Generalized Wreath Product Research Articles

Symmetry simplifications of space types in configuration interaction induced by orbital degeneracy

AbstractSymmetry simplifications are introduced in configuration interaction (CI) by reducing the number of symmetry‐allowed space types if there is degeneracy in some of the molecular orbitals by constructing the unique space types. A new symmetry group which we call the configuration symmetry group is defined and is shown to be expressible as a generalized wreath product group. Generating functions are derived for enumerating the equivalence classes of space types. A double coset method is expounded which constructs the representatives of all equivalence classes of space types using the cycle index of generalized wreath product and the double cosets of label subgroup with generalized wreath product in the symmetric group Sn, if n is twice the number of occupied and virtual orbitals. Method is illustrated with CI using the localized orbitals of polyenes, CI in benzene, and atomic CI for several reference states.

Generating functions for the nuclear spin statistics of nonrigid molecules

Generating functions are developed for the nuclear spin species and nuclear spin statistical weights of nonrigid molecules as the trace of spin projection operators. These generating functions are obtained in terms of the GCCI’s (generalized character cycle indices) of the PI groups of nonrigid molecules which are expressible as generalized wreath products. The GCCI’s of generalized wreath products can be obtained in terms of the GCCI’s of the composing groups. Thus, the method developed here does not require the character table of the P group of nonrigid molecules. From these generating functions the nuclear spin statistical weights of the rovibronic levels and nuclear spin species of nonrigid molecules can be obtained easily. The method is illustrated with several examples of nonrigid molecules containing up to 3 601 989 nuclear spin functions. Application to molecular electric beam deflection studies of weakly bound complexes such as ammonia dimer is also discussed.

Graph theoretical characterization of NMR groups, nonrigid nuclear spin species and the construction of symmetry adapted NMR spin functions

NMR group defined by Woodman is shown to be the generalized wreath product group of the NMR graph which is constructed with protons as vertices and edges representing nuclear couplings. The NMR graph can be expressed as a generalized composition of its quotient graph and various types. It is shown that the NMR group is the automorphism group of the NMR graph that preserves the coupling matrix defined here. Using the representation theory of generalized wreath product groups developed in an earlier paper of the present author, the character tables of NMR groups can be obtained. The character tables of certain NMR groups are presented. The quotient graph of the NMR graph is shown to be the composite particle graph. The automorphism group of the quotient NMR graph is the symmetry group of the composite particles, preserving the NMR coupling constants. An iterative algorithm is formulated for obtaining the automorphism group of the composite particle tree. A combinatorial approach is expounded for identifying the nuclear spin species of nonrigid and rigid molecules. The coalescence diagrams describing the effect of nonrigidity on nuclear spin species are introduced and obtained to exemplify the utility of the proposed method. It is shown that using the automorphism group of the quotient graph, the symmetry adapted composite particle spin functions can be constructed. We illustrate this method with 2,3-dimethylbutane. The nuclear spin Hamiltonian matrix of 2,3-dimethylbutane of order 214×214 is reduced to a matrix of order 64×64 by the composite particle method. This matrix can be blocked in the symmetry adapted basis set constructed through the automorphism group of the quotient graph into 40 matrices of order 1×1 and 12 matrices of order 2×2.

The symmetry groups of nonrigid molecules as generalized wreath products and their representations

The symmetry groups of nonrigid molecules first defined by Longuet–Higgins, in the most general cases are generalized wreath product groups. We first outline the representation theory of wreath product groups. Then the representation theory of generalized wreath product groups is developed. Several illustrative examples of NMR groups are offered. The character tables of several nonrigid molecular groups are presented. An example of a nonrigid triphenyl molecule is given to illustrate the use of the generalized wreath product in deriving optical selection rules.

A generalized wreath product method for the enumeration of stereo and position isomers of polysubstituted organic compounds

A generalized wreath product method for the enumeration of stereo and position isomers of polysubstituted organic compounds

A generalized wreath product method for the enumeration of stereo and position isomers of polysubstituted organic compounds

Ageneralized wreath product group is developed in the root-to-root product formalism for the enumeration of stereo and position isomers of polysubstituted organic compounds. The methods expounded here are used for enumerating the NMR signals of polysubstituted organic compounds.

Enumeration of internal rotation reactions and their reaction graphs

Threefold and twofold internal rotation reactions and their reaction graphs are enumerated using the generalized wreath product method developed by the author in an earlier paper. The correspondence between reaction directed graphs (digraphs) and finite topologies on isomers is established. It is shown that the reaction digraphs can be represented by Borel fields. Atropisomerism in polyphenyl compounds is discussed. Applications to spontaneous generation of optical activity and NMR spectroscopy are considered. Borel fields are enumerated by bumping squares of the upper rows of Young diagrams starting from the Young diagram containing just one row.

Generalized wreath products and the lattice of normal subgroups of a group

Generalized wreath products and the lattice of normal subgroups of a group

The deficiency of wreath products of groups

If G is a finite group then d( G) denotes the minimal number of generators of G. Let G p be the class of groups containing all finite p-groups G which has a presentation with d( G) = dim H 1( G, Z p ) generators and r( G) = dim H 2( G, Z p ) relations. In this article it is shown that G p is closed under general wreath products.

Generalized wreath products viewed as sets with valuation

Generalized wreath products viewed as sets with valuation

The characterization of generalized wreath products

The main USC’ of wreath products in group theory has been to supply examples of groups with special properties, and applications of this sort are too numerous and well known to be listed here. It is well known that the wreath product has a certain universal property, which we may describe in the following way. If G is a transitive permutation group on a set S, if C is a G-congruence on S, and if x E S, then G induces, in a natural way, a permutation group