Chirality Fittingness Research Articles

Type-Itemized Enumeration of Five Types of Stereoisograms and Their Simplified Diagrams for Characterizing Cubane Derivatives

Abstract For discussing stereochemistry of cubane derivatives, the concept of stereoisograms controlled by the RS-stereoisomeric group $\boldsymbol{{O}}_{\boldsymbol{{h}}\tilde{\boldsymbol{{\sigma}}}\skew2\hat{\boldsymbol{{I}}}}$ (order 98) is formulated by the extension of the point group Oh (order 48). The vertical directions of a stereoisogram are concerned with chirality as the first kind of handedness, which is controlled by rotations and reflections of Oh. The horizontal directions of a stereoisogram are concerned with RS-stereogenicity as the second kind of handedness, which is controlled by an RS-stereogenic group $\boldsymbol{{O}}_{{\tilde{\boldsymbol{{\sigma}}}}}$ (order 48). The diagonal directions of a stereoisogram are concerned with sclerality/asclerality, which is controlled by an LR-permutation group $\boldsymbol{{O}}_{{\skew2\hat{\boldsymbol{{I}}}}}$ (order 48). These groups are characterized by combined-permutation representations (CPRs), which are used to calculate respective cycle indices with chirality fittingness (CI-CFs) for enumeration under the GAP system. Enumerations are conducted under O, Oh, $\boldsymbol{{O}}_{{\tilde{\boldsymbol{{\sigma}}}}}$, $\boldsymbol{{O}}_{{\skew2\hat{\boldsymbol{{I}}}}}$, and $\boldsymbol{{O}}_{\boldsymbol{{h}}\tilde{\boldsymbol{{\sigma}}}\skew2\hat{\boldsymbol{{I}}}}$. Then, the enumeration results are discussed in terms of simplified diagrams. It has been proven that there are two main categories for characterizing the appearance of five types of stereoisograms. The first category is an ascleral case which is characterized by the presence of type-I and type-IV stereoisograms. In contrast, the second category is a scleral case which is characterized by the presence of type-II, type-III, and type V stereoisograms. There also exist minor cases in which the coexistence of the first and second categories is observed.

Standard mark table and standard USCI-CF table of the point group D6h. Unification of candidates derived from different D6h-skeletons

Combined-permutation representations (CPRs) for characterizing -skeletons (a benzene skeleton, a Haworth-projected skeleton, a superphane skeleton, and a coronene skeleton) are constructed by starting from respective sets of generators, where the permutation of each generator is combined with a mirror-permutation of 2-cycle to treat both achiral and chiral substituents under the GAP system. Thereby, the CPR of degree 8 (= 6 + 2) for the benzene skeleton, the CPR of degree 14 (= 12 + 2) for the Haworth-projected skeleton, the CPR of degree 14 (= 12 + 2) for the superphane skeleton, the CPR of degree 14 (= 12 + 2) for the coronene skeleton are generated to give primary mark tables (tables of marks) based on these CPRs. These primary mark tables generated by the GAP system are different in the sequence of subgroups from each other, although they stem from the same point group . They are unified into a single standard mark table by means of a newly-devised GAP function MarkTableforUSCI. Moreover, another newly-devised GAP function constructUSCITable is employed to construct a standard USCI-CF (unit-subduced-cycle-index-with-chirality-fittingness) table concordantly. After a set of PCI-CFs (partial cycle indices with chirality fittingness) is calculated for each skeleton, symmetry-itemized combinatorial enumeration is conducted by means of the PCI method of Fujita’s USCI approach (S. Fujita, Symmetry and Combinatorial Enumeration in Chemistry, Springer-Verlag, Berlin-Heidelberg, 1991).

