The Hellmann-Feynman theorem applied to long-range forces

The relations between the Hellmann-Feynman forces in laboratory fixed (L-) and relative (R-) coordinate systems are clarified. In the usualL-coordinate system, the force is interpreted as force on nucleus, while in theR-coordinate system, it means force on whole particles consisting of the electrons and nuclei of each interacting subsystem. From a perturbation theoretical viewpoint, the concept of the force on whole particles correctly corresponds to the perturbation energy and is superior to the force on the nucleus.     

Origin and meaning of the Fermi contact interaction

The Fermi-contact interaction (FCI) can easily be derived from 1st order perturbation theory applied to the non-relativistic wave equation for a spin-(1/2) particle of Lévy-Leblond, with the nuclear spin described by the field of an external magnetic dipole, and it results from the fact that the turn-over-rule for the operator is only valid if the derivatives implicit in are taken in the distribution sense. If one avoids to apply the turn-over-rule, the FCI is obtained without the need to introduce a -function. It is also shown that the formulation of a magnetic point dipole as the limit of an extended nucleus directly leads to the FCI. Traditional methods of the derivation of the FCI are analyzed in the light of this new interpretation. It is then explained why the perturbation expansions in powers of the magnetic moment of the nucleus necessarily diverges, but that the expression for the 1st order energy on which the concept of the FCI is based, can nevertheless be justified by means of the Hellmann-Feynman theorem with a correction term if singular wave functions are involved. Finally some comments on a theory beyond first order are made.Dedicated to Professor J. Koutecký on the occasion of his 65th birthday     

Multireference perturbation CI I. Extrapolation procedures with CAS or selected zero-order spaces

We discuss the “three class” approximation to full multireference perturbation CI, which greatly reduces the computational effort by restricting the summation of diagrams to determinants belonging to a subspace of the zero-order space. In the framework of the CIPSI algorithm, we propose a new extrapolation procedure allowing recovery of the full “two class” results. The new procedure is applied to complete active spaces (CAS) and to individually selected zero-order spaces. Comparison with a full two class calculation on a CAS shows a reduction of computer time of one or two orders of magnitude in the tests presented here, with an accuracy in the order of 0.1 kcal/mol. Our procedure can thus compete with the CASPT2 algorithm, specifically conceived to deal with CAS. In the case of selected zero-order spaces, the speed-up is less dramatic but the method still retains its advantages. Received: 12 June 1997 / Accepted: 31 July 1997     

A study of a nonprimitive model for electrolyte solutions based on perturbation theory

The equation for the Helmholtz free energy for systems of small anisotropic molecules and ions is deduced by substituting the complete expression for various potential energies (including repulsive, dispersive, electrostatic, and induced energies) into the perturbation expansion. The equation is applied to pure water. The relative dielectric constant is set at unity. Based on the equal chemical potentials of equilibrated vapor and liquid phases, the molecular parameters of water are regressed from the densities of saturated vapor in the temperature range of 0 to 370°C. The ARD of regression is 1.16%. These parameters are used to predict the heat of vaporization and densities of saturated vapor and liquid phases of water in the same temperature range. The ARDs of prediction are 4.5% and 9.8%, respectively. The equation is used to correlate the osmotic coefficients of twelve 1:1 electrolyte solutions. The relative dielectric constant is set at unity. The parameters (Soft-sphere diameter and dispersive constant) of seven ions (Na⁺, K⁺, Rb⁺, Cs⁺, Cl^–, Br^–, and I^–) are obtained. The total average absolute deviation between calculated and experimental values of the osmotic coefficient is 0.041. The parameters of ions can keep constant in different systems.     

A theoretical determination of the electronic spectrum of formaldehyde

Summary The electronically excited states of formaldehyde are examined by means of multiconfigurational second-order perturbation (CASPT2) theory with extended ANO-type basis sets. The calculations comprised five valence excited states plus all singlet 3s, 3p, and 3d members of the Rydberg series converging on the first ionization. The computed vertical excitation energies were found to be within 0.2 eV of the available experimental energies. Full geometry optimization has been performed for five valence excited states. Assuming a planar geometry, the 0-0 transition for the valence¹A₁( *) state is calculated to appear near 7.9 eV, close to the (n _y 3p) region. This state is, however, not planar and the true adiabatic energy is 7.4 eV, which is 2.3 eV below the corresponding vertical transition.     

