From van't Hoff to Unified Perspectives in Molecular Structure and Computer Oriented Representation Dr. Johann Gasteiger, Paul D. Gillespie, Ph.D., Dr. Dieter Marquarding, and Prof. Dr. Ivar Ugi * Laboratorium ffir Organische Chemie der Technischen Universitiit Mfinchen Contents .1 Introduction ..................................................... 3 2. Constitution ..................................................... 4 2.1. Formulas, Lists and Matrices .................................. 4 2.2. Constitutional Symmetry ...................................... 6 2.3. Sequentially Ordered Sets of Atoms ............................. 6 3. Stereoisomers .................................................... 9 3.1. Rigid Configurations .......................................... 10 3.2. Ligands and Skeletons ........................................ 11 3.3. Conformations ............................................... 11 3.4. CC-Matrices ................................................. 12 4. Permutation Isomers and Chiral Configurations ....................... 13 4.1. From Biot to van't Hoff and beyond ............................ 14 4.2. Transitivity Prerequisites of Homochirality and Heterochirality ... 15 4.3. Superposition of Monocentric Configurations ..................... 16 4.4. Aschan's Criticism of van't Hoff ................................ 18 4.5. Cyclic and Polycyclic Configurations ............................ 19 4.6. Chirality Elements ............................................ 20 5. Stereochemical Nomenclature System ............................... 25 5.1. Ambiguities of the Classical D,L-Nomenclature ................... 25 5.2. CIP-Rules and (R,S)-Nomenclature ............................. 26 * and Department of Chemistry, Rice University, Houston, Texas J. Gasteiger, P. D. Gillespie, D. ±Niarq~aarding, and I. Ugl 5.3. Classical and Modern Nomenclature Systems of Bicen~ric Config- urations ..................................................... 27 5.4. Permutation Isomers and Index Mappings ....................... 27 5.4.1. CIP Ligand Indices and RASI ............................ 28 5.4.2. Skelctal Indices and Symmetry ........................... 29 5.5. Permutation Descriptors ...................................... 30 5.6. Parity Descriptors ............................................ 13 6. Simultaneous Computer Oriented Representation of the Molecular Con- stitutional and Stereochemical Features and their Interconversions ....... 13 6.1. Canonical Matrix Pairs and their Transformations ................ 33 6.2. Representation of the Stereochemical Aspect by Parity Vectors and van't HofFs Concept .......................................... 34 7. References ....................................................... 34 Abbreviations AC-Matrix Atom ytivitcennoc matrix ASI Atomic sequence index BE-Matrix Bond and nortcele matrix BPR Berry noitatoroduesp CC-Matrix noitarugifnoC and conformation matrix CIP-Rules Sequence selur fo Cahn, Ingold and Prelog EM Ensemble fo molecules FIEM Family fo ensembles isomeric fo molecules PI Permutation noitaziremosi RASI evitaleR atomic sequence index (mostly fo eht a-atom fo a )dnagil R-Matrix Reaction matrix TR Turnstile rotation (TR) 2 Double TR, i. e., sequence of two TR with the same pair-trio combination without intermediate stop after the first TR. Unified Perspectives in Molecular Structure and Computer Oriented Representation Introduction The interdependence of chemical constitution and the spatial arrange- ment of atoms in molecules suggests the simultaneous treatment of constitutional and stereochemical problems. Traditional separation of these latter aspects is more than likely due to the fact that the concepts and methodology of the two chemical disciplines differ enough to justify individual approaches. A complete and detailed description of molecular structure includes statements concerning the metric coordinatesa) of atomic nuclei supple- mented by electron density distribution data. Although a large amount of data is involved, such representation is not particularly suitable for the chemically relevant structural features of molecules. The structure of a molecule is given by the three-dimensional distri- bution of atomic cores and valence electrons. This structure has been elucidated for many molecules with the use of X-ray or electron diffrac- tion data. Chemical properties of molecules are observed under conditions which permit internal motions. Such observations yield views which may differ markedly as a function of time. Thus, observable properties are determined from equilibrated ensembles of species differing in geom- etry and energy. This demands a representation invariant against changes of the latter features. Thus, it behooves one to use problem oriented models for molecules, neglecting properties irrelevant for a given problem. Such models rep- resent equivalence classes of molecules with a common characteristic feature. One must keep in mind, however, the limitations contained in the model's approximations. For the majority of chemical