Matrix elements for Morse oscillators

Abstract

We have derived exact analytic expressions for the following matrix elements for a Morse oscillator: 〈m‖exp[γ(r−r0)]‖n〉, 〈m‖(r−r0)exp[−a(r−r0)]‖n〉, and 〈m‖(r−r0)exp[−2a(r−r0)]‖n〉, where a is the Morse range parameter and γ is an arbitrary constant. We have found that several of the commonly used expressions for various diagonal matrix elements of the Morse oscillator involve considerable roundoff error. Due to near cancellation of terms the result may be many orders of magnitude lower than any of the terms. We obtain simple yet accurate asymptotic approximations for the diagonal elements which avoid these problems. These approximations are obtained for 〈m‖(r−r0)‖m〉, 〈m‖(r−r0)2‖m〉, and the cases mentioned above.

A resonance enhanced multiphoton ionization study of the gerade excited states of Xe2 with a Xe 1S+Xe* 6s[3/2]1 dissociation limit

“…The vЉϭ0 vibrational wave function of the ground state is calculated analytically using known Morse parameters. 21,22 The vibrational wave functions associated with the 1 g state can be calculated if a value of R e is assumed. The relevant Franck-Condon factors can now be computed numerically, and by introducing Gaussian and/or Lorentzian line shapes the experimental spectrum can be simulated.…”

“…The vibrational wave functions of the excited states are obtained at energies starting just above the dissociation limit to a thousand wave numbers above it. The ground state wave functions are again analytically calculated using known Morse parameters, 21,22 the continuum vibrational wave functions of the excited states are calculated using a numerical integration scheme based on the Numerov method. 23 In the analyses of the excitation spectra it is assumed that the atomic signal is proportional to the FranckCondon factors between the ground and excited state vibrational wave functions of the dimer.…”

Statistical Properties of Klauder–perelomov Coherent States for the Morse Potential

“…Since its introduction, the Morse oscillator has proved very useful for various problems in diverse fields of physics and chemistry (diatomic and polyatomic molecular systems, quantum chemistry, spectroscopy, chemical bonds) [2,3,4,5,6,7] (and the references therein).…”

Photon-added coherent states for the Morse oscillator

“…the depth of the potential energy well). The Morse potential is one of the most simple and "realistic" three-parameter anharmonic potential models, particularly used in specific calculations in spectroscopy [2], in diatomic molecule vibration and scattering [3][4][5], and in other fields (e.g. in the description of vibrations of polyatomic molecules by representing each bond in the molecule by a Morse potential [6]).…”