Quantum chemical study of solvent and substituent effects on the 1,5-hydride shift in 2,6-dimethyl-2-heptyl cations

The mechanism of the degenerate 1,5-hydride shift in 2,6-dimethyl-2-heptyl cations has been investigated using ab initio MP2 and density functional theory (DFT) hybrid (B3LYP) calculations. The potential-energy profile for the 1,5-hydride shift consists of three minima corresponding to two equivalent acyclic carbocations and one symmetrically mu-hydrido-bridged carbocation, while two equivalent unsymmetrically hydrido-bridged carbocations were located as transition-state structures. The calculated relative energy differences between acyclic carbocations and symmetrically mu-hydrido-bridged structure are significantly affected by introduction of alkyl and (CH2)n-substituents at the C4 position of the 2,6-dimethyl-2-heptyl cation structure. DFT self-consistent isodensity polarizable continuum method (SCI-PCM) and MP2 PCM continuum methods have been used to calculate the effect of solvation on geometries and relative energies of the species involved in the 1,5-hydride shift. It is found that relative energies of acyclic and mu-hydrido-bridged carbocation structures as well as the energy barriers for 1,5-hydride shifts are in accord with experimental data if solvation effects are taken into account.     

Sandwich semigroups in locally small categories I: foundations

Fix (not necessarily distinct) objects i and j of a locally small category S, and write \(S_{ij}\) for the set of all morphisms \(i\rightarrow j\). Fix a morphism \(a\in S_{ji}\), and define an operation \(\star _a\) on \(S_{ij}\) by \(x\star _ay=xay\) for all \(x,y\in S_{ij}\). Then \((S_{ij},\star _a)\) is a semigroup, known as a sandwich semigroup, and denoted by \(S_{ij}^a\). This article develops a general theory of sandwich semigroups in locally small categories. We begin with structural issues such as regularity, Green’s relations and stability, focusing on the relationships between these properties on \(S_{ij}^a\) and the whole category S. We then identify a natural condition on a, called sandwich regularity, under which the set \({\text {Reg}}(S_{ij}^a)\) of all regular elements of \(S_{ij}^a\) is a subsemigroup of \(S_{ij}^a\). Under this condition, we carefully analyse the structure of the semigroup \({\text {Reg}}(S_{ij}^a)\), relating it via pullback products to certain regular subsemigroups of \(S_{ii}\) and \(S_{jj}\), and to a certain regular sandwich monoid defined on a subset of \(S_{ji}\); among other things, this allows us to also describe the idempotent-generated subsemigroup \(\mathbb E(S_{ij}^a)\) of \(S_{ij}^a\). We also study combinatorial invariants such as the rank (minimal size of a generating set) of the semigroups \(S_{ij}^a\), \({\text {Reg}}(S_{ij}^a)\) and \(\mathbb E(S_{ij}^a)\); we give lower bounds for these ranks, and in the case of \({\text {Reg}}(S_{ij}^a)\) and \(\mathbb E(S_{ij}^a)\) show that the bounds are sharp under a certain condition we call MI-domination. Applications to concrete categories of transformations and partial transformations are given in Part II.     

Continuous time random walks and the Cauchy problem for the heat equation

We deal with anomalous diffusions induced by continuous time random walks - CTRW in ?ⁿ. A particle moves in ?ⁿ in such a way that the probability density function u(·, t) of finding it in region Ω of ?ⁿ is given by ∫_Ωu(x, t)dx. The dynamics of the diffusion is provided by a space time probability density J(x, t) compactly supported in {t ≥ 0}. For t large enough, u satisfies the equation

$$u\left( {x,t} \right) = \left[ {\left( {J - \delta } \right)*u} \right]\left( {x,t} \right)$$

, where δ is the Dirac delta in space-time. We give a sense to a Cauchy type problem for a given initial density distribution f. We use Banach fixed point method to solve it and prove that under parabolic rescaling of J, the equation tends weakly to the heat equation and that for particular kernels J, the solutions tend to the corresponding temperatures when the scaling parameter approaches 0.

     

Bases of Feigin–Stoyanovsky’s type subspaces for $$C_\ell ^{(1)}$$

In this paper, we construct combinatorial bases of Feigin–Stoyanovsky’s type subspaces of standard modules for level k affine Lie algebra \(C_\ell ^{(1)}\). We prove spanning by using annihilating field \(x_\theta (z)^{k+1}\) of standard modules. In the proof of linear independence, we use simple currents and intertwining operators whose existence is given by fusion rules.     