problems, those models are the most useful which relate to the concept hierarchy of empirical formula, con- stitution, configuration and conformation: ! Any empirical formula All compounds ¢ Same empirical formula I ": Constitutional isomers different constitution ¢ )a Cartesian or polar coordinates or the combination of bond lengths, angles and dihedral angles may be used in this context. J. Gasteiger, P. D. Gillespie, D. Marquarding, and I. Ugi + Same constitution different configuration -.' Stereoisomers and/or conformation + Same configuration -~ Conformers different conformation 2. Constitution In the first half of the last century, the concept of isomerism was discover- ed, i. e., that there are distinguishable compounds with the same empirical formula. This led to the conclusion that within molecules, the atoms can be connected in different ways, hence, to the concept of constitution. The chemical constitution of a molecule is given by a family of binary relations, namely, the covalently connected pairs of atoms. The model of a covalent single bond is a pair of atom cores held together by a pair of electrons, with the electron density being notably different from zero everywhere between the cores. The number of electron pairs that participate in a bond is its formal bond order. Some molecules exist where the bonding electrons cannot be assigned to atom pairs, but belong to more than two cores, e.g. in the polyboranes. In these cases the model concept of covalently bound atom pairs as a rep- resention basis for chemical constitution using binary relations can be sustained by the assignment of fractional bond orders. Sometimes it is advantageous to include the distribution of the free valence electrons in the constitution. 2.1. Formulas, Lists and Matrices The constitution is customarily represented by constitutional formulas, i.e. by labelled graphs )1 whose nodes are the atoms, and whose connect- ing lines are the bonds. Nodes are labelled by chemical element symbols and can carry further symbols for electrons and electrical charges. The information contained in a constitutional formula can also be given in terms of a connectivity list. Such a list tells which atom is con- nected to which (the formal bond orders are indicated, if needed). The statement that a C-atom is connected to another C-atom does not suf- fice, if there is more than one C--C-bond, but the bonded partners must be indicated individually. The connectivity list of 1 serves as an illu- stration. 4 Unified Perspectives in Molecular Structure and Computer Oriented Representation H 5 . • I 4 2 :Cli--C3--C ~N : "' ] H ~ Cll--C3, or just 1--3 N2 --C4 2--4 C3 --C4 3--4 C3 --H5 3--5 C3 --H6 3--6 Fig. 1. Connectivity list of I Rapid increase in the use of computers in Chemical Documentation and in the solution of other chemical problems lendisn creasing importance to connectivity lists and their corresponding atom connectivity matrices (AC-matrices) ,2 as well as the associated bond and electron matrices (BE-matrices) .a 123456 o o oo .El= 0011 3, C 3100 4, C 0100 5, H 0100 6, H Fig. 2. BE-matrix of 1 (The AC-matrix is obtained by replacing the diagonal entries by the element symbols) In the AC-matrices the off-diagonal entries e,j are the formal bond orders between atom pairs (A,, Aj). BE-matrices are obtained fromthe AC-matrices by augmentation with diagonal entries eu indicating the numbers of free valence electrons at the atoms As. The indices 1 .... , n can be assigned to the n atoms of a constitution in n K different ways. Accordingly there are up to n! different but equivalent connectivity lists, or AC- and BE-matrices, respectively. The direct identification and comparison of such representations is essential to Chemical Documen- tation. Only by uniquely assigning atomic indices can this be accomplish- ed. Such a unique indexing of the atoms affords unique connectivity lists, or matrices, respectively, which may be termed canonical rep- resentation )b of chemical constitution. b) The canonical form of a mathematical expression is a standardized one, partic- ularly well-suited for a given purpose. J. Gasteiger, P. D. Gillespie, D. Marquarding, and I. Ugi 2.2. Constitutional Symmetry An indexing procedure must take into account constitutional symmetry, i.e. within the indexed set, the indices of constitutionally equivalent atoms (such as the three H-atoms of a methyl group) and bonds must be recognizable as equivalence classes with permutable numbers. A canonical connectivitv list is invariant against permutations of constitutionally equivalent atom indices. It suffices for representing equivalence classes of constitutionally equivalent bonds (e.g. the C--H bonds of benzene) the use of a single symbol. This leads to abbreviated connectivity lists. The atoms of a chemical constitution are then equivalent if they belong to the same chemical element (and isotope) and, further, have the same set of connectivities and neighbors. The expression "constitutional symmetry" refers to the permutability of equivalently labelled atoms within a constitution. For example, in benzene, all H- and C-atoms are constitutionally equivalent. In chlorobenzene, however, only the o- and m-atoms are pairwise equivalent. Note here that the individual resonance formulas of delocalized bond systems are not distinguishable. lC lC 2~ 2b NMR spectroscopy derives its usefulness, in large measure, from its ability to detect constitutional symmetry. If constitutional symmetry is not taken into account in synthetic design computer programs, a re- dundancy of pathways may result. 