# Investigating Some Diatomic Molecules Bounded by the Two-Dimensional Isotropic Oscillator plus Inverse Quadratic Potential in an External Magnetic Field

Authors: O. J. Oluwadare; E. O. Ilesanmi; T. O. Abiola; O. Olubosede; E. A. Odo; S. O. Ajibade; K. J. Oyewumi
Journal: Advances in High Energy Physics
 (2022)

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## Abstract

We investigate the nonrelativistic magnetic effect on the energy spectra, expectation values of some quantum mechanical observables, and diamagnetic susceptibility for some diatomic molecules bounded by the isotropic oscillator plus inverse quadratic potential. The energy eigenvalues and normalized wave functions are obtained via the parametric Nikiforov-Uvarov method. The expectation values square of the positionr2, square of the momentum p2, kinetic energy T, and potential energy V are obtained by applying the Hellmann-Feynman theorem, and an expression for the diamagnetic susceptibility X is also derived. Using the spectroscopic data, the low rotational and low vibrational energy spectra, expectation values, and diamagnetic susceptibility X for a set of diatomic molecules (I2, H2, CO, and HCl) for arbitrary values, Larmor frequencies are calculated. The computed energy spectra, expectation values, and diamagnetic susceptibility X were found to be more influenced by the external magnetic field strength and inverse quadratic potential strength g than the vibrational frequencies and the masses of the selected molecules.


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## Body

## 1. Introduction

Several studies in quantum mechanics, solid state physics, condensed matter physics, nuclear physics, chemical physics, molecular physics, and other related areas have proven to an outstanding degree that potential models are very important models for stimulating atomic and molecular interaction since it is capable of predicting and describing some behavior of atoms and molecules. It also provides an insight into the understanding of molecular spectra, vibrations and dynamics [1, 2], spin-orbit interaction, relativistic corrections and diamagnetic susceptibility [3, 4], optical properties [5, 6], interband light absorption and interband optical transitions [7, 8], energy and relativistic effects in weakly bound nuclei [9–11], external magnetic fields and/or Aharonov-Bohm flux fields [12–17], interactions between the magnetic and electric fields [18], thermal and/or thermodynamic properties [19–23], spin and pseudospin symmetries [24], and two-body effects [25–28] among others.One of the important potential models in this regard is the so-called isotropic oscillator plus inverse quadratic potential (IOPIQP) or anharmonic oscillator potential, which has been explored by some authors in both the relativistic and nonrelativistic domains of quantum mechanics [29–32], [33, 34]. An isotropic oscillator (three-dimensional harmonic oscillator) plus inverse quadratic potential may be defined [29–34] as
(1)Vr=12μω2r2+gr2,where g is the potential strength, μ represents the mass of the vibrating molecules, and ω is the angular frequency with which the molecules vibrate in the presence of a magnetic field. Oyewumi [29] employed the hyperradial equation for the isotropic harmonic oscillator plus inverse quadratic potential and presented the normalized hyperradial and hyperangular solutions, and the solutions depend on the dimension as well as the potential parameters. The hidden symmetries and thermodynamic properties for a harmonic oscillator plus inverse square potential have been exposed by Dong et al. [30], while Arda and Sever reported the exact solutions of Schrödinger for this potential within the framework of the Laplace transform technique [31].In the same vein, Abdelmadjid [32] also studied the exact nonrelativistic quantum spectrum systems for the isotropic harmonic oscillator plus inverse quadratic potential within the formalisms of both Boopp’s shift method and standard perturbation theory in both noncommutativity of the two-dimensional real space and phase (NC-2D: RSP) and presented the exact corrections for the spectrum and the associated noncommutative anisotropic Hamiltonian. Again, due to the unflinching interest, Abdelmadjid [33] looked into the effect of both noncommutativity of the three-dimensional space and phase on the Schrödinger equation with an isotopic harmonic oscillator plus inverse quadratic potential and reported the exact degenerated spectrum associated for noncommutative space and phase.Furthermore, Dianawati et al. [34] investigated the Schrödinger equation with quantum deformation for a three-dimensional harmonic oscillator plus inverse quadratic potential via the hypergeometric method. The energy spectra which were calculated and visualized by MATLAB R2013a were found to depend on the quantum deformation and quantum number.It is in the light of the relevance of this potential model that we are motivated to examine the two-dimensional radial Schrödinger equation with the isotropic harmonic oscillator plus inverse quadratic potential in an external magnetic field via the parametric Nikiforov-Uvarov method, obtain the eigensolutions, and discuss the behavior of energy spectra, expectation values of some quantum mechanical observables, and diamagnetic susceptibility for some selected diatomic molecules bounded by this interaction potential model.The sensitivity of the bounded molecules in an external magnetic field (using arbitrary values of Larmor frequencies), low rotational and vibrational levels, and inverse quadratic potential strength would be adequately investigated. The situation where Larmor frequencyωL=0 implies the absence of an external magnetic field, whereas Larmor frequencies ωL>0 indicate the presence of an external magnetic field. The case of the low vibrational energy level n=0,1,2,3, low rotational energy level m=0,+1, and inverse quadratic potential strengths g=0,1 would be examined.Other methods that can be used to solve the aforementioned bound state problems include the wave function ansatz method [13], asymptotic iteration method [16], formula method [17], Euler-Maclaurin approximation [23], Laplace transform technique [31, 35], and supersymmetric approach [36], among others. In Section 2, we give a review of the parametric Nikiforov-Uvarov method. Section 3 contains nonrelativistic eigensolutions, expectation values of some quantum mechanical observables, and diamagnetic susceptibility of the isotropic harmonic oscillator plus inverse quadratic potential in an external magnetic field. The results are discussed extensively in Section 4, while the concluding remarks are given in Section 5.


