Quantum mechanical single molecule partition function from path integral Monte Carlo simulations

An algorithm for calculating the partition function of a molecule with the path integral Monte Carlo method is presented. Staged thermodynamic perturbation with respect to a reference harmonic potential is utilized to evaluate the ratio of partition functions. Parallel tempering and a new Monte Carlo estimator for the ratio of partition functions are implemented here to achieve well converged simulations that give an accuracy of 0.04 kcal/mol in the reported free energies. The method is applied to various test systems, including a catalytic system composed of 18 atoms. Absolute free energies calculated by this method lead to corrections as large as 2.6 kcal/mol at 300 K for some of the examples presented.     

Alpha,gamma-diamino acids: asymmetric synthesis of new constrained 6-amino-3-azabicyclo[3.2.1]octane-6-carboxylic acids

The synthesis of two new diastereomeric 6-amino-3-azabicyclo[3.2.1]octane-6-carboxylic acids exo- and endo-8,9 is reported using exo- and endo-norbornene amino acids as chiral building blocks. This method provides a fast access to optically pure amino acids 8 and 9 which can be considered both alpha,gamma- and alpha,delta-diamino acids containing sterical constraints and characterized by alpha,alpha-disubstitution.     

Efficient reconstruction of complex free energy landscapes by multiple walkers metadynamics

Recently, we have introduced a new method, metadynamics, which is able to sample rarely occurring transitions and to reconstruct the free energy as a function of several variables with a controlled accuracy. This method has been successfully applied in many different fields, ranging from chemistry to biophysics and ligand docking and from material science to crystal structure prediction. We present an important development that speeds up metadynamics calculations by orders of magnitude and renders the algorithm much more robust. We use multiple interacting simulations, walkers, for exploring and reconstructing the same free energy surface. Each walker contributes to the history-dependent potential that, in metadynamics, is an estimate of the free energy. We show that the error on the reconstructed free energy does not depend on the number of walkers, leading to a fully linear scaling algorithm even on inexpensive loosely coupled clusters of PCs. In addition, we show that the accuracy and stability of the method are much improved by combining it with a weighted histogram analysis. We check the validity of our new method on a realistic application.     

On l-adic representations attached to Hilbert and Picard modular surfaces

In this paper we compute the l-adic Lie algebra of a product of l-adic representations associated to a Hilbert modular surface and a Picard modular surface.     

Growth diagrams for the Schubert multiplication

We present a partial generalization of the classical Littlewood-Richardson rule (in its version based on Schützenberger's jeu de taquin) to Schubert calculus on flag varieties. More precisely, we describe certain structure constants expressing the product of a Schubert and a Schur polynomial. We use a generalization of Fomin's growth diagrams (for chains in Young's lattice of partitions) to chains of permutations in the so-called k-Bruhat order. Our work is based on the recent thesis of Beligan, in which he generalizes the classical plactic structure on words to chains in certain intervals in k-Bruhat order. Potential applications of our work include the generalization of the S₃-symmetric Littlewood-Richardson rule due to Thomas and Yong, which is based on Fomin's growth diagrams.     

Maximum principles and symmetry results for a class of fully nonlinear elliptic PDEs

This paper is concerned with a class of boundary value problems for fully nonlinear elliptic PDEs involving the p-Hessian operator. We first derive a maximum principle for a suitable function involving the solution u(x) and its gradient. This maximum principle is then applied to obtain some sharp estimates for the solution and the magnitude of its gradient. We also investigate some symmetry properties of Ω or u(x) under specific boundary condition or geometry of Ω.     

On the order of convergence of Voronovskaja-type formulas

In this work general quantitative estimates of Voronovskaja-type formulas in the space of functions of several variables that are twice continuously differentiable and in the space of functions with weighted second-order derivatives are established.     

Group 12 metal aryl selenolates. Crystal and molecular structure of [2-(Et2NCH2)C6H4]2Se2 and [2-(Me2NCH2)C6H4Se]2M (M = Zn, Cd)

Diorganodiselenide [2-(Et₂NCH₂)C₆H₄]₂Se₂ (1) was obtained by hydrolysis/oxidation of the corresponding [2-(Et₂NCH₂)C₆H₄]SeLi derivative. The treatment of [2-(Et₂NCH₂)C₆H₄]₂Se₂ with elemental sodium in THF resulted in [2-(Et₂NCH₂)C₆H₄]SeNa (2). Reactions between alkali metal selenolates [2-(R₂NCH₂)C₆H₄]SeM′ (R = Me, Et; M′ = Li, Na) and MCl₂ (M = Zn, Cd) in a 2:1 molar ratio resulted in the [2-(R₂NCH₂)C₆H₄Se]₂M species [R = Me, M = Zn (3), Cd (4); R = Et, M = Zn (5), Cd (6)]. The new compounds were characterized by multinuclear NMR (¹H, ¹³C, ⁷⁷Se, ¹¹³Cd) and mass spectrometry. The crystal and molecular structures of 1, 3 and 4 revealed monomeric species stabilized by N → Se (for 1) and N → M (for 3 and 4) intramolecular interactions.     

Synthesis of fused dihydro-pyrimido[4,3-d]coumarins using Biginelli multicomponent reaction as key step

A new efficient approach for the synthesis of functionalized dihydro-pyrimido[4,3-d]coumarins is described. Use of benzyl- or tert-butyl acetoacetate and various salicyl aldehydes in typical Biginelli multicomponent reactions provides esters that can be easily deprotected to yield the corresponding hydroxy acids. Annelation of the latter leads to the target dihydro-pyrimido[4,3-d]coumarins. Formation of some unexpected oxo-bridged tetrahydro-pyrimidines is also described. Spectral properties of the synthesized dihydro-pyrimido[4,3-d]coumarins indicate the necessity to revisit some compounds previously reported as products of direct one-pot Biginelli condensation involving 4-hydroxycoumarin as dicarbonyl component.