2.3. Sequentially Ordered Sets of Atoms The sequence rules of Cahn, Ingold and Prelog (CIP rules) )4 have been defined for establishing sequential order in the ligand set of asymmetric C-atoms. Attempts to adopt these rules for indexing the atoms of a chemical constitution would lead to an indexing procedure requiring quite a cumbersome computer algorithm and attendant loss of efficiency. Since the CIP rules take into account formal bond orders, the assignment of atomic indices to delocalized bond systems by some modified set of the former would depend upon the choice of resonance formula (see e.g. the indexing of 2c or 2d). 6 Unified Perspectives in Molecular Structure and Computer Oriented Representation ' H ~H ! 1C 1C s s H ~ H 9 OlH n H Hn~'-HlO 2tH ZlH 2c 2d By the use of rules relying primarily upon atomic and coordination numbers it is possible to achieve an unambiguous ordering of atoms taking into account the bond orders indirectly via the coordination numbers. This extends to delocalized bond systems as well. It is possible to consider first the coordination numbers, and then the atomic numbers, or to proceed in the reverse order. If there are more differences in atomic numbers than in coordination numbers the first mode is more effective, while the existence of many atoms of the same element with different coordination number favors the use of the second. Most atoms of a given chemical element have the same coordination num- ber, particularly in organic compounds. If the H-atoms are, however, disregarded, one obtains constitutional formulas which can be indexed quite well by primary consideration of the degrees of the nodesc), i.e., the number of immediate covalent neighbors. The Morgan algorithm )5 is a device used by the Chemical Abstracts Service for assigning indices to the nodes of constitutional formulas whose H-atoms have been omitted, i. e. their reduced graphs. If tile indices of all atoms are needed, e.g. for the representation of stereochemistry, an additional procedure is needed for establishing those for the H-atoms. Constitutional symmetry is not indicated directly by Morgan indices. A recently developed computer program )r3 for synthetic design on the basis of a mathematical model requires the transformation of arbitrary BE-matrices into the canonical forms. It was found that the atomic sequence indices (ASI)6) are particularly well-suited for this purpose. The algorithm for assigning the ASI depends initially on atomic numbers, and secondlyo n coordination numbers. The following procedure for ordering atoms of a constitution is a translation of the ideas and cri- teria used in the ASI algorithm design. a) The atoms are collected in chemical element equivalence classes by the atomic number; the higher the element number of an atom, the lower its index. In isotopically labelled compounds the mass number is also taken into account. b) Atoms belonging to the same chemical element are distinguished according to the atomic numbers of their ~-atoms, i.e., their immediate )e One might well as remove II~ univalent atoms, 7 J. Gasteiger, P. D. Gillespie, D. Marquarding, and I. Ugi covalent neighbors. For each atom of an element equivalence class the atomic numbers of its a-atoms are written in a row in descending order from left to right. The number of rows resulting is the number of atoms in the element equivalence class, and the number of columns is the maximum coordination number of the chemical element considered. If an atom has fewer a-atoms than its maximum coordination number in the given compound, the missing a-atoms are represented by phantom atoms with atom numbers x, written in the empty spaces of the row on the right hand side of the known atomic numbers. Now, the entries of the first column are compared, with x being rated higher than any known atomic number. The atom corresponding to tile row with highest first entry obtains the lowest available index of the element equivalence class. The atom belonging to the row with the second highest first entry receives the next index etc. Those indices of the element equivalence class which have not been assigned by the first column, are given to the remaining atom by analyzing the second, third, etc. columns in a similar position. c) If the comparison of the a-atom set does not permit complete assignment of indices, the p-atoms are considered, i.e., the atoms that are connected to a given atom via a sequence of two covalent bonds, and then the y-atom (3-bond neighbor) etc. are taken into account