Implementation of Morse potential for approximation of vibrational terms of diatomic molecules

Abstract

We analysed the problem of approximation of the potential function of a diatomic molecule by a Morse model function with constant anharmonicity ν x using the Birge-Sponer extrapolation. The analysis of the approximations used in the derivation of the Morse equation shows that the solution of this problem is ambiguous. A scheme for optimizing the selection of initial parameters is proposed, which is illustrated by examples taken from the literature. The advantages of delineation of anharmonicity in the excitation of vibrational levels by deviations of the value ν x from the constant value according to Morse are demonstrated. An attempt is made to use the dimensionless anharmonicity parameter x* as a universal characteristic of the shape features of the electronic term of the molecule. Keywords: Morse potential, diatomic molecule, Birge-Sponer extrapolation, anharmonicity, electronic terms, vibrational structure.

Approximation of electronic term of diatomic molecule by the Morse function. The role of anharmonicity

“…In particular the overestimated value of the bond energy for M1, which is caused by imprecision of the equation ( 4), which is correct only for the Morse oscillator, i.e. for the exact equation (2). Similarly, for the M2 model, for which 𝐷 𝑒 is fixed, the change in the anharmonicity coefficient, carried out due to the factor 𝑥 𝑒 ′ , is realized using equation ( 4), which significantly changes the entire system of vibrational levels and leads to a significant broadening of the potential well.…”

“…The HCl molecule was several times used by the authors of textbooks and monographs to illustrate the accuracy of the estimate of the Morse dissociation energy (M1), the results do not differ significantly (17%, 15%, 20%). Other hydrogen halide molecules also likely belong to this group [2].…”

“…(1), the eigenvalues (vibrational energies) of this potential are approximately (series is limited by two terms): 𝐸(𝑣) ℎ𝑐 ⁄ = 𝐺(𝑣) = 𝜔 𝑒 (𝑣 + ½) − 𝜔 𝑒 𝑥 𝑒 (𝑣 + ½) 2 𝑣 = 0,1,2 … 𝑣 𝑚 (2),…”

“…where 𝜔 𝑒 , harmonic frequency, cm −1 , and 𝜔 𝑒 𝑥 𝑒 , anharmonicity, cm −1 , are parameters which can be calculated from two experimental vibrational frequencies 𝜔(0 − 1) and 𝜔(0 − 2), and 𝑣 𝑚 is maximum vibrational quantum number, which defines the bond energy 𝐷 𝑒 . Coefficients in (2) expressed in terms of parameters of (1) and the reduced mass 𝜇, are:…”

“…In the current communication the main attention is focused on the development of the original results of the authors' recent work on the features of the Morse approximation [1][2][3]. They include the existence of two independent interpretations M1 and M2 of the solution of the Schrödinger equation for an alternative approximations of the potential function of the molecule 𝑈(𝑟), the introduction of the empirical anharmonicity function 𝜔 е 𝑥(𝑣), which characterizes the anharmonicity of the potential well in the vicinity of the vibrational quantum number 𝑣, the development of the approach for estimating the deviations of the models M1 and M2 from the real term 𝑈(𝑟).…”