The numerical integration of a molecular integral using two different techniques

A two-electron integral which commonly occurs in molecular calculations is evaluated numerically using the different methods of Boys and Conroy and the results are discussed.This paper was presented during the session on numerical integration methods for molecules of the 1970 Quantum Theory Conference in Nottingham. It has been revised in the light of the interesting discussion which followed.     

Synthesis of azaspirocyclic compounds via organoiron complexes. Potential synthetic routes to histrionicotoxin and cephalotaxus alkaloids.

Reaction of cyclohexadienylium-Fe(CO)₃ complexes (7) and (8) with sodiomalono-nitrile gives the carbo-spirocyclic derivatives (9) and (10), whilst reaction with benzylamine and removal of iron leads to the azaspirocyclic enones (15) and (16), respectively.     

Multivalued groups, their representations and Hopf algebras

This paper introduces the concept ofn-valued groups and studies their algebraic and topological properties. We explore a number of examples. An important class consists of those that we calln-coset groups; they arise as orbit spaces of groupsG modulo a group of automorphisms withn elements. However, there are many examples that do not arise from this construction. We see that the theory ofn-valued groups is distinct from that of groups with a given automorphism group. There are natural concepts of the action of ann-valued group on a space and of a representation in an algebra of operators. We introduce the (purely algebraic) notion of ann-Hopf algebra and show that the ring of functions on ann-valued group and, in the topological case, the cohomology has ann-Hopf algebra structure. The cohomology algebra of the classifying space of a compact Lie group admits the structure of ann-Hopf algebra, wheren is the order of the Weyl group; the homology with dual structure is also ann-Hopf algebra. In general the group ring of ann-valued group is not ann-Hopf algebra but it is for ann-coset group constructed from an abelian group. Using the properties ofn-Hopf algebras we show that certain spaces do not admit the structure of ann-valued group and that certain commutativen-valued groups do not arise by applying then-coset construction to any commutative group.     

Anion induced and inhibited circumrotation of a [2]catenane

The first example of a catenane capable of performing circumrotation via an anion switching methodology is described; of particular interest is a conformational locking mechanism which results from chloride coordination in the catenane binding cavity.     

Unusual odd-electron fragments from even-electron protonated prodiginine precursors using positive-ion electrospray tandem mass spectrometry

Reports of anticancer and immunosuppressive properties have spurred recent interest in the bacterially produced prodiginines. We use electrospray tandem mass spectrometry (ES-MS/MS) to investigate prodigiosin, undecylprodiginine, and streptorubin B (butyl-meta-cycloheptylprodiginine) and to explore their fragmentation pathways to explain the unusual methyl radical loss and consecutive fragment ions that dominate low-energy collision-induced dissociation (CID) mass spectra. The competition between the formation of even-electron ions and radical ions is examined in detail. Theoretical calculations are used to optimize the structures and calculate the energies of both reactants and products using the Gaussian 03 program. Results indicate that protonation occurs on the nitrogen atom that initially held no hydrogen, thus allowing formation of a pseudo-seven-membered ring that constitutes the most stable ground state [M + H](+) structure. From this precursor, experimental data show that methyl radical loss has the lowest apparent threshold but, alternatively, even-electron fragment ions can be formed by loss of a methanol molecule. Computational modeling indicates that methyl radical loss is the more endothermic process in this competition, but the lower apparent threshold associated with methyl radical loss points to a lower kinetic barrier. Additionally, this characteristic and unusual loss of methyl radical (in combination with weaker methanol loss) from each prodiginine is useful for performing constant neutral loss scans to quickly and efficiently identify all prodiginines in a complex biological mixture without any clean-up or purification. The feasibility of this approach has been proven through the identification of a new, low-abundance prodigiosin analog arising from Hahella chejuensis.     

Conjugacy languages in groups

We study the regularity of several languages derived from conjugacy classes in a finitely generated group G for a variety of examples including word hyperbolic, virtually abelian, Artin, and Garside groups. We also determine the rationality of the growth series of the shortlex conjugacy language in virtually cyclic groups, proving one direction of a conjecture of Rivin.