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Computational and numerical developments
Published in Seán M. Stewart, R. Barry Johnson, Blackbody Radiation, 2016
Seán M. Stewart, R. Barry Johnson
Here (s)k = s(s + 1)(s + 2) … (s + k − 1) is the Pochhammer symbol [51]. Note that since s is a positive integer the Pochhammer symbol can be written in terms of factorials as
Accuracy of XH-stretching intensities with the Deng–Fan potential
Published in Molecular Physics, 2019
Emil Vogt, Daniel S. Sage, Henrik G. Kjaergaard
The Schrödinger equation for a diatomic Deng–Fan oscillator is [15] where D is the dissociation energy and a is an inherent parameter of the Deng–Fan potential. The Deng–Fan potential provides an interesting solution to the mathematically incorrect boundary condition of the Morse potential as . As , the denominator of the Deng–Fan potential approaches zero, thereby making , i.e. the correct limit. Furthermore, the fraction approaches one when, meaning that the potential correctly approaches the constant D. Solving Equation (9) yields the exact analytical bound state energy levels, where , , and are all constants (see Ref. [20] or the supporting information, SI, for the derivation of the equations included in this section). The corresponding orthonormalDeng–Fan wavefunctions are where and The normalisation constant is and the hypergeometric function is defined as [32] where the Pochhammer symbol is equivalent to the rising factorial . Equation (14) reduces to a vth-order polynomial in y, since is zero for integers where n>v [22]. Parameters of the Deng–Fan potential can be obtained from experimental or calculated transition frequencies using [24] Unlike Equation (7), this equation is not linear in v, and a nonlinear least-squares optimisation is therefore necessary. In this work, the parameters D and a are determined by solving Equation (15) for the first two transition frequencies obtained for the ab initio Morse potential. The equilibrium bond distance is determined as the bond distance for the optimised geometry. We fix this parameter when Equation (15) is fitted to Morse transitions frequencies. Contrary to the Deng–Fan energy levels, the Morse energy levels do not depend on . Nevertheless, this parameter is needed to calculate oscillator strengths for both potentials as reflects the relative position of the potential and the DMF.