Symmtry-itemized enumeration of compounds derived from different $${\varvec{I}}_{h}$$-skeletons by means of combined-permutation representations and newly-developed GAP functions

Combined-permutation representations (CPRs) for characterizing $${\varvec{I}}_{h}$$-skeletons (a dodecahedral skeleton 1 having 20 vertices, an icosahedral skeleton 2 having 12 vertices, and a $$\hbox {C}_{{60}}$$-fullerene skeleton 3 having 60 vertices) are constructed by starting from respective sets of generators, where the permutation of each generator is combined with a mirror-permutation of 2-cycle to give the CPR of degree 22 (= 20 + 2) for 1, the CPR of degree 14 (= 12 + 2) for 2, and the CPR of degree 62 (= 60 + 2) for 3. Mark tables (tables of marks) of these CPRs are different in the sequence of subgroups from each other when they are produced as primary mark tables by the GAP system. On the other hand, the GAP functions MarkTableforUSCI and constructUSCITable, which have been previously developed to systematize the concordant construction of a standard mark table and a standard USCI-CF (unit-subduced-cycle-index-with-chirality-fittingness) table, are capable of constructing the standard mark table and the standard USCI-CF table even if we start from any of these CPRs. After a set of PCI-CFs (partial cycle indices with chirality fittingness) is calculated for each skeleton, symmetry-itemized combinatorial enumeration is conducted by means of the PCI method of Fujita’s USCI approach (Fujita in Symmetry and combinatorial enumeration in chemistry, Springer, Berlin, 1991). Construction of the CPR of degree 7 (= 5 + 2) for characterizing the $${\varvec{I}}_{h}$$ group is also discussed by starting from the alternating group $$\hbox {A}_{{5}}$$ isomorphic to the point group $${\varvec{I}}$$.

Standardization of Mark Tables and USCI-CF (Unit Subduced Cycle Indices with Chirality Fittingness) Tables Derived from Different D3h-Skeletons

Abstract Combined-permutation representations (CPRs) for characterizing D3h-skeletons (i.e., a cyclopropane skeleton, a trigonal bipyramidal skeleton, an iceane skeleton, and so on) are constructed by starting from respective sets of generators, where the permutation of each generator is combined with a mirror-permutation of 2-cycles to give the CPR of degree 8 (= 6 + 2) for the cyclopropane skeleton, the CPR of degree 7 (= 5 + 2) for the trigonal bipyramidal skeleton, the CPR of degree 14 (= 12 + 2) for the iceane skeleton, and so on. Mark tables (tables of marks) of these CPRs are different in the alignment of subgroups from each other when they are produced as primary mark tables by the GAP system. On the other hand, the GAP functions MarkTableforUSCI and constructUSCITable, which have been previously developed to systematize the concordant construction of a standard mark table and a standard USCI-CF (unit-subduced-cycle-index-with-chirality-fittingness) table, are capable of constructing the standard mark table and the standard USCI-CF table even if we start from any of these CPRs. After a set of PCI-CFs (partial cycle indices with chirality fittingness) is calculated for each skeleton by means of the newly-developed GAP functions, symmetry-itemized combinatorial enumeration is conducted by means of the PCI method of Fujita’s USCI approach (S. Fujita, Symmetry and Combinatorial Enumeration in Chemistry, Springer-Verlag, Berlin-Heidelberg, 1991).

Stereoisomerism Accompanied with Bond Rotations. Characterization of Propane Derivatives by Correlation Diagrams of Epimeric Stereoisograms

