Rotation and vibration of diatomic molecule oscillator with hyperbolic potential function

The explicit expressions of energy eigenvalues and eigenfunctions of bound states for a three-dimensional diatomic molecule oscillator with a hyperbolic potential function are obtained approximately by means of the hypergeometric series method. Then for a one-dimensional system, the rigorous solutions of bound states are solved with a similar method. The eigenfunctions of a one-dimensional diatomic molecule oscillator, expressed in terms of the Jacobi polynomial, are employed as an orthonormal basis set, and the analytic expressions of matrix elements for position and momentum operators are given in a closed form.     

An improved approximation scheme for the centrifugal term and the Hulthén potential

We present a new approximation scheme for the centrifugal term to solve the Schrödinger equation with the Hulthén potential for any arbitrary l -state by means of a mathematical Nikiforov-Uvarov (NU) method. We obtain the bound-state energy eigenvalues and the normalized corresponding eigenfunctions expressed in terms of the Jacobi polynomials or hypergeometric functions for a particle exposed to this potential field. Our numerical results of the energy eigenvalues are found to be in high agreement with those results obtained by using the program based on a numerical integration procedure. The s -wave (l = 0analytic solution for the binding energies and eigenfunctions of a particle are also calculated. The physical meaning of the approximate analytical solution is discussed. The present approximation scheme is systematic and accurate.     

Approximate solutions of the Wei Hua oscillator using the Pekeris approximation and Nikiforov–Uvarov method

The approximate analytical bound-state solutions of the Schrödinger equation for the Wei Hua oscillator are carried out in N-dimensional space by taking Pekeris approximation scheme to the orbital centrifugal term. Solutions of the corresponding hyper-radial equation are obtained using the conventional Nikiforov–Uvarov (NU) method.     

The scattering states of the generalized Hulthén potential with an improved new approximate scheme to the centrifugal term

This paper finds the approximate analytical scattering state solutions of the arbitrary l-wave Schr?dinger equation for the generalized Hulthén potential by taking an improved new approximate scheme for the centrifugal term. The normalized analytical radial wave functions of the l-wave Schr?dinger equation for the generalized Hulthén potential are presented and the corresponding calculation formula of phase shifts is derived. Some useful figures are plotted to show the improved accuracy of the obtained results and two special cases for the standard Hulthén potential and Woods--Saxon potential are also studied briefly.     

Frequency-mixing scheme for the production of tunable narrowband XUV radiation (91–95 nm): application to precision spectroscopy and predissociation in diatomic molecules

Tunable narrowband extreme ultraviolet radiation in the range 91–95 nm is produced by sum-frequency mixing of the outputs of a visible pulsed dye amplifier (seeded by a ring dye laser) and of a seeded second-harmonic Nd:YAG laser and subsequent frequency tripling in a gas jet of xenon. The capability of this scheme to provide tunable narrowband extreme ultraviolet radiation is demonstrated in several spectroscopic studies. The bandwidth of this system (0.01 cm^-1) is deduced from a recording of absorption spectra of the 4p⁵(²P_1/2)6d, J=1 line in krypton. The applicability of the system for gas-phase molecular spectroscopic studies is demonstrated in recordings of the Werner bands (4,0) in H₂ and (5,0) in D₂ at unprecedented absolute accuracy. Line-broadening studies are performed on the b¹_u⁺,v=5 valence state in N₂, yielding a lifetime of 210±25 ps. A singlet–triplet perturbation, giving rise to an accidental predissociation in an excited ¹ Rydberg state in carbon monoxide at an excitation energy of 107680 cm^-1, is analyzed in high resolution. PACS 42.65.Ky; 32.80.Rm; 33.20.Ni; 33.80.Gj; 42.60.By     

The supersymmetric WKB approximation of ring-shaped oscillator potential in r and θ dimensions with spherical polar coordinates

According to calculation of the energy spectrum of ring-shape oscillator potential by using the supersymmetric WKB approximation, it is shown that the energy spectrum of some noncentral separable potentials can be exactly obtained in r and θ dimensions by above method.     

