Transport equations for chiral fermions to order ? and electroweak baryogenesis: Part I

This is the first in a series of two papers. In this first part, we use the Schwinger-Keldysh formalism to derive semiclassical Boltzmann transport equations, accurate to order ?, for massive chiral fermions, scalar particles, and for the corresponding CP-conjugate states. Our considerations include complex mass terms and mixing fermion and scalar fields, such that CP-violation is naturally included, rendering the equations particularly suitable for studies of baryogenesis at a first order electroweak phase transition. We provide a quantitative criterion in which case the reduction to the diagonal kinetic equations in the mass eigenbasis is justified, leading to a quasiparticle picture even in the case of mixing scalar or fermionic particles. Within the approximations we make, it is possible to first study the Boltzmann equations without the collision term. In a second paper [Ann. Phys. xxx (2004) xxx] we discuss the collision terms and reduce the Boltzmann equations to fluid equations.     

Dynamics of superhorizon photons during inflation with vacuum polarization

We study asymptotic dynamics of photons propagating in the polarized vacuum of a locally de Sitter Universe. The origin of the vacuum polarization is fluctuations of a massless, minimally coupled, scalar, which we model by the one-loop vacuum polarization tensor of scalar electrodynamics. We show that late time dynamics of the electric field on superhorizon scales approaches that of an Airy oscillator. The magnetic field amplitude, on the other hand, asymptotically approaches a nonvanishing constant (plus an exponentially small oscillatory component), which is suppressed with respect to the initial (vacuum) amplitude. This implies that the asymptotic photon dynamics is more intricate than that of a massive photon obeying the local Proca equation.     

Mostar index

We propose and investigate a new bond-additive structural invariant as a measure of peripherality in graphs. We first determine its extremal values and characterize extremal trees and unicyclic graphs. Then we show how it can be efficiently computed for large classes of chemically interesting graphs using a variant of the cut method introduced by Klav?ar, Gutman and Mohar. Explicit formulas are presented for several classes of benzenoid graphs and Cartesian products. At the end we state several conjectures and list some open problems.     

A highly enantio- and diastereoselective catalytic intramolecular Stetter reaction

A highly enantio- and diastereoselective intramolecular Stetter reaction has been developed. Subjection of alpha,alpha-disubstituted Michael acceptors to an asymmetric intramolecular Stetter reaction results in a highly enantioselective conjugate addition and a diastereoselective proton transfer. Available evidence suggests the diastereoselective protonation occurs via intramolecular delivery to the sterically more hindered face of the enolate. The scope of the trisubstituted Michael acceptors has been examined and found to be broad with respect to the size of the alpha-substituent and nature of the Michael acceptor. Aliphatic and aromatic aldehydes were examined and found to afford the desired product in good overall yield with high enantio- and diastereoselectivity.     

Slater orbital molecular integrals with numerical fourier transform methods. I. (coplanar) multicenter exchange integrals over 1s orbitals

Slater orbital r₁₂^?1 integrals are calculated with a numerical Fourier-transform method based on a formulation first given by Bonham, Peacher and Cox. Spherical wave expansions are introduced that decouple the Feynman integrations for the charge distribution Fourier transforms. The Feynman integrals are evaluated semianalytically, and their properties are analyzed in detail. The final computational step involves a numerical integration over charge distribution quantities. Results for (coplanar) multicenter exchange integrals over 1s orbitals are given. As long as the charge distributions are overlapping considerably, the method gives good results, even when these distributions are highly asymmetric. The method as presently implemented fails when highly disconnected charge distributions are involved.     

Molecular orbital resonance theory: A new approach to the treatment of quantum chemical problems

An alternative approach to the treatment of the quantum chemical problems combining both, the MO and VB theory, is proposed. This approach retains the concept of resonance from the VB method, but it treats each particular bond in the MO sense. The method is illustrated with a few examples. Relative stabilities of benzene, pentalene and cyclobutadiene are derived. A Hückel (4m + 2) rule is derived for the annulenes. The charge polarisation in the case of the pentalene molecule is explained. A distortion of the pentalene molecule is considered and it is shown that within this approach the distortion depends on the charge polarisation.     