## 2. Review of the Parametric Nikiforov-Uvarov Method

The parametric Nikiforov-Uvarov method is a straightforward, consistent, and efficient analytical technique for analyzing second-order linear differential equations arising from bound state problems. The choice of this method is due to the fact that it has been proven to an outstanding degree and is well reported to give excellent results in comparison with other methods in the literature [12]. According to Nikiforov and Uvarov [37], the second-order linear differential equation reduces to the generalized equation of hypergeometric type [37, 38]. With an appropriate coordinate transformation z=zr, the equation takes the form
(2)ψnl′′z+τ¯zσzψnl′z+σ¯zσ2zψnlz=0,where σz and σ¯z are polynomials, at most in the second degree, and τ¯z is a first-degree polynomial.To solve equation (2), one needs to break the wave function Ψz into parts as
(3)ψnlz=ϕzyz.Therefore, equation (2) reduces to the hypergeometric-type equation:
(4)σzy′′z+τzy′z+λyz=0,where
(5)τz=τ¯z+2πz,satisfies the condition τ′z<0, has a negative derivative, and is related to the function ϕs by
(6)πz=σzddzlnϕz.The parameterλ is defined by
(7)λ=λn=−nτ′z−nn−12σ′′n=0,1,2,⋯.The energy eigenvalues can be calculated from equation (7). In order to calculate the energy eigenvalues, we need first to determine λ by using the first derivative of πz and defining
(8)λ=k+π′z.By solving the resulting quadratic equation forπz, we obtain the following expression:
(9)πz=σ′−τ¯2±σ′−τ¯22−σ¯+kσ.Here,πz is a polynomial with the parameter z and the prime denotes the first derivative of the functions σz and τz, respectively. The determination of k is the essential point in the calculation of πz. It can be obtained by setting the discriminant of the square root to zero [37]; therefore, a general quadratic expression for k can be obtained. On substitution of the values of k, π′z, τ′z, and σ′′ into equations (7) and (8) and equating (7) and (8), one can evaluate the energy equation for any potential. The wave function ϕs in equation (2) satisfies the condition
(10)ϕ′zϕz=πzσz,which can be evaluated using the Rodrigues relation. The polynomial solutions ynz are given by
(11)ynz=Cnρzdndznσnzρz,where Cn is a normalization constant and the weight function ρz satisfies the following relation:
(12)ddsσzρz=τzρz.A more generalized form of equation (2) for any potential may be presented as [39]
(13)ψnl′′z+β1−β2zz1−β3zψnl′z+−ρ2z2+ρ1z−ρ0z21−β32ψnlz=0,which satisfies the wave functions of equation (3). By comparing equation (2) with equation (13), we have the following polynomials:
(14)τ¯z=β1−β2z,σz=z1−β3z,σ¯z=−ρ2z2+ρ1z−ρ0.Substituting equation (13) into equation (9), one obtains
(15)πz=β4+β5z±β6−kβ3z2+β7+kz+β8,with the following parametric constants:
(16)β4=121−β1,β5=12β2−2β3,β6=β52+ρ2,β7=2β4β5−ρ1,β8=β42+ρ0.According to the Nikiforov-Uvarov method, the discriminant of equation (15) must be set to zero so that the expression for k can be quadratically obtained as
(17)k±=−β7+2β3β8±2β8β9,β9=β3β7+β3β8+β6.Since the negative value ofk (that is k−) gives the bound state solution, we consider
(18)k−=−β7+2β3β8−2β8β9.Inserting equation (18) into (15), we have
(19)πz=β4+β5z−β9+β3β8z−β8,having its first derivative as
(20)π′z=β5−β9+β3β8.Putting equations (14) and (15) into equation (5), one obtains
(21)τz=β1+2β4+2β5−β2z−2β9+β3β8z−β8,and its first derivative becomes
(22)τ′z=−β2−2β5−2β9+β3β8.By applying equation (16) to equation (22), we have
(23)τ′z=−2β3−2β9+β3β8<0.By applying equations (18) and (20) in equation (8), we have
(24)λ=−β7+2β3β8−2β8β9+β5−β9+β3β8.With equations (14) and (22), the parameter λn in equation (7) becomes
(25)λn=β2n−2nβ5+2nβ9+β3β8+nn−1β3n=0,1,2,⋯.Equating equations (24) and (25), one obtains the bound state energy equation for any potential [37–40] as
(26)β2n−2n+1β5+2n+1β9+β3β8+nn−1β3+β7+2β3β8+2β8β9=0.Using equations (10)–(12), the wave function parameters can be evaluated as
(27)ρz=zβ101−β3zβ11,φz=zβ121−β3zβ13,β12>0,β13>0,ynlz=Pnlβ10,β111−2β3z,β10>−1,β11>−1,in such a way that the associated wave function in equation (3) becomes
(28)ψnlz=Nnlzβ121−β3zβ13Pnlβ10,β111−2β3z,where Pnlμ,vx, μ>−1, v>−1, and x∈−1,1, are Jacobi polynomials with the following parametric constants:
(29)β10=β1+2β4+2β8−1>−1,β11=β2−2β5+2β9+β3β8>−1,β3≠0,β12=β4+β8>0,β13=β5−β9−β3β8,β3≠0.By considering a special case whereβ3=0, then the associated wave function reduces to the form [39, 40]
(30)limβ3⟶0Pnlβ10,β111−2β3z=Lnlβ10β11z,limβ3⟶01−β3zβ13=eβ13z,ψnlz=Nnlzβ12eβ13zLnlβ10β11z,where Lnlβ10z is well known as a Laguerre polynomial.


## 3. Nonrelativistic Eigensolutions of the Isotropic Oscillator plus Inverse Quadratic Potential in an External Magnetic Field

