Characterization of supramolecular hidden chirality of hydrogen-bonded networks by advanced graph set analysis

Article abstract of DOI:10.1002/chem.201303770

Supramolecular hidden chirality of hydrogen-bonded (HB) networks of primary ammonium carboxylates was exposed by advanced graph set analysis from a symmetric viewpoint in topology. The ring-type HB (R-HB) networks are topologically regarded as faces, and therefore exhibit prochirality and positional isomerism due to substituents attached on the faces. To describe the symmetric properties of the faces, additional symbols, Re (right-handed or clockwise), Si (left-handed or anticlockwise), and m (mirror), were proposed. According to the symbols, various kinds of faces were classified based on the symmetry. This symmetry consideration of the faces enables us to precisely evaluate supramolecular chirality, especially its handedness, of 0D-cubic, 1D-ladder and 2D-sheet HB networks that are composed of the faces. The 1D-ladder and 2D-sheet HB networks generate chirality by accumulating the chiral faces in 1D and 2D manners, respectively, whereas 0D-cubic HB networks generate chirality based on combinations of eight kinds of faces, similar to the chirality of dice. Supramolecular hidden chirality: Conventional graph set analysis was advanced by incorporating symmetry. Under this analysis, ring-type hydrogen-bonded (HB) networks are considered as faces with distinguishable front and back sides (see figure). This consideration led to the clarification of not only prochirality on the faces, but also the handedness of HB supramolecular architectures. Copyright

Full text of DOI:10.1002/chem.201303770

DOI: 10.1002/chem.201303770
Full Paper
&
Supramolecular Chemistry
Characterization of Supramolecular Hidden Chirality of Hydrogen-
Bonded Networks by Advanced Graph Set Analysis
Toshiyuki Sasaki,[a] Yoko Ida,[a] Ichiro Hisaki,*[a] Tetsuharu Yuge,[a] Yoshiaki Uchida,[b]
Norimitsu Tohnai,[a] and Mikiji Miyata*[a, c]
Abstract: Supramolecular hidden chirality of hydrogen-
bonded (HB) networks of primary ammonium carboxylates
was exposed by advanced graph set analysis from a symmet-
ric viewpoint in topology. The ring-type HB (R-HB) networks
are topologically regarded as faces, and therefore exhibit
prochirality and positional isomerism due to substituents at-
tached on the faces. To describe the symmetric properties of
the faces, additional symbols, Re (right-handed or clockwise),
Si (left-handed or anticlockwise), and m (mirror), were pro-
posed. According to the symbols, various kinds of faces
were classified based on the symmetry. This symmetry con-
sideration of the faces enables us to precisely evaluate
supramolecular chirality, especially its handedness, of 0D-
cubic, 1D-ladder and 2D-sheet HB networks that are com-
posed of the faces. The 1D-ladder and 2D-sheet HB networks
generate chirality by accumulating the chiral faces in 1D and
2D manners, respectively, whereas 0D-cubic HB networks
generate chirality based on combinations of eight kinds of
faces, similar to the chirality of dice.
graph set analysis in mathematics.[6] However, it still remains
challenging to clarify both chirality generation and handedness
determination in the HB networks. This situation led us to the
idea that the graph set analysis may be reconsidered from
a view point of topological symmetry to evaluate the chirality
of the HB networks (Figure 1c).
Introduction
Organic crystals have been getting so much attention due to
their potential for comprehensively creating chemical and
physical functions.[1] It is well-known that the functions depend
on not only molecular structures but also molecular arrange-
ments.[2] Therefore, a large number of investigations has been
devoted to control molecular arrangements in crystals through
non-covalent interactions.[3] The powerful concept emerged as
supramolecular synthons for crystal engineering and enabled
us to construct zero- to three-dimensional (0D to 3D) architec-
tures (Figure 1a).[4] Among the synthons, hydrogen-bonding
(HB) groups with robust and anisotropic nature are quite
useful for designing the low-dimensional structures (Fig-
ure 1b).[5] Such HB networks are systematically designated by
We encounter supramolecular chirality (SMC) in the molecu-
lar assemblies of organic crystals. In particular, it fascinates us
that even achiral molecules generate the SMC due to their
asymmetric molecular arrangements.[7] However, it is mysteri-
ous that there are no general rules for determining the hand-
edness of the SMC that originates from the arrangements. So
far, our group struggled to make clear handedness of the SMC
of twofold helical assemblies in organic crystals.[8] Nevertheless,
it still remains insufficient to correlate between the arrange-
ments and chiral properties. This is surely because flexible non-
covalent interactions in length, directionality, and reversibility
construct various complicated supramolecular architectures,
making it difficult to evaluate the SMC. Chemists wait for
widely applicable rules for the SMC to further develop crystal
engineering in organic materials.
[a] T. Sasaki, Y. Ida, Dr. I. Hisaki, Dr. T. Yuge, Dr. N. Tohnai, Prof. Dr. M. Miyata
Department of Material and Life Science
Graduate School of Engineering, Osaka University
2-1 Yamadaoka, Suita, Osaka 565-0871 (Japan)
Fax: (+81)6-6879-7404
E-mail: hisaki@mls.eng.osaka-u.ac.jp
miyata@mls.eng.osaka-u.ac.jp
As a model system to set up a general rule for determining
the handedness of the SMC, we focused on the chirality of the
HB networks composed of primary ammonium carboxylates
owing to the following three reasons. Firstly, carboxylic and
amino groups are essential in life because they are the build-
ing blocks of amino acids and proteins. These groups are
useful in various applications such as chiral synthesis[9] and
separation.[10] Secondly, these groups are useful for controlling
molecular arrangements to form supramolecular low-dimen-
sional assemblies. Thirdly, it is noteworthy that anisotropic
nature of the HBs yields diverse networks with topological
[b] Prof. Dr. Y. Uchida
Department of Mathematics
Kobe Pharmaceutical University
4-19-1 Motoyama Kitamachi, Higashinada
Kobe, Hyogo 658-8558 (Japan)
Fax: (+81)78-441-7582
[c] Prof. Dr. M. Miyata
Present address: The Institute of Scientific and Industrial Research
Osaka University, 8-1 Mihogaoka
Ibaraki, Osaka 567-0047 (Japan)
Supporting information for this article is available on the WWW under
http://dx.doi.org/10.1002/chem.201303770.
