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

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

Canadian Journal of Chemistry ◽

10.1139/v93-046 ◽

1993 ◽

Vol 71 (3) ◽

pp. 313-317 ◽

Cited By ~ 25

Author(s):

Mahmoud Korek ◽

Hafez Kobeissi

Keyword(s):

Vibrational Energy ◽

Rotational Constant ◽

Centrifugal Distortion ◽

Jones Potential ◽

Similar Good ◽

Centrifugal Distortion Constants ◽

Schrödinger Perturbation ◽

First Time ◽

Analytical Expressions

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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Non-relativistic eigensolutions of molecular and heavy quarkonia interacting potentials via the Nikiforov Uvarov method

Canadian Journal of Physics ◽

10.1139/cjp-2020-0039 ◽

2020 ◽

Vol 98 (12) ◽

pp. 1125-1132 ◽

Cited By ~ 3

Author(s):

Ekwevugbe Omugbe

Keyword(s):

Vibrational Energy ◽

Energy Levels ◽

Rotational Constant ◽

Special Cases ◽

Radial Schrödinger Equation ◽

Interacting Systems ◽

Centrifugal Distortion Constants ◽

Vibrational Energy Levels ◽

The Masses ◽

Uvarov Method

In this paper, the analytical eigensolutions of the radial Schrödinger equation are obtained using the Nikiforov Uvarov method. Special cases of the modeled potential were discussed. The energy eigenvalues expressions for both quarkonia and diatomic molecular interacting systems are obtained. Furthermore, the masses of the charmonium and bottomonium mesons were computed. The vibrational energy levels, inertia rotational constant, and first two centrifugal distortion constants of the Kratzer–Fues oscillator were obtained in closed form. The results obtained are in excellent agreement with the results in the literature.

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A canonical approach for computing the eigenvalues of the Schrödinger equation for double-well potentials

Canadian Journal of Physics ◽

10.1139/p00-072 ◽

2000 ◽

Vol 78 (11) ◽

pp. 969-976 ◽

Cited By ~ 4

Author(s):

M Korek ◽

K Fakhreddine

Keyword(s):

Integral Equation ◽

Wave Function ◽

Schrödinger Equation ◽

Volterra Integral Equation ◽

Schrodinger Equation ◽

High Accuracy ◽

Present Formulation ◽

Numerical Application ◽

Canonical Approach ◽

Double Well Potential

The problem of obtaining the eigenvalues of the Schrödinger equation for a double-well potential function is considered. By replacing the differential Schrödinger equation by a Volterra integral equation the wave function will be given by [Formula: see text] where the coefficients ai are obtained from the boundary conditions and the fi are two well-defined canonical functions. Using these canonical functions, we define an eigenvalue function F(E) = 0; its roots E1, E2, ... are the eigenvalues of the corresponding double-well potential. The numerical application to analytical potentials (either symmetric or asymmetric) and to a numerical potential of the (2)1 [Formula: see text] state of the molecule Na2 shows the validity and the high accuracy of the present formulation. PACS No.: 03.65Ge

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