For a charged particle moving in a uniform magnetic field, the Hamiltonian of the system may be defined ([16] and the references therein) as
(31)H=12μp+ecA2+Vr,where m is the mass of the charged particle, e is the electronic charge, p is the momentum of the charged particle, A=1/2B×r is the vector potential in the symmetric gauge, c is the velocity of light, and Vr is the cylindrical potential representing the potential in equation (1). The Hamiltonian for this system can be evaluated, in the CGS system and in atomic units ℏ=e=1, as
(32)H=12μ−i∇+12B×r2+Vr,and the Schrödinger equation yields
(33)Hφ=12μ−i∇+12B×r2φ+Vrφ=i∂tφ=Eφ.Since this problem involves two dimensions, therefore, it is sufficient enough to study in polar coordinatesr,ϕ within the plane and to employ the following ansatz for the eigenfunction:
(34)φr,ϕ=eimϕ2πRrr,m=0,±1,±2,⋯.Consequently, the radial wave functionRr must satisfy the following radial Schrödinger equation [16, 17]:
(35)d2Rrdr2+2E−VeffrRr=0,with the effective potential Veffr defined as
(36)Veffr=mωL+12ωL2r2+m2−1/42r2+Vr,where ωL=B/2c, m, and E symbolize the Larmor frequency, the eigenvalue of the angular momentum, and the energy spectra of the vibrating molecules, respectively. By using the Vr as the isotropic oscillator plus inverse quadratic potential (IOPIQP), the effective potential influenced by an external magnetic field becomes
(37)Veffr=mωL+m2−1/4/2+gr2+12ωL2+μω2r2,where μ and ω represent the mass and angular frequency of the vibrating molecules bounded by the IOPIQP, and the molecular constants for the selected diatomic molecules in this study are displayed in Table 1. Inserting equation (37) into equation (35) and applying a variable z=r2, equation (35) can be transformed as
(38)R′′z+12zR′z+−ρ1z2+ρ2z−ρ3z2Rz=0.Table 1
Molecular parameters for this study [21].
MoleculesVibrational frequenciesω×1013s−1Massμ in a.m.u.CO6.4716.8606719HCl8.8140.9801045I20.64263.45223502H212.9600.50391Comparing equation (38) with equation (13), we obtain the following analytical expressions:
(39)β1=12,β2=β3=0,ρ1=ωL2+μω24,ρ2=E−mωL2,ρ3=m2+2g−1/44.Using equations (16) and (29), other values of parametric constants βii=4,5,6,⋯ and their analytical values required for the derivation of energy eigenvalues and eigenfunctions are obtained and displayed in Table 2.Table 2
The values of parametric constants required for the derivation of energy eigenvalues and eigenfunctions.
Parametric constantsAnalytical valuesβ414β50β6ωL2+μω24β7mωL−E2β8m2+2g4β9ωL2+μω24β10m2+2g>−1β11ωL2+μω2β121+2m2+2g4>0β13−ωL2+μω24Using the analytical values in Table2 for the parametric constants βii=1,2,3,⋯ in equations (26) and (30), the energy eigenvalues and the normalized radial eigenfunctions for the IOPIQP in the presence of an external magnetic field are obtained, respectively, as
(40)E=mωL+ωL2+μω22n+1+m2+2g,(41)Rr=2n!γ2δ+2n+2δ+1!1/2r2δ+3/2e−1/2γr2Ln2δ+1γr2,where δ=−1/2+m2/4+g/2, γ=ωL2+μω2, and Ln2δ+1γr2 is the associated Laguerre polynomial.


### 3.1. Expectation Values (r−2, p2, T, and V) of the Isotropic Oscillator plus Inverse Quadratic Potential in an External Magnetic Field

The Hellmann-Feynman theorem (HFT) is one of the useful techniques for obtaining expectation values of some quantum mechanical observables for any arbitrary values of quantum numbers [41, 42]. Suppose that the Hamiltonian Hα for a particular quantum mechanical system depends on parameter α such that Enlα and ψnmα are the eigenvalues and the eigenfunctions, respectively. Therefore, the Hellmann-Feynman theorem (HFT) states that
(42)∂Enmα∂α=ψnmα∂Hα∂αψnmα,provided that the normalized eigenfunctions ψnmα are continuous, differentiable with respect to parameter α. The effective Hamiltonian of the isotropic oscillator plus inverse quadratic potential in an external magnetic field is given as
(43)H=−12d2dr2+mωL+m2−1/4/2+gr2+12ωL2+μω2r2.To find the expectation value ofr2, we let α=ω such that equation (42) becomes
(44)∂Enmω∂ω=ψnω∂Hω∂ωψnω.Taking the first derivative of the effective HamiltonianHω in equation (43) with respect to vibrational frequency ω, one obtains
(45)∂Hω∂ω=μωr2.Taking the first derivative of the energy eigenvaluesEω in equation (40) with respect to vibrational frequency ω, we have
(46)∂Enmω∂ω=μω2n+1+m2+2gωL2+μω2.With equations (45) and (46) in equation (44), it is easy for one to evaluate r2 as
(47)r2=2n+1+m2+2gωL2+μω2.In principle and withα=μ in equation (44), the expectation values of p2, T, and V can be obtained, respectively, as
(48)p2=−μ2ω22n+1+m2+2gωL2+μω2,T=−μω22n+1+m2+2g2ωL2+μω2,V=mωL+ωL2+μω22n+1+m2+2g+μω2/22n+1+m2+2gωL2+μω2.


