Model studies on the systematics and interpolation of Franck-Condon factor arrays

A simple harmonic oscillator approximation is used to derive approximate algebraic expressions for low vibrational quantum number Franck-Condon factors for band systems for which molecular data are sparse. These simple expressions involve a transition parameter u which is characteristic of the electronic transition involved. The expressions provide Franck-Condon factors which agree quite well with RKR values when these are available. The formulae are used to explain quantitatively the dependence on u of the systematic properties of Franck-Condon factor arrays and to provide a method for their interpolation. The formulae are also used to describe quantitatively the dependence on u of the gross appearance of molecular band systems in emission and absorption.     

A New Expression for Multidimensional Franck-Condon Integrals

A new closed form expression for the Franck-Condon integrals for overlap between arbitrary multidimensional harmonic oscillators has been exactly derived by employing the generating functions method. The calculations lead to the deduction of some general rules whereby an arbitrary multidimensional Franck-Condon integral can be expressed as sums of products of the Hermite polynomials. Copyright 1999 Academic Press.     

Overlaps of deformed and non-deformed harmonic oscillator basis states

A systematic approach for expanding non-deformed harmonic oscillator basis states in terms of deformed ones, and vice versa, is presented. The objective is to provide analytical results for calculating these overlaps (transformation brackets) between deformed and non-deformed basis states in spherical, cylindrical, and Cartesian coordinates. These overlaps can be used for reducing the complexity of different research problems that employ three-dimensional harmonic oscillator basis states, for example as used in coherent state theory and the nuclear shell-model, especially within the context of ab initio symmetry-adapted no-core shell model.     

Harmonic oscillator fiducials for Hermite-Gaussian laser beams

This paper exploits the identical functional forms of quantum mechanical harmonic oscillators and Hermite-Gaussian laser beams to investigate the probability densities of large-order modes. The classical limits of a two-dimensional harmonic oscillator provide corresponding integration limits for the photon probability densities of the laser-beam modes to determine the fraction of photons detected therein. The probabilities of detecting photons within the ‘classical limits’ of Hermite-Gaussian laser beams exhibit a power-law dependence in the limit of large-order modes and asymptotically approach unity, in agreement with the Correspondence Principle. The classical limits for large-order modes are shown to include all of the nodes and peaks for Hermite-Gaussian laser beams; Sturm's theorem provides a direct proof that is further illustrated by examination of the concavity and convexity of Hermite functions.     

Analytical relationship for the cranking inertia

The wave functions of a spheroidal harmonic oscillator without spin-orbit interaction are expressed in terms of associated Laguerre and Hermite polynomials. The pairing gap and Fermi energy are found by solving the BCS system of two equations. Analytical relationships for the matrix elements of inertia are obtained as a function of the main quantum numbers and potential derivative. They may be used to test complex computer codes developed in a realistic approach of the fission dynamics. Results given for the ²⁴⁰Pu nucleus are compared with a hydrodynamical model. The importance of taking into account the correction term due to the variation of the occupation number is stressed.     

Frank–Condon principle and adjustment of optical waveguides with nonhomogeneous refractive indices

The adjustment of two different selfocs is considered using both exact formulas for the mode-connection coefficients expressed in terms of Hermite polynomials of several variables and a qualitative approach based on the Frank–Condon principle. Several examples of the refractive-index dependence are studied and illustrative plots for these examples are presented. The connection with the tomographic approach to quantum states of a two-dimensional oscillator and the Frank–Condon factors is established.     

变形谐振子势的能谱方程

讨论了3种变形谐振子势:左右两边不同参数的谐振子势、左边方形势加右边谐振子势和谐振子势中间加δ势中的能量本征态函数.这些函数都可以由厄米函数表示.由波函数及其一次导数在原点的衔接条件,得到了能谱方程.     

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.     

Exact Solutions of Schrödinger Equation for the Position-Dependent Effective Mass Harmonic Oscillator

A one-dimensional harmonic oscillator with position-dependent effective mass is studied. We quantize the oscillator to obtain a quantum Hamiltonian, which is manifestly Hermitian in configuration space, and the exact solutions to the corresponding Schrödinger equation are obtained analytically in terms of modified Hermite polynomials. It is shown that the obtained solutions reduce to those of simple harmonic oscillator as the position dependence of the mass vanishes.     

An alternative factorization of the quantum harmonic oscillator and two-parameter family of self-adjoint operators

We introduce an alternative factorization of the Hamiltonian of the quantum harmonic oscillator which leads to a two-parameter self-adjoint operator from which the standard harmonic oscillator, the one-parameter oscillators introduced by Mielnik, and the Hermite operator are obtained in certain limits of the parameters. In addition, a single Bernoulli-type parameter factorization, which is different from the one introduced by M.A. Reyes, H.C. Rosu, and M.R. Gutiérrez [Phys. Lett. A 375 (2011) 2145], is briefly discussed in the final part of this work.     

谐振子薛定谔方程的超对称解法

通过构造哈密顿量与谐振子系统哈密顿量对易的超对称系统,量子谐振子的性质就可以通过对超对称系统的研究来得到.利用超对称系统的性质,在没有用到厄米多项式的情况下,给出了谐振子本征函数中展开系数间的递推关系,由递推关系可以直接得到本征函数.此方法下得到的归一化本征函数与用厄米多项式表达的本征函数完全相同,并且本征函数的宇称可以明显的显示出来.     