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symmetry (Figure 1c). The com-
ponent ring-type HB (R-HB) net-
works are recognized as faces
with prochirality and positional
isomerism of substituents. This
advanced analysis serves as
a powerful method for charac-
terizing the SMC of the low-
dimensional HB networks com-
posed of primary ammonium
carboxylates.
Results and Discussion
Nine 0D-cubic HB networks by
crystallographic studies
First, we focused on supramolec-
ular clusters with the 0D-cubic
HB networks generated from
bulky triphenylmethyl groups.
Inverted-micellar [4+4] ion pairs
are met in salts of triphenyl-
methyl amine (TMPA) with sul-
fonic acids[13] or phosphonic
acids[14] as well as salts of triphe-
nylacetic acid (TPAA) with pri-
mary amines.[11,15] Among them,
the carboxylate salts are attrac-
tive enough to form asymmetric
cubic HB networks, which are
analogues of water octamers.[16]
So far, eleven salts of TPAA with
primary amines (Figure 2a) were
Figure 1. a) Models of 0D clusters, 1D columns, 2D sheets, and 3D crystals from atoms and/or molecules assem-
bled by non-covalent interactions. b) Supramolecular architectures composed of primary ammonium carboxylates
with ring-type hydrogen-bonded (R-HB) networks: (i) 0D-cubic, (ii) 1D-ladder, and (iii) 2D-sheet HB networks. c) R-
HB networks with (i) denotations in the conventional graph set analysis, (ii) prochirality of the R-HB networks, and
(iii) supramolecular chirality of the R-HB networks on the basis of prochirality and positional isomerism of substitu- prepared in methanol as raw
ents.
products and were recrystallized
from non-polar solvents as
mixed solutions of chloroform/
SMC. Our previous studies exposed hidden SMC of 0D-cubic[11]
and 1D-ladder HB networks.[8b,d–g] The latter 1D networks, con-
sisting of twofold helical assemblies that display the helical
SMC and its handedness is judged from tiltness toward helical
axes, termed as supramolecular-tilt-chirality (STC).[8] However,
this STC method cannot be applied to the former 0D-cubic as
well as 2D-sheet HB networks, requiring a more general
method for defining the SMC of diverse low-dimensional as-
semblies.
hexane and toluene/hexane. Such a selection of non-polar sol-
vents originates from the idea that the cubic clusters may be
formed in non-polar circumstances as inverted micelles.
The resulting single crystals were subjected to X-ray crystal-
lographic analysis (the Supporting Information). The resultant
crystal data display a difference of the following two kinds of
oxygen atoms of carboxylate anions (Figure 2b). The one is
designated as O(a) with two HBs towards different ammonium
cations, whereas the other is labeled as O(b) with one HB.
Such oxygen atoms induce diverse symmetry of the resulting
cubic HB networks. As a result, the networks are topologically
classified into three achiral types (D2d, S4, and CS) and three
pairs of enantiomeric types (C2/C2’, C1(a)/C1(a)’, and C1(b)/C1(b)’) re-
garding point group symmetry described by Schçnflies nota-
tion (Figure 2c).
So-called graph set analysis provides a general and quite
powerful method for characterizing the HB networks, and has
been applied to a large number of crystal structures.[6,12] Con-
ventionally, the analysis deals with types of the HB networks
(chain, ring, intramolecular, and other finite patterns) as well as
numbers of donor and acceptor atoms in the focused HB net-
works based on an undirected graph. Accordingly, symmetric
properties, especially SMC, are ignored for simplification.
This article presents an advancement of the graph set analy-
sis based on directed graph from a viewpoint of topological
All of the nine symmetries were successfully confirmed
(Table 1), giving the first experimental proof for the theoretical
expectation based on topology.[11] Each 0D-cubic HB network
with the corresponding symmetry was obtained from the
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The enantiomeric networks C2/
C2’, C1(a)/C1(a)’ and C1(b)/C1(b)’ were
confirmed as racemates in the
crystals. These results suggest
that the symmetries of the 0D-
cubic HB networks depend on
the component amines in size,
shape, and chirality.
The conventional graph set
analysis for the 0D-cubic HB
network
The conventional graph set anal-
ysis with some mathematical
concepts was applied to the
cubic HB networks.[6c] The net-
work of TPAA·1 (refcode:
MIBTOH) is taken as an example
for a procedure of the graph set
analysis (Figure 3). Two kinds of
oxygen atoms are discriminated
(Figure 3a) to yield the corre-
sponding topological cubic net-
work (Figure 3b). A perspective
view of the cubic is drawn, in-
volving a front face with a large,
blue-lined square as well as
a back face with a small, red-
lined square (Figure 3c). The HBs
are denoted as vectors a, b, and
c, with an arrow from donor to
acceptor, like H16!O1, H17!
O2, and H18!O2, respectively.
The cubic network is opened up
as a view from outside to inside
of the cube (Figure 3d). The R-
HB networks of the faces are de-
scribed as the ordered assem-
blies of the vectors in the way of
Figure 2. Supramolecular clusters with the 0D-cubic HB networks composed of primary ammonium carboxylates.
a) Salts list of triphenylacetic acid (TPAA) with primary amines. b) The structures of enantiomeric TPAA·3 clusters.