### 3.2. Diamagnetic Susceptibility of the Isotropic Oscillator plus Inverse Quadratic Potential in an External Magnetic Field

The diamagnetic susceptibility is given [3, 21] as
(49)X=−Nze26μc2r2,where N is the Avogadro number, z is the atomic number, e is the electronic charge, c is the speed of light, and μ is the effective mass of the vibrating molecules in this study. It has been found that diamagnetism is a fundamental magnetic phenomenon that explains the tendency of electric charges to partially shield the interior of a body from an external magnetic field and that diamagnetic materials possess magnetic effects due to an external field that alters electron motion within the atoms [21]. Using equation (47) in (49) defines the diamagnetic susceptibility for the isotropic oscillator plus inverse quadratic potential in an external magnetic field as
(50)χ=−Nze26μc22n+1+m2+2gωL2+μω2,where all the symbols have been explained accordingly. The corresponding magnetic moment μB can be expressed as
(51)μB=−e26μc2r2B=−2e2ωL6μc2n+1+m2+2gωL2+μω2.


## 3.1. Expectation Values (r−2, p2, T, and V) of the Isotropic Oscillator plus Inverse Quadratic Potential in an External Magnetic Field

The Hellmann-Feynman theorem (HFT) is one of the useful techniques for obtaining expectation values of some quantum mechanical observables for any arbitrary values of quantum numbers [41, 42]. Suppose that the Hamiltonian Hα for a particular quantum mechanical system depends on parameter α such that Enlα and ψnmα are the eigenvalues and the eigenfunctions, respectively. Therefore, the Hellmann-Feynman theorem (HFT) states that
(42)∂Enmα∂α=ψnmα∂Hα∂αψnmα,provided that the normalized eigenfunctions ψnmα are continuous, differentiable with respect to parameter α. The effective Hamiltonian of the isotropic oscillator plus inverse quadratic potential in an external magnetic field is given as
(43)H=−12d2dr2+mωL+m2−1/4/2+gr2+12ωL2+μω2r2.To find the expectation value ofr2, we let α=ω such that equation (42) becomes
(44)∂Enmω∂ω=ψnω∂Hω∂ωψnω.Taking the first derivative of the effective HamiltonianHω in equation (43) with respect to vibrational frequency ω, one obtains
(45)∂Hω∂ω=μωr2.Taking the first derivative of the energy eigenvaluesEω in equation (40) with respect to vibrational frequency ω, we have
(46)∂Enmω∂ω=μω2n+1+m2+2gωL2+μω2.With equations (45) and (46) in equation (44), it is easy for one to evaluate r2 as
(47)r2=2n+1+m2+2gωL2+μω2.In principle and withα=μ in equation (44), the expectation values of p2, T, and V can be obtained, respectively, as
(48)p2=−μ2ω22n+1+m2+2gωL2+μω2,T=−μω22n+1+m2+2g2ωL2+μω2,V=mωL+ωL2+μω22n+1+m2+2g+μω2/22n+1+m2+2gωL2+μω2.


## 3.2. Diamagnetic Susceptibility of the Isotropic Oscillator plus Inverse Quadratic Potential in an External Magnetic Field

The diamagnetic susceptibility is given [3, 21] as
(49)X=−Nze26μc2r2,where N is the Avogadro number, z is the atomic number, e is the electronic charge, c is the speed of light, and μ is the effective mass of the vibrating molecules in this study. It has been found that diamagnetism is a fundamental magnetic phenomenon that explains the tendency of electric charges to partially shield the interior of a body from an external magnetic field and that diamagnetic materials possess magnetic effects due to an external field that alters electron motion within the atoms [21]. Using equation (47) in (49) defines the diamagnetic susceptibility for the isotropic oscillator plus inverse quadratic potential in an external magnetic field as
(50)χ=−Nze26μc22n+1+m2+2gωL2+μω2,where all the symbols have been explained accordingly. The corresponding magnetic moment μB can be expressed as
(51)μB=−e26μc2r2B=−2e2ωL6μc2n+1+m2+2gωL2+μω2.