Experimental and theoretical study of the IR spectrum and barriers to internal rotation of 4-hydroxypropyl-4′-cyanobiphenyl

The IR absorption spectrum of polycrystalline 4-hydroxypropyl-4′-cyanobiphenyl is measured in a KBr pellet over the range 400–4000 cm^?1. The structure of the molecule and the frequencies and the intensities of the bands in the spectrum are calculated in the approximation of the B3LYP hybrid density functional with the 6-31 G(d) and 6-31+G(d) basis sets. The normal vibrations are reliably assigned by the method of linear scaling of frequencies, which made it possible to reproduce the experimental IR spectrum with high accuracy. The barriers to internal rotation of phenyl rings and three barriers for the hydroxypropyl radical are calculated.     

Free vibration analysis of a rotating hub–functionally graded material beam system with the dynamic stiffening effect

A comprehensive dynamic model of a rotating hub–functionally graded material (FGM) beam system is developed based on a rigid–flexible coupled dynamics theory to study its free vibration characteristics. The rigid–flexible coupled dynamic equations of the system are derived using the method of assumed modes and Lagrange's equations of the second kind. The dynamic stiffening effect of the rotating hub–FGM beam system is captured by a second-order coupling term that represents longitudinal shrinking of the beam caused by the transverse displacement. The natural frequencies and mode shapes of the system with the chordwise bending and stretching (B–S) coupling effect are calculated and compared with those with the coupling effect neglected. When the B–S coupling effect is included, interesting frequency veering and mode shift phenomena are observed. A two-mode model is introduced to accurately predict the most obvious frequency veering behavior between two adjacent modes associated with a chordwise bending and a stretching mode. The critical veering angular velocities of the FGM beam that are analytically determined from the two-mode model are in excellent agreement with those from the comprehensive dynamic model. The effects of material inhomogeneity and graded properties of FGM beams on their dynamic characteristics are investigated. The comprehensive dynamic model developed here can be used in graded material design of FGM beams for achieving specified dynamic characteristics.     

Symmetry and Fourier analysis of the ab initio-determined torsional variation of structural and Hessian-related quantities for application to vibration–torsion–rotation interactions in CH3OH

The aim of the present paper is to investigate the use of quantum chemistry calculations to obtain the torsional dependence of various structural and vibrational-force-field-related quantities that could help in estimating the vibration–torsion–rotation interaction terms needed to treat perturbations observed in the spectra of methanol-like molecules. We begin by using the Gaussian suite of programs to determine the steepest-descent path from a stationary point at the top of the internal rotation potential barrier in methanol to the equilibrium structure at the bottom of the barrier. This procedure requires determining the gradient ?V of the potential (as calculated in mass-weighted Cartesian coordinates) along the internal rotation path. In addition, we use the Gaussian suite to calculate the Hessian ??V along this path and to generate from these second derivatives the 3N ? 7 small-amplitude vibrational frequencies and the 3N Cartesian vibrational displacements for each of these vibrations. We then symmetrize the internal coordinates used in presenting the structures, gradients, Hessians and vibrational displacements along the path to take into account the periodic variation of the behavior of the three methyl hydrogen atoms H_i as they pass in turn through the C_s-plane of the HOC frame. The symmetrized linear combinations of the CH_i stretches, of the OCH_i bends, and of the HOCH_i dihedral angles of the methyl group depend on the internal rotation angle γ and they are determined by considering coordinate transformations from the G₆ permutation-inversion group appropriate for internally rotating methanol. This symmetrization procedure permits us to explore the feasibility of expressing the structures, gradients, Hessians, and vibrational displacement vectors along the internal rotation path as short Fourier series in γ, which is one of the main goals of this paper. In summary, we find that the symmetrized structures, gradients, and Hessians, as well as nine of the 11 projected vibrational frequencies and the vibrational displacement vectors for the three vibrations occurring primarily in the HOC frame can be expressed by short Fourier series expansions to their precision in the Gaussian output, and that these series involve only sin 3nγ or only cos 3nγ terms, as required by G₆ symmetry considerations. A preliminary discussion is given of why short Fourier expansions fail for the projected frequencies of the two methyl asymmetric stretches, and for the vibrational displacement vectors of the methyl group vibrational modes. Looking more closely at the symmetrized and projected 3N × 3N Hessian, we find algebraically that only elements in the (3N ? 7) × (3N ? 7) small-amplitude-vibrational block of the Hessian are useful for spectroscopic problems. Non-zero elements in the rest of the 3N × 3N symmetrized and projected Hessian cannot be converted into quantities needed for perturbation studies.     