Abstract The stereoisomerism of propane derivatives with bond rotations is investigated by means of correlation diagrams of stereoisograms, where the diagrams are governed by epimeric RS-stereoisomeric groups formulated as subgroups of stereoisomeric groups. A molecular-symmetry group GCσ for specifying the eight positions of a propane skeleton is obtained as a wreath product with chirality fittingness (C2v[C3v, C∞v]), where bond rotations are taken into consideration. The group GCσ of order 36 is extended to give the corresponding RS-stereoisomeric group $\boldsymbol{{G}}_{\boldsymbol{{C}}\boldsymbol{{\sigma}}\tilde{\boldsymbol{{\sigma}}}\skew2\hat{\boldsymbol{{I}}}}$ of order 72, which characterizes the global symmetry of the propane skeleton. Then the group $\boldsymbol{{G}}_{\boldsymbol{{C}}\boldsymbol{{\sigma}}\tilde{\boldsymbol{{\sigma}}}\skew2\hat{\boldsymbol{{I}}}}$ is further extended to give a stereoisomeric group H of order 288 for treating chiral and achiral proligands (or $\ddot{\boldsymbol{{H}}}$ of order 144 for treating achiral proligands only) by considering epimerizations at the three RS-stereogenic carbon centers (for Ci, i = 1,2,3). The groups H or $\ddot{\boldsymbol{{H}}}$ are useful to investigate thestereoisomerism of propane derivatives. An epimeric RS-stereoisomeric group $\boldsymbol{{G}}_{\boldsymbol{{C\boldsymbol{{\sigma}}} }}^{\tilde{{\boldsymbol{{\sigma}}}}_{\boldsymbol{{i}}}}$ (or $\ddot{\boldsymbol{{G}}}_{\boldsymbol{{C\boldsymbol{{\sigma}}} }}^{\tilde{{\boldsymbol{{\sigma}}}}_{\boldsymbol{{i}}}}$) is selected as a subgroup of H (or $\ddot{\boldsymbol{{H}}}$) in order to show the local RS-stereogenicity at a given RS-stereogenic center (Ci), where the element $\tilde{\sigma }_{i}$ represents an epimerization at the Ci atom. Such an epimeric RS-stereoisomeric group $\boldsymbol{{G}}_{\boldsymbol{{C\boldsymbol{{\sigma}}} }}^{\tilde{{\boldsymbol{{\sigma}}}}_{\boldsymbol{{i}}}}$ (or $\ddot{\boldsymbol{{G}}}_{\boldsymbol{{C\boldsymbol{{\sigma}}} }}^{\tilde{{\boldsymbol{{\sigma}}}}_{\boldsymbol{{i}}}}$) corresponds to a correlation diagram of stereoisograms for showing RS-stereoisomeric relationships at the Ci atom, i.e., enantiomeric, RS-diastereomeric, and holantimeric relationships. Two RS-diastereomers of each pair are pairwise specified by a pair of RS-stereodescriptors of the CIP system.

Hierarchical enumeration of octahedral complexes by using combined-permutation representations

Hierarchical enumerations of octahedral derivatives are conducted in accord with the hierarchy of groups for characterizing an octahedral skeleton, i.e., point groups ( $${\varvec{O}}$$ and $${\varvec{O}}_{h}$$ ; orders 24 and 48) $$\subset $$ RS-stereoisomeric group ( $${\varvec{O}}_{h\widetilde{\sigma }\widehat{I}}$$ ; order 96) $$\subset $$ stereoisomeric group ( $$\widetilde{{\varvec{O}}}_{h\widetilde{\sigma }\widehat{I}} \,(= {\varvec{S}}^{[6]}_{\sigma \widehat{I}})$$ ; order 1440) $$=$$ isoskeletal group ( $$\widetilde{\widetilde{{\varvec{O}}}}_{h\widetilde{\sigma }\widehat{I}} \,(= {\varvec{S}}^{[6]}_{\sigma \widehat{I}})$$ ; order 1440). The corresponding cycle indices with chirality fittingness (CI-CFs) are calculated by using combined-permutation representations (Fujita in MATCH Commun Math Comput Chem 76:379–400, 2016). Then, a set of three ligand-inventory functions for 3D enumeration is introduced into the CI-CFs for the enumerations under the point groups and the RS-stereoisomeric group, while a single ligand-inventory function for 2D (graph) enumeration is introduced into the CI-CFs for the enumerations under the stereoisomeric group and the isoskeletal group. The expansion of the resulting equations gives generating functions, in which the coefficients of respective terms show the numbers of octahedral derivatives. They are discussed by drawing isomer-classification diagrams after they are categorized into octahedral derivatives with achiral proligands, those with achiral and chiral proligands, and those with chiral proligands. Type-I and type-V stereoisograms are drawn to demonstrate the conceptual distinction between RS-stereogenicity and stereogenicity, where the combination of configuration indices and C/A-descriptors is discussed.

