Chemistry Reference
In-Depth Information
with the choice of the grid to be analyzed in accordance with (10). Applied to the
20 random grids from above, the POF gives at most 0.034% features and the
minimum value is 0.001%. Up to one digit, there are 0.0% features in these grids
which is expected for random grids. For an application to experimental data, see
Sect. 3.
2.2 The Improper Probability Density Approach: e
gross
and |
F|
D
As mentioned earlier, there is another way to perceive the fractal distribution of the
residual density. Suppose it is known how the residual density is distributed in the
unit cell from a probability or frequency point of view, i.e., it is known how many
times each residual density value appears. This is a one-dimensional function and
therefore of less information content than the whole three-dimensional residual
density distribution in the unit cell. If this function is normalized, i.e., if the area of
the histogram or under the curve is chosen to equal unity by an appropriate choice
of the normalization constant, the function may serve as a probability density
function (p.d.f.).
Now, it is important to realize that this reduced knowledge is already sufficient
to calculate all properties we are interested in via the usual formulas of statistical
expectation values. As an example, we take
e
gross
and define it now from the
statistical point of view rather than in (5). This leads to
1
2
V
e
gross
¼
h
jr
0
ð
r
Þj
i
(15)
and
1
rjhi¼
p
ðr
0
Þ rjj
d
r
0
:
(16)
1
If the p.d.f.
p
(
r
0
) is known, it can be substituted into (16) and the result into (15)
for the calculation of
e
gross
.
There is a nice application of (15) and (16) for the limiting case we are most
interested in, that is when the refinement is at or close to the optimum and the p.d.f.
is a Gaussian:
1
s
r
0
2
2
s
2
dp
e
p
ðr
0
Þ
dp
¼
p
:
(17)
2
p
Substituted in (16), this yields after integration:
r
2
p
rjhi¼
s:
(18)