## 4. Results and Discussions

In order to verify the reliability, validity, and consistency of our results, using the molecular constants in Table1 [21], we present the computed results for the energy spectra, expectation values r2, p2, T, and V, and diamagnetic susceptibility X for the selected molecules bounded by the isotropic oscillator plus inverse quadratic potential with varying Larmor frequencies ωL for the case of the low vibrational energy level n=0,1,2,3, low rotational energy level m=0,+1, and inverse quadratic potential strength g=0,1 in Tables 3–11. Figures 1–4 show the variation of expectation values for some quantum mechanical observables as a function of Larmor frequencies ωL for various rotational energy levels n=0,1,2,3 with g=m=1 for the selected molecules in Table 2. Figure 5 shows the variation of diamagnetic susceptibility as a function of Larmor frequencies ωL for various rotational energy levels n with g=m=z=e=1 for the molecules in Table 1. In all the calculations, we have also employed the following recently used conversions: 1a.m.u.=1.66×10−27kg, c=3.00×108m/s, and N=6.02×1023 moles. All our results are in their standard units.Table 3
Energy spectra for CO molecules bounded by the isotropic oscillator plus inverse quadratic potential for arbitrary Larmor frequenciesωL..
nm=0,ωL=0m=1,ωL=0m=0,ωL=5m=1,ωL=5m=0,ωL=10m=1,ωL=10g=006.9057213.81148.5257822.051612.152734.3055120.717227.622925.577339.103136.458258.6109234.528641.434342.628956.154760.763782.9164348.348855.245759.680573.206285.0691107.222g=1016.671918.866820.583128.292929.339343.2019130.483332.678237.634645.344453.644867.5074244.294746.489654.686262.396077.950291.8128358.106260.301171.737779.4475102.256116.118Table 4
Energy spectra for HCl molecules bounded by the isotropic oscillator plus inverse quadratic potential for arbitrary Larmor frequenciesωL.
nm=0,ωL=0m=1,ωL=0m=0,ωL=5m=1,ωL=5m=0,ωL=10m=1,ωL=10g=003.555197.110396.135117.270210.613231.2263110.665614.220818.405329.540431.839552.4527217.776021.331230.675541.810653.065973.6790324.886428.441642.945754.080874.292294.9054g=108.58309.7129714.811421.761425.622538.9957115.693416.823427.081634.031646.848860.2221222.803823.933739.351846.301868.075181.4484329.914231.044151.622058.572089.3015102.675Table 5
Energy spectra for I2 molecules bounded by the isotropic oscillator plus inverse quadratic potential for arbitrary Larmor frequencies ωL.
nm=0,ωL=0m=1,ωL=0m=0,ωL=5m=1,ωL=5m=0,ωL=10m=1,ωL=10g=002.083594.167185.4167715.833510.214830.429516.250778.3343716.250326.667130.644350.859210.418012.501527.083837.500651.073871.2886314.585116.668737.917448.334171.503391.7181g=105.030235.6924813.077219.798924.660637.907219.197429.8596623.910830.632445.090158.3368213.364614.026834.744341.465965.519778.7663317.531818.194045.577852.299585.949299.1958Table 6
Energy spectra for H2 molecules bounded by the isotropic oscillator plus inverse quadratic potential for arbitrary Larmor frequencies ωL.
