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Analytic expression of the rotation harmonics in the vibration–rotation wave function of a diatomic molecule
Author(s) -
Kobeissi Hafez,
Korek Mahmoud
Publication year - 1982
Publication title -
international journal of quantum chemistry
Language(s) - English
Resource type - Journals
SCImago Journal Rank - 0.484
H-Index - 105
eISSN - 1097-461X
pISSN - 0020-7608
DOI - 10.1002/qua.560220104
Subject(s) - harmonics , diatomic molecule , rotation (mathematics) , physics , function (biology) , wave function , mathematical analysis , vibration , computation , atomic physics , schrödinger equation , solid harmonics , mathematical physics , quantum mechanics , classical mechanics , mathematics , molecule , geometry , voltage , evolutionary biology , algorithm , biology
By using the vibration–rotation canonical functions, we show that the wave function for a vibration–rotation level ( v , J ) can be represented by ψ vJ ( r ) = ψ v ( r )+Σ l = 0 ∞λ l ϕ l ( r ) with λ = J ( J + 1). The radial Schrödinger equation can be written ( H v + λ H J )ψ v λ = ( E v + Σ l = 1 ∞λ l ε l )ψ v λ , where ε 1 , ε 2 , ε 3 ,… stand for the rotational constants B v , D v , H v ,…. The radial equation being satisfied for any value of λ, the rotation “harmonics” ϕ 1 , ϕ 2 ,… are found to be the solutions of a set of inhomogenous differential equations of the form H ϕ l = E v ϕ l + f l ( r )ψ v . An analytic expression of the harmonics ϕ i is given for any potential. The numerical application shows that, for a given r , the harmonics decrease in absolute values like B v , D v , H v ,… and that the agreement between the values of ψ vJ deduced from the computation of the harmonics on one hand, and the direct computation on the other hand, is very satisfactory.