On the Integrability of q-Oscillators Based on Invariants of Discrete Fourier Transforms

A discretization of the quantum mechanical phase space is presented in the context of q-noncommutative structures. We give two generalizations of the Heisenberg algebra in the arising lattice phase space. In contrast to ordinary quantum mechanics, there is, a priori, no systematic approach to an integrable oscillator Hamiltonian in lattice quantum mechanics. This is the central obstacle to deal with in this Letter. To do so, we show how in general the integrability of the harmonic oscillator is related to the Fourier transform between momentum and space variables. This will be done in both cases, the continuous and the discrete one. As an application, we finally obtain an integrable lattice Hamiltonian for the harmonic oscillator with generalized Hermite eigenfunctions.     

S2O‾自由基光电子能谱的Franck-Condon分析

本文考虑多振动模混合和热带效应,凭借谐振子模型,推得计算两维-四振动模Franck-Condon重叠积分的解析表示,且应用于S2O‾ 自由基光电子能谱的理论研究。对于S2O( 1A′) – S2O‾( 2A″) 光脱附过程,结合分子轨道从头算和密度泛函理论,计算Franck-Condon因子,从而得到电子跃迁振动谱线的相对强度,理论上得到的光电子能谱与实验上观测到的能谱达到较好的一致;进一步在光谱模拟过程中,拟合实验能谱得到可靠的负离子自由基S2O‾电子态（ 2A″）的几何结构参数：键长R(SS) = 2.008 +/-0.005Å 和 R(SO) = 1.519+/-0.005Å.     

A new formula to calculate Franck-Condon factors for displaced and distorted harmonic oscillators

A new formula has been derived to calculate Franck-Condon factors for two harmonic oscillators with different equilibrium positions and different vibrational frequencies. The derivation is simple and straightforward, and a new expression for calculating the matrix element 〈v|xⁿ|v′〉 has also been derived.     

Synchronization of phase oscillators with heterogeneous coupling: A solvable case

We consider an extension of Kuramoto’s model of coupled phase oscillators where oscillator pairs interact with different strengths. When the coupling coefficient of each pair can be separated into two different factors, each one associated to an oscillator, Kuramoto’s theory for the transition to synchronization can be explicitly generalized, and the effects of coupling heterogeneity on synchronized states can be analytically studied. The two factors are respectively interpreted as the weight of the contribution of each oscillator to the mean field, and the coupling of each oscillator to that field. We explicitly analyze the effects of correlations between those weights and couplings, and show that synchronization can be completely inhibited when they are strongly anti-correlated. Numerical results validate the theory, but suggest that finite-size effect are relevant to the collective dynamics close to the synchronization transition, where oscillators become entrained in synchronized frequency clusters.     

Franck-Condon integral for bound-free transitions between harmonic and flat potentials using exact wavefunctions

The spectral features of electronic transitions between a bound upper state and a relatively flat lower state may be modeled using harmonic oscillator and particle-in-a-box wavefunctions. Evaluation of the Franck-Condon overlap integral yields an expression for the spectral intensity as a function of transition energy which displays the characteristic broad and narrow fluctuations. The very good agreement between these results and those obtained from a semiclassical WKB treatment supports the applicability of the WKB method to problems involving realistic potentials.     

An approach to polyatomic Franck-Condon integrals: Application to the photoelectron spectrum of water

An analytical method, based on harmonic oscillator expansions for each normal coordinate and the neglect of the Duschinsky A matrix, is presented for the prediction of Franck-Condon factors in polyatomic molecules from ab initio electronic wavefunctions. The method is tested against earlier theoretical results for H₂ and LiH and then applied to predict the vibrational structure for the first band of the photoelectron spectrum of water.     

Predissociation and the vibrational spectrum of molecular oxygen in an intense laser field. A Schumann-Runge band interval

The probabilities of predissociation and vibronic transitions between the states of the oxygen molecule in the Schumann-Runge band in the presence of a strong laser field are examined. The interaction of the molecule with the laser field is described using the rotating wave approximation. The predissociation probabilities for the avoided crossing of two adiabatic molecular terms are calculated within the framework of the Landau-Zener model. The energies of the vibrational states in the laser field are determined by diagonalization of the adiabatic Hamiltonian in the harmonic oscillator basis set. The predissociation thresholds are determined and the Franck-Condon factors are calculated as functions of the frequency and intensity of the external electromagnetic field.     

Quantum damped oscillator I: Dissipation and resonances

Quantization of a damped harmonic oscillator leads to so called Bateman’s dual system. The corresponding Bateman’s Hamiltonian, being a self-adjoint operator, displays the discrete family of complex eigenvalues. We show that they correspond to the poles of energy eigenvectors and the corresponding resolvent operator when continued to the complex energy plane. Therefore, the corresponding generalized eigenvectors may be interpreted as resonant states which are responsible for the irreversible quantum dynamics of a damped harmonic oscillator.     

Supersymmetric Pair of q-Deformed Nonlocal Operators

A simple version of the q-deformed calculus is used to generate a pair ofq-nonlocal, second-order difference operators by means of deformed counterpartsof Darboux intertwining operators for the Schrödinger—Hermite oscillators atzero factorization energy. These deformed nonlocal operators may be consideredas supersymmetric partners and their structure contains contributions originatingin both the Hermite operator and the quantum harmonic oscillator operator. Thereare also extra ±x contributions. The undeformed limit, in which allq-nonlocalities wash out, corresponds to the usual supersymmetric pair of quantum mechanicalharmonic oscillator Hamiltonians. The more general case of negative factorizationenergy is briefly discussed as well.