Their chiral HB networks are schematically drawn. A positive direction of the carboxylate groups is defined as
O(a) with two HBs to O(b) with one HB, and represented by blue and red arrows for clockwise and anticlockwise,
respectively. c) Nine types of networks, (i) D2d, (ii) S4, (iii) CS, (iv) C2/C2’, (v) C1(a)/C1(a)’ and (vi) C1(b)/C1(b)’ in Schçnflies no-
tation, classified according to topological analysis. The individual HBs are labeled for graph set analysis by using
the labels of atoms in crystal structures of TPAA·1 (refcode: MIBTOH), TPAA·4 (refcode: IRUQUH), TPAA·3 (refcode:
GIVFEX), TPAA·5 (refcode: MIBVEZ), TPAA·8 (CCDC no. 809494), TPAA·3 (refcode: MIBVAV) registered in the Cam-
bridge Structural Database.[17] The precise procedures are described in the Supporting Information. Blue solid and
dotted lines represent vertical and horizontal HBs of the front HB networks, respectively. Black solid lines repre-
sent lines connecting front and back HB networks. Pink solid and dotted lines represent vertical and horizontal
HBs of the back HB networks, respectively. The arrows on the lines indicate the direction from donors to accept-
ors. Oxygen, carbon, and nitrogen atoms are represented as red, black, and light-blue balls, respectively.
~ ~
~ ~
Rabac for front, Rabac for back,
~
~
~
~
Rabac for right, Rabac for left,
~ ~
~ ~
Rbcbc for top, Rbcbc for bottom
(Figure 3e).
The analyses for the other net-
works are briefly shown in Fig-
ure 2c. The precise procedures
and results of the conventional
graph set analysis are summar-
ized in the Supporting Informa-
amines involving linear alkyl groups (1, 2) for D2d, tert-butyl (4)
and phenyl groups (7, 9, 10, 11) for S4, isobutyl group (3) for
CS, groups with bulky substituents around a carbons (5, 6) for
tion. As a result, the R-HB networks are classified into the fol-
lowing three types; R42ð8Þ, R43ð10Þ, and R44ð12Þ (Figure 4a(i)).
Namely, various combinations of these three R-HBs construct
all of nine kinds of the cubic HB networks.
C2/C2’, a group with quite bulky substituents (8) for C1(a)/C1(a)’
,
and the isobutyl group (3) for C1(b)/C1(b)’. Even TPAA·10 and
TPAA·11 involving chiral amines yielded achiral S4 networks.
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metry in topology, enabling us to distinguish rectus (Re)- and
sinister (Si)-faces, as in the case of molecular chirality.[18] Thus,
we can trace two types of oxygen atoms of the carboxylate
anions from O(a) to O(b) clockwise and anticlockwise to char-
acterize prochiral symmetry, Re and Si, of the R-HB networks,
respectively. The R34ð10Þ HB networks have the face F2 with
R34(10Re) and F2’ with R43ð10SiÞ. Similarly, the R44ð12Þ HB net-
works have F3 with R44(12Re), F3’ with R44ð12SiÞ, and F4 with
R44ð12mÞ.
Table 1. Topological symmetry of the 0D-cubic HB networks in connec-
tion with countercations of TPAA.
Topology Countercations (refcode or CCDC no.)
D2d
S4
CS
1 (MIBTOH),[a] 2 (MIBTUN)[a]
4 (IRUQUH),[a] 7 (810275), 9 (810265), 10 (810277), 11 (810276)
3 (GIVFEX)[a]
C2/C2’
5 (MIBVEZ),[a] 6 (810274)
C1(a)/C1(a)’ 8 (809494)
C1(b)/C1(b)‘ 3 (MIBVAV)[a]
In this manner, the advanced graph set analysis results in six
different kinds of the faces, denoted as F1, F2, F2’, F3, F3’, and
F4 (Figure 4a(ii)). These faces combine in various ways to yield
the nine cubic HB networks. Figure 4b shows their combina-
tions for chiral networks; C2/C2’, C1(a)/C1(a)’ and C1(b)/C1(b)’. Detailed
opened-up cubes of the R-HB networks are described by
means of the advanced graph set denotations in Figures S4
and S5 (the Supporting Information).
[a] See ref. [11].
Handedness determination of the 0D-cubic HB networks
based on priority rules: Face-vertex method
At last we can discuss the chiral properties of the cubic HB
networks. For such a purpose, we need to recall the enantio-
meric relationship between a pair of dice (Figure S6, the Sup-
porting Information),[19] whose chirality originates from assem-
bly manners of six faces, called topological diversity. Their
handedness is distinguishable by selecting a vertex and tracing
the surrounding three faces clockwise or anticlockwise accord-
ing to the priority sequence: one>two>three. Therefore, it is
indispensable to characterize eight vertices of the cubic HB
networks and select the most favorable vertex among them.
This procedure requires ordering of the above-mentioned
six faces, F1, F2, F2’, F3, F3’, and F4. We can define a priority
sequence of these faces by referring to the Cahn–Ingold–
Prelog (CIP) rule for MC (Figure 5a).[20] According to the stan-
dard sub rule 1 and 2 in the sequence rule, the higher total
atomic number in the faces precedes the lower number, result-
ing in the order; 8<10<12 (Figure 5a(i)). Based on the stan-
dard sub rule 5, the sequence of Si, Re, and m is settled as an
alphabetical order, resulting in the order: Si<Re<m (Fig-
ure 5a(ii)). Since the former rule is prior to the latter, a combina-
tion of these two rules leads to the following priority se-
quence: 8 m<10Si<10Re<12Si<12Re<12m. Accordingly,
the six faces have the sequence, F1(R24ð8mÞ)<F2’(R43ð10SiÞ)<
F2(R34(10Re)) <F3’(R44ð12SiÞ)<F3(R44(12Re))<F4(R44ð12mÞ), as
shown in Figure 5a(iii). For simplicity these six faces are denot-
ed by the symbols of number one to four with black or white
color.
Figure 3. Graph set representations for the cubic HB networks of TPAA·1
(refcode: MIBTOH). a) An independent ion pair (i) and a supramolecular clus-
ter (ii). b) The motif of the D2d cubic HB network. c) Perspective drawing of
the cubic network. d) An opened-up cube of the cubic HB networks and
graph set representations of each R-HB network; e) Graph set descriptions
of the six R-HB networks.
