Molecular Spinless Energies of the Modified Rosen-Morse Potential Energy Model

Abstract

We solve the Klein-Gordon equation with the modified Rosen-Morse potential energy model. The bound state energy equation has been obtained by using the supersymmetric shape invariance approach. The relativistic vibrational transition frequencies for the $6^1{\Pi}_u$ state of the $^7Li_2$ molecule have been computed by using the modified Rosen-Morse potential model. The calculated relativistic vibrational transition frequencies are in good agreement with the experimental RKR values.

Introduction

There has been the continuous interest in investigating solutions of the Klein-Gordon equation with some diatomic molecule potential energy models, such as the Morse potential,1,2 Rosen-Morse potential,3-5 Manning-Rosen potential, 6 Pöschl-Teller potential,7 - 9 and Deng-Fan potential.10,11 As far as we known, the authors in these works did not quantitatively investigated relativistic rotation-vibrational energies for actual diatomic molecules. In recent years, one of present authors and co-workers12-15 studied solutions of the Klein-Gordon equation with the improved Manning- Rosen potential, improved Rosen-Morse potential, improved Tietz potential, and Morse potential, and calculated relativistic vibrational transition frequencies for the a3Σu+ state of the 7Li2 molecule, the 33Σg+ state of the Cs2 molecule, the C1Пu state of the Na2 molecule, and the X1Σ+ state of the ScI molecule.

In 2012, Zhang et al.16 introduced the effect of inner-shell radii of two atoms for diatomic molecules into the original Rosen-Morse potential,17 and proposed a modified Rosen- Morse potential energy model,

where , De denotes the dissociation energy, re denotes the equilibrium bond length, and Ke presents the equilibrium harmonic vibrational force constant. The parameter K is a dimensionless constant, K = 4.00.18 The ability of a potential energy model to reproduce the potential energy curve is judged by agreement with the known potential energy curve determined by the Rydberg-Klein-Rees (RKR) approach.19-21 The modified Rosen-Morse potential is superior to the Morse potential and original Rosen-Morse potential in reproducing potential energy curves for six molecule states examined in Ref..16 Recently, with the use of the standard function analysis method, Tang et al.22 solved the Schrödinger equation with the modified Rosen-Morse potential model to obtain rotation-vibrational energy spectra for a diatomic molecule, and calculated the vibrational energy levels calculated for the 61Πu state of the 7Li2 molecule. In terms of the RKR method and the multireference configuration interaction approach, many authors have carried out on investigation on the vibrational levels and the interaction potential curves for the lithium dimer.23-25

In this work, we employ the basic concept of the supersymmetric shape invariance approach to study the bound state solutions of the Klein-Gordon equation with the modified Rosen-Morse potential energy model. We also attempt to calculate the relativistic vibrational transition frequencies of the 61Πu state of the 7Li2 molecule and compare the present calculated values with the RKR data.

 

Bound State Solutions

The Klein-Gordon equation with a scalar potential S(r) and a vector potential V(r) for the nuclear motion of a diatomic molecule with reduced mass μ is given by12

where ▽2 is the Laplace operator, E denotes the relativistic energy of the quantum system, c is the speed of light, and = h/2π, h is the Planck constant. We express the wave function as Ψ(r,θ,φ) (uvJ(r)/r)YJm = (θ,φ), where YJm (θ,φ) is the spherical harmonic function. Substituting this expression into Eq. (2), we obtain the radial part of the Klein-Gordon equation,

where v and J are the vibrational and rotational quantum numbers, respectively. Taking the equal scalar and vector potentials, S(r) = V(r), Eq. (3) turns to the form

Under the nonrelativistic limit, Eq. (4) becomes a Schrödinger equation with the interaction potential 2V(r). In order to make the interaction potential as V(r), not 2V(r) in nonrelativistic limit, we take the scheme proposed by Alhaidari et al.26 to rescale the scalar potential S(r) and vector potential V(r), and write Eq. (4) in the form of

Considering the scalar and vector potentials as the modified Rosen-Morse potential, S(r) = V(r) = UMRM(r), we obtain the following second-order Schrödinger-like equation,

where λ = . In the case of J = 0, we can exactly solve the above equation. When J ≠ 0 , one can only approximately solve it. We apply the Pekeris approximation scheme to deal with the centrifugal term.27 The Pekeris approximation approach has been widely used to investigate the analytical solutions of the Klein-Gordon equation with various molecular potential models.11-13 We replace the centrifugal potential energy term by the following form22

where r=(J(J+1)c2)/, and the coefficients c0, c1, and c2 are given by22

Substituting approximation expression (7) into Eq. (6), we rewrite it in the following form

where the parameters A, B, εvJ are defined as

With the use of the supersymmetric shape invariance approach, 28-30 we solve Eq. (11). The ground-state wave function u0,J(r) is written as

where W(r) is called a superpotential in supersymmetric quantum mechanics.28 Substituting expression (15) into Eq. (11) yields the following relation satisfied by the superpotential W(r),

where ε0, J presents the ground-state energy. We take the superpotential W(r) in the form

where C1 and C2 are two constants. Substituting the above expression into expression (15), we obtain the following expression for the ground-state wave function u0, J(r)

For the bound state solutions, the wave function uvJ(r) should satisfy the boundary conditions: uvJ (∞) = 0 and uvJ (0) is limitary. These regularity conditions leads us to have C1 < 0 and C2 < 0.

