Geometry of Discrete-Time Spin Systems

Classical Hamiltonian spin systems are continuous dynamical systems on the symplectic phase space \((S^2)^n\). In this paper, we investigate the underlying geometry of a time discretization scheme for classical Hamiltonian spin systems called the spherical midpoint method. As it turns out, this method displays a range of interesting geometrical features that yield insights and sets out general strategies for geometric time discretizations of Hamiltonian systems on non-canonical symplectic manifolds. In particular, our study provides two new, completely geometric proofs that the discrete-time spin systems obtained by the spherical midpoint method preserve symplecticity. The study follows two paths. First, we introduce an extended version of the Hopf fibration to show that the spherical midpoint method can be seen as originating from the classical midpoint method on \(T^*\mathbf {R}^{2n}\) for a collective Hamiltonian. Symplecticity is then a direct, geometric consequence. Second, we propose a new discretization scheme on Riemannian manifolds called the Riemannian midpoint method. We determine its properties with respect to isometries and Riemannian submersions, and, as a special case, we show that the spherical midpoint method is of this type for a non-Euclidean metric. In combination with Kähler geometry, this provides another geometric proof of symplecticity.     

Extension of bounded holomorphic functions¶in convex domains

The E. Amar and G. Henkin theorem on the bounded extendability of bounded holomorphic functions from certain closed complex submanifolds of strictly pseudoconvex domains to the whole domain is generalized to the case of finite type convex domains and their intersections with affine linear hyperplanes. Suitable integral operators of Berndtsson–Andersson type are constructed and estimated for this purpose. Received: 7 July 2000     

Reconstruction of the finite size canonical ensemble from incomplete micro-canonical data

In this paper we discuss how partial knowledge of the density of states for a model can be used to give good approximations of the energy distributions in a given temperature range. From these distributions one can then obtain the statistical moments corresponding to e.g. the internal energy and the specific heat. These questions have gained interest apropos of several recent methods for estimating the density of states of spin models. As a worked example we finally apply these methods to the 3-state Potts model for cubic lattices of linear order up to 128. We give estimates of e.g. latent heat and critical temperature, as well as the micro-canonical properties of interest.      

Ruthenium-catalyzed asymmetric transfer hydrogenation of ketones in ethanol

A ruthenium catalyst formed in situ by combining [Ru(p-cymene)Cl₂]₂ and an amino acid hydroxy-amide was found to catalyze efficiently the asymmetric reduction of aryl alkyl ketones under transfer hydrogenation conditions using ethanol as the hydrogen donor. The secondary alcohol products were obtained in moderate to good yields and with good to excellent enantioselectivity (up to 97% ee).     

Titanium isopropoxide as efficient catalyst for the aza-Baylis-Hillman reaction. Selective formation of alpha-methylene-beta-amino acid derivatives

The direct formation of alpha-methylene-beta-amino acid derivatives is achieved using the aza version of the Baylis-Hillman protocol. The products are readily formed in a three-component one-pot reaction between arylaldehydes, sulfonamides, and alpha,beta-unsaturated carbonyl compounds. The reaction is efficiently catalyzed by titanium isopropoxide and 2-hydroxyquinuclidine in the presence of molecular sieves. The protocol allows for structural variation of the substrates, tolerating electron-poor and electron-rich arylaldehydes and various Michael acceptors.     

Asymmetric hydrogenation of tri-substituted alkenes with Ir-NHC-thiazole complexes

An efficient chiral N-heterocyclic carbene ligand for the Ir-catalyzed asymmetric hydrogenation of largely unfunctionalized tri-substituted olefins has been developed. The Ir-NHC-thiazole catalyst is able to reduce a large variety of substrates with excellent conversions and good enantioselectivities with ee’s ranging from 34% to 90%, depending on the geometry around the double bond of the substrates.     

The importance of alkali cations in the [{RuCl2(p-cymene)}2]-pseudo-dipeptide-catalyzed enantioselective transfer hydrogenation of ketones

We studied the role of alkali cations in the [{RuCl2(p-cymene)}2]-pseudo-dipeptide-catalyzed enantioselective transfer hydrogenation of ketones with isopropanol. Lithium salts were shown to increase the enantioselectivity of the reaction when iPrONa or iPrOK was used as the base. Similar transfer-hydrogenation systems that employ chiral amino alcohol or monotosylated diamine ligands are not affected by the addition of lithium salts. These observations have led us to propose that an alternative reaction mechanism operates in pseudo-dipeptide-based systems, in which the alkali cation is an important player in the ligand-assisted hydrogen-transfer step. DFT calculations of the proposed transition-state (TS) models involving different cations (Li+, Na+, and K+) confirm a considerable loosening of the TS with larger cations. This loosening may be responsible for the fewer interactions between the substrate and the catalytic complex, leading to lower enantiodifferentiation. This mechanistic hypothesis has found additional experimental support; the low ee obtained with [BnNMe3]OH (a large cation) as base can be dramatically improved by introducing lithium cations into the system. Also, the complexation of Na+, K+, and Li+ cations by the addition of [15]crown-5 and [18]crown-6 ethers and cryptand 2.1.1 (which selectively bind to these cations and, thus, increase their bulkiness), respectively, to the reaction mixture led to a significant drop in the enantioselectivity of the reaction. The lithium effect has proved useful for enhancing the reduction of different aromatic and heteroaromatic ketones.     

Olefin epoxidation with bis(trimethylsilyl) peroxide catalyzed by inorganic oxorhenium derivatives. Controlled release of hydrogen peroxide.

The replacement of organometallic rhenium species (e.g., CH(3)ReO(3)) by less expensive and more readily available inorganic rhenium oxides (e.g., Re(2)O(7), ReO(3)(OH), and ReO(3)) can be accomplished using bis(trimethylsilyl) peroxide (BTSP) as oxidant in place of aqueous H(2)O(2). Using a catalytic amount of a proton source, controlled release of hydrogen peroxide helps preserve sensitive peroxorhenium species and enables catalytic turnover to take place. Systematic investigation of the oxorhenium catalyst precursors, substrate scope, and effects of various additives on olefin epoxidation with BTSP are reported in this contribution.