Thermal buckling and natural vibration of the beam with an axial stick–slip–stop boundary

As a first attempt to study the dynamics of a heated structure with complicated boundaries, this paper deals with the thermal buckling and the natural vibration of a simply supported slender beam, which is subject to a uniformly distributed heating and has a frictional sliding end within a clearance. This sliding end is initially at a stick status under the friction force, but may be slightly slipping due to the thermal expansion of the beam until the sliding end contacts a stop, i.e., the bound of the clearance. The material properties of the beam are temperature-independent for low temperature, but temperature-dependent for high temperature. For each case, the analytic solutions for the critical buckling temperature and the natural frequencies of the heated beam are derived first. Then, discussions are made to reveal the effects of beam parameters, such as the ratio of beam length to beam thickness, the ratio of clearance to beam length and the temperature-dependent material properties, on the critical buckling temperature and the fundamental natural frequency of the heated beam. The study shows that both friction force and clearance have significant influences on the critical buckling temperature and the fundamental natural frequency of the beam. When the friction force is not very large, the clearance can greatly increase the critical buckling temperature. These conclusions enable one to properly design the stick–slip–stop boundary so as to improve the mechanical performance of the beam in thermal environments.     

The Chern–Simons term induced at high temperature and the quantization of its coefficient

By perturbative calculations of the high-temperature ground-state axial vector current of fermion fields coupled to gauge fields, an anomalous Chern–Simons topological mass term is induced in the three-dimensional effective action. The anomaly in three dimensions appears just in the ground-state current rather than in the divergence of ground-state current. In the Abelian case, the contribution comes only from the vacuum polarization graph, whereas in the non-Abelian case, contributions come from the vacuum polarization graph and the two triangle graphs. The relation between the quantization of the Chern–Simons coefficient and the Dirac quantization condition of magnetic charge is also obtained. It implies that in a (2+1)-dimensional QED with the Chern–Simons topological mass term and a magnetic monopole with magnetic charge g present, the Chern–Simons coefficient must be also quantized, just as in the non-Abelian case. Received: 7 April 1999 / Published online: 3 November 1999     

The high-resolution infrared spectrum of Si2H6: rotation–torsion analysis of the ν5 and ν7 fundamentals,and torsional splittings in the degenerate vibrational states

The infrared active ν₇ and ν₅ fundamentals of disilane, coupled by an x,y Coriolis interaction, have been analysed on a Fourier transform spectrum between 2120 and 2225?cm^?1, at the Doppler limited spectral resolution. A Fermi resonance with 2ν ₂?+?ν₉ affects the Δ K?=?1 side of ν₇, and both ν₇ and ν₅ show the effects of several additional localized perturbations. Line splittings in the ν₅ transitions are not observed, showing that the torsional splitting in the ν₅ excited state and in the vibrational ground state are almost equal. The intrinsic torsional splitting of ν₇ is found to be smaller than in the ground vibrational state by 0.0085?cm^?1. This splitting value and those found for the other two infrared active degenerate fundamentals, ν₈ and ν₉, follow the trend expected from our theoretical predictions. Exploratory numerical calculations show that the decrease of the torsional splittings, observed in the fundamental degenerate vibrational states of disilane, can actually be accounted for by the head–tail and torsional Coriolis coupling of all the degenerate vibrational fundamentals, in several torsional states.     

