"Nonintegral" expressions for the diatomic centrifugal distortion constants

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.

On nonintegral E corrections in perturbation theory: application to the perturbed Morse oscillator

1994 ◽  
Vol 72 (1-2) ◽  
pp. 80-85 ◽  
Author(s):  
Hafez Kobeissi ◽  
Majida Kobeissi ◽  
Chafia H. Trad
Keyword(s):  
Perturbation Theory ◽  
Good Accuracy ◽  
Morse Oscillator ◽  
Numerical Application ◽  
Correction Formula ◽  
Theory Application ◽  
Nonhomogeneous Differential Equation ◽  
Particular Solution ◽  
New Formulation ◽  
Schrödinger Perturbation

A new formulation of the Rayleigh–Schrödinger perturbation theory is applied to the derivation of the vibrational eigenvalues of the perturbed Morse oscillator (PMO). This formulation avoids the conventional projection of the Ψ corrections on the basis of unperturbed eigenfunctions [Formula: see text], or the projection of the nonhomogeneous Schrödinger equations on [Formula: see text], it gives simple expressions for each E correction [Formula: see text] free of summations and integrals. When the PMO is characterized by the potential U = UM + UP (where UM is the unperturbed Morse potential), the eigenvalue of a vibrational level ν is given by: [Formula: see text]. According to the new formulation the correction £, [Formula: see text] is given by [Formula: see text], where σp(r) is a particular solution of the nonhomogeneous differential equation y″ + f y = sp; here [Formula: see text], sp is well known for each p: for p = 0, [Formula: see text]; for [Formula: see text]. For the numerical application one single routine is used, that of integrating y″ + f y = s, where the coefficients are known as well as the initial values. An example is presented for the Huffaker PMO of the (carbon monoxide) CO-X1Σ+ state. The vibrational eigenvalues Eν are obtained to a good accuracy (with p = 4) even for high levels. This result confirms the validity of this new formulation and gives a semianalytic expression for the PMO eigenvalues.

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

1993 ◽  
Vol 71 (3) ◽  
pp. 313-317 ◽  
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.

New analytical expression for the rotational factor in Raman transitions

1995 ◽  
Vol 73 (9-10) ◽  
pp. 559-565 ◽  
Author(s):  
M. Korek ◽  
H. Kobeissi
Keyword(s):  
Direct Calculation ◽  
Matrix Elements ◽  
Numerical Application ◽  
Raman Transitions ◽  
The Matrix ◽  
Present Formalism ◽  
Vibration Wave ◽  
Schrödinger Perturbation ◽  
Analytical Expressions ◽  
Rotational Factor

The matrix elements of the polarizability anisotropy γ in the Raman spectra of diatomic molecules are investigated. These matrix elements are given by [Formula: see text] where Gνν′(m) is the rotational factor with m = [(J′(J′ + 1) − J(J + 1)]/2 and J′ − J = ±2. By using a nonconventional approach to the Rayleigh–Schrödinger perturbation theory the rotational factor can be written as Gνν′(m) = A0 + A1m + A2m2 where the coefficients A0, A1, and A2 are given by simple analytical expressions in terms of the integrals [Formula: see text] and [Formula: see text] where Y stands for Ψ(0) (the pure vibration wave function), or Ψ(0) (the first rotational perturbative correction to Ψ(0), or Ψ(2) (the second correction). A numerical application is presented for the ground states of CO and H2 molecules. A comparison with a numerical and direct calculation of the rotational factor Gνν′(m) shows the accuracy of the present formalism.

PREDICTION OF THE TRANSPORT PROPERTIES OF SF6 WITH O2, CO2, CF4, N2 AND CH4 MIXTURES AT LOW DENSITY BY THE INVERSION METHOD (PART II)

2004 ◽  
Vol 03 (01) ◽  
pp. 69-90 ◽  
Author(s):  
BEHZAD HAGHIGHI ◽  
ALIREZA HASSANI DJAVANMARDI ◽  
MOHAMAD MEHDI PAPARI ◽  
MOHSEN NAJAFI
Keyword(s):  
Potential Energy ◽  
Diffusion Coefficients ◽  
Potential Energy Function ◽  
Model Potential ◽  
Inversion Method ◽  
Low Density ◽  
Initial Model ◽  
Lennard Jones ◽  
Inversion Procedure ◽  
And Diffusion

Viscosity and diffusion coefficients for five equimolar binary gas mixtures of SF 6 with O 2, CO 2, CF 4, N 2 and CH 4 gases are determined from the extended principle of corresponding states of viscosity by the inversion technique. The Lennard–Jones 12-6 (LJ 12-6) potential energy function is used as the initial model potential required by the technique. The obtained interaction potential energies from the inversion procedure reproduce viscosity within 1% and diffusion coefficients within 5%.

33S Nuclear Hyperfine Structure in the Rotational Spectrum of Thiazole

1993 ◽  
Vol 48 (12) ◽  
pp. 1219-1222 ◽  
Author(s):  
U. Kretschmer ◽  
H. Dreizler
Keyword(s):  
Fourier Transform ◽  
Hyperfine Structure ◽  
Molecular Beam ◽  
Microwave Spectroscopy ◽  
Coupling Constants ◽  
Rotational Spectrum ◽  
Centrifugal Distortion ◽  
Quadrupole Coupling ◽  
Nuclear Quadrupole ◽  
Centrifugal Distortion Constants

Abstract We investigated the 33S nuclear quadrupole coupling of thiazole- 33S in natural abundance by molecular beam Fourier transform microwave spectroscopy. In addition the 14N nuclear quadrupole coupling could be analyzed with high precision. We derived the rotational constants A = 8529.29268 (70) MHz, B = 5427.47098 MHz, and C = 3315.21676 (26) MHz, quartic centrifugal distortion constants and the quadrupole coupling constants of 33S χaa = 7.1708 (61) MHz and χbb= -26.1749 (69) MHz and of 14N χ aa = -2.7411 (61) MHz and χbb = 0.0767 (69) MHz.

Sign in / Sign up

Export Citation Format

Share Document