Chemistry Reference
In-Depth Information
There are (
n
y
1)(
n
z
1) faces of cuboids in one
x
-layer and (
n
x
1) of those
layers, which we chose to match (2|
h
|
max
1), resulting in the appearance of the
residual density value zero to be at most (2|
h
|
max
1) for the
x
-direction. Similar formulas can be derived for the
y
- and
z
-directions, leading in
total to
1)(
n
y
1)(
n
z
N
max
ðr
0
¼
0
Þ¼ð
2
h
jj
max
1
Þð
n
y
1
Þð
n
z
1
Þ
þð
2
k
jj
max
1
Þð
n
z
1
Þð
n
x
1
Þ
þð
2
l
jj
max
1
Þð
n
x
1
Þð
n
y
1
Þ
(11)
which is the maximum number. We are, however, interested in the average number.
This is derived from (11) by taking the half, because every time a border is crossed
between two adjacent residual density grid values, there is a chance of ½ that the
residual density sign changes. If it changes, the residual density value zero has been
assumed at least once somewhere between the grid points (although the value itself
need not appear as a grid value).
1
2
N
max
ðr
0
¼
h
N
ðr
0
¼
0
Þ
i ¼
0
Þ:
(12)
Taking (8)-(12) together, it is possible to calculate the expectation value of
d
f
(0)
with a pocket calculator from the maximum indices in
h
,
k
, and
l
. For example, for
|
h
|
max
¼
|
k
|
max
¼
|
l
|
max
¼
25 one obtains:
log
ð
0
:
5
3
49
3
Þ
¼
2
:
83721
:
(13)
49
3
1
3
log
ð
3
Þ
50 Gaussian distributed
random numbers was generated 20 times with mean value zero. The grid was
analyzed with the Residual Density Analysis software jnk2RDA [
10
], which addi-
tionally to the numbers
e
gross
and
e
net
gives
d
f
(0) and a plot of
d
f
(
r
0
). The mean
value and standard deviation were:
d
f
(0)
To test the derived formula, a grid of 50
50
¼
2.83723
0.00041, which is in good
agreement with the predicted value from (13).
2.1.8 Quantification of Features: Percentage of Features
The results from the last paragraph can be employed to quantify the amount of
features. It was shown that the value
d
f
(0) scatters, although only little, around its
mean value. Therefore, the ratio of the actual value and the expected value may give
values slightly over 100%, which is counterintuitive.
It is therefore suggested to use the following definition as a measure of the POF
in the residual density distribution:
;
d
f
ð
0
Þ
POF
¼
100 1
(14)
h
d
f
ð
0
Þ
i
