Generalized Approach to Relaxed Force Constants

The main purpose of potential constants, i. e., force and compliance constants, is to provide a quantitative understanding of bonding relationships. Compliance constants have been shown to possess several mathematical advantages over conventional force constants.^1-3 Compliance constants, however, are parameters which decrease in magnitude as bond strength increases; consequently, their use as bond strength parameters is not particularly satisfying. The desirability of a measure of bond strength which increases in magnitude as the bond becomes stronger, and at the same time possesses all the advantages of compliance constants is self evident. For this reason, the relaxed force constant T_ii=1/C_ii was introduced by Jones some time ago.⁴ In the present communication, this approach is generalized to include relaxed interaction force constants. A brief discussion of their meaning is also presented.     

Determination of Water Force Constants with Fuzzy Methods

The paper proposes a new approach to the determination of moleculare force fields from observed vibrational frequencies for the case, that observed frequencies are only given in more or less sharp expectation ranges (intervals). Starting from a physically reasonable force constant space, we determine force fields so that the calculated eigenfrequencies are near by the experimental ones. However, in contrast to classical least squaress methods, we will model the uncertainty expressed by the statement “near by” with special triangle functions where the experimental values stand in the head of the triangles. In the sense of the fuzzy set theory, these functions are interpreted as membership functions. Within the force constant space we carry out Normal Coordinate Analyses in the usual way and determine a force field so that the eigenfrequencies have the least possible deviation from the experimental values. The deviation is evaluated with the help of the triangle membership functions. Hence, the selected force field with least deviation is called a fuzzy-optimal one. The water molecule is selected as a model case, because its least-squares force field is precisely known. Following the Born-Oppenheimer-Approximation, we treat all water isotopes in the same manner and select fuzzy-optimal force fields so that the evaluations for each water isotope are approximately similar and as high as possible. We will see that the fuzzy approach is always applicable, even if the assumptions for application of least squares methods are failed. This is a main advantage of the fuzzy method.

     

Calculation of the Interatomic Force Constants of Pyrazole

In the frame of a more complete study on the tautomery of the pyrazolines-5-ones, the authors determined the normal vibration modes of pyrazole. They calculate the interatomic force constants by means of a “simplex” optimisation procedure (1).     

Renormalization Group Approach to Generalized Cosmological Models

We revisit here the problem of generalized cosmology using renormalization group approach. A complete analysis of these cosmologies, where specific models appear as asymptotic fixed-points, is given here along with their linearized stability analysis.     

Scaled DFT Force Constants and Vibrational Spectrum of Cyclopropylamine

Abstract

The molecular structure and harmonic vibrational frequencies of cyclopropylamine have been calculated using the B3LYP density functional method with the 6‐31G(2d,2p) basis set. The scaled DFT force field gives very good reproduction of the experimental vibrational frequencies. Several of the vibrational fundamental modes assigned previously are reassigned on the basis of the B3LYP/6‐31G(2d,2p) method and the scaled force field calculations. The optimized scaling factors were used to scale the B3LYP/6‐31G(2d,2p) force field of cyclopropane and cyclopropylamine (–ND₂) molecules.     

Binary Bell Polynomials Approach to Generalized Nizhnik-Novikov-Veselov Equation

The elementary and systematic binary Bell polynomials method is applied to the generalized Nizhnik-Novikov-Veselov (GNNV) equation. The bilinear representation, bilinear Bäcklund transformation, Lax pair and infinite conservation laws of the GNNV equation are obtained directly, without too much trick like Hirota's bilinear method.     

Force Constants,Coriolis Coupling Constants,and Mean Amplitudes of Vibration of InCl6 3-

Abstract

The general quadratic force constants, coriolis coupling constants and mean amplitudes of vibration have been evaluated for InCl₆ ^3- using recent vibrational data. The results are employed to study the trend of variation in the isoelectronic sequences e.g. InCl₆ ^3-, SnCl₆ ^2- and SbCl₆ ^?.

In the present communication we report the force constants, coriolis coupling constants and mean amplitudes of vibration for InCl₆ ^3- which have not been reported so far. The results are used to study the trend of variation of the molecular parameters in the isoelectronic sequence e.g. InCl₆ ^3-, SnCl₆ ^2- and SbCl₆ ^?. The fundamental frequencies of InCl₆ ^3-, employed in this computation, have been reported on the basis of octahedral symmetry, from infrared and Raman Spectral Studies¹.     

Approach to Generalized Synchronization with Application to Chaos-Based Secure Communication

A constructive theorem is established for generalized synchronization (GS) related to C¹ diffeomorphic transformations of unidirectionally coupled dynamical arrays. The theorem provides some interpretations about the underlying mechanism of various GS phenomena in nature. As a direct application of the theorem, a chaos-based secure Internet communication scheme is proposed. Moreover, a cellular neural network (CNN) of Chen's chaotic circuits with GS property is designed and studied. Numerical simulation shows that this Chen's CNN has high security and is fast and reliable for secure Internet communications.     