Hierarchical enumeration of Kekulé’s benzenes, Ladenburg’s benzenes, Dewar’s benzenes, and benzvalenes by using combined-permutation representations

The group hierarchy for each skeleton of ligancy 6 is formulated to be: point group (PG $${\varvec{G}}_{\sigma }$$ ) $$\subseteq $$ RS-stereoisomeric group (RS-SIG $${\varvec{G}}_{\sigma \widetilde{\sigma }\widehat{I}}$$ ) $$\subseteq $$ stereoisomeric group (SIG $$\widetilde{{\varvec{G}}}_{\sigma \widetilde{\sigma }\widehat{I}}$$ ) $$\subseteq $$ isoskeletomeric group (ISG $$\widetilde{\widetilde{{\varvec{G}}}}_{\sigma \widetilde{\sigma }\widehat{I}}$$ = $${\varvec{S}}^{[6]}_{\sigma \widehat{I}}$$ ), where we start from the PG $${\varvec{G}}_{\sigma }$$ = $${\varvec{D}}_{6h}$$ for the Kekule benzene skeleton, from the PG $${\varvec{G}}_{\sigma }$$ = $${\varvec{D}}_{3h}$$ for the Ladenburg benzene skeleton, from the PG $${\varvec{G}}_{\sigma }$$ = $${\varvec{C}}_{2v}$$ for the Dewar benzene skeleton, or from the PG $${\varvec{G}}_{\sigma }$$ = $${\varvec{C}}_{2v}$$ for the benzvalene skeleton. After these groups are constructed as combined-permutation representations, the calculation of the respective cycle indices with chirality fittingness (CI-CFs) and the introduction of ligand-inventory functions are conducted to give generation functions for 3D-based enumerations (for PGs and RS-SIGs) and 2D-based enumerations (for SIGs and ISGs). The enumeration results are discussed by means of isomer-classification diagrams, in which equivalence classes under enantiomerism (for PGs), RS-stereoisomerism (for RS-SIGs), stereoisomerism (for SIGs), and isoskeletomerism (for ISGs) are illustrated schematically. The implicit connotations of the conventional terms “skeletal isomerism”, “positional isomerism”, and “constitutional isomerism” are discussed, where the effects of the concept of isoskeletomerism are emphasized.

Simplified Enumeration of Benzene Isomsers Based on the GAP System

After the development of a GAP command for calculating cycle indices with chirality fittingness (CICFs), benzene derivatives have been enumerated as 3D molecular entities.

Type-Itemized Enumeration of RS-Stereoisomers of Octahedral Complexes

Stereoisograms of octahedral complexes are classified into five types (type I--typeV) under the action of the corresponding RS-stereoisomeric group. Their enumeration is accomplished in a type-itemized fashion, where Fujita's proligand method developed originally for combinatorial enumeration under point groups (S. Fujita, Theor. Chem. Acc., 113, 73--79 (2005)) is extended to meet the requirement of Fujita's stereoisogram approach. The cycle index with chirality fittingness (CI-CF) of the point group O_h is modulated by taking account of the CI-CF for calculating type-V quadruplets contained in stereoisograms. The modulated CI-CF is combined with a CI-CF of the maximum chiral point group (O), a CI-CF of the maximum RS-permutation group, a CI-CF of the maximum ligand-reflection group, and a CI-CF of the RS-stereoisomeric group, so as to generate CI-CFs for evaluating type-I to type-V quadruplets. By introducing ligand-inventory functions into the CI-CFs, the numbers of quadruplets ofoctahedral complexes are obtained and shown in tabular forms. Several stereoisograms for typical complexes are depicted. Their configuration indices and C/A-descriptors are discussed on the basis of Fujita's stereoisogram approach.

