216
10 Importance of Molecular Crystals
that the distribution of electron density mostly resembles those assuming the finite
thickness of the decoration layers at both sides of the ideal G surface [95–97] and
that the aromatic core parts form rods of jungle gyms. That is, in turn, the terminal
methyl groups decorate the G surface. The successful analysis again indicates the
importance of systematics as an indispensable tool in studying complex systems like
molecular ensembles (Sect. 9.4.2).
The so-called maximum entropy method (MEM) was employed to reveal the
molecular aggregation. The MEM is a methodology to estimate the probability distribution p(x) of the system under consideration based on the statistical argument.
The entropy I here is not thermodynamic, but information entropy defined like
I = −
p(x) log 2 p(x)dx.
(10.3)
The essence of the argument is that the most probable (and consequently reliable)
p(x) should realize the maximum uncertainty under the constraint that is compatible
with known information (input data). Diffraction crystallography usually treats the
electron density as the un-normalized probability density based on its property that
cannot be negative as in the usual probability [98, 99]. If no information is available,
the most reliable electron density consequently results in the uniform one everywhere.
Thus, we can characterize the result of the MEM as the electron density with the
least features compatible with the constraint exerted by experimental information.
The constraint adopted in the analysis of the Gyroid phase was
1
n
n
i=1
(G ci − G oi )
2
[σ(G oi )] 2 ≤ 1,
(10.4)
where the indices i distinguish experimental diffractions, and σ is experimentally estimated error for G. The subscript “c” and “o” stand for “calculated” and “observed,”
respectively. In the standard use of the MEM in the single-crystal crystallography,
the atomic detail of the crystal structure is a priori known, and the distribution of
the electron density is precisely refined. On the other hand, in the case of the Gyroid
phase, a priori information is only the space group, I a3d. Thus, G was set equal to
the structure factors F only for two prominent diffractions and their absolute values
|F| for others [88].
In the description of the Gyroid phase based on the G surface, the body diagonal
of the cubic unit cell passes four flat points (with vanishing Gaussian curvature)
on the surface. The flat points serve as junctions of jungle gyms. Four maxima and
minima would exist in this length, accordingly. However, this expectation was correct
only for a limited range of chain-length. In reality [88], this holds in the short-chain
regime in the phase diagram against the chain length but does not in the long-chain
regime. In the latter, the maximum splits into two, with the separation shorter than the
length of the molecular core part. Further, the split maxima continue while exhibiting
10 Importance of Molecular Crystals
that the distribution of electron density mostly resembles those assuming the finite
thickness of the decoration layers at both sides of the ideal G surface [95–97] and
that the aromatic core parts form rods of jungle gyms. That is, in turn, the terminal
methyl groups decorate the G surface. The successful analysis again indicates the
importance of systematics as an indispensable tool in studying complex systems like
molecular ensembles (Sect. 9.4.2).
The so-called maximum entropy method (MEM) was employed to reveal the
molecular aggregation. The MEM is a methodology to estimate the probability distribution p(x) of the system under consideration based on the statistical argument.
The entropy I here is not thermodynamic, but information entropy defined like
I = −
p(x) log 2 p(x)dx.
(10.3)
The essence of the argument is that the most probable (and consequently reliable)
p(x) should realize the maximum uncertainty under the constraint that is compatible
with known information (input data). Diffraction crystallography usually treats the
electron density as the un-normalized probability density based on its property that
cannot be negative as in the usual probability [98, 99]. If no information is available,
the most reliable electron density consequently results in the uniform one everywhere.
Thus, we can characterize the result of the MEM as the electron density with the
least features compatible with the constraint exerted by experimental information.
The constraint adopted in the analysis of the Gyroid phase was
1
n
n
i=1
(G ci − G oi )
2
[σ(G oi )] 2 ≤ 1,
(10.4)
where the indices i distinguish experimental diffractions, and σ is experimentally estimated error for G. The subscript “c” and “o” stand for “calculated” and “observed,”
respectively. In the standard use of the MEM in the single-crystal crystallography,
the atomic detail of the crystal structure is a priori known, and the distribution of
the electron density is precisely refined. On the other hand, in the case of the Gyroid
phase, a priori information is only the space group, I a3d. Thus, G was set equal to
the structure factors F only for two prominent diffractions and their absolute values
|F| for others [88].
In the description of the Gyroid phase based on the G surface, the body diagonal
of the cubic unit cell passes four flat points (with vanishing Gaussian curvature)
on the surface. The flat points serve as junctions of jungle gyms. Four maxima and
minima would exist in this length, accordingly. However, this expectation was correct
only for a limited range of chain-length. In reality [88], this holds in the short-chain
regime in the phase diagram against the chain length but does not in the long-chain
regime. In the latter, the maximum splits into two, with the separation shorter than the
length of the molecular core part. Further, the split maxima continue while exhibiting