# Approximate Solutions of Schrodinger Equation with Some Diatomic Molecular Interactions Using Nikiforov-Uvarov Method

Authors: Ituen B. Okon; Oyebola Popoola; Cecilia N. Isonguyo
Journal: Advances in High Energy Physics
 (2017)

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## Abstract

We used a tool of conventional Nikiforov-Uvarov method to determine bound state solutions of Schrodinger equation with quantum interaction potential called Hulthen-Yukawa inversely quadratic potential (HYIQP). We obtained the energy eigenvalues and the total normalized wave function. We employed Hellmann-Feynman Theorem (HFT) to compute expectation valuesr-2, r-1, T, and p2 for four different diatomic molecules: hydrogen molecule (H2), lithium hydride molecule (LiH), hydrogen chloride molecule (HCl), and carbon (II) oxide molecule. The resulting energy equation reduces to three well-known potentials which are as follows: Hulthen potential, Yukawa potential, and inversely quadratic potential. The bound state energies for Hulthen and Yukawa potentials agree with the result reported in existing literature. We obtained the numerical bound state energies of the expectation values by implementing MATLAB algorithm using experimentally determined spectroscopic constant for the different diatomic molecules. We developed mathematica programming to obtain wave function and probability density plots for different orbital angular quantum number.


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## Body

## 1. Introduction

The study of diatomic molecules is very significant and applicable in many areas of chemical and physical sciences. The excitation of atoms of some diatomic molecules especially the homonuclear diatomic molecules is the principle used in spectrophotometric technique. Diatomic molecules contains two atoms per molecule and can either be homonuclear if it contains two atoms of the same kind per molecule or be heteronuclear if its contains two atoms of different kind per molecule [1–5]. Bound state solutions of relativistic and nonrelativistic wave equation arouse a lot of interest for decades. Schrodinger wave equations constitute nonrelativistic wave equation while Klein-Gordon and Dirac equations constitute the relativistic wave equations [6–10]. Bound state solutions predominantly have negative energies because the energy of the particle is less than the maximum potential energy [11]. The quantum interaction potential (HYIQP) can be used to compute the bound state energies for both homonuclear and heteronuclear diatomic molecules. Other potentials have been used in studying bound state solutions like the following: Hulthen, Poschl-Teller, Eckart, Coulomb, Hylleraas, pseudoharmonic, and scarf II potentials and many others [6, 12–19]. These potentials are studied with some specific methods and techniques like the following: asymptotic iteration method, Nikiforov-Uvarov method, supersymmetric quantum mechanics approach, formular method, exact quantisation, and many more [19–29]. This article is divided into seven sections. Section 1 is the introduction; Section 2 is the brief introduction of conventional Nikiforov-Uvarov method. In Section 3, we presented the radial solution to Schrodinger wave equation using the proposed potential and obtained both the energy eigenvalue and their corresponding normalized wave function. In Section 4, we have deductions of three well-known potentials from the proposed potential. The numerical results of two of the potentials are compared to that of existing literature. In Section 5, we applied HFT to compute the expectation values for different diatomic molecules. In Section 6, we present numerical results of the expectation values by implementing MATLAB algorithm. Section 7 gives the analytical solutions of finding the normalization constant using confluent hypergeometric function while Section 8 gives the general conclusion of the article.The proposed quantum interaction potential is given by(1)Vr=-V0e-2αr1-e-2αr-Ae-αrr+Br2+C,where V0 is the potential depth, α is the adjustable parameter known as the screening parameter. A, B, and C are all spectroscopic constants which are molecular bond length, molecular constant, and potential range, respectively.Figure1 shows the graph of quantum interaction potential for various values of the screening parameter α which decays exponentially. Meanwhile, Figure 2 shows the graph of individual potentials plotted in the same scale with the quantum interaction potential (HYIQP). From this MATLAB plot, it can be seen that HYIQP is best suitable in describing bound state energies of diatomic molecules. In elementary quantum mechanics, the wave function implicitly described the behaviour of quantum mechanical systems. The developed potential model is used to study the behaviour of four diatomic molecules, namely, hydrogen, carbon (II) oxide, lithium hydride, and hydrogen chloride molecules. The wave function and probability density plots, gotten from mathematica programming as seen in Figures 3–8 for wave function plots and Figure 9 for probability density plots for orbital angular quantum numbers l=0,1,2,3,4, and 5, respectively, is very significant in investigating the behaviour of diatomic molecules. This plot shows that carbon (II) oxide with maximum peak is very flammable and a highly toxic gas though industrially it is a good reducing agent. Lithium hydride possesses similar characteristic being a toxic and poisonous gas. However, the various plot shows that hydrogen chloride and hydrogen molecules are less reactive as compared to lithium hydride and carbon (II) oxide molecules.Figure 1
The graph of the quantum interaction potential (HYIQP) for various value ofα.Figure 2
The graph of the HYIQP as compared to that of individual potentials.F = HYIQP potential, F1 = Hulthen potential, F2 = Yukawa potential, and F3 = inversely quadratic potential.Figure 3
Wave function plot for the interaction potential with orbital angular quantum numberl=0.Figure 4
Wave function plot for the interaction potential with orbital angular quantum numberl=1.Figure 5
Wave function plot for the interaction potential with orbital angular quantum numberl=2.Figure 6
Wave function plot for the interaction potential with orbital angular quantum numberl=3.Figure 7
Wave function plot for the interaction potential with orbital angular quantum numberl=4.Figure 8
Wave function plot for the interaction potential with orbital angular quantum numberl=5.Figure 9
Probability density plots for the interaction potential with orbital angular quantum numberl=0,1,2,3,4, and 5.