Type-itemized enumeration of quadruplets of RS-stereoisomers: II—cycle indices with chirality fittingness modulated by type-V quadruplets

Fujita’s proligand method developed originally for combinatorial enumeration under point groups (Fujita in Theor Chem Acc 113:73–79, 2005) has been extended to meet combinatorial enumeration under Fujita’s stereoisogram approach (Fujita in J Org Chem 69:3158–3165, 2004). A new succinct method for type-itemized enumeration of quadruplets of RS-stereoisomers has been developed on the basis of cycle indices with chirality fittingness (CI–CF) modulated by type-V quadruplets, where CI–CFs for the subgroups of the corresponding RS-stereoisomeric group are used to calculate CI–CFs of the respective types (type I to type V). By introducing ligand-inventory functions to these CI–CFs, generating functions have been obtained to give the numbers of inequivalent quadruplets, which are itemized with respect to five types. The method using a modulated CI–CF is applied to enumerations based on an oxirane skeleton, an allene skeleton, and a tetrahedral skeleton. The relationship of RS-stereoisomers versus stereoisomers is discussed by referring to the enumeration results of oxirane derivatives.

Type-itemized enumeration of quadruplets of RS-stereoisomers. I. Cycle indices with chirality fittingness modulated by type-IV quadruplets

Type-itemized enumeration of quadruplets of RS-stereoisomers. I. Cycle indices with chirality fittingness modulated by type-IV quadruplets

Symmetry-itemized enumeration of RS-stereoisomers of allenes. I. The fixed-point matrix method of the USCI approach combined with the stereoisogram approach

After the RS-stereoisomeric group \(\mathbf{D}_{2d\widetilde{\sigma }\widehat{I}}\) of order 16 has been defined by starting point group \(\mathbf{D}_{2d}\) of order 8, the isomorphism between \(\mathbf{D}_{2d\widetilde{\sigma }\widehat{I}}\) and the point group \(\mathbf{D}_{4h}\) of order 16 is thoroughly discussed. The non-redundant set of subgroups (SSG) of \(\mathbf{D}_{2d\widetilde{\sigma }\widehat{I}}\) is obtained by referring to the non-redundant set of subgroups of \(\mathbf{D}_{4h}\). The coset representation for characterizing the orbit of the four positions of an allene skeleton is clarified to be \(\mathbf{D}_{2d\widetilde{\sigma }\widehat{I}}(/\mathbf{C}_{s\widetilde{\sigma }\widehat{I}})\), which is closely related to the \(\mathbf{D}_{4h}(/\mathbf{C}_{2v}^{\prime \prime \prime })\). According to the unit-subduced-cycle-index (USCI) approach (Fujita, Symmetry and combinatorial enumeration of chemistry. Springer, Berlin 1991), the subduction of \(\mathbf{D}_{2d\widetilde{\sigma }\widehat{I}}(/\mathbf{C}_{s\widetilde{\sigma }\widehat{I}})\) is examined so as to generate unit subduced cycle indices with chirality fittingness (USCI-CFs). Then, the fixed-point matrix method of the USCI approach is applied to the USCI-CFs. Thereby, the numbers of quadruplets are calculated in an itemized fashion with respect to the subgroups of \(\mathbf{D}_{2d\widetilde{\sigma }\widehat{I}}\). After the subgroups of \(\mathbf{D}_{2d\widetilde{\sigma }\widehat{I}}\) are categorized into types I–V, type-itemized enumeration of quadruplets is conducted to illustrate the versatility of the stereoisogram approach.

