Rearrangements and Tunneling Splittings in Small Water Clusters

Recent far-infrared vibration-rotation tunneling (FIR-VRT) experiments pose new challenges to theory because the interpretation and prediction of such spectra requires a detailed understanding of the potential energy surface (PES) away from minima. In particular we need a global description of the PES in terms of a complete reaction graph. Hence all the transition states and associated mechanisms which might give rise to observable tunneling splittings must be characterized. It may be possible to guess the detailed permutations of atoms from the transition state alone, but experience suggests this is unwise. In this contribution a brief overview of the issues involved in treating the large amplitude motions of such systems will be given, with references to more detailed discussions and some specific examples. In particular we will consider the effective molecular symmetry group, the classification of rearrangement mechanisms, the location of minima and transition states and the calculation of reaction pathways. The application of these theories to small water clusters ranging from water dimer to water hexamer will then be considered. More details can be found in recent reviews.Comment: 15 pages, 5 figures. This paper was prepared in August 1997 for the proceedings volume of the NATO-ASI meeting on "Recent Theoretical and Experimental Advances in Hydrogen Bonded Clusters" edited by Sotiris Xantheas, which has so far not appeare

Embeddings of Sz(32) in E_8(5)

We show that the Suzuki group Sz(32) is a subgroup of E_8(5), and so is its automorphism group. Both are unique up to conjugacy in E_8(F) for any field F of characteristic 5, and the automorphism group Sz(32):5 is maximal in E_8(5)

Lie algebras generated by extremal elements

We study Lie algebras generated by extremal elements (i.e., elements spanning inner ideals of L) over a field of characteristic distinct from 2. We prove that any Lie algebra generated by a finite number of extremal elements is finite dimensional. The minimal number of extremal generators for the Lie algebras of type An, Bn (n>2), Cn (n>1), Dn (n>3), En (n=6,7,8), F4 and G2 are shown to be n+1, n+1, 2n, n, 5, 5, and 4 in the respective cases. These results are related to group theoretic ones for the corresponding Chevalley groups.Comment: 28 page

A Poset Connected to Artin Monoids of Simply Laced Type

Let W be a Weyl group whose type is a simply laced Dynkin diagram. On several W-orbits of sets of mutually commuting reflections, a poset is described which plays a role in linear representatons of the corresponding Artin group A. The poset generalizes many properties of the usual order on positive roots of W given by height. In this paper, a linear representation of the positive monoid of A is defined by use of the poset

BMW algebras of simply laced type

It is known that the recently discovered representations of the Artin groups of type A_n, the braid groups, can be constructed via BMW algebras. We introduce similar algebras of type D_n and E_n which also lead to the newly found faithful representations of the Artin groups of the corresponding types. We establish finite dimensionality of these algebras. Moreover, they have ideals I_1 and I_2 with I_2 contained in I_1 such that the quotient with respect to I_1 is the Hecke algebra and I_1/I_2 is a module for the corresponding Artin group generalizing the Lawrence-Krammer representation. Finally we give conjectures on the structure, the dimension and parabolic subalgebras of the BMW algebra, as well as on a generalization of deformations to Brauer algebras for simply laced spherical type other than A_n.Comment: 39 page

Tangle and Brauer Diagram Algebras of Type Dn

A generalization of the Kauffman tangle algebra is given for Coxeter type Dn. The tangles involve a pole or order 2. The algebra is shown to be isomorphic to the Birman-Murakami-Wenzl algebra of the same type. This result extends the isomorphism between the two algebras in the classical case, which in our set-up, occurs when the Coxeter type is of type A with index n-1. The proof involves a diagrammatic version of the Brauer algebra of type Dn in which the Temperley-Lieb algebra of type Dn is a subalgebra.Comment: 33 page