Advanced graph set analysis for discriminating the R-HB
networks as prochiral faces
It is natural that the most predominant vertex is composed
of three faces with number 1, 2, or 3, which have their priority
sequence; 1>2>3 (Figure 5b). Such a predominant vertex in-
volves a clockwise rotation of the faces (Black 1 (F4)!White 2
(F3’)!Black 3 (F2)) to yield a supR-assembly as well as an anti-
clockwise rotation (White 1 (F4)!Black 2 (F3)!White 3 (F2’))
to yield a supS-assembly. The symbols, supR and supS, correspond
to right- and left-handedness of SMC, respectively. The super-
The above-mentioned R-HB networks are further classified into
subgroups on the basis of prochirality, since the R-HB networks
are recognized as faces from a topological viewpoint (Fig-
ure 4a(ii)). The R24ð8Þ network has the face F1 with mirror sym-
metry (m) due to a lack of O(b), and the F1 is denoted as
R24ð8mÞ.The R43ð10Þ and R44ð12Þ networks exhibit prochiral sym-
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Figure 4. Graph set analysis on six R-HB networks. a) (i) Conventional graph set analysis and (ii) advanced graph set analysis. b) All possible combinations of
three enantiomeric R-HB networks of the cubic HB networks around vertices: Na, Nb, Nc, Nd, Ca, Cb, Cc, and Cd. The vertices, Na/Ca and Nb/Cb are located on the
upper and lower side of the front face, respectively, whereas Nc/Cc and Nd/Cd are located on the upper and lower side of the back face, respectively.
graph set analysis provides the common vertex that discrimi-
nates handedness of the cubic HB networks (insets of Fig-
ure 5b). The above-mentioned method may be termed as the
face-vertex method.
Chirality analysis on the 1D-ladder HB networks based on
the advanced graph set analysis
We retrieved crystal structures of primary ammonium carboxy-
lates with the 1D-ladder HB networks from the Cambridge
Structural Database (CSD)[17] and found six topologically differ-
ent networks (L1’/L1 to L5) (Figure 6a).[21] L1’/L1 are enantio-
meric, whereas L2 to L5 are achiral. These networks were sub-
jected to the conventional graph set analysis with some math-
ematical concepts in the same way as the 0D-cubic HB net-
works (for typical examples see the Supporting Information).
Subsequently, the advanced graph set analysis was applied for
chirality discrimination. As a result, it was found that the
R-HB networks are further classified owing to positional iso-
merism of substituents of the organic ions (Figure 6b). Such
isomerism provided the faces; F5’/F5, F6, F7, and F8’/F8, in ad-
dition to the faces: F1, F2’/F2, F3’/F3, F4, as described below
(Figure 6c–f).
Figure 5. Definition of handedness on the 0D-cubic HB networks. a) Priority
sequence of the R-HB networks based on standard sub rules of the CIP rule,
and the R-HB networks are symbolized as black/white 1, black 2, white 2,
black 3, white 3, and black/white 4 for R44ð12mÞ, R44ð12Re), R44ð12Si), R34ð10Re),
R34ð10SiÞ, and R24ð8mÞ, respectively. b) Handedness determination by tracing
the three R-HB networks based on their priority sequence. The HB networks,
C2, C1(a), C1(b) and C2’, C1(a)’, C1(b)’ were defined as left-handed (supS) and right-
handed (supR), respectively.
script sup is attached to the symbols, R and S, which represent
right- and left-handedness of MC, respectively.[8e–g]
In the cubic networks, we are able to determine prochirality
(Re or Si) of the faces by ignoring the substituents of the or-
ganic ions. This is due to the fact that the closed 0D-cubic net-
works have the substituents on their exclusive front-sides out
of the cubic (Figure 4a). In contrast, the unclosed 1D-ladder
networks do not have such exclusive sides, but exhibit posi-
tional isomerism of the substituents on the basis of front and
back sides of the prochiral faces (Figure 6b). First of all, we ex-
plain this isomerism in the R-HB network, R24ð8Þ. The face F1
has two substituents of the ammonium ions on the front-side
Finally, we characterized eight vertices consisting of four ver-
tices with ammonium groups (Na,b,c,d) and four vertices with
carboxylate groups (Ca,b,c,d) (Figure 4b). The cubic network with
C2/C2’ has such a predominant vertex at Nb, Nc, Ca, or Cd. The
cubic with C1(a)/C1(a)’ has the vertex at Nb. The cubic with C1(b)
/
C1(b)’ has one at Nb and Ca. Among them, we find only one
common vertex, Nb, which enables us to determine handed-
ness of the cubic networks. In this manner, the advanced
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Figure 6. Graph set analysis on the 1D-ladder HB networks based on the advanced graph set analysis. a) Schematic representations of the 1D-ladder HB net-
works, L1’/L1 to L5 composed of the R-HB networks F5’/F5, F6, F7, and F8’/F8, which are confirmed in crystal structures retrieved from the Cambridge Struc-
tural Database.[17] b) Positions of substituents on prochiral faces. c)–f) R-HB networks by the advanced graph set analysis based on prochirality and positional
isomerism of the substituents: c) F1 and F7, d) F2’/F2 and F5’/F5, e) F3’/F3 and F6, f) F4 and F8’/F8. The R-HB networks in (i) are observed on the 0D-cubic
HB networks, whereas those in (ii) are observed on the 1D-ladder HB networks considering positional isomerism of their substituents. Symbols with “_” mean
that the directions are defined as viewed from the back side of the R-HB networks. g) Handedness determination of the chiral 1D-ladder HB networks L1’/L1
based on the advanced graph set analysis (top) and supramolecular-tilt-chirality method[8] (bottom).
but no substituent on the back side, resulting in mirror sym-
metry noted as R24ð8mÞ (Figure 6c(i)). On the other hand, the
face F7 has one substituent on the front side as well as the
other substituent on the back side, yielding inversion symme-
try (i) noted as R24ð8mÞ (Figure 6c(ii)).
the faces with all the substituents located on the front sides
(Figure 6e(i)). On the other hand, the face F6 has each one of
the substituents of two ammonium and two carboxylate locat-
ed on the front side as well as the other on the back side,
yielding inversion symmetry as R44ð12iÞ (Figure 6e(ii)). The face
F4 has all of the substituents located on the front side, yielding
mirror symmetry as R44ð12mÞ (Figure 6 f(i)). The racemic faces
F8’/F8 have one carboxylate on the front side and the other
one on the back side (Figure 6 f(ii)). Their prochirality is first
evaluated by the carboxylate on the front side, and subse-
quently by the carboxylate on the back side. The R-HB net-
works of the faces F8’/F8 are denoted as R44ð12ReReÞ and
R44ð12SiSi), respectively. The latter symbol “_” attached to “Si”
and “Re” means that the carboxylates on the backside have
their rotational direction from O(a) to O(b) viewed from the
back side.