Symmetry-itemized enumeration of quadruplets of RS-stereoisomers: I—the fixed-point matrix method of the USCI approach combined with the stereoisogram approach

The RS-stereoisomeric group $$\mathbf{T}_{d\widetilde{\sigma }\widehat{I}}$$ is examined to characterize quadruplets of RS-stereoisomers based on a tetrahedral skeleton and found to be isomorphic to the point group $$\mathbf{O}_{h}$$ of order 48. The non-redundant set of subgroups (SSG) of $$\mathbf{T}_{d\widetilde{\sigma }\widehat{I}}$$ is obtained by referring to the non-redundant SSG of $$\mathbf{O}_{h}$$ . The coset representation for characterizing the orbit of the four positions of the tetrahedral skeleton is clarified to be $$\mathbf{T}_{d\widetilde{\sigma }\widehat{I}}(/\mathbf{C}_{3v\widetilde{\sigma }\widehat{I}})$$ , which is closely related to the $$\mathbf{O}_{h}(/\mathbf{D}_{3d})$$ . According to the unit-subduced-cycle-index (USCI) approach (Fujita in Symmetry and combinatorial enumeration in chemistry. Springer, Berlin, 1991), the subdution of $$\mathbf{T}_{d\widetilde{\sigma }\widehat{I}}(/\mathbf{C}_{3v\widetilde{\sigma }\widehat{I}})$$ is examined so as to generate unit subduced cycle indices with chirality fittingness (USCI-CFs). The fixed-point matrix method of the USCI approach is applied to the USCI-CFs. Thereby, the numbers of quadruplets are calculated in an itemized fashion with respect to the subgroups of $$\mathbf{T}_{d\widetilde{\sigma }\widehat{I}}$$ . After the subgroups of $$\mathbf{T}_{d\widetilde{\sigma }\widehat{I}}$$ are categorized into types I–V, type-itemized enumeration of quadruplets is conducted to illustrate the versatility of the stereoisogram approach.

Symmetry-Itemized Enumeration of Cubane Derivatives as Three-Dimensional Entities by the Elementary-Superposition Method of the USCI Approach

Abstract Cubane derivatives with a given chemical formula and a given symmetry are enumerated by applying the elementary-superposition method (S. Fujita, Theor. Chim. Acta1992, 82, 473–498) of the unit-subduced-cycle-index (USCI) approach to a cubane skeleton of the point group Oh. A cycle index (CI) for a regular or an irregular case of chiral and achiral ligands is calculated in accord with such a given chemical formula. The CI is superposed elementarily onto respective subduced cycle indices with chirality fittingness (SCI-CFs), which are calculated by starting from unit subduced cycle indices with chirality fittingness (USCI-CFs). Thereby, the numbers of fixed points (derivatives) are obtained with respect to the respective SCI-CFs to give a fixed-point vector (FPV). The resulting FPV is multiplied by an inverse matrix of the mark table of Oh to give an isomer-counting vector (ICV), which contains the numbers of 3D-structural isomers in an itemized fashion with respect to point-group symmetries. The concept of prochirality is examined by using cubane derivatives enumerated by the ICV.

Symmetry-Itemized Enumeration of Cubane Derivatives as Three-Dimensional Entities by the Partial-Cycle-Index Method of the USCI Approach

Abstract By placing chiral and achiral proligands on the eight positions of a cubane skeleton of Oh, resulting cubane derivatives are counted as three-dimensional (3D) structural isomers by the partial-cycle-index (PCI) method of the unit-subduced-cycle-index (USCI) approach (S. Fujita, Symmetry and Combinatorial Enumeration in Chemistry, Springer-Verlag, 1991; S. Fujita, Diagrammatical Approach to Molecular Symmetry and Enumeration of Stereoisomers, University of Kragujevac, Kragujevac, 2007), where the numbers of 3D structural isomers with given constitutional formulas are itemized with the subsymmetries of Oh. For this purpose, partial cycle indices with chirality fittingness (PCI-CFs) for respective subgroups of Oh are calculated by using unit subduced cycle indices with chirality fittingness (USCI-CFs) and an inverse matrix of the mark table of Oh. After introducing ligand-inventory functions into the PCI-CFs, expansions of the resulting equations provide us with generating functions for giving the numbers of 3D structural isomers. A Maple program source for counting cubane derivatives as 3D structural isomers is given as an example of practical calculation for listing the resulting data in tabular forms.