## 2. Conventional Nikiforov-Uvarov Method

The NU method is based on reducing second-order linear differential equation to a generalized equation of hypergeometric type [30, 31]. This method provides exact solutions in terms of special orthogonal functions as well as corresponding energy eigenvalues. The NU method is applicable to both relativistic and nonrelativistic equations. With appropriate coordinate transformation S=S(x) the equation can be written as(2)ψ′′s+τ~sσsψ′s+σ~sσ2sψs=0.τ~(s) is a polynomial of degree one while σs and σ~s are polynomials of at most degree two.In order to find the exact solution to (2), we set the wave function as(3)Ψs=ϕsχs and substituting (3) into (2) reduces (2) into hypergeometric type.(4)σsχ′′s+τsχ′s+λχs=0,where the wave function ψ(s) is defined as the logarithmic derivative(5)ϕ1sϕs=πsσs,where π(s) is at most polynomial of degree one.Likewise, the hypergeometric type functionχ(s) in (4) for a fixed n is given by the Rodrigue relation(6)χns=Bnρsdndsnσ′′sρs.Bn is the normalization constant and the weight function ρ(s) satisfies the condition(7)ddsσsρs=τsρssuch that(8)τs=τ~s+2πs.In order to accomplish the conditions imposed on the weight function ρ(s), it is necessary that the classical orthogonal polynomials τ(s) be equal to zero and its derivative be less than zero; that is,(9)dτsds<0.Therefore, the function π(s) and the parameter λ required for the NU method are defined as follows:(10)πs=σ′-τ~2±σ′-τ~22-σ~+kσ,(11)λ=k+π′s.The k-values in (10) are possible to evaluate if the expression under the square root must be square of polynomials. This is possible, if and only if its discriminant is zero. With this, a new eigenvalue equation becomes(12)λ=λn=ndτds-nn-12d2σds2,n=0,1,2,…,where τ(s) is as defined in (8) and, on comparing (11) and (12), we obtain the energy eigenvalues.