in analogy to b), until all neighbors have been considered, or until all indices have been assigned. If there are tings the procedure ends at the latest when the cycle is completed. d) Atoms which cannot be distinguished by a--c are constitutionally equivalent and correspond to an interval of equivalent indices. The indices are permutable within any interval of indices belonging to constitutionally equivalent atoms, as long as the a-atoms of these atoms do not also belong to constitutionally equivalent classes. In the latter case, additional rules are needed for deciding between equivalent assignment of indices. lP 3a Unified Perspectives in Molecular Structure and Computer Oriented Representation In hexamethylene tetramine 3a all atoms belonging to the same chemical element are constitutionally equivalent, and their indices belong to the following equivalence intervals N: {1,...,4} C: {5,..., }01 H: {11, ..., 22} Within one of these intervals there is free choice of indices, but once this is decided the other assignments are no longer random. The arbitrary assignment of indices to the N-atoms determines the indices of the adjacent C-atoms and these in turn the order of the H-atoms. It is further possible to describe a given constitution by mapping the indexed set of atoms onto the independently indexed graph. The indexed set of atoms must be partially symmetrized with regard to the element equivalence classes of the atoms and the indices of the graph with respect to its symmetry. That constitution is defined as the reference constitution in which the atomic indices a and the graph indices g match. The permutational descriptor of the given chemical constitution is then a permutation of atom indices (or the inverse permutation of graph indices) which transforms the reference constitution into the considered constitution. Such a procedure corresponding to the permutation-nomenclature system of configuration )7 is in accordance with the fact that a chemical constitution is a representative of a product D, ×... × Da of the double cosets De of pairs of subgroups symmetric groups Sn, .~ The latter correspond to permutations of atoms of the same element and the sym- metry of the adjacent subset of graph points whose degree is compatible with the coordination number of the element equivalence class of the atom. Despite its mathematical clarity this permutation nomenclature is of no practical value, due to its cumbersome nature, requiring a rather complicated procedure for generating the graphs with standard indices. 3. Stereoisomers It is customary to describe the essential spatial features of stereoisomers by their configurations and conformations. J. Gasteiger, P. D. Gillespie, D. Marquarding, and I. Ugi Tile traditional limitation of the stereoisomer concept to isomeric com- pounds with different configurations stems from the idea that configu- rations are not interconvertible under the observations conditions.S) A definition of stereoisomers on this basis is questionable as its application is then dependent upon observation conditions. It also fails to account for the fact that many mobile interconversions of configurations (e.g. configurations involving tricoordinate Nitrogen or pentacoordinate Phosphorus) are known, as well as thermally stable conformations (e.g. the atropisomers). Present usage prefers considering any constitutionally equivalent molecules which are configurationally and/or conformationally distin- guishable as stereoisomers. In the case of a mobile equilibrium of inter- convertible conformations, that configurational data suffices which form class characteristics of the whole equilibrium ensemble of con- formations. In the main, configurational data suffices, because it is possible to manufacture models of energetically preferred conformations by finding the corresponding flexional and torsional potential energy minimum. The present classification methods for stereoisomers are the product of a long historical development whose discussion is useful for understanding modern views. 3.1. Rigid Configurations Rigid configurations are given by the distribution of a ligand set on the ligand sites of an achiral molecular skeleton )o with a characteristic sym- metry and coordination number, i.e., the number of ligands that it can carry. Molecular skeletons can be monocentric, corresponding to the valence state of the atoms (e.g. sp ,3 dsp ,3 d~ sp3), or they can be poly- centric, composed of monocentric units, such as in 4, a skeleton that corresponds to cyclo-propane. L I 4 In this model, idealized skeletons are assumed, deformations of the skeleton by ligands being neglected. The ligands have a static or dynamic symmetry about their skeletal bond axes which is compatible with the skeletal symmetry. Herei t should be noted that the conceptual dissection of molecules into skeleton and ligands has been a standard procedure developed by stereochemists quite some time ago. )d For reasons which are given below the present discussion is confined to con- figurations of achiral skeletons, 10