Next, we move to the second R-HB network, R34ð10Þ (Fig-
ure 6d). Both of the racemic faces F2’/F2 and F5’/F5 have
a common denotation, R43(10Re) and R34ð10SiÞ, on the basis of
prochirality of the faces, but the substituents of ammoniums
and carboxylates are located on the different sides of the
faces. So that the 0D and 1D networks have a common
method, we first put the substituent of the carboxylate on the
front side recognized as the prochiral (Re)- or (Si)-face, and sub-
sequently discriminate a location of two substituents of the
amines. The faces F2’/F2 have two substituents of the amines
on the front side (Figure 6d(i)), whereas the faces F5’/F5 have
one substituent on the front side as well as the other one on
the back side (Figure 6d(ii)).
Figure 6 g (top) shows the enantiomeric 1D-ladder HB net-
works, L1’/L1, composed of only chiral faces. Namely, the L1
network consists of (Re)-faces F5, and is defined as a right-
handed network (supR). On the other hand, the L1’ is composed
of (Si)-faces F5’, and defined as a left-handed network (supS). As
Figure 6e and 6 f show the third R-HB network, R44ð12Þ. Like
the racemic faces F2’/F2, the racemic faces F3’/F3 are discrimi-
nated as R44(12Re) and R44ð12SiÞ on the basis of prochirality of
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previously reported, the L1 and L1’ networks consist of two-
fold helical assemblies (Figure 6 g (bottom)). The former is de-
termined as a right-handed helix (supP) and the latter a left-
handed helix (supM) by means of the SMTC method. It can be
seen from Figure 6 g that both the SMTC method and the ad-
vanced graph set analysis provide the same handedness. On
the other hand, the 1D-ladder HB networks, L2 to L5, are achi-
ral, since the 1D networks are composed of a mixture of achiral
faces (F6 and F7 for L2), only achiral faces (F6 for L3), enantio-
meric, and achiral faces (F5’/F5, F6 and F7 for L4 as well as F6,
F7, and F8’/F8 for L5).
In this way, the advanced graph set analysis is based on the
fact that the R-HB networks have prochirality and positional
isomerism of substituents. In other words, discrimination of
front and back sides of the prochiral faces in the cubic and
ladder led to the discovery of new symmetric properties, that
is, the exposure of the hidden topological SMC.
Chirality analysis on the 2D-sheet HB networks based on
the advanced graph set analysis
The advanced graph set analysis was applied to the seven 2D-
sheet HB networks: S1’/S1, S2, S3, S4, S5, and S6, as described
below. A precise procedure of the conventional analysis is de-
scribed in the Supporting Information.
So that the 0D, 1D, and 2D networks have a common
method for determining handedness of chirality, we preferen-
tially fix the front and back side of the face and put the sub-
stituents of the carboxylate on the front side. Subsequently,
we judge locations of the substituents as inside or outside in
a hexagon (Figure 7e). For example, we explain the R-HB net-
work, R56ð16Þ. In appearance, this facial network seems to have
mirror symmetry with the R-HB network as R56ð16mÞ (Fig-
ure 7 f(i)), since the substituents of the two carboxylic acids
direct towards the front side, and one is right-handed whereas
the other is left-handed. However, actually, the R-HB network is
chiral due to prochirality of the face and positional isomerism
of the substituents. Figure 7 f(ii) schematically shows the chiral
locations in which two substituents of the carboxylates are put
on the front side and are inclined to the same direction. In
other words, one is located outside of the hexagonal whereas
the other is inside. Based on the substituent located on the
front and outside, prochirality of the face is determined from
a clockwise or anticlockwise rotation of O(a) to O(b) in the cor-
responding hexagon. Such prochirality and positional isomer-
ism provide denotations of R56ð16SiRein)/R56ð16ReSiin) for the
enantiomeric R-HB networks of the faces F9’/F9, respectively.
The symbol “in” represents that the carboxylic acid located
inside of the ring is used for discriminating clockwise or anti-
clockwise rotations from the O(a) to O(b) for prochirality.
The same method affords the enantiomeric R-HB networks
denoted as R56ð16SiSiinÞ/R65ð16ReRein) for the faces F10’/F10 (Fig-
ure 7 f(iii)). The other enantiomeric faces F15’/F15 have the no-
tation as R66ð18SiSiinRein)/R66ð18ReReinSiin) (Figure 7 f(viii)). On the
other hand, the achiral R-HB networks have the faces: F11,
F12, F13, F14, and F16, designated as R44ð12iÞ, R86ð20iÞ, R42ð8iÞ,
R88ð24iÞ, and R48ð16iÞ, as shown in Figure 7 f(iv), 7f(v), 7f(vi),
7f(vii), and 7f(ix), respectively.