## 3. Radial Solution of Schrodinger Equation

The Schrodinger wave equation is given by(13)d2ψrdr2+2μħ2E-Vr-ħ2ll+12μr2ψr=0.Substituting (1) into (13) gives(14)d2ψrdr2+2μħ2E+V0e-2αr1-e-2αr+Ae-αrr-Br2-C-ħ2ll+12μr2ψr=0.Equation (14) can only be solved analytically to obtain exact solution if the angular orbital momentum number l=0. However, for l>0 (14) can only be solve by using some approximations to the centrifugal term. Greene Aldrich approximation is best suitable for (14).Let us define Greene Aldrich approximation as(15)1r2=4α2e-2αr1-e-2αr2⟹1r=2αe-αr1-e-2αr.Substituting (15) into (14) with the transformation s=e-2αr gives(16)d2ψsds2+1-ss1-sdψsds+1s21-s2-ε21-s2+δ2s1-s+σ1s1-s-σ2s-σ31-s2-sll+1ψs=0,where (17)ε2=-μE2ħ2α2,δ2=μV02ħ2α2,σ1=μAħ2α,σ2=2μBħ2,σ3=μc2ħ2α2.Simplifying (16) further reduced to (18)d2ψsds2+1-ss1-sdψsds+1s21-s2-ε2+δ2+σ1+σ3s2+2ε2+δ2+σ1-σ2+2σ3-ll+1s-ε2+σ3ψs=0.Comparing (18) to (2) we obtained the following:(19)σs=s1-s,σ~s=-ε2+δ2+σ1+σ3s2+2ε2+δ2+σ1-σ2+2σ3-ll+1s-ε2+σ3,τ~s=1-s.Then, using (10), the polynomial equation π(s) then becomes(20)πs=-s2±14+ε2+δ2+σ1+σ3-ks2-2ε2+δ2+σ1-σ2+2σ3-ll+1-ks-ε2+σ3To find the value of k we consider the discriminant such that b2-4ac=0(21)a=14+ε2+δ2+σ1+σ3-k,b=-2ε2+δ2+σ1-σ2+2σ3-ll+1-k,c=-ε2+σ3.Hence, (22)k1=-σ2-σ1-δ2+ll+1+ε2+σ34σ2+4ll+1+1,k2=-σ2-σ1-δ2+ll+1-ε2+σ34σ2+4ll+1+1.Substituting the values of k1 and k2 into (20), then the four values of π(s) are given below.22aπs=-s2±124σ2+4ll+1+1+ε2+σ3s-ε2+σ3,k1=-σ2-σ1-δ2+ll+1+ε2+σ34σ2+4ll+1+1124σ2+4ll+1+1-ε2+σ3s-ε2+σ3,k2=-σ2-σ1-δ2+ll+1-ε2+σ34σ2+4ll+1+1.π(s) has four solutions and one of the solutions satisfied bound state condition which is22bπs=-s2-124σ2+4ll+1+1-ε2+σ3s-ε2+σ3.dτ(s)/ds<0 is the condition for bound state solution. Using (8) we have it that(23)τs=1-2s-2124σ2+4ll+1+1-ε2+σ3s-ε2+σ3such that (24)τ′s=-2-2124σ2+4ll+1+1-ε2+σ3<0which satisfies the bound state condition. However using (11) (25)λ=k+π′s=-12-124σ2+4ll+1+1-ε2+σ3+ε2+σ34σ2+4ll+1+1-σ2-σ1-δ2+ll+1.Using (12)(26)λn=n2+n+n4σ2+4ll+1+1-2nε2+σ3.Equating (25) and (26) gives the result(27)ε2=n2+n+1/2+n+1/24σ2+4ll+1+1+σ2-σ1-δ2+ll+12n+1+4σ2+4ll+1+12-σ3.Substituting parameters of (17) into (27) gives(28)Enl=-2ħ2α2μ2μB/ħ2-μA/ħ2α-μV0/2ħ2α2+ll+1+n2+n+1/2+n+1/28μB/ħ2+4ll+1+11+2n+8μB/ħ2+4ll+1+12+cEquation (28) is the energy for the combined potential.


### 3.1. Calculation of the Wave Function

By using (5), but(29)σs=s-s2,πs=-s2-124σ2+4ll+1+1+ε2+σ3s-ε2+σ3,then(30)πsσs=-121-s-1s1-s124σ2+4ll+1+1+ε2+σ3s-ε2+σ3,(31)ϕ1sϕs=πsσs=-121-s-1s1-s124σ2+4ll+1+1+ε2+σ3s-ε2+σ3,(32)ϕ1sϕs=-121-s-1s1-s124σ2+4ll+1+1+ε2+σ3s-ε2+σ3.Taking integral of (32) gives(33)∫ϕ1sϕsds=-12∫11-sds-124σ2+4ll+1+1∫11-sds-ε2+σ3∫11-sds+ε2+σ3∫1s+11-sdswhich then leads to(34)ln⁡ϕs=ln⁡1-s1/2+ln⁡1-s1/24σ2+4ll+1+1+ln⁡1-sε2+σ3+ln⁡sε2+σ3-ln⁡1-sε2+σ3.Equation (34) can further be reduced to(35)ln⁡ϕs=ln⁡1-s1/2+1/24σ2+4ll+1+1·sε2+σ3.The integration constant is ignored since the final equation is to be expressed in terms of Rodrique relation with normalization constant. Taking exponent of (35) gives(36)ϕs=1-s1/2+1/24σ2+4ll+1+1sε2+σ3.Equation (36) gives the first part of the wave function. To determine the second part of the wave function, we first of all calculate the weight function.