Dynamic polarizability,dispersion coefficient C6 and dispersion energy in the effective fragment potential method

The development of a fragment–fragment dispersion energy expression, for the general effective fragment potential (EFP2) method is presented. C₆ dispersion coefficients, expressed in terms of the dynamic polarizabilties over the imaginary frequency range (α(iν)), were calculated for a set of homo and hetero dimers. Using these coefficients the dispersion energy has been calculated. The dispersion energy is expressed using a simple London series expansion terminated after the n=6 term and implemented using distributed localized molecular orbitals (LMOs). The EFP2 dispersion energy is compared to symmetry adapted perturbation theory (SAPT) values. From this comparison, it is apparent that one needs to include higher order terms in the dispersion energy. Adding an estimated C₈ term to the C₆ energy greatly improves the agreement with the benchmark SAPT energies.     

Sonochemical effectiveness factor (eUS) in the reactors for wastewater treatment by sono-Fenton oxidation: Novel considerations

A comprehensive algorithm was recently proposed for calculation of the sonochemical effectiveness factor and wastewater treatment modeling. The presented approach implies that ultrasound is an auxiliary source of free radicals in Fenton type reactions; introduction of ultrasound represents an enhancement of pollutant degradation rates. The sonochemical effectiveness factor was introduced in kinetic models as the e_US factor (Gr?i? et al., 2012 [1]). As a substantial follow-up, this study presents novel considerations. The e_US factor was modeled as a function of employed frequency, actual cavitation-related power intensity of ultrasound and a portion of the cavitationally active zone, i.e. dimensionless active volume. The effect of temperature was disregarded in the present model considerations. Cavitationally active zone in reactors was determined based on the erosion of aluminum foil, resulting in cone-shaped space arising from transducer.In the present study, sonochemical treatment of industrial wastewater containing HCOONa as organic pollutant was performed using different equipment: ultrasonic baths (UB1, UB2 and UB3), cylindrical reactor with homogenizer (HCR) and three-frequency hexagonal cell, i.e. ultrasonic pilot reactor prototype (PP). Explored frequency range was from 20 to 120 kHz. Homogeneous and heterogeneous Fenton-type sonochemical processes, US/Fe(II)(FeSO₄,aq.)/H₂O₂ and US/Fe(II)(steel-plate)/H₂O₂, respectively, applied to industrial wastewater were investigated in terms of mineralization kinetics. Newly modeled e_US factor was introduced in corresponding kinetic models and the overall model was validated. Kinetic parameters of Fenton process were treated as independent of ultrasound, since e_US factor consists of cavitation-related phenomena responsible for the mineralization rate enhancement. In average, a 21% increase of mineralization efficiency was achieved using a single frequency, while more than 70% increase can be achieved by combining 20, 68 and 120 kHz in PP.     

Exact approaches for solving robust prize-collecting Steiner tree problems

In the Prize-Collecting Steiner Tree Problem (PCStT) we are given a set of customers with potential revenues and a set of possible links connecting these customers with fixed installation costs. The goal is to decide which customers to connect into a tree structure so that the sum of the link costs plus the revenues of the customers that are left out is minimized. The problem, as well as some of its variants, is used to model a wide range of applications in telecommunications, gas distribution networks, protein–protein interaction networks, or image segmentation.     

Theoretical scrutinization of nine benzoic acid dimers: Stability and energy decomposition analysis

Aromatic carboxylic acids are able to form diverse dimers and multimers due to their hydrogen bond donor and acceptor cites, as well as the aromatic rings. In this work, we examine nine benzoic acid dimers stabilized by hydrogen bonding and stacking interactions. Interacting quantum atoms methodology revealed that dominant attractive interactions in all of them, including hydrogen bonded systems, are due to exchange-correlation. Coulomb interactions are significant only in the most stable dimer with a double hydrogen bond, although the corresponding energy term is almost two times lower compared to the nonclassical one. Since interacting quantum atoms approach treats monomers binding by considering electronic energy only, in order to examine dissociation kinetics we performed density functional theory-based molecular dynamics simulations of selected stacked dimers: in 40% of the studied systems at 300 K thermal energy was sufficient to overpower barrier for dissociation within 1 ps, which resulted in the separation of the monomers, whereas 20% of them remained in the stacked position even after 5 ps. These results highlight the importance of noncovalent interactions, particularly weak stacking interactions, on the structure and dynamics of carboxylic acids and their derivatives.     

A survey of isotopic composition (2H, 3H, 18O) of groundwater from Vojvodina

Isotopes of hydrogen (³H, ²H) and oxygen (¹⁸O) are perfect candidates for groundwater tracers. A survey of isotopic composition of 34 groundwater samples and one Lake from Vojvodina region (Serbia) is presented here. Tritium activity concentration and stable isotope composition (δ²H, δ¹⁸O), as well as deuterium excess, were determined. The groundwater samples lie on the groundwater regression line. Minor deviations and a few lower deuterium excess values indicate waters recharged in a different climate regime and subjected to evaporation, respectively. According to the obtained results, most of the analyzed groundwater can be characterized as modern waters, recharged mostly from precipitation.