In terms of expression (17), we can construct a pair of supersymmetric partner potentials U−(r) and U+(r),

Making a comparison of Eq. (19) with Eq. (16), we have the following three relationships

Solving Eqs. (22) and (23), we obtain

Substituting expression into expressions (19) and (20) and using Eq. (22), we can rewrite the two supersymmetric partner potentials U−(r) and U+(r) as follows

With the use of expressions (26) and (27), we can yield the following relationship

where a0 = C2, a1 is a function of a0, i.e., a1 = h(a0) = a0 + αλ, and the reminder R(a1) is independent of . Eq. (28) implies that the supersymmetric partner potentials U−(r) and U+(r) possess the shape invariance. Using the shape invariance approach,29 one can exactly determine their energy spectra. The energy spectra of the potential U−(r) are given by

where the quantum number v = 0, 1, 2, ···. From Eqs. (11), (16) and (26), we obtain the following relationship for εvJ ,

Substituting expressions (30) and (21) into Eqs. (31) and considering C1 = A/2C2 − C2/2λ, we obtain the following expression for εvJ,

Substituting expression (25) into expression (32), we have

Substituting expressions (12) and (13) into expression (33) and using , we obtain the relativistic rotation-vibrational energy eigenvalue equation for the diatomic molecule in the presence of equal scalar and vector modified Rosen-Morse potential energy models,

where v = 0, 1, 2, 3,..., λ = , and we have made a replacement for

When J = 0, we obtain the relativistic vibrational energy eigenvalue equation for the diatomic molecule with equal scalar and vector modified Rosen-Morse potentials,

With the use of the superpotential given in expression (17) and the ground-state wave function given in expression (18), one can determine the excited state wave functions by employing the explicit recursion operator approach. 31,32

 

Discussion

The force constant ke is defined as the second derivative of the potential energy function U(r) for a diatomic molecule, namely Using this definition and the relation , we deduce the following expression for the potential parameter α appearing in the modified Rosen- Morse potential,

where ωe denotes the equilibrium harmonic vibrational reciprocal wavelength, and W is the Lambert W function, which satisfies

We consider the 61Πu state of the 7Li2 molecule. The experimental values of De, re, and ωe for the 7Li2 61Πu state are taken from the literature23: cm−1, Å, and cm−1. Taking these experimental data as inputs and applying expression (36), we can determine the value of the parameter α in the modified Rosen-Morse potential. The experimental RKR data points for the 61Πu state of 7Li2 are shown in Figure 1, which also contains the curves reproduced by employing the four three-parameter potential models: modified Rosen- Morse potential, Rosen-Morse potential,17,33 Morse potential,34 and Frost-Musulin potential.33,35 The Rosen-Morse, Morse, and Frost-Musulin potential functions are given by, respectively

Fig. 1.(Color online) RKR data points and four empirical potential energy models for the 61Πu state of the 7Li2 molecule.

An outlook for the range covered by the experimental RKR data points tells us that the modified Rosen-Morse potential is superior to the Rosen-Morse, Morse, and Frost-Musulin potential functions for the 61Πu state of the 7Li2 molecule.

An available potential function should well model the experimental RKR potential curve, and satisfy the Lippincott criterion, i.e., an average absolute deviation of less than 1% of the dissociation energy De.36 The average absolute deviation is defined as where is the number of experimental data points, Uexp(r) and Ucalc(r) are the experimentally determined potential and the empirical potential, respectively. The average absolute deviations of the modified Rosen-Morse potential, Rosen- Morse potential, Morse potential, and Frost-Musulin potential for the 61Πu state of the 7Li2 molecule from the RKR potential reported by Grochola et al.23 are 0.648% of De, 1.053% of De, 1.901% of De, and 2.64% of De, respectively. These deviation values show that the modified Rosen-Morse potential is best for the examined four potential models in reproducing the potential energy curve for the 61Πu state of the 7Li2 molecule.

By employing energy eigenvalue Eq. (35), we calculate relativistic vibrational transition frequencies for the 61Πu state of the 7Li2 molecule. The present calculated values are given in Table 1, in which we also present the RKR values taken from the literature.23 From Table 1, we observe that the relativistic vibrational transition frequencies obtained by using the modified Rosen-Morse potential model are in good agreement with the RKR data.

Table 1.A comparison of the calculated relativistic vibrational transition frequencies and experimental RKR values for the 61Πu state of the 7Li2 molecule (in units of cm−1)

 

Conclusions

In this work, we have investigated the bound state solutions of the Klein-Gordon equation with the Morse potential energy model. The energy eigenvalue equation has been obtained using the supersymmetric shape invariance method. We calculate the relativistic vibrational transition frequencies for the 61Πu state of the 7Li2 molecule. The relativistic vibrational transition frequencies predicted with the modified Rosen-Morse potential model are in good agreement with the experimental RKR data.