### 3.2. Calculation of Weight Function

Using (7)(37)ddsσsρs=τsρs⟹∫ρ′sρsds=∫τs-σ′sσsds.Substituting the parameters gives(38)ln⁡ρs=∫1-2s-21/24σ2+4ll+1+1+ε2+σ3s+2ε2+σ3-1+2ss1-sds.Integrating (38) gives(39)ρs=1-s-2ε2+σ3-4σ2+4ll+1+1s4ε2+σ3.Rewriting (39) in its Rodrigue form by making use of (6) gives(40)Yns=Bnss-4ε2+σ31-s--2ε2+σ3-4σ2+4ll+1+1×dndsnsn+4ε2+σ31-sn+-2ε2+σ3-4σ2+4ll+1+1.Let us define standard associated Laguerre polynomial as(41)χns=Bnss-υ1-s-μdndsnsn+υ1-sn+μ=Pnμ+υ,μ1-s.Then rewriting (40) in terms of (41) gives the second part of the wave function as(42)Yns=BnsPn2ε2+σ3-44σ2+4ll+1+1,-2ε2+σ3-44σ2+4ll+1+11-s.Hence the total wave function is given by(43)Ψns=ϕsYns=BnsPn2ε2+σ3-44σ2+4ll+1+1,-2ε2+σ3-44σ2+4ll+1+11-s×1-s1/2+1/24σ2+4ll+1+1sε2+σ3.Substituting (17) into (43) reduced it to(44a)Ψns=ϕsYns=BnsPn2μc/2ħ2α2-μE/2ħ2α2-48μB/ħ2+4ll+1+1,-2μc/2ħ2α2-μE/2ħ2α2-48μB/ħ2+4ll+1+11-e-2αr×1-e-2αr1/2+1/28μB/ħ2+4ll+1+1e-2αrμc/2ħ2α2-μE/2ħ2α2.Equation 44 is the total wave function for the proposed potential. Equation 44 can further be reduced to(44b)Ψns=ϕsYns=BnsPn2μc/2ħ2α2-μE/2ħ2α2-48μB/ħ2+4ll+1+1,-2μc/2ħ2α2-μE/2ħ2α2-48μB/ħ2+4ll+1+11-s×1-s1/2+1/28μB/ħ2+4ll+1+1sμc/2ħ2α2-μE/2ħ2α2.The normalized wave function of (44b) is given as(44c)Ψns=1s1sPn2μc/2ħ2α2-μE/2ħ2α2-48μB/ħ2+4ll+1+1,-2μc/2ħ2α2-μE/2ħ2α2-48μB/ħ2+4ll+1+11-s×1-s1/2+1/28μB/ħ2+4ll+1+1sμc/2ħ2α2-μE/2ħ2α2,where the normalization constant is given asBn=1/s1, where(44d)s1=--1nΓn-2ξ1-4∧+12Γn+2ξ1-4∧+122αΓn+1Γ1-2ξ1-4∧×Γn+r+1-8∧Γn+1-8∧∑p=0n∑r=0n-1p+r2-n-r2-spp!n-p!r!n-r!Γn+2ξ1-4∧-p+1Γp-2ξ1-4∧+1Γr+2ξ1-4∧+1×Γα0+1Γ∧+2α0Γα0+∧+2.The detail of the analytical solution to the normalization constant is given in Section 7.


## 3.1. Calculation of the Wave Function

By using (5), but(29)σs=s-s2,πs=-s2-124σ2+4ll+1+1+ε2+σ3s-ε2+σ3,then(30)πsσs=-121-s-1s1-s124σ2+4ll+1+1+ε2+σ3s-ε2+σ3,(31)ϕ1sϕs=πsσs=-121-s-1s1-s124σ2+4ll+1+1+ε2+σ3s-ε2+σ3,(32)ϕ1sϕs=-121-s-1s1-s124σ2+4ll+1+1+ε2+σ3s-ε2+σ3.Taking integral of (32) gives(33)∫ϕ1sϕsds=-12∫11-sds-124σ2+4ll+1+1∫11-sds-ε2+σ3∫11-sds+ε2+σ3∫1s+11-sdswhich then leads to(34)ln⁡ϕs=ln⁡1-s1/2+ln⁡1-s1/24σ2+4ll+1+1+ln⁡1-sε2+σ3+ln⁡sε2+σ3-ln⁡1-sε2+σ3.Equation (34) can further be reduced to(35)ln⁡ϕs=ln⁡1-s1/2+1/24σ2+4ll+1+1·sε2+σ3.The integration constant is ignored since the final equation is to be expressed in terms of Rodrique relation with normalization constant. Taking exponent of (35) gives(36)ϕs=1-s1/2+1/24σ2+4ll+1+1sε2+σ3.Equation (36) gives the first part of the wave function. To determine the second part of the wave function, we first of all calculate the weight function.