nm=0,ωL=0m=1,ωL=0m=0,ωL=5m=1,ωL=5m=0,ωL=10m=1,ωL=10g=003.748317.496626.2489917.498010.679431.3588111.244914.993218.747029.995932.038252.7176218.741622.489931.244942.493953.397174.0765326.238229.986543.742954.991974.755995.4353g=109.0492210.240615.086422.072525.782439.1767116.545817.737227.584434.570547.141260.5355224.042525.233840.082347.068568.500081.8943331.539132.730452.580359.566589.8588103.253Table 7
Expectation valuesr2 for CO, HCl, I2, and H2 molecules bounded by the isotropic oscillator plus inverse quadratic potential for arbitrary Larmor frequencies ωL with g=m=1.
COHClnωL=0ωL=5ωL=10ωL=0ωL=5ωL=1000.3956220.3204460.2248100.7684670.4453150.25742110.6852370.5550280.3893821.3310200.7713080.44586620.9748520.7896110.5539541.8935801.0973000.63431131.2644701.0241900.7185262.4561401.4232900.822756I2H201.311220.5043690.2674610.7288750.4371990.25582412.271100.8735930.4632561.2624500.7572510.44310023.230981.2428200.6590511.7960201.0773000.63037734.190871.6120400.8548462.3296001.3973500.817653Table 8
Expectation valuesp¯2 for CO, HCl, I2, and H2 molecules bounded by the isotropic oscillator plus inverse quadratic potential for arbitrary Larmor frequencies ωL with g=m=1.
COHClnωL=0ωL=5ωL=10ωL=0ωL=5ωL=100-2.14868e-25-1.74039e-25-1.22098e-25-1.58027e-26-9.15745e-27-5.29359e-271-3.72163e-25-3.01445e-25-2.11479e-25-2.73712e-26-1.58612e-26-9.16877e-272-5.29457e-25-4.28850e-25-3.00861e-25-3.89396e-26-2.25649e-26-1.30440e-263-6.86752e-25-5.56256e-25-3.90242e-25-5.0508e-26-2.92686e-26-1.69191e-26I2H20-5.99593e-25-2.30637e-25-1.22304e-25-8.56614e-27-5.13820e-27-3.00659e-271-1.03852e-24-3.99475e-25-2.11837e-25-1.48370e-26-8.89963e-27-5.20756e-272-1.47746e-24-5.68313e-25-3.01369e-25-2.11078e-26-1.26611e-26-7.40853e-273-1.91639e-24-7.37151e-25-3.90902e-25-2.73787e-26-1.64225e-26-9.60951e-27Table 9
Expectation valuesT¯ for CO, HCl, I2, and H2 molecules bounded by the isotropic oscillator plus inverse quadratic potential for arbitrary Larmor frequencies ωL with g=m=1.
COHClnωL=0ωL=5ωL=10ωL=0ωL=5ωL=100-9.43338-7.64086-5.36046-4.85649-2.81426-1.626821-16.3391-13.2344-9.28460-8.41168-4.87444-2.817742-23.2448-18.8279-13.2087-11.9669-6.93462-4.008663-30.1505-24.4214-17.1329-15.5221-8.99480-5.19957I2H20-2.84624-1.09482-0.580571-5.12029-3.07129-1.797141-4.92983-1.89629-1.00558-8.86860-5.31962-3.112742-7.01342-2.69775-1.43059-12.6169-7.56796-4.428343-9.09701-3.49922-1.85559-16.3652-9.81600-5.74390Table 10
Expectation valuesV¯ for CO, HCl, I2, and H2 molecules bounded by the isotropic oscillator plus inverse quadratic potential for arbitrary Larmor frequencies ωL with g=m=1.
COHClnωL=0ωL=5ωL=10ωL=0ωL=5ωL=10028.300235.933748.562314.569524.575740.6225149.017358.578876.791925.235038.906063.0398269.734581.2238105.02235.900653.236485.4571390.4516103.869133.25146.566267.5668107.874I2H208.5387220.893738.487815.360925.143840.9738114.789532.528759.342326.605839.890163.6483221.040344.163780.196937.850754.636586.3227327.291055.7987101.05149.095769.3828108.997Table 11
Diamagnetic susceptibilitiesX for CO, HCl, I2, and H2 molecules bounded by the isotropic oscillator plus inverse quadratic potential for arbitrary Larmor frequencies ωL with g=m=z=e=1.
