Error bars and quadratically convergent methods for the numerical simulation of the Hartree-Fock equations

Summary. This paper presents an a posteriori error analysis of the discretization methods used in computational quantum chemistry on the Hartree-Fock equations. Upper and lower bounds for the energy are obtained from any discrete approximation strategy of the solution and the estimator proposed is shown to possess further approximation virtues.     

The kinetics and mechanism of the pyrolysis elimination of methyl 4-bromocrotonate

The pyrolysis of methyl 4-bromocrotonate in the temperature range 300–340°C and pressure range 74–170 torr has been shown to be homogeneous, unimolecular, and to follow a first-order rate law. The reaction was carried out in a static system, seasoned with allyl bromide, and in the presence of the radical chain exhibitor toluene. The rate coefficients are represented by the Arrhenius expression: log k₁(s^?1) = (13.30 ± 0.66) ? (185.2 ± 7.5) kJ mol^?1 (2.303RT)^?1. The carbomethoxy group appears to provide anchimeric assistance in the process of dehydrobromination and lactone products formation. The partial rates for the parallel reaction have been estimated, reported, and discussed. The pyrolysis elimination is explained in terms of an intimate ion pair-type of mechanism.     

Arbres distingués,bi-arbres et théorèmes de relèvement

We introduce the notion of distinguished tree relation and give applications. We prove in ZFC an old conjecture of A.V. Ostrovsky about the image of a Borel space under a compact covering mapping. We also prove that if $\text{?}{}_{\mathrm{\xi +1}}{}^{\mathrm{L}}$ is countable then any compact covering mapping from a Δ¹₁ space onto a Σ⁰_1+ξ+2 or Π⁰_1+ξ+2 space is inductively perfect. The converse of last statement is shown. To cite this article: G. Debs, J. Saint Raymond, C. R. Acad. Sci. Paris, Ser. I 336 (2003).     

On the coupled BEM and FEM for a nonlinear exterior Dirichlet problem in R2

Summary In this paper we apply the coupling of boundary integral and finite element methods to solve a nonlinear exterior Dirichlet problem in the plane. Specifically, the boundary value problem consists of a nonlinear second order elliptic equation in divergence form in a bounded inner region, and the Laplace equation in the corresponding unbounded exterior region, in addition to appropriate boundary and transmission conditions. The main feature of the coupling method utilized here consists in the reduction of the nonlinear exterior boundary value problem to an equivalent monotone operator equation. We provide sufficient conditions for the coefficients of the nonlinear elliptic equation from which existence, uniqueness and approximation results are established. Then, we consider the case where the corresponding operator is strongly monotone and Lipschitz-continuous, and derive asymptotic error estimates for a boundary-finite element solution. We prove the unique solvability of the discrete operator equations, and based on a Strang type abstract error estimate, we show the strong convergence of the approximated solutions. Moreover, under additional regularity assumptions on the solution of the continous operator equation, the asymptotic rate of convergenceO (h) is obtained.The first author's research was partly supported by the U.S. Army Research Office through the Mathematical Science Institute of Cornell University, by the Universidad de Concepción through the Facultad de Ciencias, Dirección de Investigación and Vicerretoria, and by FONDECYT-Chile through Project 91-386.     

Substituent effects in the gas-phase eliminations of alpha;-substituted alkyl chlorides the pyrolyses of 2-chloro-3-methylbutane and pinacolyl chloride

The gas-phase pyrolyses of 2-chloro-3-methylbutane and pinacolyl chloride in the temperature range of 345-390°C and pressure range of 60-220 mm Hg are homogeneous and unimolecular in a seasoned reaction vessel. The temperature dependence of the rate constants is given by the Arrhenius equations and respectively. The Wagner-Meerwein rearrangement was not observed in these eliminations. Some correlations for the pyrolysis of α-substituted alkyl chlorides are presented and discussed. These facts are consistent with the heterolytic nature of the cyclic transition state mechanisms for these reactions.     

Nondegenerate two-photon down conversion: Coherent inputs and nonclassical effects

We study the nondegenerate two-photon down conversion described by a quantum trilinear Hamiltonian. The idler mode is initially prepared in the vacuum while the pump (laser) and the signal mode are prepared in coherent states which at high intensities resemble classical inputs. Such setup with a coherent signal mode allows us to scan the dynamics from the regime of the down conversion (empty signal) up to the frequency conversion (highly excited signal). The analysis concentrates on the entanglement properties of the modes which are compared with their other statistical properties such as squeezing and antibunching to give a more complete characterization of the modes. We show that the single mode nonclassical effects (squeezing and antibunching) disappear when an initial signal intensity highly exceeds that of the pump. In this regime the numerical results are confirmed by approximate analytical calculations. We point out that initially comparable intensities of the signal and pump mode lead to the effect of the spontaneous disentanglement of the signal mode from others and to the production of its squeezed and sub-Poissonian state which is pure to a good approximation.On leave from: Institute of Physics, Slovak Acad. Sci., Dúbravská cesta 9, 842 28 Bratislava, Slovakia.This work was supported in part by the Grant Agency of the Slovak Acad. Sci. Grant No. GA SAV 2/999143/93. We thank Dr. V. Buek for stimulating discussion. The financial support from the Max-Planck Society for one of us (I.J.) is acknowledged.