Enumeration of Three-Dimensional Structures Derived from a Dodecahedrane Skeleton under a Restriction Condition. II. The Restricted-Partial-Cycle-Index (RPCI) Method Based on Restricted Subduced Cycle Indices

The restricted-partial-cycle-index (RPCI) method for combinatorial enumeration under the restriction of no adjacency of ligands has been developed as a restricted version of the partial-cycle-index (PCI) method of the unit-subduced-cycle-index (USCI) approach (S. Fujita, "Symmetry and Combinatorial Enumeration in Chemistry," Springer-Verlag (1991)). To take account of the restriction condition, (unrestricted) subduced cycle indices with chirality fittingness (SCI-CFs) of the USCI approach are converted into restricted subduced cycle indices with chirality fittingness (RSCI-CFs). Then, restricted partial cycle indices with chirality fittingness (RPCI-CFs) are derived from the RSCI-CFs, just as partial cycle indices with chirality fittingness (PCI-CFs) are derived from the SCI-CFs in the USCI approach. The resulting RPCI-CFs provide generating functions for restricted enumerations. The RPCI method using such RPCI-CFs is applied to enumeration of dodecahedrane derivatives under the restriction of no adjacency of ligands. Several enumerated derivatives are depicted and their symmetries are discussed to comprehend stereochemical properties such as pseudoasymmetry, sphericity, and prochirality.

Symmetry-Itemized Enumeration of Cubane Derivatives as Three-Dimensional Entities by the Fixed-Point Matrix Method of the USCI Approach

Abstract Cubane derivatives with chiral and achiral proligands are counted as three-dimensional (3D) structural isomers by the fixed-point matrix method of the unit-subduced-cycle-index (USCI) approach (S. Fujita, “Symmetry and Combinatorial Enumeration in Chemistry,” Springer-Verlag, 1991). The numbers of such 3D structural isomers are itemized with respect to their point-group symmetries, which are subgroups of the point group Oh for a cubane skeleton. For this purpose, the full list of unit subduced cycle indices with chirality fittingness (USCI-CFs) of Oh is constructed in a tabular form. Fixed-point vectors or fixed-point matrices, which are calculated by starting from USCI-CFs, are used to count 3D structural isomers in the form of isomer-counting matrices. A Maple program source for counting cubane derivatives as 3D structural isomers is given as an example of practical calculation.

Combinatorial enumeration of three-dimensional trees as stereochemical models of alkanes: an approach based on Fujita’s proligand method and dual recognition as uninuclear and binuclear promolecules

Three-dimensional (3D) trees, which are defined as a 3D extension of trees, are enumerated by Fujita’s proligand method (Fujita, Theor. Chem. Acc. 113 (2005) 73–79 and 80–86; 115 (2006) 37–53). Such 3D-trees are dually recognized as uninuclear promolecules and as binuclear ones. The 3D-trees regarded as uninuclear promolecules are enumerated to give the gross number of 3D-trees, which suffers from redundancy due to contaminants. To evaluate the number of such contaminants, the 3D-trees are alternatively enumerated as binuclear promolecules. Cycle indices with chirality fittingness (CI-CFs) composed of three kinds of sphericity indices (SIs), i.e., a d for homospheric cycles, c d for enantiospheric cycles, and b d for hemispheric cycles are obtained for evaluating promolecules of the two kinds. The CI-CFs for the uninuclear promolecules and those for the related binuclear promolecules are compared in terms of the dichotomy between balanced 3D-trees and unbalanced 3D-trees. Thereby, the redundancy due to such contaminants is deleted effectively so as to give the net number of 3D-trees. The validity of this procedure is proved in three ways, all of which are based on the respective modes of the correspondence between uninuclear promolecules and binuclear ones. In order to enumerate 3D-trees by following this procedure, the CI-CFs are converted into functional equations by substituting the SIs for a(x d ), c(x d ), and b(x d ). Thereby, the numbers of 3D-trees or equivalently those of alkanes as stereoisomers are calculated under various conditions and collected up to 20 carbon content in a tabular form. Now, the stereochemical problems (on the number of stereoisomers) by van’t Hoff (van’t~Hoff, Arch. Neerlandaises des Sci. Exactes et Nat, 9 (1874) 445–454”) and by LeBel (LeBel, Bull. Soc. Chim. Fr. (2), 22 (1874) 337–347) and the enumeration problems (on the number of trees) by Cayley (Cayley, Philos. Mag. 47 (1874) 444–446), both initiated in the 1870s, have been solved in a common theoretical framework, which satisfies both chemical and mathematical requirements.