## 3.2. Calculation of Weight Function

Using (7)(37)ddsσsρs=τsρs⟹∫ρ′sρsds=∫τs-σ′sσsds.Substituting the parameters gives(38)ln⁡ρs=∫1-2s-21/24σ2+4ll+1+1+ε2+σ3s+2ε2+σ3-1+2ss1-sds.Integrating (38) gives(39)ρs=1-s-2ε2+σ3-4σ2+4ll+1+1s4ε2+σ3.Rewriting (39) in its Rodrigue form by making use of (6) gives(40)Yns=Bnss-4ε2+σ31-s--2ε2+σ3-4σ2+4ll+1+1×dndsnsn+4ε2+σ31-sn+-2ε2+σ3-4σ2+4ll+1+1.Let us define standard associated Laguerre polynomial as(41)χns=Bnss-υ1-s-μdndsnsn+υ1-sn+μ=Pnμ+υ,μ1-s.Then rewriting (40) in terms of (41) gives the second part of the wave function as(42)Yns=BnsPn2ε2+σ3-44σ2+4ll+1+1,-2ε2+σ3-44σ2+4ll+1+11-s.Hence the total wave function is given by(43)Ψns=ϕsYns=BnsPn2ε2+σ3-44σ2+4ll+1+1,-2ε2+σ3-44σ2+4ll+1+11-s×1-s1/2+1/24σ2+4ll+1+1sε2+σ3.Substituting (17) into (43) reduced it to(44a)Ψns=ϕsYns=BnsPn2μc/2ħ2α2-μE/2ħ2α2-48μB/ħ2+4ll+1+1,-2μc/2ħ2α2-μE/2ħ2α2-48μB/ħ2+4ll+1+11-e-2αr×1-e-2αr1/2+1/28μB/ħ2+4ll+1+1e-2αrμc/2ħ2α2-μE/2ħ2α2.Equation 44 is the total wave function for the proposed potential. Equation 44 can further be reduced to(44b)Ψns=ϕsYns=BnsPn2μc/2ħ2α2-μE/2ħ2α2-48μB/ħ2+4ll+1+1,-2μc/2ħ2α2-μE/2ħ2α2-48μB/ħ2+4ll+1+11-s×1-s1/2+1/28μB/ħ2+4ll+1+1sμc/2ħ2α2-μE/2ħ2α2.The normalized wave function of (44b) is given as(44c)Ψns=1s1sPn2μc/2ħ2α2-μE/2ħ2α2-48μB/ħ2+4ll+1+1,-2μc/2ħ2α2-μE/2ħ2α2-48μB/ħ2+4ll+1+11-s×1-s1/2+1/28μB/ħ2+4ll+1+1sμc/2ħ2α2-μE/2ħ2α2,where the normalization constant is given asBn=1/s1, where(44d)s1=--1nΓn-2ξ1-4∧+12Γn+2ξ1-4∧+122αΓn+1Γ1-2ξ1-4∧×Γn+r+1-8∧Γn+1-8∧∑p=0n∑r=0n-1p+r2-n-r2-spp!n-p!r!n-r!Γn+2ξ1-4∧-p+1Γp-2ξ1-4∧+1Γr+2ξ1-4∧+1×Γα0+1Γ∧+2α0Γα0+∧+2.The detail of the analytical solution to the normalization constant is given in Section 7.


## 4. Deductions from the Proposed Potential

### 4.1. Hulthen Potential

SettingA=B=C=0 in (1) results in Hulthen potential given as(45)Vr=-V0e-2αr1-e-2αr.The energy of this potential is given as(46)Enl=-2ħ2α2μ-μV0/2ħ2α2+ll+1+n2+n+1/2+n+1/24ll+1+11+2n+4ll+1+12.However(4l(l+1)+1)=2l+1, and then (46) becomes (47a)Enl=-2ħ2α2μ-μV0/2ħ2α2+n+ln+l+2+12n+l+12.According to Arda and Sever, 2012, the standard Hulthen potential is given by(47b)Enl=-β2ħ22mn+ln+l+2+1-2mV0/β2α22n+l+12.Settingα=β/2 and μ=m in (47a) results in (47b). Equation (47b) is computed from parameters taken from Arda and Sever, 2012 [32], for comparison. However, the adopted parameters are V0=β=δ and μ=1.0. Table 18 shows the bound state solution of Hulthen potential of the present work in comparison to other works reported in existing literature.


### 4.2. Yukawa Potential

SettingV0=B=C=0 in (1) then the potential reduced to Yukawa potential.(48)Vr=-Ae-αrr.The corresponding energy equation for this potential is given as(49a)Enl=-2ħ2α2μ-μA/ħ2α+ll+1+n2+n+1/2+n+1/24ll+1+11+2n+4ll+1+12.Equation (49a) can further be simplified as (49b)Enl=-ħ2α2μμA/ħ2α-n+12-l2n+l+2n+l+12.Hamzavi et al. [33] present the eigenenergy equation for Yukawa potential as (49c)Enl=-a22mmV0/a-n+12-l2n+l+2n+l+12.Equations (49b) and (49c) are the same with just variation in parameter where (50)Enl=a=α,2m=μ,A=V0,ħ=1.