Topological diversity of the 2D-sheet HB networks
One or more regular polygons assemble to form 2D planes, as
exemplified in Figure 7a, 7b, and 7d. Regular tessellation by
using a single kind of polygon, such as a triangle, tetragon,
and hexagon, forms the plane according to Pythagoras. More
kinds of polygons are known to perform eight kinds of Archi-
medean tessellations.[22]
We retrieved crystal structures of primary ammonium car-
boxylates with the 2D-sheet HB networks from the CSD. On
this occasion, the triangle-containing tessellations were exclud-
ed due to the fact that they are not formed by an alternative
combination of the primary amines and carboxylic acids in
a 1:1 molar ratio. As a result, we found three kinds of tessella-
tions with seven topologically different 2D-sheet HB networks,
S1’/S1 and S2 to S6 (Figure 7b and 7c).[9c,21a,23] The first tessel-
lation is composed of regular hexagons, which form honey-
comb structures (Figure 7b(i)) with the corresponding hexago-
nal HB network (Figure 7b(ii)). Four topologically different HB
networks, S1’/S1 (Figure 7c(i)), S2 (Figure 7c(ii)), and S3 (Fig-
ure 7c(iii)), were confirmed. The second tessellation is com-
posed of regular tetragons and octagons (Figure 7b(iii)) with
the corresponding HB network (Figure 7b(iv)), including S4
(Figure 7c(iv)) and S5 (Figure 7c(v)). The third tessellation con-
sists of combinations of tetragons, hexagons, and octagons
(Figure 7b(v)), whose HB network (Figure 7b(vi)) includes S6 in
specific topology (Figure 7c(vi)).
The enantiomeric 2D-sheet HB networks, S1/S1’, are com-
posed of only chiral faces (Figure 7c(vii)). Namely, the S1 net-
work consists of (Re)-faces F9, and is defined as a right-handed
2D network (supR). On the other hand, the S1’ is composed of
(Si)-faces F9’, and defined as a left-handed 2D network (supS). It
is noteworthy that the S1/S1’ has translational transformation
but no other symmetric operations. This is the simplest chirali-
ty originating from molecular arrangements. On the other
hand, the 2D-sheet HB networks, S2 to S6, are achiral, since
the 2D networks are composed of racemic faces (F9/F9’ for S2
(Figure 7c(viii)) and F10/F10’ for S3 (Figure 7c(ix))), only achiral
faces (F11 and F12 for S4 (Figure 7c(x)), F13 and F14 for S5
(Figure 7c(xi))), as well as achiral and racemic faces (F11, F16
and F15/F15’ for S6 (Figure 7c(xii))).
In addition, we did not find the 2D tessellations composed
of only regular tetragons as well as combinations of regular
tetragons, hexagons, and dodecahedrons (Figure 7d). This is
because the tessellations with the regular tetragons have four
sides around a vertex, demanding four hydrogen bonds by an
ammonium cation or a carboxylate anion. This seems to be im-
possible, since an ammonium cation with three protons does
not usually form four hydrogen bonds. Similarly, the 2D tessel-
lations with dodecahedrons are not considered to be stable
due to the large R-HB networks.
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Figure 7. Tessellations and 2D-sheet HB networks of primary ammonium carboxylates. a) Tessellations including regular triangles. b) Tessellations from hexa-
gons (i), combinations of tetragons and octagons (iii), as well as combinations of tetragons, hexagons, and octagons (v), with the corresponding 2D HB net-
works, (ii), (iv), as well as (vi), respectively. c) Schematic representations of the 2D-sheet HB networks, which are confirmed in crystal structures retrieved from
the Cambridge Structural Database.[17] The 2D network b(ii) includes three kinds of sheets; S1’/S1 (i), S2 (ii), and S3 (iii) with the R-HB networks: F9’/F9, F9’/F9,
and F10’/F10, respectively. The 2D network b(iv) includes two kinds of sheets, S4 (iv) and S5 (v) with F11 and F12, F13, and F14, respectively. The 2D network
b(vi) has S6 (vi) with F11, F15’/F15, and F16. d) Unfavorable tessellations for the 2D-sheet HB networks of primary ammonium carboxylates. e) An example of
positions of two carboxylates as viewed from front and back sides on prochiral faces. f) The advanced graph set analysis of the component R-HB networks on
the basis of prochirality and positional isomerism of multiple carboxylate groups: (i) presumed network with mirror symmetry, (ii) F9’/F9, (iii) F10’/F10, (iv) F11,
(v) F12, (vi) F13, (vii) F14, (viii) F15’/F15, and (ix) F16. Symbols with “_” and “in” represent directions that are defined from back side and inside of the R-HB net-
works on the basis of the carboxylate anions, respectively.
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fined. All calculations were performed using CrystalStructure crys-
tallographic software package[26] except for refinement, which was
performed using SHELXL-97. Crystallographic data for this paper
has been deposited at the Cambridge Crystallographic Data Centre
under deposition number CCDC-810274 (TPAA·6), CCDC-810275
(TPAA·7), CCDC-809494 (TPAA·8), CCDC-810265 (TPAA·9) CCDC-
810277 (TPAA·10), and CCDC-810276 (TPAA·11) contain the
supplementary crystallographic data for this paper. These data can
be obtained free of charge from The Cambridge Crystallographic
Data Centre via www.ccdc.cam.ac.uk/data_request/cif. Their crystals
parameters are shown in Tables S2 and S3 in the Supporting
Information.
Conclusion
We have advanced the conventional graph set analysis from
a symmetric viewpoint in topology by focusing on the R-HB
networks with prochirality and positional isomerism of sub-
stituents. The advanced analysis was applied to the 0D-cubic,
1D-ladder, and 2D-sheet HB networks, and exposed conven-
tionally hidden SMC, leading to handedness determination of
the SMC, supR and supS.
The chiral 0D, 1D, and 2D HB networks consist of assemblies
of the chiral and/or achiral R-HB networks. The chiral 0D-cubic
HB networks consist of the six R-HB networks, which are or-
dered with priority rules. The common vertex with three pre-
dominant R-HB networks plays a decisive role in determining
the handedness of the SMC, like chirality of dice in daily life.
The chiral 1D-ladder HB networks consist of twofold helices of
only chiral R-HB networks. Handedness of the ladder networks
are defined as helical chirality, like supramolecular-tilt-chirality
of twofold helical facial molecules.[8] On the other hand, the
chiral 2D-sheet HB networks consist of only chiral hexagonal R-
HB networks. Handedness of the sheet networks is determined
as translational chirality.
