Point interactions for 3D sub-Laplacians

In this paper we show that, for a sub-Laplacian Δ on a 3-dimensional manifold M, no point interaction centered at a point $q_0\in M$ exists. When M is complete w.r.t. the associated sub-Riemannian structure, this means that Δ acting on $C_0^{\infty }(M?\{q_0\})$ is essentially self-adjoint in $L^2(M)$. A particular example is the standard sub-Laplacian on the Heisenberg group. This is in stark contrast with what happens in a Riemannian manifold N, whose associated Laplace-Beltrami operator acting on $C_0^{\infty }(N?\{q_0\})$ is never essentially self-adjoint in $L^2(N)$, if $\dim ?N\le 3$. We then apply this result to the Schrödinger evolution of a thin molecule, i.e., with a vanishing moment of inertia, rotating around its center of mass.     

X-ray diffraction study of mesogenic 4-methoxyphenyl-4′-n-nonyloxybenzoate

4-methoxyphenyl-4′-n-nonyloxybenzoate have been investigated by single crystal X-ray structural analysis. Crystal packing of the compounds does not represent a precursor of the mesophse although the typical for liquid crystal compounds separation of crystal packing on aromatic and aliphatic areas is observed here. Possible reason for monotropic properties are discussed.     

Domain-based local pair natural orbital methods within the correlation consistent composite approach

Ab initio composite approaches have been utilized to model and predict main group thermochemistry within 1 kcal mol⁻¹, on average, from well-established reliable experiments, primarily for molecules with less than 30 atoms. For molecules of increasing size and complexity, such as biomolecular complexes, composite methodologies have been limited in their application. Therefore, the domain-based local pair natural orbital (DLPNO) methods have been implemented within the correlation consistent composite approach (ccCA) framework, namely DLPNO-ccCA, to reduce the computational cost (disk space, CPU (central processing unit) time, memory) and predict energetic properties such as enthalpies of formation, noncovalent interactions, and conformation energies for organic biomolecular complexes including one of the largest molecules examined via composite strategies, within 1 kcal mol⁻¹, after calibration with 119 molecules and a set of linear alkanes. © 2019 Wiley Periodicals, Inc.     

On the domain of fractional Laplacians and related generators of Feller processes

In this paper we study the domain of the generator of stable processes, stable-like processes and more general pseudo- and integro-differential operators which naturally arise both in analysis and as infinitesimal generators of Lévy- and Lévy-type (Feller) processes. In particular we obtain conditions on the symbol of the operator ensuring that certain (variable order) Hölder and Hölder–Zygmund spaces are in the domain. We use tools from probability theory to investigate the small-time asymptotics of the generalized moments of a Lévy or Lévy-type process ${(X_t)}_{t\ge 0}$,

$\underset{t\to 0}{\lim }?\frac{1}{t}({\mathbb{E}}^{x}f(X_t)?f(x)),x\in {\mathbb{R}}^d,$

for functions f which are not necessarily bounded or differentiable. The pointwise limit exists for fixed $x\in {\mathbb{R}}^d$ if f satisfies a Hölder condition at x. Moreover, we give sufficient conditions which ensure that the limit exists uniformly in the space of continuous functions vanishing at infinity. As an application we prove that the domain of the generator of ${(X_t)}_{t\ge 0}$ contains certain Hölder spaces of variable order. Our results apply, in particular, to stable-like processes, relativistic stable-like processes, solutions of Lévy-driven SDEs and Lévy processes.     

Validating additive correction schemes against gradient-based extrapolations

The use of additive correction schemes to obtain structures and vibrational frequencies of increasingly larger molecules is becoming more common. Such approaches, based on the cubic extrapolation formula applied directly to the quantity of interest, have been successfully validated only at the highest levels of computational accuracy: for coupled cluster methods with comparably large basis sets. Here, a systematic validation of geometries and vibrational frequencies is carried out, including more affordable and relevant levels of theory, such as the Møller-Plesset perturbation theory applied with smaller basis sets. Comparisons of such additive schemes against the more rigorous gradient-based extrapolation are presented. The cbs () routine of the open-source quantum-chemistry package Psi4 has been extended for this purpose. The results confirm that geometries and frequencies of covalently bound species obtained with additive correction schemes are in an excellent agreement with the results of gradient-based extrapolations. However, when applied to systems involving noncovalent interactions, the errors due to such schemes are significantly larger. In general, we propose the application of gradient-based extrapolations, as they incur no extra cost compared to additive schemes.     

Ill-posedness of the Camassa–Holm and related equations in the critical space

We prove norm inflation and hence ill-posedness for a class of shallow water wave equations, such as the Camassa–Holm equation, Degasperis–Procesi equation and Novikov equation etc., in the critical Sobolev space $H^{3/2}$ and even in the Besov space $B_{p,r}^{1+1/p}$ for $p\in [1,\infty ],r\in (1,\infty ]$. Our results cover both real-line and torus cases (only real-line case for Novikov), solving an open problem left in the previous works ([5], [14], [16]).     