Distorted wave response of ultrasonic annular stator incorporating non-uniform geometry

The traveling wave ultrasonic stator is normally fabricated with teeth. The tooth geometry improves the driving speed, but it creates natural frequency splitting and mode contamination, especially a distorted traveling wave. A dynamic model of a stepped-plate periodic stator is developed to examine the distortion. The stator is treated as an annular supported by a thin mid plate, and the support stiffness is formulated by using equivalent energy principle. The effects of the tooth and mid plate on the natural frequency and vibration mode are examined by using the perturbation method. The rules governing the frequency splitting, frequency perturbation as well as mode contamination are also identified. The traveling wave response and elliptical trace on stator surface are obtained by using the mode superposition method and they are proved to be distorted due to the tooth geometry. The response at the repeated doublets becomes coupled forward and backward traveling waves, but that at the split doublets becomes coupled forward traveling, standing and backward traveling waves. The results indicate that the tooth mass instead of the stiffness decreases the vibration amplitude and driving speed of the dominant wave, but their effects are different at the repeated and split doublets. Inspection of the model implies that the distortion can be suppressed by using a suitable combination of the wavenumber, tooth count, tooth height and occupying fraction. Numerical calculations are carried out to demonstrate the tooth geometry effect on the transient waveform, driving speed and elliptical trace. The optimization of the tooth geometry that can help achieve a purer traveling wave is discussed.     

Equivalent nonlinear beam model for the 3-D analysis of shear-type buildings: Application to aeroelastic instability

Tower buildings can be very sensitive to dynamic actions and their dynamic analysis is usually carried out numerically through sophisticated finite element models. In this paper, an equivalent nonlinear one-dimensional shear–shear–torsional beam model immersed in a three-dimensional space is introduced to reproduce, in an approximate way, the dynamic behavior of tower buildings. It represents an extension of a linear beam model recently introduced by the authors, accounting for nonlinearities generated by the stretching of the columns. The constitutive law of the beam is identified from a discrete model of a 3D-frame, via a homogenization process, which accounts for the rotation of the floors around the tower axis. The macroscopic shear strain in the equivalent beam is produced by the bending of columns, accompanied by negligible rotation of the floors. A coupled nonlinear shear–torsional mechanical model is thus obtained. The coupling between shear and torsion is related to a non-symmetric layout of the columns, while mechanical nonlinearities are proportional to the slenderness of the columns. The model can be used for the analysis of the response of tower buildings to any kind of dynamic and static excitation. A first application is here presented to investigate the effect of mechanical and aerodynamic coupling on the critical galloping conditions and on the postcritical behavior of tower buildings, based on a quasi-steady model of aerodynamic forces.     

Group inverse and core inverse in Banach and C*-algebras

Abstract

In this paper, we have focused our study on the acute perturbation of the group inverse for the elements of Banach algebra with respect to the spectral radius. We also give perturbation analysis for the core inverse in C*- algebra. The perturbation bounds for the core inverse under some conditions are presented. Additionally, this paper extends the results obtained in [11, 14].     

The string of variable density: Perturbative and non-perturbative results

We obtain systematic approximations for the modes of vibration of a string of variable density, which is held fixed at its ends. These approximations are obtained iteratively applying three theorems which are proved in the paper and which hold regardless of the inhomogeneity of the string. Working on specific examples we obtain very accurate approximations which are compared both with the results of WKB method and with the numerical results obtained with a collocation approach. Finally, we show that the asymptotic behaviour of the energies of the string obtained with perturbation theory, worked to second order in the inhomogeneities, agrees with that obtained with the WKB method and implies a different functional dependence on the density that in two and higher dimensions.     

Equation of state of zero-temperature quark matter

We outline the key elements of a recent calculation aimed at determining the equation of state of deconfined (but unpaired) quark matter at zero temperature and high density, using finite quark masses. The computation is performed in perturbation theory up to three loops, and necessitates the development and application of some novel computational tools. In this talk, we introduce the basic features of these new techniques and review the main sources of motivation for considering finite mass effects in perturbation theory.