Acknowledgements
This work was financially supported by Grant-in-Aid for Scien-
tific Research B (24350072) and Grant-in-Aid for Scientific Re-
search on Innovative Areas (24108723) from MEXT and JSPS,
Japan. T.S. thanks Grant-in-Aid for JSPS Fellows (25763), the
GCOE Program of Osaka University and Grants for Excellent
Graduate Schools, MEXT, Japan. We are grateful to Dr. K. Miura,
Dr. S. Baba, and Dr. N. Mizuno for helpful discussions and
crystallographic data collection at the BL38B1 (proposal
nos. 2012A1570 and 2013A1605) in the SPring-8, JASRI.
The key point for these handedness determinations lies in
the consideration of the low-dimensional HB networks as as-
semblies of elemental R-HB networks. This indicates that sym-
metry characterization of elemental supramolecular modules
(the R-HB networks in this case) enables us to expose the
hidden chirality of supramolecular architectures (the low-di-
mensional HB networks in this case). Such a method would be
hopeful for characterizing SMC of various supramolecular archi-
tectures, and contribute to establishing the general rules for
handedness determination on SMC, as well as designing chiral
supramolecular architectures.
Keywords: chirality
·
graph set
·
hydrogen bonds
·
supramolecular chemistry · X-ray diffraction
[1] a) L. Addadi, Z. Berkovitch-Yellin, I. Weissbuch, J. v. Mil, L. J. W. Shirnon,
M. Lahav, L. Leiserowitz, Angew. Chem. 1985, 97, 476–496; Angew.
Chem. Int. Ed. Engl. 1985, 24, 466–485; b) J. S. Brooks, Chem. Soc. Rev.
2010, 39, 2667–2694; c) J. T. A. Jones, T. Hasell, X. Wu, J. Bacsa, K. E.
Jelfs, M. Schmidtmann, S. Y. Chong, D. J. Adams, A. Trewin, F. Schiffman,
F. Cora, B. Slater, A. Steiner, G. M. Day, A. I. Cooper, Nature 2011, 474,
367–371.
[2] a) G. R. Desiraju, Crystal Engineering: The Design of Organic Solids, Elsevi-
er, Amsterdam, 1989; b) J. Haleblian, W. Mccrone, J. Pharm. Sci. 1969,
58, 911–929.
Experimental Section
[3] a) B. Moulton, M. J. Zaworotko, Chem. Rev. 2001, 101, 1629–1658; b) A.
Priimagi, G. Cavallo, P. Metrangolo, G. Resnati, Acc. Chem. Res. 2013, 46,
2686–2695; c) G. R. Desiraju, J. Am. Chem. Soc. 2013, 135, 9952–9967.
[4] a) G. R. Desiraju, Angew. Chem. 1995, 107, 2541–2558; Angew. Chem.
Int. Ed. Engl. 1995, 34, 2311–2327; b) P. Ganguly, G. R. Desiraju, CrystEng-
Comm 2010, 12, 817–833.
[5] a) L. J. Prins, D. N. Reinhoudt, P. Timmerman, Angew. Chem. 2001, 113,
2446; Angew. Chem. Int. Ed. 2001, 40, 2382; b) K. Sada, T. Tani, S. Shinkai,
Synlett 2006, 2364–2374.
[6] a) M. C. Etter, Acc. Chem. Res. 1990, 23, 120–126; b) J. Bernstein, R. E.
Davis, L. Shimoni, N. L. Chang, Angew. Chem. 1995, 107, 1689–1708;
Angew. Chem. Int. Ed. Engl. 1995, 34, 1555–1573; c) J. Grell, J. Bernstein,
G. Tinhofer, Acta Crystallogr. Sect. B 1999, 55, 1030–1043.
Formation of the clusters with cubic hydrogen-bonded net-
works
Molar ratio (1:1) of TPAA and various primary amines were dis-
solved in methanol. Evaporation of the solutions yielded powdered
organic salts. The salts were dissolved in a mixture of non-polar
solvents such as toluene/hexane, benzene/hexane, or chloroform/
hexane. The single crystals were obtained by evaporation of the
solvents for 8 to 24 h.
Data collection
[7] a) D. B. Amabilino, Chirality at the Nanoscale: Nanoparticles, Surfaces,
Materials and More, Wiley-VCH, Weinheim, 2009; b) J. Jacques, A. Collet,
S. H. Wilen, Enantiomers, Racemates, and Resolutions, Krieger, Malabar,
1991; c) T. Matsuura, H. Koshima, J. Photochem. Photobiol. C 2005, 6, 7–
24; d) C. Dryzun, D. Avnir, Chem. Commun. 2012, 48, 5874–5876.
[8] a) M. Miyata, N. Tohnai, I. Hisaki, Acc. Chem. Res. 2007, 40, 694–702;
b) T. Yuge, T. Sakai, N. Kai, I. Hisaki, M. Miyata, N. Tohnai, Chem. Eur. J.
2008, 14, 2984–2993; c) I. Hisaki, T. Sasaki, K. Sakaguchi, W.-T. Liu, N.
Tohnai, M. Miyata, Chem. Commun. 2012, 48, 2219–2221; d) I. Hisaki, T.
Sasaki, N. Tohnai, M. Miyata, Chem. Eur. J. 2012, 18, 10066–10073, and
references therein; e) T. Sasaki, I. Hisaki, S. Tsuzuki, N. Tohnai, M. Miyata,
CrystEngComm 2012, 14, 5749–5752; f) T. Sasaki, I. Hisaki, T. Miyano, N.