COHClnωL=0ωL=5ωL=10ωL=0ωL=5ωL=100-3.87265e+31-3.13677e+31-2.20061e+31-5.26560e+32-3.05133e+32-1.76387e+321-6.70762e+31-5.43304e+31-3.81157e+31-9.12028e+32-5.28506e+32-3.05511e+322-9.54260e+31-7.72932e+31-5.42252e+31-1.29750e+33-7.51880e+32-4.34635e+323-1.23776e+32-1.00256e+32-7.03348e+31-1.68297e+33-9.75253e+32-5.63759e+32I2H20-1.38779e+31-5.33822e+30-2.83079e+30-9.71393e+32-5.82668e+32-3.40944e+321-2.40372e+31-9.24607e+30-4.90308e+30-1.68250e+33-1.00921e+33-5.90532e+322-3.41966e+31-1.31539e+31-6.97536e+30-2.39361e+33-1.43575e+33-8.40121e+323-4.43559e+31-1.70618e+31-9.04765e+30-3.10472e+33-1.86229e+33-1.08971e+33Figure 1
The expectation value of the square of the positionr2 as a function of Larmor frequencies ωL for various numbers of states n with g=m=1.Figure 2
The expectation value of the square of the momentump2 as a function of Larmor frequencies ωL for the molecules (H2, HCl, CO, and I2) and for various numbers of states n with g=m=1.Figure 3
The expectation value of kinetic energyT as a function of Larmor frequencies ωL for various numbers of states n with g=m=1.Figure 4
The expectation value of potential energyV as a function of Larmor frequencies ωL for the molecules (H2, HCl, CO, and I2) and for various numbers of states n with g=m=1.Figure 5
The diamagnetic susceptibilitiesX as a function of Larmor frequencies ωL for the molecules (H2, HCl, CO, and I2) and various numbers of states n with g=m=z=e=1.It was observed that energy spectra increase with the increase in the magnetic field strength (Larmor frequency), vibrational level, and inverse quadratic potential strengthg for all the selected molecules. See Tables 3–6. This observation suggests that the energy spectra of the selected molecules would be affected significantly by the external magnetic field. The expectation value r2which is positive increases with increasing rotational energy level n but decreases monotonically towards zero with the increase in magnetic field strengths for all the selected molecules for g=m=1. See Table 7 and Figure 1. The expectation value p2which is negative decreases with increasing rotational energy level n but increases with the increase in magnetic field strengths and tends to converge at a very high magnetic field strength (Larmor frequencies ωL>10) for all the selected molecules for g=m=1. See Table 8 and Figure 2.The expectation valueTwhich is negative decreases with increasing rotational energy level n but increases monotonically with the increase in magnetic field strengths and tends to converge at a very high magnetic field strength (Larmor frequencies ωL>10) for all the selected molecules for g=m=1. See Table 9 and Figure 3. The expectation value Vwhich is positive increases with increasing rotational energy level n as well as the magnetic field strength. A clear divergence is noticeable at all values of magnetic field strength for all the selected molecules for g=m=1. See Table 10 and Figure 4.The diamagnetic susceptibilityXwhich is negative increases monotonically with increasing rotational energy level n as well as the magnetic field strength and tends to converge at any ωL>10 for all the selected molecules for g=m=z=e=1. See Table 11 and Figure 5.