### 4.3. Inversely Quadratic Potential

SettingV0=A=C=0 in (1) then the potential reduced to inversely quadratic potential. (51)Vr=Br2.The corresponding energy for this potential is given as(52)Enl=-2ħ2α2μ2μB/ħ2+ll+1+n2+n+1/2+n+1/28μB/ħ2+4ll+1+11+2n+8μB/ħ2+4ll+1+12.


## 4.1. Hulthen Potential

SettingA=B=C=0 in (1) results in Hulthen potential given as(45)Vr=-V0e-2αr1-e-2αr.The energy of this potential is given as(46)Enl=-2ħ2α2μ-μV0/2ħ2α2+ll+1+n2+n+1/2+n+1/24ll+1+11+2n+4ll+1+12.However(4l(l+1)+1)=2l+1, and then (46) becomes (47a)Enl=-2ħ2α2μ-μV0/2ħ2α2+n+ln+l+2+12n+l+12.According to Arda and Sever, 2012, the standard Hulthen potential is given by(47b)Enl=-β2ħ22mn+ln+l+2+1-2mV0/β2α22n+l+12.Settingα=β/2 and μ=m in (47a) results in (47b). Equation (47b) is computed from parameters taken from Arda and Sever, 2012 [32], for comparison. However, the adopted parameters are V0=β=δ and μ=1.0. Table 18 shows the bound state solution of Hulthen potential of the present work in comparison to other works reported in existing literature.


## 4.2. Yukawa Potential

SettingV0=B=C=0 in (1) then the potential reduced to Yukawa potential.(48)Vr=-Ae-αrr.The corresponding energy equation for this potential is given as(49a)Enl=-2ħ2α2μ-μA/ħ2α+ll+1+n2+n+1/2+n+1/24ll+1+11+2n+4ll+1+12.Equation (49a) can further be simplified as (49b)Enl=-ħ2α2μμA/ħ2α-n+12-l2n+l+2n+l+12.Hamzavi et al. [33] present the eigenenergy equation for Yukawa potential as (49c)Enl=-a22mmV0/a-n+12-l2n+l+2n+l+12.Equations (49b) and (49c) are the same with just variation in parameter where (50)Enl=a=α,2m=μ,A=V0,ħ=1.


## 4.3. Inversely Quadratic Potential

SettingV0=A=C=0 in (1) then the potential reduced to inversely quadratic potential. (51)Vr=Br2.The corresponding energy for this potential is given as(52)Enl=-2ħ2α2μ2μB/ħ2+ll+1+n2+n+1/2+n+1/28μB/ħ2+4ll+1+11+2n+8μB/ħ2+4ll+1+12.


## 5. Hellmann-Feynman Theorem

Hellmann-Feynman theorem (HFT) is commonly used in the calculation of intermolecular forces in molecules [34]. However, in order to engage HFT in calculating the expectation values, one then needs to promote the fixed parameters which appear in the Hamiltonian to be continuous variable in order to ease the mathematical purpose of taking the derivatives. This theorem states that if Hamiltonian H for a particular quantum mechanical system is given as a function of some parameters q, then let E(q) and Ψ(q) be the eigenvalues and the eigenfunctions of Hamiltonian H(q), respectively; then (53)∂Enq∂q=Ψq∂Hq∂qΨq.However, the Hamiltonian which contains the effective potential can be expressed as(54)H=-ħ22μd2dr2+ħ2ll+12μr2-v0e-2αr1-e-2αr-Ae-αrr+Br2+C.Let us recall the energy equation as given in (28).


### 5.1. Expectation Value ofr-2

Substitutingq=l into (53) then(55)∂Enl∂l=Ψl∂Hl∂lΨl.Taking the partial derivative of (28) with respect to l gives (56)∂Enl∂l=-4ħ2α2μ1+2n+8μBħ2+4ll+1+12l+1+4n+22l+1-2μB/ħ2-μA/ħ2α-μV0/2ħ2α2+ll+14l+28μB/ħ2+4ll+1+1+n2+n+1/24l+28μB/ħ2+4ll+1+1+n+124l+2×1+2n+8μBħ2+4ll+1+12-1×2μB/ħ2-μA/ħ2α-μV0/2ħ2α2+ll+1+n2+n+1/2+n+1/28μB/ħ2+4ll+1+11+2n+8μB/ħ2+4ll+1+1.Taking the partial derivative of (54) with respect to l gives(57)Ψl∂Hl∂lΨl=ħ22μ2l+1r-2.Equating (56) to (57) gives the expectation values of r-2 for different orbital quantum number. Hence,(58)r-2=-8α22l+11+2n+8μBħ2+4ll+1+12l+1+4n+22l+1-2μB/ħ2-μA/ħ2α-μV0/2ħ2α2+ll+14l+28μB/ħ2+4ll+1+1+n2+n+1/24l+28μB/ħ2+4ll+1+1+n+124l+2×1+2n+8μBħ2+4ll+1+12-1×2μB/ħ2-μA/ħ2α-μV0/2ħ2α2+ll+1+n2+n+1/2+n+1/28μB/ħ2+4ll+1+11+2n+8μB/ħ2+4ll+1+1.


