Curl condition for a four-state Born-Oppenheimer system employing the Mathieu equation

When a group of four states forms a subspace of the Hilbert space, i.e., appears to be strongly coupled with each other but very weakly interacts with all other states of the entire space, it is possible to express the nonadiabatic coupling (NAC) elements either in terms of s or in terms of electronic basis function angles, namely, mixing angles presumably representing the same sub-Hilbert space. We demonstrate that those explicit forms of the NAC terms satisfy the curl conditions--the necessary requirements to ensure the adiabatic-diabatic transformation in order to remove the NAC terms (could be often singular also at specific point(s) or along a seam in the configuration space) in the adiabatic representation of nuclear SE and to obtain the diabatic one with smooth functional form of coupling terms among the electronic states. In order to formulate extended Born-Oppenheimer (EBO) equations [J. Chem. Phys. 2006, 124, 074101] for a group of four states, we show that there should exist a coordinate independent ratio of the gradients for each pair of ADT/mixing angles leading to zero curls and, thereafter, provide a brief discussion on its analytical validity. As a numerical justification, we consider the first four eigenfunctions of the Mathieu equation to demonstrate the interesting features of nonadiabatic coupling (NAC) elements, namely, the validity of curl conditions and the nature of curl equations around CIs.     

Sustainable Synthesis of α-Hydroxycarboxylic Acids by Manganese Catalyzed Acceptorless Dehydrogenative Coupling of Ethylene Glycol and Primary Alcohols**

Herein, we report a straightforward synthesis of valuable α-hydroxycarboxylic acid molecules via an acceptorless dehydrogenative coupling of ethylene glycol and primary alcohols. A bench-stable manganese complex catalyzed the reaction, which is scalable, with the product being isolated with high yields and selectivities under mild conditions. The protocol is environmentally benign, producing water and hydrogen gas as the only byproducts. Methanol can also be used as a C1 source for producing the platform molecule lactic acid, with a high turnover of >10⁴. The methodology was also used to functionalize alcohols derived from natural products and fatty acids. Furthermore, it was applied for synthesizing α-amino acid, α-thiocarboxylic acid, and several drugs and bioactive molecules, including endogenous metabolites, Danshensu, Enalapril, Lisinopril, and Rosmarinic acid. Preliminary mechanistic studies were performed to shed light on the mechanism involved in the reaction.     

Simple Chemical Route Synthesis of Fe2O3 Nanoparticles and its Application for Adsorptive Removal of Congo Red from Aqueous Media: Artificial Neural Network Modeling

Nanocrystalline Fe₂O₃ powder was synthesized by a simple chemical route involving FeCl₃ and NaOH. The Fe₂O₃ powder thus prepared was characterized using x-ray diffraction study, scanning electron microscopy, and Fourier transform infrared spectroscopy. The adsorption properties of crystalline Fe₂O₃ powder have been investigated with an aim to explore a possible low cost and efficient way to remove Congo red (CR) from waste water. Fe₂O₃ powder was found as an excellent adsorbent for CR from aqueous medium. Adsorption capacity as much as 203.66 mg g^?1 is reported at room temperature. Effect of different experimental parameters such as reaction pH, initial CR dye concentration, adsorbent dose, and reaction temperature were studied on adsorption capacity of Fe₂O₃ powder and modeled by artificial neural network (ANN). Optimal ANN structure (4–5–1) shows minimum mean squared error (MSE) of 0.00235 and determination coefficient (R²) of 0.991 with Levenberg–Marquardt algorithm. Isotherm analysis of experimental data exhibited better fit to the Langmuir isotherm. The adsorption process was found to follow second-order kinetics as depicted by the analysis of experimental results. Thermodynamic study shows that the adsorption process is endothermic, spontaneous, and thermodynamically favorable in the temperature range of 27°C to 60°C.     

Charge and/or spin limits for black holes at a non-commutative scale

We study the cosmological dynamics for R ^p exp(λ R) gravity theory in the metric formalism, using dynamical systems approach. Considering higher-dimensional FRW geometries in case of an imperfect fluid which has two different scale factors in the normal and extra dimensions, we find the exact solutions, and study its behaviour and stability for both vacuum and matter cases. It is found that stable solutions corresponding to accelerated expansion at late times exist, which can describe the inflationary era of the Universe. We also study the evolution of scale factors both in the normal and extra dimensions for different values of anisotropy parameter and the number of extra dimensions for such a scenario.     

Rate processes with dynamical disorder: a direct variational approach

Using path integral approach, we develop variational approximations to the calculation of survival probability for rate processes with dynamical disorder. We derive both upper and lower bounds to the survival probability using Jensen's inequality. The inequalities involve the use of a trial action for which the path integrals can be evaluated exactly. Any parameter in the trial action can be varied to optimize the bounds. We have also derived a lower bound to the rate of the process. As a simple illustration, we apply the method to the problem of a particle undergoing Brownian motion in a harmonic potential well, in the presence of a delta function sink, for which one can calculate the exact survival probability numerically. The calculation confirms the two inequalities. The method should be very useful in similar but more complex problems where even numerical solution is not possible.