Data enhanced Hammett-equation: reaction barriers in chemical space

It is intriguing how the Hammett equation enables control of chemical reactivity throughout chemical space by separating the effect of substituents from chemical process variables, such as reaction mechanism, solvent, or temperature. We generalize Hammett''s original approach to predict potential energies of activation in non aromatic molecular scaffolds with multiple substituents. We use global regression to optimize Hammett parameters ρ and σ in two experimental datasets (rate constants for benzylbromides reacting with thiols and ammonium salt decomposition), as well as in a synthetic dataset consisting of computational activation energies of ∼2400 S_N2 reactions, with various nucleophiles and leaving groups (–H, –F, –Cl, –Br) and functional groups (–H, –NO₂, –CN, –NH₃, –CH₃). Individual substituents contribute additively to molecular σ with a unique regression term, which quantifies the inductive effect. The position dependence of substituents can be modeled by a distance decaying factor for S_N2. Use of the Hammett equation as a base-line model for Δ-machine learning models of the activation energy in chemical space results in substantially improved learning curves reaching low prediction errors for small training sets.

We generalize Hammett''s original approach to predict potential energies of activation in non aromatic molecular scaffolds with multiple substituents.     

Topological transitivity of billiards in polygons

Consider a billiard in a polygon QR² having all angles commensurate with . For the majority of initial directions, density of every infinite semitrajectory in configuration space is proved. Also proved is the typicality of polygons for which some billiard trajectory is dense in phase space.Translated from Matematicheskie Zametki, Vol. 18, No. 2, pp. 291–300, August, 1975.     

2-Isoxazolines with an electron-acceptor substituent at C<Subscript>(5)</Subscript> in reactions with nucleophilic reagents

Depending on the substituent at position 5 of the heterocycle, reaction of 2-isoxazolines with K-selectride, molybdenum hexacarbonyl, dimsyl sodium, and butyl lithium gave 1,3-cyclo-decomposition, aromatization of the isoxazoline ring, or reduction of the functional groups in the substituted isoxazolines. __________ Translated from Khimiya Geterotsiklicheskikh Soedinenii, No.3, 447–455, March, 2007.     

A propos de la propriété de Bergman

In a recent preprint, George M. Bergman has investigated the following property:for any generating set X of the group G there exists an integer n such that any element of G is a product of n elements of $\text{X}\cup {\text{X}}^{\mathrm{?1}}$. We will say in this case that G has the Bergman property.We have solved some of the questions asked in the above mentioned preprint and have found it suitable to investigate this property in a more general context, in particular for rings (essentially Boolean rings). To cite this article: A. Khelif, C. R. Acad. Sci. Paris, Ser. I 342 (2006).     

Equations in free groups are not finitely approximable

We give an equation in any free group of rank that has a solution in any finite quotient of , but has no solution in .

     

Variational optimization of effective atom centered potentials for molecular properties

In plane wave based electronic structure calculations the interaction of core and valence electrons is usually represented by atomic effective core potentials. They are constructed in such a way that the shape of the atomic valence orbitals outside a certain core radius is reproduced correctly with respect to the corresponding all-electron calculations. Here we present a method which, in conjunction with density functional perturbation theory, allows to optimize effective core potentials in order to reproduce ground-state molecular properties from arbitrarily accurate reference calculations within standard density functional calculations. We demonstrate the wide range of possible applications in theoretical chemistry of such optimized effective core potentials (OECPs) by means of two examples. We first use OECPs to tackle the link atom problem in quantum mechanics/molecular mechanics (QM/MM) schemes proposing a fully automatized procedure for the design of link OECPs, which are designed in such a way that they minimally perturb the electronic structure in the QM region. In the second application, we use OECPs in two sample molecules (water and acetic acid) such as to reproduce electronic densities and derived molecular properties of hybrid (B3LYP) quality within general gradient approximated (BLYP) density functional calculations.     

The return of the linear search problem

The linear search problem concerns a search on the real line for a point selected at random according to a given probability distribution. The search begins at zero and is made by a continuous motion with constant speed, first in one direction and then the other. The problem is to determine when it is possible to devise a “best” search plan. In former papers the best plan has been selected according to the criterion of minimum expected path length. In this paper we consider a more general, nonlinear criterion for a “best” plan and show that the substantive requirements of the earlier results are not affected by these changes.     

Interval exchange transformations and some special flows are not mixing

An interval exchange transformation (I.E.T.) is a map of an interval into itself which is one-to-one and continuous except for a finite set of points and preserves Lebesgue measure. We prove that any I.E.T. is not mixing with respect to any Borel invariant measure. The same is true for any special flow constructed by any I.E.T. and any “roof” function of bounded variation. As an application of the last result we deduce that in any polygon with the angles commensurable with π the billiard flow is not mixing on two-dimensional invariant manifolds. The author is partially supported by grant NSF MCS 78-15278.