Diatomic centrifugal distortion constants for large orders at any level: application to the  state

The determination of the centrifugal distortion constants (CDC) of a diatomic molecule is sought for high orders. When the vibrational energy e0 = Ev is known for a vibrational level v, the use of Rayleigh–Schrödinger perturbation theory gives the rotational constant [Formula: see text] and the CDC,[Formula: see text] [Formula: see text] [Formula: see text] where Φn(r) is the solution of the nth rotational Schrödinger equation. The problem of the determination of a function Φn is solved by deriving exact analytical expressions for the initial values Φn(r0) and [Formula: see text] at an arbitrary "origin" r0, the determination of any Φn(r) becoming as easy as that of Φn(r) when e0 is known; that of en becomes as easy as that of [Formula: see text] The application of the present formulation to the model Lennard–Jones potential function allows the numerical computation of Dv, Hv, Lv, Mv, Nv, Ov, Pv, Qv for low and high v; the CDC beyond Mv are given for the first time; higher order CDC may be reached. The results for the four lowest order constants are in good agreement with those from previously confirmed methods. Appropriate tests for all orders show that the present method provides an elegant and competitive solution to the diatomic CDC problem even for large orders and high levels (near dissociation). Similar good results are obtained for an RKR potential of the [Formula: see text] state bounded by 109 levels.

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Solution of the Rovibrational Schrödinger Equation of a Molecule Using the Volterra Integral Equation

Advances in Physical Chemistry ◽

10.1155/2018/1487982 ◽

2018 ◽

Vol 2018 ◽

pp. 1-11 ◽

Cited By ~ 8

Author(s):

Mahmoud Korek ◽

Nayla El-Kork

Keyword(s):

Integral Equation ◽

Wave Function ◽

Volterra Integral Equation ◽

Vibrational Energy ◽

Excited Electronic State ◽

Rotational Constant ◽

Vibrational Wave Function ◽

Oppenheimer Approximation ◽

Centrifugal Distortion Constants ◽

Schrödinger Perturbation

By using the Rayleigh-Schrödinger perturbation theory the rovibrational wave function is expanded in terms of the series of functions ϕ0,ϕ1,ϕ2,…ϕn, where ϕ0 is the pure vibrational wave function and ϕι are the rotational harmonics. By replacing the Schrödinger differential equation by the Volterra integral equation the two canonical functions α0 and β0 are well defined for a given potential function. These functions allow the determination of (i) the values of the functions ϕι at any points; (ii) the eigenvalues of the eigenvalue equations of the functions ϕ0,ϕ1,ϕ2,…ϕn which are, respectively, the vibrational energy Ev, the rotational constant Bv, and the large order centrifugal distortion constants Dv,Hv,Lv….. Based on these canonical functions and in the Born-Oppenheimer approximation these constants can be obtained with accurate estimates for the low and high excited electronic state and for any values of the vibrational and rotational quantum numbers v and J even near dissociation. As application, the calculations have been done for the potential energy curves: Morse, Lenard Jones, Reidberg-Klein-Rees (RKR), ab initio, Simon-Parr-Finlin, Kratzer, and Dunhum with a variable step for the empirical potentials. A program is available for these calculations free of charge with the corresponding author.

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"Nonintegral" expressions for the diatomic centrifugal distortion constants

Canadian Journal of Physics ◽

10.1139/p95-047 ◽

1995 ◽

Vol 73 (5-6) ◽

pp. 339-343

Author(s):

Hafez Kobeissi ◽

Chafia H. Trad

Keyword(s):

Model Potential ◽

Centrifugal Distortion ◽

Numerical Application ◽

Lennard Jones ◽

Particular Solution ◽

New Formulation ◽

Centrifugal Distortion Constants ◽

Schrödinger Perturbation ◽

Analytical Expressions ◽

Nonhomogeneous Equation

The problem of the centrifugal distortion constants (CDC), Dν, Hν, … for a diatomic molecule is considered. It is shown that a new formulation of the standard Rayleigh–Schrödinger perturbation theory can give simple and compact analytical expressions of the CDC (up to any order). Thus, the constants e1 = Bν, e2 = −Dν, e3 = Hν,…, en,… are all of the form en = lim σn(r)/σ0(r) as r → ∞. σ0 is the particular solution of the nonhomogeneous equation y″ + k(Eν – U)y = s, with s = ψν, where (Eν, ψν) is the eigenvector corresponding to the rotationless potential U(r) and to the vibrational level ν; and where σ0(0) = σ′0(0) = 0. σn is the particular solution of the above equation, where s is known for each order of n. The numerical application to the standard Lennard–Jones model potential shows that good results are obtained for Dν, Hν, Lν,…,Oν, Pν, for ν = 0 to 22, which is only at 2 × 10−4 of the well depth. The program uses one routine (the integration of the equation y″ + fy = s) repeated for different s; it is quite simple and gives no difficulties at the boundaries and there is no need to use any mathematical or numerical artifices.

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