Study on heterogeneous equilibria in saturated solutions of sparingly soluble diprotic acids

Heterogeneous equilibria in saturated solutions of sparingly soluble diprotic acids: 1,2-, 1,3- and 1,4-benzenedicarboxylic acid, and octanedioic acid, respectively, at ionic strengthI=1 (NaCl) and the temperature of 25 °C, have been investigated. The corresponding equilibrium constants were determined by the application of three independent methods: the formation function method, the method of protons bound to the acid in the solution and the solubility method. The results obtained by these three different methods are in good agreement.

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Untersuchung heterogener Gleichgewichte in gesättigten Lösungen schwerlöslicher diprotischer Säuren

Zusammenfassung Es wurden die heterogene Gleichgewichte in gesättigten Lösungen schwerlöslicher diprotischer Säuren (Benzol-1,2-, -1,3- und-1,4-dicarbonsäuren und Oktandisäure) bei einer IonenstärkeI=1 (NaCl) und der Temperatur von 25 °C untersucht. Die entsprechende Gleichgewichtskonstanten wurden mit drei unabhängigen Methoden bestimmt: mit der Methode der Bildungsfunktion, der Methode der an die Säure in Lösung gebundenen Protonen, bzw. mit der Methode der Löslichkeitsbestimmung. Die erhaltenen Ergebnisse stehen in guter Übereinstimmung.

     

Augmentation of the Generalised n×n Eigenvalue Equation to a Generalised (n+1)×(n+1) Eigenvalue Equation

Generalised n×n eigenvalue equation B| _i = _i S ^b | _i (i=1,...,n) where B and S ^b are n×n Hermitian matrices while S ^b is in addition positive definite is considered. This equation is augmented to a generalised (n+1)(n+1) eigenvalue equation H| _k = _k S| _k (k=1,...,n+1) where Hermitian matrices H and S represent matrices B and S ^b , respectively, augmented by one additional row and one additional column. It is shown how the eigenvalues _k and the eigenvectors | _k of the augmented eigenvalue equation can be expressed in terms of the eigenvalues _i and the eigenvectors | _i of the original eigenvalue equation. Operation count to obtain by this method all augmented eigenvalues and eigenvectors is of the order O(n ²). Unless matrices involved are of some special kind such as sparse matrices or alike, this operation count is one order of magnitude smaller than operation count required by other presently known methods. In many practical cases operation count to obtain a single selected eigenvalue and/or eigenvector by this method is of the order O(n). In the case of the generalised eigenvalue equation, all other methods usually require again O(n ³) operations, even if only a single eigenvalue and/or eigenvector is required. Thus in many cases of interest operation count to obtain a selected eigenvalue and/or eigenvector by this method is two orders of magnitude smaller than operation count required by other methods.     

Vibrational isotope effect by the low rank perturbation method

Mathematical formalism of the low rank perturbation method (LRP) is applied to the vibrational isotope effect in the harmonic approximation. A pair of two n-atom isotopic molecules A and B which are identical except for isotopic substitutions at atomic sites is considered. Relations which express vibrational frequencies _k and normal modes _k of the perturbed isotopic molecule B in terms of the vibrational frequencies _i and normal modes _i of the unperturbed molecule A are derived. In these relations complete specification of the unperturbed normal modes _i is not required. Only amplitudes | _i of normal modes _i at sites affected by the isotopic substitution are needed.     

Mechanochemistry for Synthesis

Mechanochemical solvent‐free reactions by milling, grinding or other types of mechanical action have emerged as a viable alternative to solution chemistry. Mechanochemistry offers not only a possibility to eliminate the need for bulk solvent use, and reduce the generation of waste, but it also unlocks the door to a different reaction environment in which synthetic strategies, reactions and molecules previously not accessible in solution, can be achieved. This Minireview examines the potential of mechanochemistry in chemical and materials synthesis, by providing a cross‐section of the recent developments in using ball milling for the formation of molecules and materials based on covalent and coordination bonds.