a Fourier series.
However, while
wR
1
is a global measure,
e
gross
can also be evaluated locally on a
residual density grid.
This result gives a different view on the definition of the fractal dimension of the
residual density distribution. The RDA plot can also be interpreted as the logarithm
of an improper (not normalized to unity) probability density function.
3 Applications
...
In the following, some examples of applications of the RDA are given. As these
examples represent different states of development of the theory, they are not at the
same stage. For example, in Sec. 2.2 “the improper probability density approach:
