Calculation of the multimode Franck–Condon factors based on the coherent state method

The equivalence of two ways for the calculation of overlap integrals, i.e. the Sharp–Rosenstock generating function method and the Doktorov coherent state method, has been proved. On the basis of the generating function of the overlap integrals, a new closed form expression for the Franck–Condon integrals for overlap multidimensional harmonic oscillators has been exactly derived. In addition, some useful analytical expressions for the calculations of the multimode Franck–Condon factors have been given.     

Computationally efficient recurrence relations for one-dimensional Franck–Condon overlap integrals

Calculations based on analytical expressions for the harmonic oscillator Franck–Condon factors often yield numerically unstable and erroneous results for large values of the oscillator quantum numbers. This instability arises from inherent machine precision limits and large number round-off associated with the products and ratios of factorial and gamma functions in these expressions; the analytical expressions themselves are exact. This paper presents, first, efficient, exact recurrence relations to evaluate Franck–Condon factors for the harmonic oscillator model. The recurrence relations, which are similar to those originally found by Manneback, Wagner and Ansbacher avoid the direct use of the factorial and gamma functions. Second, a variational strategy for the evaluation of Franck–Condon factors for the Morse oscillator is proposed. The Schrödinger equation for the Morse model is solved variationally with a large enough basis set of one-dimensional harmonic oscillator functions to get good agreement with the analytic eigenvalues of the Morse potential itself. The eigenvectors of this analysis are then used together with the associated harmonic oscillator Franck–Condon overlap matrix elements to evaluate the overlap for the Morse potential. This approach allows one, in principle, to estimate Franck–Condon overlap up to states near to the dissociation limit of the Morse oscillator.     

Transform method in resonance Raman scattering with quadratic Franck–Condon and Herzberg–Teller interactions

A new method of considering quadratic vibronic interactions and deviations from the Condon approximation in resonance Raman scattering (RRS) is proposed. The method is used for the generalization of the transform law between absorption and Raman excitation profiles, derived earlier for a basic model. In the case of arbitrary mixing of modes with similar frequencies, a simple generalization of the transform law for first-order RRS is obtained. For an arbitrary change of frequencies without mode mixing the transform law for the nth-order RRS is given. The equations are valid for an arbitrary number of modes and also take temperature effects into account.     

A Method for Calculating the Franck–Condon Factors in the Tomographic Probability Representation of Quantum Mechanics

We introduce a new approach to calculate the Franck–Condon factors in polyatomic molecules based on the tomographic probability representation of quantum mechanics. This approach is implemented to calculate the transition probabilities in various systems under an instantaneous change of frequency and equilibrium position of nuclei in a molecule by an external force. The problem is considered within the framework of the linear approximation for different types of the Dushinsky matrix and for any quantity of atoms in a molecule.     

Robust,multidimensional mesh-motion based on Monge–Kantorovich equidistribution

Mesh-motion (r-refinement) grid adaptivity schemes are attractive due to their potential to minimize the numerical error for a prescribed number of degrees of freedom. However, a key roadblock to a widespread deployment of this class of techniques has been the formulation of robust, reliable mesh-motion governing principles, which (1) guarantee a solution in multiple dimensions (2D and 3D), (2) avoid grid tangling (or folding of the mesh, whereby edges of a grid cell cross somewhere in the domain), and (3) can be solved effectively and efficiently. In this study, we formulate such a mesh-motion governing principle, based on volume equidistribution via Monge–Kantorovich optimization (MK). In earlier publications [1], [2], the advantages of this approach with regard to these points have been demonstrated for the time-independent case. In this study, we demonstrate that Monge–Kantorovich equidistribution can in fact be used effectively in a time-stepping context, and delivers an elegant solution to the otherwise pervasive problem of grid tangling in mesh-motion approaches, without resorting to ad hoc time-dependent terms (as in moving-mesh PDEs, or MMPDEs [3], [4]). We explore two distinct r-refinement implementations of MK: the direct method, where the current mesh relates to an initial, unchanging mesh, and the sequential method, where the current mesh is related to the previous one in time. We demonstrate that the direct approach is superior with regard to mesh distortion and robustness. The properties of the approach are illustrated with a hyperbolic PDE, the advection of a passive scalar, in 2D and 3D. Velocity flow fields with and without flow shear are considered. Three-dimensional grid, time-step, and nonlinear tolerance convergence studies are presented which demonstrate the optimality of the approach.     

Comparison of methods for calculating Franck–Condon factors beyond the harmonic approximation: how important are Duschinsky rotations?

Three different approaches for calculating Franck–Condon factors beyond the harmonic approximation are compared and discussed in detail. Duschinsky effects are accounted for either by a rotation of the initial or final wavefunctions – which are obtained from state-specific configuration-selective vibrational configuration interaction calculations – or by a rotation of the underlying multi-dimensional potential energy surfaces being determined from explicitly correlated coupled-cluster approaches. An analysis of the Duschinsky effects in dependence on the rotational angles and the anisotropy of the wavefunction is provided. Benchmark calculations for the photoelectron spectra of ClO₂, HS^?₂ and ZnOH^? are presented. An application of the favoured approach for calculating Franck–Condon factors to the oxidation of Zn(H₂O)⁺ and Zn₂(H₂O)⁺ demonstrates its applicability to systems with more than three atoms.     

Sound quality evaluation of vehicle suspension shock absorber rattling noise based on the Wigner–Ville distribution

In this paper, a new sound metric–Sound Metric based on the Wigner–Ville distribution (SMWVD) – was developed to investigate the relationship between subjective evaluations and vehicle suspension shock absorber rattling noise, which substantially affects passengers’ psychological and physiological perceptions. A complete vehicle road test was conducted to measure the interior shock absorber noises, which were subjectively evaluated by 20 jurors. Conventional psychoacoustic indices, i.e. loudness, sharpness, roughness and fluctuation strength, were used to calculate the correlation coefficients between the objective and subjective evaluations, and then, the results were compared with the performance of the SMWVD. The results show that conventional sound metrics have poor relationships with the subjective ratings, while the SMWVD displayed a high correlation of >0.9 between the objective evaluation and the subjective evaluation. These results indicate that the SMWVD can be used to estimate the rattling noise index of a suspension shock absorber without jury evaluation.     

Franck–Condon Factors and r‐Centroids for the B–, C–, F–, and G–X Band Systems of YF Molecule

The Franck–Condon factors and r‐centroids, which are very closely related to relative transition probabilities, have been evaluated by a more reliable numerical integration procedure for the B¹π–X¹Σ⁺, C¹Σ⁺–X¹Σ⁺, F¹Σ⁺–X¹Σ⁺, and G¹π–X¹Σ⁺ band systems of the YF molecule, using suitable potentials.     

Immersed boundary method for the simulation of 2D viscous flow based on vorticity–velocity formulations

A new immersed boundary method based on vorticity–velocity formulations for the simulation of 2D incompressible viscous flow is proposed in present paper. The velocity and vorticity are respectively divided into two parts: one is the velocity and vorticity without the influence of the immersed boundary, and the other is the corrected velocity and the corrected vorticity derived from the influence of the immersed boundary. The corrected velocity is obtained from the multi-direct forcing to ensure the well satisfaction of the no-slip boundary condition at the immersed boundary. The corrected vorticity is derived from the vorticity transport equation. The third-order Runge–Kutta for time stepping, the fourth-order finite difference scheme for spatial derivatives and the fourth-order discretized Poisson for solving velocity are applied in present flow solver. Three cases including decaying vortices, flow past a stationary circular cylinder and an in-line oscillating cylinder in a fluid at rest are conducted to validate the method proposed in this paper. And the results of the simulations show good agreements with previous numerical and experimental results. This indicates the validity and the accuracy of present immersed boundary method based on vorticity–velocity formulations.     

Exact solutions of Dyson–Schwinger equations for iterated one-loop integrals and propagator-coupling duality

The Hopf algebra of undecorated rooted trees has tamed the combinatorics of perturbative contributions, to anomalous dimensions in Yukawa theory and scalar φ³ theory, from all nestings and chainings of a primitive self-energy subdivergence. Here we formulate the nonperturbative problems which these resummations approximate. For Yukawa theory, at spacetime dimension d=4, we obtain an integrodifferential Dyson–Schwinger equation and solve it parametrically in terms of the complementary error function. For the scalar theory, at d=6, the nonperturbative problem is more severe; we transform it to a nonlinear fourth-order differential equation. After intensive use of symbolic computation we find an algorithm that extends both perturbation series to 500 loops in 7 minutes. Finally, we establish the propagator–coupling duality underlying these achievements making use of the Hopf structure of Feynman diagrams.     

