A new treatment of the vibration–Rotation eigenvalue problem for a diatomic molecule

Abstract

The problem of the determination of the vibration-rotation eigenvalue in diatomic molecules is considered. An eigenvalue equation totally independent from the eigenfunction is written for any potential, analytical or numerical. This equation uses uniquely the vibration-rotation canonical functions; its resolution is reduced to that of a simple and classical numerical problem. Examples of numerical applications for analytical (Morse) and numerical potentials are presented. It is shown that the vibrational eigenvalues deduced from the eigenvalue equation are within 10-6 cm-' of the exact values. Comparison with conventional methods are presented and discussed.

Theoretical study of the low‐lying electronic states of the molecular ion KRb+

“…The vibration–rotation wave function ψ vJ of a given electronic state of a diatomic molecule is the solution of the radial Schrödinger equation: where r is the internuclear distance, R ( r ) = 1/ r 2 , U ( r ) is the rotationless potential characterizing the electronic state, v and J are, respectively, the vibrational and rotational quantum numbers, and 2μ/ħ is a given constant for the considered molecule. By using the canonical functions approach 10, 11 Eq. (1) can be replaced by a set of functions: where ϕ 0 is the pure vibrational wave function, ϕ n are the rotation harmonics, e 0 = E v is the pure vibrational eigenvalue 4, and e n are given by 12–14 with e 1 = B v is the rotational constant and e n are the centrifugal distortion constants D v , H v , … .…”

Theoretical study of the low-lying electronic states of the RbCs+molecular ion

“…During the past 12 years the canonical functions approach (Kobeissi et al 1983) has been used to solve many problems in molecular physics such as the calculation of the rotational and centrifugal distortion constants (Kobeissi et al 1989, Kobeissi and Korek 1993, Korek and Kobeissi 1991, 1992, 1993, Korek 1999, the rovibrational matrix elements in infrared transitions (Kobeissi and Korek 1994, Korek and Kobeissi 1994, 1998, Korek 1997, Korek et al 2000c, Raman transitions (Korek and Kobeissi 1995, Korek et al 2000d and the bound states of the coupled-channel Schrödinger equation (Fakhreddine et al 1999) for different types of potentials, either empirical or of the RKR-type.…”

Numerical comparison of the Cooley and Kobeissi methods for solving the one‐dimensional Schrödinger equation

“…Although the CFM is fundamentally less efficient than the Cooley algorithm, it does offer some advantages in practical applications, as it avoids unnecessary integration in the nonclassical regions. There are doubtless other methods capable of better limiting accuracy [lo], including perhaps other implementations of the CFM [ 5 ] ; but the Numerov integrator can effortlessly provide better than part-per-million reliability, which is more than adequate for most applications. In fact, even for cases where closed-form expressions for + are known, the Numerov generator, requiring only four slow operations (multiplications or divisions) per step [ 101, may still be the easiest and most efficient way to produce a tabulated wave function.…”