## 5. Concluding Remarks

We have studied the two-dimensional radial Schrödinger equation with the isotropic harmonic oscillator plus inverse quadratic potential in an external magnetic field via the parametric Nikiforov-Uvarov method. The energy eigenvalue equation, normalized wave function, expressions for expectation values square of the positionr2, square of the momentum p2, kinetic energy T, and potential energy V, and diamagnetic susceptibility X for the interaction potential model have been obtained. The computed results for the energy spectra, expectation values r2, p2, T, and V, and diamagnetic susceptibility X for some diatomic molecules bounded by the isotropic harmonic oscillator plus inverse quadratic potential for low vibrational and rotational levels are found to have strong dependence on the magnetic field strengths as well as the inverse quadratic potential strength g.The expectation valuesr2, p2, and T and diamagnetic susceptibility X as a function of Larmor frequencies ωL for all the molecules tend to converge at a very high magnetic field strength ωL>10. This may signify a case where low vibrational and rotational energy levels cease to have any significant effect on the expectation values of r2, p2, and T and diamagnetic susceptibility X for all the molecules, despite the increasing magnetic field strength. A divergence is noticeable for the expectation value of potential energy V which signifies that the rotational and vibration energy levels will continue to have a meaningful effect as long as the magnetic field strength increases. Also, as the magnetic field strength increases, diamagnetic susceptibility X increases, which may likely increase the tendency of the magnetic field to shield or alter the vibrational motion of the molecules bounded by the potential model in this study.Using the results obtained in this work, one can study the thermodynamic properties of the system, Shannon entropy, and Fisher information for the first and second excited states. And this would be our focus in the subsequent research work.



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