Tohnai, K. Morimoto, H. Sato, S. Tsuzuki, M. Miyata, Nat. Commun. 2013,
X-ray diffraction data were collected on a Rigaku RAXIS-RAPID
imaging plate diffractometer with a 2D area detector using graph-
ite monochromated CuKa radiation (l=1.54178 ꢁ). Lattice parame-
ters were obtained by least-squares analysis from reflections for
three oscillation images. Structure solution was achieved by direct
methods, SIR2004[24] or SHELXS97.[25] The structures were refined
by a full-matrix least-squares procedure with all the observed re-
flections based on F2. All non-hydrogen atoms were refined with
anisotropic displacement parameters, and hydrogen atoms were
placed in idealized positions with isotropic displacement parame-
ters relative to the connected non-hydrogen atoms, and not re-
Chem. Eur. J. 2014, 20, 2478 – 2487
www.chemeurj.org
2486
ꢀ 2014 Wiley-VCH Verlag GmbH & Co. KGaA, Weinheim

Full Paper
4, 1787; g) T. Sasaki, Y. Ida, A. Tanaka, I. Hisaki, N. Tohnai, M. Miyata,
CrystEngComm 2013, 15, 8237–8240.
[17] http://www.ccdc.cam.ac.uk/products/csd/.
[18] V. Prelog, G. Helmchen, Angew. Chem. 1982, 94, 614–631; Angew. Chem.
Int. Ed. Engl. 1982, 21, 567–583.
[9] a) M. Leibovitch, G. Olovsson, J. R. Scheffer, J. Trotter, Pure Appl. Chem.
1997, 69, 815–823; b) J. R. Scheffer, Can. J. Chem. 2001, 79, 349–357;
c) A. Tanaka, K. Inoue, I. Hisaki, N. Tohnai, M. Miyata, A. Matsumoto,
Angew. Chem. 2006, 118, 4248–4251; Angew. Chem. Int. Ed. 2006, 45,
4142–4145.
[10] a) L. M. Pasteur, C. R. Acad. Sci. 1853, 37, 162–166; b) K. Kodama, Y. Ko-
bayashi, K. Saigo, Chem. Eur. J. 2007, 13, 2144–2152; c) Y. Imai, M. Take-
shita, T. Sato, R. Kuroda, Chem. Commun. 2006, 1070–1072; d) A. Ichika-
wa, H. Ono, T. Echigo, Y. Mikata, CrystEngComm 2011, 13, 4536–4548;
e) K. Megumi, F. N. B. M. Arif, S. Matsumoto, M. Akazome, Cryst. Growth
Des. 2012, 12, 5680–5685.
[11] T. Yuge, N. Tohnai, T. Fukuda, I. Hisaki, M. Miyata, Chem. Eur. J. 2007, 13,
4163–4168.
[12] For example, see W. D. S. Motherwell, G. P. Shields, F. H. Allen, Acta Crys-
tallogr. Sect. B 1999, 55, 1044–1056.
[13] a) N. Tohnai, Y. Mizobe, M. Doi, S. Sukata, T. Hinoue, T. Yuge, I. Hisaki, Y.
Matsukawa, M. Miyata, Angew. Chem. 2007, 119, 2270–2273; Angew.
Chem. Int. Ed. 2007, 46, 2220–2223; b) A. Yamamoto, T. Hamada, I.
Hisaki, M. Miyata, N. Tohnai, Angew. Chem. 2013, 125, 1753–1756;
Angew. Chem. Int. Ed. 2013, 52, 1709–1712; c) A. Yamamoto, T. Hasega-
wa, T. Hamada, T. Hirukawa, I. Hisaki, M. Miyata, N. Tohnai, Chem. Eur. J.
2013, 19, 3006–3016; d) A. Yamamoto, T. Hirukawa, I. Hisaki, M. Miyata,
N. Tohnai, Tetrahedron Lett. 2013, 54, 1268–1273.
[19] M. Gardner, The New Ambidextrous Universe, W. H. Freeman and Compa-
ny, New York, 1999 (for chirality of dice, see pp. 29–30).
[20] a) R. S. Cahn, C. K. Ingold, J. Chem. Soc. 1951, 612–622; b) R. S. Cahn,
C. K. Ingold, V. Prelog, Experientia 1956, 12, 81–124; c) R. S. Cahn, C. K.
Ingold, V. Prelog, Angew. Chem. 1966, 78, 413–447; Angew. Chem. Int.
Ed. Engl. 1966, 5, 385–415.
[21] a) S. Nagahama, K. Inoue, K. Sada, M. Miyata, A. Matumoto, Cryst.
Growth Des. 2003, 3, 247–256; b) K. Kinbara, Y. Tagawa, K. Saigo, Tetra-
hedron: Asymmetry 2001, 12, 2927–2930; c) K. Kinbara, A. Kai, Y. Maeka-
wa, Y. Hashimoto, S. Naruse, M. Hasegawa, K. Saigo, J. Chem. Soc. Perkin
Trans. 2 1996, 247–253.
[22] B. Grꢃnbaum, G. C. Shephard, Tilings and Patterns, W. H. Freeman, New
York, 1986.
[23] a) A. Matsumoto, T. Kunisue, A. Tanaka, N. Tohnai, K. Sada, M. Miyata,
Chem. Lett. 2004, 33, 96–97; b) R. Basaran, S.-Q. Dou, A. Weiss, Ber. Bun-
senges. Phys. Chem. 1992, 96, 35–44; c) R. Basaran, S.-Q. Dou, A. Weiss,
Ber. Bunsenges. Phys. Chem. 1991, 95, 46–57.
[24] A. Altomare, M. Burla, M. Camalli, G. Cascarano, C. Giacovazzo, A. Gua-
gliardi, A. Moliterni, G. Polidori, R. Spagna, J. Appl. Crystallogr. 1999, 32,
115.
[25] G. M. Sheldrick, Acta Crystallogr. Sect. A 2008, 64, 112–122.
[26] Crstal Structure 3.8.: Crystal Structure Analysis Package; Rigaku and
Rigaku Americas: The Woodlands, TX, 2000–2007.
[14] T. Yuge, N. Kai, I. Hisaki, M. Miyata, N. Tohnai, Chem. Lett. 2007, 36,
1390–1391.
[15] T. Yuge, I. Hisaki, M. Miyata, N. Tohnai, CrystEngComm 2008, 10, 263–
266.
[16] S. McDonald, L. Ojamꢂe, S. J. Singer, J. Phys. Chem. A 1998, 102, 2824–
2832.
Received: September 26, 2013
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