Inducement of ferromagnetic–metallic phase and magnetoresistance behavior in charged ordered monovalent-doped Pr0.75Na0.25MnO3 manganite by Ni substitution

In this communication, the study on the effect of Ni²⁺ substitution on structural, magnetic and electrical transport properties were performed in Pr_0.75Na_0.25Mn_1-xNi_xO₃ (x = 0–0.10) ceramics synthesized using conventional solid-state method. X-ray diffraction patterns showed that all samples were present in single phase and crystallized in orthorhombic structure with Pnma space group. Rietveld refinement analysis revealed unit cell volume slight increase with increase Ni concentration, thereby indicating partial substitution of Ni²⁺ at Mn³⁺. The presence majority of Ni²⁺ states in the compound were confirmed by X-ray photoelectron spectrum. Tolerance factor calculation suggested that Ni substitution exerted no strong effect on structural distortion. For un-doped sample (x = 0), AC susceptibility (χ′) against temperature (T) curve showed paramagnetic (PM)–antiferromagnetic(AFM) behavior at Neel temperature (T_N) of approximately 170 K. Furthermore, resistivity (ρ) against temperature (T) curve showed an insulating behavior for the whole measured temperature range. The χ′ against T curve of x = 0 sample showed broad peak at approximately 218 K which was attributed to the onset of charge ordered (CO) state. No such broad peak was observed in Ni-substituted samples which indicated the weakening of CO state. Moreover, χ′ measurements exhibited successful inducement of PM–FM transition with Curie temperature (T_C), decreasing from 132 K (x = 0.02) to 92 K (x = 0.08). Electrical resistivity measurement on samples (x = 0.02–0.08) displayed inducement of metal–insulator transition, where transition temperature (T_MI) decreased and resistivity increased, with x before re-entrant insulating behavior at x = 0.10. Notably, upturn resistivity was observed below 40 K for x = 0.06 and 0.08 samples. The suppression of CO state and inducement of ferromagnetic-metallic (FMM) state beginning from x = 0.02 sample was attributed to the reduced degree of Jahn–Teller distortion and Coulomb interaction among Mn ions, as well as the presence of ferromagnetic superexchange (FM SE) interaction among Ni²⁺–O–Mn⁴⁺ which improved the alignment charge carrier spins and induced the double-exchange (DE) interaction among Mn³⁺–O–Mn⁴⁺. The decrease in T_C and T_MI with increased x may be due to the enhanced AFM SE interactions of Mn³⁺–O–Mn³⁺, Mn⁴⁺–O–Mn⁴⁺ and Ni²⁺–O–Ni²⁺ which decreased the FM SE interaction of Ni²⁺–O–Mn⁴⁺. Consequently, the effective DE interaction was decreased. In addition, the decreased metallic behavior and re-entrant insulating behavior for x = 0.10 sample was due to the strong AFM interaction between Ni²⁺ ions which consequently contributed to the suppression of FM SE and DE interactions. The observed upturn resistivity below 40 K for x = 0.06 and 0.08 samples was attributed to the Kondo-like effect which resulted from the interaction between itinerant conduction electron spin and localized spin impurity.     

Solid-State NMR Spectroscopy: A Powerful Technique to Directly Study Small Gas Molecules Adsorbed in Metal–Organic Frameworks

Metal–organic frameworks (MOFs) have shown great potential in gas separation and storage, and the design of MOFs for these purposes is an on-going field of research. Solid-state nuclear magnetic resonance (SSNMR) spectroscopy is a valuable technique for characterizing these functional materials. It can provide a wide range of structural and motional insights that are complementary to and/or difficult to access with alternative methods. In this Concept article, the recent advances made in SSNMR investigations of small gas molecules (i.e., carbon dioxide, carbon monoxide, hydrogen gas and light hydrocarbons) adsorbed in MOFs are discussed. These studies demonstrate the breadth of information that can be obtained by SSNMR spectroscopy, such as the number and location of guest adsorption sites, host–guest binding strengths and guest mobility. The knowledge acquired from these experiments yields a powerful tool for progress in MOF development.     

Probing molecular environments with a fictitious isotopic dipole

A HD-like (HD: mono-deuterated hydrogen molecule) isotopic dipole moment is proposed as a sensible probe for molecular environments, in particular for electrostatic fields and polarizable (reactive) sites of molecules. Fictitious nuclear masses are chosen in order to yield a rigid dipole with a small appropriate magnitude. Upon subtracting the Born-Oppenheimer energy, the interaction is reduced to field-dipole-like and dipole-polarizability-like terms, the last one being particularly informative since connected to potentially reactive sites. Possible asymmetries of this term appear as signatures of charged sites in the molecule. The field strength and orientation are easily obtained by identifying the minimum field-dipole energy configuration and flipping the dipole from it. Tests with hydrogen, water, benzene, and chlorobenzene molecules confirm the good performance of the method. In an application to test the present models for hydrogen activation by a frustrated Lewis pair, the full potential of the method is assessed.