### 5.2. Expectation Value ofr-1

Taking the partial derivative of (28) with respect to A gives (59)∂Enl∂A=4α1+2n+8μB/ħ2+4ll+1+122μBħ2-μAħ2α-μV02ħ2α2+ll+1+n2+n+12+n+128μBħ2+4ll+1+1.Also, taking the partial derivative of (54) with respect to A gives(60)ΨA∂HA∂AΨA=-e-αrr-1.Equating (59) to (60) then gives (61)r-1=-4αeαr1+2n+8μB/ħ2+4ll+1+122μBħ2-μAħ2α-μV02ħ2α2+ll+1+n2+n+12+n+128μBħ2+4ll+1+1.


### 5.3. Expectation Values forT and p2

Taking the partial derivative of (28) with respect to μ, this then gives(62)∂Enμ∂μ=-4ħ2α2μ1+2n+8μBħ2+4ll+1+12Bħ2-Aħ2α-v02ħ2α2+n+1/24Bħ28μB/ħ2+4ll+1+1-2μB/ħ2-μA/ħ2α-μV0/2ħ2α2+ll+14Bħ28μB/ħ2+4ll+1+1+n2+n+1/24Bħ28μB/ħ2+4ll+1+1+n+1/24Bħ2×1+2n+8μBħ2+4ll+1+12-1×2μB/ħ2-μA/ħ2α-μV0/2ħ2α2+ll+1+n2+n+1/2+n+1/28μB/ħ2+4ll+1+11+2n+8μB/ħ2+4ll+1+1+2ħ2α2μ22μB/ħ2-μA/ħ2α-μV0/2ħ2α2+ll+1+n2+n+1/2+n+1/28μB/ħ2+4ll+1+11+2n+8μB/ħ2+4ll+1+12.However, taking the partial derivative of (54) with respect to μ gives(63)Ψμ∂HA∂AΨμ=-ħ22μ2d2dr2-ħ22μ2ll+1r2=1μ-ħ22μd2dr2-ħ22μll+1r2⟹-1μħ22μd2dr2+ħ22μll+1r2⟹-1μT=-1μH-V.Hence,(64)Ψμ∂HA∂AΨμ=-1μT⟹-1μH-V.From the relation T=p2/2μ, substituting for T in (64) gives(65)Ψμ∂HA∂AΨμ=-1μH-V=-12μ2p2.Equating (62) to (64) gives the expectation value of 〈T〉. Therefore,(66)T=4ħ2α21+2n+8μBħ2+4ll+1+12Bħ2-Aħ2α-v02ħ2α2+n+1/24Bħ28μB/ħ2+4ll+1+1-2μB/ħ2-μA/ħ2α-μV0/2ħ2α2+ll+14Bħ28μB/ħ2+4ll+1+1+n2+n+1/24Bħ28μB/ħ2+4ll+1+1+n+1/24Bħ2×1+2n+8μBħ2+4ll+1+12-1×2μB/ħ2-μA/ħ2α-μV0/2ħ2α2+ll+1+n2+n+1/2+n+1/28μB/ħ2+4ll+1+11+2n+8μB/ħ2+4ll+1+1-2ħ2α2μ2μB/ħ2-μA/ħ2α-μV0/2ħ2α2+ll+1+n2+n+1/2+n+1/28μB/ħ2+4ll+1+11+2n+8μB/ħ2+4ll+1+12.Also, equating (62) to (65) gives 〈p2〉. Thus,(67)p2=8ħ2α2μ1+2n+8μBħ2+4ll+1+12Bħ2-Aħ2α-v02ħ2α2+n+1/24Bħ28μB/ħ2+4ll+1+1-2μB/ħ2-μA/ħ2α-μV0/2ħ2α2+ll+14Bħ28μB/ħ2+4ll+1+1+n2+n+1/24Bħ28μB/ħ2+4ll+1+1+n+1/24Bħ2×1+2n+8μBħ2+4ll+1+12-1×2μBħ2-μAħ2α-μV02ħ2α2+ll+1+n2+n+12+n+128μBħ2+4ll+1+1×1+2n+8μBħ2+4ll+1+1-1-4ħ2α22μBħ2-μAħ2α-μV02ħ2α2+ll+1+n2+n+12+n+128μBħ2+4ll+1+1×1+2n+8μBħ2+4ll+1+1-12.