Numbers of Monosubstituted Alkanes as Stereoisomers

Monosubstituted alkanes as stereoisomers, not as constitutional isomers, are regarded as planted three-dimensional (3D) trees, which are defined as a 3D extension of planted trees (graphs). They are thus recognized as 3D-objects (planted promolecules) and enumerated by Fujita's proligand method (Fujita S (2005) Theor. Chem. Acc. 113:73–79, 113:80–86, Fujita S (2006) Theor. Chem. Acc. 115:37–53). By starting from three kinds of sphericity indices, i.e., ad for homospheric cycles, cd for enantiospheric cycles, and bd for hemispheric cycles, cycle indices with chirality fittingness (CI-CFs) are obtained to enumerate planted 3D-trees or equivalently monosubstituted alkanes as stereoisomers. Functional equations a(x), c(x2), and b(x) for recursive calculations are derived from the CI-CFs and programmed in three ways by means of the Maple programming language. The three recursive procedures for calculating the numbers of planted 3D-trees are executed to give identical results, which are collected up to 100 carbon content in a tabular form. The results are compared with the enumeration of planted trees (as graphs).

Graphs to chemical structures. 4: Combinatorial enumeration of planted three-dimensional trees as stereochemical models of monosubstituted alkanes

Planted three-dimensional (3D) trees, which are defined as a 3D version of planted trees, are enumerated by means of Fujita’s proligand method formulated in Parts 1–3 of this series [Fujita in Theor Chem Acc 113:73–79, 80–86, 2005; Fujita in Theor Chem Acc 115:37–53, 2006]. By starting from the concepts of proligand and promolecule introduced previously [Fujita in Tetrahedron 47:31–46, 1991], a planted promolecule is defined as a 3D object in which the substitution positions of a given 3D skeleton are occupied by a root and proligands. Then, such planted promolecules are introduced as models of planted 3D-trees. Because each of the proligands in a given planted promolecule is regarded as another intermediate planted promolecule in a nested fashion, the given planted promolecule is recursively constructed by a set of such intermediates planted promolecules. The recursive nature of such intermediate planted promolecules is used to derive generating functions for enumerating planted promolecules or planted 3D-trees. The generating functions are based on cycle indices with chirality fittingness (CI-CFs), which are composed of three kinds of sphericity indices (SIs), i.e., a d for homospheric cycles, c d for enantiospheric cycles, and b d for hemispheric cycles. For the purpose of evaluating c d recursively, the concept of diploid is proposed, where the nested nature of c d is demonstrated clearly. The SIs are applied to derive functional equations for recursive calculations, i.e., a(x), c(x 2), and b(x). Thereby, planted 3D-trees or equivalently monosubstituted alkanes as stereoisomers are enumerated recursively by counting planted promolecules. The resulting values are collected up to 20 carbon content in a tabular form. Now, the enumeration problem initiated by mathematician Cayley [Philos Mag 47(4):444–446, 1874] has been solved in such a systematic and integrated manner as satisfying both mathematical and chemical requirements.