Analytical solution of the Korteweg–de Vries equation and microtubule equation using the first integral method

In this paper, the first integral method is applied to solve the Korteweg–de Vries equation with dual power law nonlinearity and equation of microtubule as nonlinear RLC transmission line. This method is manageable, straightforward and a powerful tool to find the exact solutions of nonlinear partial differential equations.     

Boundary Effects on Exact Solutions of the Lagrangian-Averaged Navier–Stokes-α Equations

In order to clarify the behavior of solutions of the Lagrangian-averaged Navier–Stokes- (LANS-) equations in the presence of solid walls, we identify a variety of exact solutions of the full equations and their boundary layer approximations. The solutions demonstrate that boundary conditions suggested for the LANS- equations in the literature⁽¹⁾ for a bounded domain do not apply in a semi-infinite domain. The convergence to solutions of the Navier–Stokes equations as 0 is elucidated for infinite-energy solutions in a semi-infinite domain, and non-uniqueness of these solutions is discussed. We also study the boundary layer approximation of LANS- equations, denoted the Prandtl- equations, and report solutions for turbulent jets and wakes. Our version of the Prandtl- equations includes an extra term necessary to conserve linear momentum and corrects an earlier result of Cheskidov.⁽²⁾     

A numerical method for solving the Vlasov–Poisson equation based on the conservative IDO scheme

We have applied the conservative form of the Interpolated Differential Operator (IDO-CF) scheme in order to solve the Vlasov–Poisson equation, which is one of the multi-moment schemes. Through numerical tests of the nonlinear Landau damping and two-stream instability, we compared the present scheme with other schemes such as the Spline and CIP ones. We mainly investigated the conservation property of the L1-norm, energy, entropy and phase space area for each scheme, and demonstrated that the IDO-CF scheme is capable of performing stable long time scale simulation while maintaining high accuracy. The scheme is based on an Eulerian approach, and it can thus be directly used for Fokker–Planck, high dimensional Vlasov–Poisson and also guiding-center drift simulations, aiming at particular problems of plasma physics. The benchmark tests for such simulations have shown that the IDO-CF scheme is superior in keeping the conservation properties without causing serious phase error.     

Analysis on accuracy improvement of rotor–stator rubbing localization based on acoustic emission beamforming method

This paper attempts to introduce an improved acoustic emission (AE) beamforming method to localize rotor–stator rubbing fault in rotating machinery. To investigate the propagation characteristics of acoustic emission signals in casing shell plate of rotating machinery, the plate wave theory is used in a thin plate. A simulation is conducted and its result shows the localization accuracy of beamforming depends on multi-mode, dispersion, velocity and array dimension. In order to reduce the effect of propagation characteristics on the source localization, an AE signal pre-process method is introduced by combining plate wave theory and wavelet packet transform. And the revised localization velocity to reduce effect of array size is presented. The accuracy of rubbing localization based on beamforming and the improved method of present paper are compared by the rubbing test carried on a test table of rotating machinery. The results indicate that the improved method can localize rub fault effectively.     

On the resolution of singularities of multiple Mellin–Barnes integrals

One of the two existing strategies of resolving singularities of multifold Mellin–Barnes integrals in the dimensional regularization parameter, or a parameter of the analytic regularization, is formulated in a modified form. The corresponding algorithm is implemented as a Mathematica code MBresolve.m     

Modeling of superconductors based on the timedependent Ginsburg–Landau equations

Results of modeling of superconductor magnetization process based on a numerical solution of the timedependent Ginsburg-Landau equations are presented. Methods of grid approximation of the equations and method of finite elements are used. Two-dimensional patterns of changes in the order parameter and supercurrent distribution in superconductors are calculated and visualized. The main results are in agreement with the well-known representations for type I and II superconductors.