Conformational analysis: Part 42. A modelling and LIS/NMR investigation of the conformations of some aromatic SO compounds

A refined Lanthanide‐Induced‐Shift Analysis (LISA), in which both the paramagnetic and the diamagnetic lanthanide induced chemical shifts are normalised separately, is used with molecular mechanics (MMFF94) and ab initio (RHF/6‐31G, RHF/6‐311G** and B3LYP/6‐311G**) calculations to investigate the conformations of methyl para‐tolyl sulphoxide 1 , sulphone 2 and sulphonate 3 and the conformational equilibria of methyl 2,4‐dimethylphenyl sulphoxide 4 and sulphone 5 . In 1 acceptable agreement factors (AFs) in the LISA analysis were only obtained when the SO/phenyl dihedral angle was increased from that (10–15°) of the computed geometries to 30–40°. The calculated geometries for 2 had the methyl group orthogonal to the phenyl ring and gave good AF in the LISA analysis. In both compounds, the AF were in the order MMFF94 ≤ B3LYP < RHF with the larger basis sets giving better agreement than the limiting 6‐31G* function. In 3 all the geometries again had the methoxy group orthogonal to the phenyl ring but, while the ab initio geometries still gave good AFs, the MMFF94 structure gave an AF which was only just acceptable. The unsymmetrical compounds 4 and 5 display conformational equilibria. In 4 two energy minima were found, both with the SMe group orthogonal to the phenyl ring, with the SO oxygen cis or trans to the 2‐methyl group. The energy difference (cis–trans) was calculated as 1.6 kcal/mol (MMFF94), 2.7 (6‐31G*), 1.9 (6‐311G**) but ?0.11 (DFT). The LISA analyses on the separate conformers gave poor agreement (ca. 3%) for the trans conformer but increasing the SO/Ph dihedral angle again gave an acceptable value (ca. 1.4%). Inclusion of even small amounts of the cis conformer in the LIS analysis gave poorer results. Thus, the LISA analysis confirms the MM and RHF prediction that the trans conformer is more stable and suggests that the conformer energy difference in solution is equal to or greater than the largest calculated value of 2.7 kcal/mol. In 5 all the calculations gave two mirror image conformations with the SMe orthogonal to the phenyl ring. The LISA analysis in this case is complicated by the fact that in the stable conformer the complexing oxygen atoms are not equivalent. This was resolved by considering both coordination sites in one Z‐matrix and varying the % coordination on each oxygen atom. This gave an acceptable AF of ca. 1.0% for all the geometries with ca. 2:1 coordination on the oxygen atoms trans and cis to the C2‐methyl. The difference in the geometries and conformer energies found by LISA for these molecules in solution compared to the theoretical values may be due to interactions with the chloroform solvent for these polar compounds. Copyright © 2010 John Wiley & Sons, Ltd.     

Exploring informal mathematical products of low achievers at the secondary school level

This article examines the notion of informal mathematical products, in the specific context of teaching mathematics to low achieving students at the secondary school level. The complex and relative nature of this notion is illustrated and some of its characteristics are suggested. These include the use of ad-hoc strategies, mental calculations, idiosyncratic ideas, everyday rather than mathematical language, non-symbolic explanations, visual justifications and common-sense based reasoning. The main argument raised in the article concerns the challenge of valuing informal mathematical products, created by low achievers, and using them within the mathematics classroom as means for advancing such students. The data draws from several research and design projects conducted in Israel since 1991. Selected examples of students’ products, gathered from low-track mathematics classrooms involved in these projects, are presented and analyzed. The analyses highlight various features of such products, and portray the possible gains of teaching approaches that legitimize, and build onwards from, informal products of low achievers.     

Feller Property and Infinitesimal Generator of the Exploration Process

We consider the exploration process associated to the continuous random tree (CRT) built using a Lévy process with no negative jumps. This process has been studied by Duquesne, Le Gall and Le Jan. This measure-valued Markov process is a useful tool to study CRT as well as super-Brownian motion with general branching mechanism. In this paper we prove this process is Feller, and we compute its infinitesimal generator on exponential functionals and give the corresponding martingale. The research of the second author was partially supported by NSERC Discovery Grants of the Probability group at Univ. of British Columbia.     

1H NMR spectra. Part 30: 1H chemical shifts in amides and the magnetic anisotropy,electric field and steric effects of the amide group

The ¹H spectra of 37 amides in CDCl₃ solvent were analysed and the chemical shifts obtained. The molecular geometries and conformational analysis of these amides were considered in detail. The NMR spectral assignments are of interest, e.g. the assignments of the formamide NH₂ protons reverse in going from CDCl₃ to more polar solvents. The substituent chemical shifts of the amide group in both aliphatic and aromatic amides were analysed using an approach based on neural network data for near (≤3 bonds removed) protons and the electric field, magnetic anisotropy, steric and for aromatic systems π effects of the amide group for more distant protons. The electric field is calculated from the partial atomic charges on the N.C═O atoms of the amide group. The magnetic anisotropy of the carbonyl group was reproduced with the asymmetric magnetic anisotropy acting at the midpoint of the carbonyl bond. The values of the anisotropies Δχ_parl and Δχ_perp were for the aliphatic amides 10.53 and ?23.67 (×10^?6 ų/molecule) and for the aromatic amides 2.12 and ?10.43 (×10^?6 ų/molecule). The nitrogen anisotropy was 7.62 (×10^?6 ų/molecule). These values are compared with previous literature values. The ¹H chemical shifts were calculated from the semi‐empirical approach and also by gauge‐independent atomic orbital calculations with the density functional theory method and B3LYP/6–31G⁺⁺ (d,p) basis set. The semi‐empirical approach gave good agreement with root mean square error of 0.081 ppm for the data set of 280 entries. The gauge‐independent atomic orbital approach was generally acceptable, but significant errors (ca. 1 ppm) were found for the NH and CHO protons and also for some other protons. Copyright © 2013 John Wiley & Sons, Ltd.     

A note on the computation of the CP-rank

The purpose of this note is to address the computational question of determining whether or not a square nonnegative matrix (over the reals) is completely positive and finding its CP-rank when it is. We show that these questions can be resolved by finite algorithms and we provide (non-polynomial) complexity bounds on the number of arithmetic/Boolean operations that these algorithms require. We state several open questions including the existence of polynomial algorithms to resolve the above problems and availability of algorithms for addressing complete positivity over the rationals and over {0, 1} matrices.     

Reassessment of doses in TLD100 by measuring the residual dose

Second readouts of TLD 100 crystals exposed to radiation are used to reassess high doses. When the second readout is performed by a regular readout cycle of 13.3 s, the accuracy of the estimations is too low to be practical for low doses in the 10–100 mSv range. A simple and efficient method was studied, which consists of expanding the heating time to 30 s. The efficiency relative to a 13.3 readout time is about three timesbetter, and also the background is lower, thus the reassessment efficiency is significantly improved.