Understanding relativistic effects of chemical bonding

To elucidate the physical origin of relativistic changes of molecular properties, exact theorems, perturbation theory, and Hartree-Fock-Slater-Pauli calculations are exploited. The relativistic molecular virial theorem offers insight into the relativistic and nonrelativistic, kinetic, and potential energy contributions to the bond energy. In general, there exist two contributions to the relativistic correction of a molecular property: the relativistic change at the nonrelativistic equilibrium geometry and the change of the nonrelativistic property due to the relativistic change of the equilibrium geometry. Sometimes the first and sometimes the second contribution is the dominant one. Accurate numerical results for H⁺₂-like systems are obtained using direct relativistic double perturbation theory. In some cases, near-degenerate perturbation theory is mandatory. Relativistic changes of chemical bond energies are often proportional to the density change in the K-shell when the bond is formed. Relativistic corrections to many properties (and also to the 1s²-correlation energy) are often proportional to Z²α². © 1996 John Wiley & Sons, Inc.     

Comments on “vibration of a rotating beam with tip mass”

In diagnostic applications of machinery vibration or noise data it is convenient to describe the whole process by a set of numbers called discriminants. Five discriminants can be constructed and measured for every vibroacoustical process. Two of them have dimensional natures and give information about process amplitude and frequency. Two others, dimensionless, give information about the amplitude and spectral spread of the process. The last one, also dimensionless, characterizes the time fluctuations of the process, and can be used to detect instability of a running machine. Numerical and experimental examples are presented.     

Relativistic perturbation theory of chemical properties

Summary Double perturbation theory is developed for the case where relativity is one perturbation and the other perturbation describes a chemically interesting observable such as molecular structure, force constant or polarizability. Relativity is treated according to Rutkowski's nonsingular perturbation approach. Expressions for four-component and two-component wave-functions and for the Hartree-Fock approximation are given. The method is applied analytically to the relativistic corrections of the electric polarizability of the H atom, and algebraically to the potential curve of the H ₂ ⁺ molecule. Second and third order double perturbation interchange relations are numerically verified. In the present formalism, terms up to third order are needed to qualitatively understand the relativistic corrections of chemical observables.     

[10b, 10c]Diazapyraceheptylene

The labile [10b, 10c]diazapyraceheptylene 1 was obtained by thermoflash dehydrogenation of a mixture of olefins 5a/b. The ¹H NMR spectrum indicates a strong influence of the 14π-electron periphery. A by-product of the synthesis of 1, i.e., pyrazino[2,1,6-cd:5,4,3-c′d′]dipyrrolizine 6, points to a rearrangement of the diazapyraceheptylene system, which may be frontier orbital controlled.     

Retractability and the fixed point property for products

LetP, Q be ordered sets and letaP. IfP \ {a} is a retract ofP and setsP and {xP:x>p} (or its dual) have the fixed point property then, for each chain complete setP,P×Q has the fixed point property if and only if (P\{a})×Q has this property. This establishes the fixed point property for some products of ordered sets which are beyond the reach of all known product theorems.The work of the first of authors was supported in part by the K.B.N. Grant No. 2 2037 92 03.     

Investigation of the IR spectra of weakly hydrogen-bonded complex Cl<Subscript>3</Subscript>CH…O(CD<Subscript>3</Subscript>)<Subscript>2</Subscript> in a cryosolution in liquid krypton

The IR absorption spectra of cryosolutions of chloroform, dimethyl ether, and their mixtures in liquefied krypton have been measured. It has been shown that, in solutions of mixtures, Cl₃CH…O(CD₃)₂ complexes with a weak hydrogen bond are formed. The spectral characteristics of individual absorption bands that refer to proton donors and to proton acceptors have been determined. From temperature measurements of the integrated intensities of the bands of monomers and of the complex (T ~ 120–160 K), the enthalpy of the complex formation has been estimated. The measurement data have been analyzed in comparison with the results of ab initio calculations in terms of the МР2/6-311++G(2df, 2pd) approximation. The analysis has been done taking into account peculiarities of the dipole moment function of chloroform and possible effects of the anharmonic interaction between the С–Н stretching vibration and the overtone of the Cl–C–H bending vibration.     

Solvent effect on the blue shifted weakly H-bound F3 CH…FCD3 complex

Solvent effect on the ν^c frequency of CH stretching vibration of the blue shifted F₃CH…FCD₃ complex has been studied in liquefied N₂, CO, Ar, Kr and Xe. In the case of Xe, the spectroscopic measurements have also been extended to the solid state. It was found that the ν^c position of the complex in the solutions studied lowers with respect to the value in the gas phase. In liquid Xe, characterized by the largest permittivity, this effect reaches its maximum value of −14.5 cm⁻¹. The ν^c frequency begins to grow again just below the freezing point of Xe, where a noticeable (15%) increase of the density of Xe occurs. The experimental results obtained for the liquid phase have been analyzed in the framework of the Onsager-like reaction field model and Polarizable Continuum Model (PCM) implemented into a standard Gaussian 98 Program.     

Random times and multiplicative systems

The present research is motivated by the recent results of Jeanblanc and Song (2011)  and . Our aim is to demonstrate, with the help of multiplicative systems introduced in Meyer (1979) [21], that for any given positive F$\mathbb{F}$-submartingale F$F$ such that _F∞=1$F_{\infty }=1$, there exists a random time τ$\tau$ on some extension of the filtered probability space such that the Azéma submartingale associated with τ$\tau$ coincides with F$F$. Pertinent properties of this construction are studied and it is subsequently extended to the case of several correlated random times with the predetermined univariate conditional distributions.