On the Generalized Phase Space Approach to Duffin-Kemmer-Petiau Particles

We present a general derivation of the Duffin-Kemmer-Petiau (D.K.P) equation on the relativistic phase space proposed by Bohm and Hiley. We consider geometric algebras and the idea of algebraic spinors due to Riesz and Cartan. The generators ^(p) of the D.K.P algebras are constructed in the standard fashion used to construct Clifford algebras out of bilinear forms. Free D.K.P particles and D.K.P particles in a prescribed external electromagnetic field are analized and general Liouville type equations for these cases are obtained. Choosing particular values for the label p we classify the different types of the D.K.P Liouville operators.     

An Algebraic Approach to the Generalized Symmetrical Double-Well Potential

We have obtained the energy eigenvalues and the corresponding eigenfunctions for the generalized double-well potential in the non-relativistic Schr?dinger equation. We have calculated the creation and annihilation operators directly from the eigenfunction and we have shown these operators satisfy the commutation relation of the SU(2) group. We have expressed the Hamiltonian in terms of the su(2) algebra. Some interesting result including the standard symmetrical double-well potential, reflectionless-type potential and V ₀tanh ²(r/d) potential are also discussed.     

Simultaneous Determination of Force Constants and Dipole Moment Derivatives of Methane

The expressions of the effective Hamiltonian and dipole moment spectroscopic parameters in the tetrahedral formalism are used simultaneously to fit the force field and dipole moment derivatives of the methane molecule. Data, the so-called “observed parameters,” are the values of the spectroscopic parameters determined from the frequencies and line strengths analyses. The ambiguities of most parameters (in the polyad scheme) are treated consistently with the Hamiltonian reduction chosen in the frequency analyses. As an illustration, the method is applied to the tetrahedralXY₄isotopic species only. The quadratic and cubic force field constants have been determined in addition to the linear and six of the seven quadratic dipole moment derivatives. The observed parameters are reproduced with a standard deviation of 4%. The results are compared with previous works and it is moreover shown that the introduction of the dipole moment data removes some correlations between the force constants.     

Perturbational Approach to Force — Free Toroidal Equilibria

Solutions of the equation curl B = γ B inside a torus-shaped region with vanishing normal component on the torus surface are sought by the perturbation method. The perturbation parameter is proportional to the curvature of the torus, i.e zero-order solutions are exact helical solutions for a straight tube.     

The GVFT Force Constants for PO4 3-/Td/

The force constants for the tetrahedral PO₄ ^3- ion have been calculated on the basis of the general valence force field with the application of the Wilson FG matrix method^1,2.     

On Generalized Relativistic Billiards in External Force Fields

In this Letter, we study generalized relativistic billiards: as a particle reflects from the boundary of the domain, its velocity is transformed as if the particle underwent an elastic collision with a moving wall, considered within the framework of the special theory of relativity. Inside the domain, the particle moves under the influence of some gravitational and nongravitational force fields.We study both periodic and 'monotone' action of the boundary. We prove that under some general conditions the invariant manifold in the velocity phase space of the generalized billiard, where the point velocity equals the velocity of light, is an exponential attractor, and for an open set of initial conditions the particle energy tends to infinity.     

Numerical Convergence of the Block-Maxima Approach to the Generalized Extreme Value Distribution

In this paper we perform an analytical and numerical study of Extreme Value distributions in discrete dynamical systems. In this setting, recent works have shown how to get a statistics of extremes in agreement with the classical Extreme Value Theory. We pursue these investigations by giving analytical expressions of Extreme Value distribution parameters for maps that have an absolutely continuous invariant measure. We compare these analytical results with numerical experiments in which we study the convergence to limiting distributions using the so called block-maxima approach, pointing out in which cases we obtain robust estimation of parameters. In regular maps for which mixing properties do not hold, we show that the fitting procedure to the classical Extreme Value Distribution fails, as expected. However, we obtain an empirical distribution that can be explained starting from a different observable function for which Nicolis et al. (Phys. Rev. Lett. 97(21): 210602, 2006) have found analytical results.     

Force Constants and Mean Amplitudes of Vibration of the Thiazyl Halides Cisn and BrSN

The force constants have been calculated for thiazylnalides CISN and BrSN employing general valence force field and Urey-Dradley force field. The computed results show some interesting features. The mean amplitudes of vibration have also been calculated at temperatures, O°K, 298.15°K and 500°K. These results will be helpful for the interpretation of the electron diffraction data whenever available.