Laser pumping and magneto-optical rotation of the light polarization plane in a cell with an antirelaxation coating of the walls: II. Calculation for atoms with the Λ scheme of sublevels

The general solution of the Boltzmann kinetic equations obtained previously by the author for the atomic density matrix in a long cylindrical cell with an antirelaxation coating of the walls under conditions of laser pumping in a magnetic field is used for the calculation of the magneto-optical rotation of the light polarization plane. It is shown that, in accordance with experimental data, the dependence of the rotation angle of the polarization plane of the pumping light on a magnetic field ranging from ?1 to 1 Oe represents a sum of two antisymmetric curves: a narrow curve with a spacing between extrema of about 2 μOe and a broad curve with a spacing of about 0.03 Oe. It is found that the characteristics (the width and the amplitude) of these two curves depend quite differently on the parameters of the problem—the pumping power, the coating Q factor, the pumping beam radius, and the cell radius. This difference is explained by the fact that the two curves are associated with different physical processes: the narrow curve characterizes the distortion caused by the magnetic field in the atomic polarization accumulated during multiple passages through the pumping beam after almost nondepolarizing collisions with the coating, while the broad curve corresponds to the distortion of the polarization formed during a single passage through the beam.     

Arbitrary <Emphasis Type="Italic">l</Emphasis>-state solutions of the Schrödinger equation for the Eckart potential with an improved approximation of the centrifugal term

Applying an improved approximation scheme to the centrifugal term, the approximate analytical solutions of the Schrödinger equation for the Eckart potential are presented. Bound state energy eigenvalues and the corresponding eigenfunctions are obtained in closed forms for the arbitrary radial and angular momentum quantum numbers, and different values of the screening parameter. The results are compared with those obtained by the other approximate and numerical methods. It is shown that the present method is systematic, more efficient and accurate.     

The effect of orientation and distance between donor and acceptor molecules on the efficiency of singlet–singlet energy transfer in Langmuir–Blodgett films

Singlet–singlet energy transfer between molecules of fluorescein and oxazine dyes in Langmuir–Blodgett films is studied experimentally. The dependence of the energy-transfer efficiency on the distance shows that the quenching of the donor fluorescence is the most efficient when the layers of the donor and acceptor molecules are in a direct contact. An increase in the distance between the donor and acceptor layers leads to a decrease in the energy-transfer efficiency. To establish the mutual orientation of the donor and acceptor molecules, quantum-chemical calculations of the energy transfer process in the donor–acceptor pair are carried out. The calculations show that the best correlation of the experimental and calculated values of the energy-transfer efficiency is observed when the interacting particles are shifted relative to each other by about ~0.12 nm in parallel planes. The presented approach can be used to estimate the relative orientation of interacting particles in multimolecular ensembles.     

Computation of vector coupling coefficients for atoms with one and two open shells in the Hartree–Fock approximation

A new way to introduce and calculate vector coupling coefficients (VCCs) is proposed for computations in a Hartree–Fock approximation of atoms with one and two open shells. In the case of atoms with one open shell (Roothaan method), the obtained formulas for VCCs are applicable to the calculation of all terms of such atoms without exception. The VCCs are also calculated for a number of configurations of atoms with two open shells (Huzinaga method). The correctness of the obtained VCCs is confirmed by numerous calculations of energies and dipole polarizabilities for atoms with one and two open shells.     

Measurement of the total spectrum of electrons and positrons in the energy range of 300–1500 GeV in the PAMELA experiment with the aid of a sampling calorimeter and a neutron detector

A method based on the use of a sampling calorimeter was developed for measuring the total energy spectrum of electrons and positrons from high-energy cosmic rays in the PAMELA satellite-borne experiment. This made it possible to extend the range of energies accessible to measurements by the magnetic system of the PAMELA spectrometer. Themethod involves a procedure for selecting electrons on the basis of features of a secondary-particle shower in the calorimeter. The results obtained by measuring the total spectrum of cosmic-ray electrons and positrons in the energy range of 300–1500 GeV by the method in question are presented on the basis of data accumulated over a period spanning 2006 and 2013.