## 5.1. Expectation Value ofr-2

Substitutingq=l into (53) then(55)∂Enl∂l=Ψl∂Hl∂lΨl.Taking the partial derivative of (28) with respect to l gives (56)∂Enl∂l=-4ħ2α2μ1+2n+8μBħ2+4ll+1+12l+1+4n+22l+1-2μB/ħ2-μA/ħ2α-μV0/2ħ2α2+ll+14l+28μB/ħ2+4ll+1+1+n2+n+1/24l+28μB/ħ2+4ll+1+1+n+124l+2×1+2n+8μBħ2+4ll+1+12-1×2μB/ħ2-μA/ħ2α-μV0/2ħ2α2+ll+1+n2+n+1/2+n+1/28μB/ħ2+4ll+1+11+2n+8μB/ħ2+4ll+1+1.Taking the partial derivative of (54) with respect to l gives(57)Ψl∂Hl∂lΨl=ħ22μ2l+1r-2.Equating (56) to (57) gives the expectation values of r-2 for different orbital quantum number. Hence,(58)r-2=-8α22l+11+2n+8μBħ2+4ll+1+12l+1+4n+22l+1-2μB/ħ2-μA/ħ2α-μV0/2ħ2α2+ll+14l+28μB/ħ2+4ll+1+1+n2+n+1/24l+28μB/ħ2+4ll+1+1+n+124l+2×1+2n+8μBħ2+4ll+1+12-1×2μB/ħ2-μA/ħ2α-μV0/2ħ2α2+ll+1+n2+n+1/2+n+1/28μB/ħ2+4ll+1+11+2n+8μB/ħ2+4ll+1+1.


## 5.2. Expectation Value ofr-1

Taking the partial derivative of (28) with respect to A gives (59)∂Enl∂A=4α1+2n+8μB/ħ2+4ll+1+122μBħ2-μAħ2α-μV02ħ2α2+ll+1+n2+n+12+n+128μBħ2+4ll+1+1.Also, taking the partial derivative of (54) with respect to A gives(60)ΨA∂HA∂AΨA=-e-αrr-1.Equating (59) to (60) then gives (61)r-1=-4αeαr1+2n+8μB/ħ2+4ll+1+122μBħ2-μAħ2α-μV02ħ2α2+ll+1+n2+n+12+n+128μBħ2+4ll+1+1.


## 5.3. Expectation Values forT and p2

Taking the partial derivative of (28) with respect to μ, this then gives(62)∂Enμ∂μ=-4ħ2α2μ1+2n+8μBħ2+4ll+1+12Bħ2-Aħ2α-v02ħ2α2+n+1/24Bħ28μB/ħ2+4ll+1+1-2μB/ħ2-μA/ħ2α-μV0/2ħ2α2+ll+14Bħ28μB/ħ2+4ll+1+1+n2+n+1/24Bħ28μB/ħ2+4ll+1+1+n+1/24Bħ2×1+2n+8μBħ2+4ll+1+12-1×2μB/ħ2-μA/ħ2α-μV0/2ħ2α2+ll+1+n2+n+1/2+n+1/28μB/ħ2+4ll+1+11+2n+8μB/ħ2+4ll+1+1+2ħ2α2μ22μB/ħ2-μA/ħ2α-μV0/2ħ2α2+ll+1+n2+n+1/2+n+1/28μB/ħ2+4ll+1+11+2n+8μB/ħ2+4ll+1+12.However, taking the partial derivative of (54) with respect to μ gives(63)Ψμ∂HA∂AΨμ=-ħ22μ2d2dr2-ħ22μ2ll+1r2=1μ-ħ22μd2dr2-ħ22μll+1r2⟹-1μħ22μd2dr2+ħ22μll+1r2⟹-1μT=-1μH-V.Hence,(64)Ψμ∂HA∂AΨμ=-1μT⟹-1μH-V.From the relation T=p2/2μ, substituting for T in (64) gives(65)Ψμ∂HA∂AΨμ=-1μH-V=-12μ2p2.Equating (62) to (64) gives the expectation value of 〈T〉. Therefore,(66)T=4ħ2α21+2n+8μBħ2+4ll+1+12Bħ2-Aħ2α-v02ħ2α2+n+1/24Bħ28μB/ħ2+4ll+1+1-2μB/ħ2-μA/ħ2α-μV0/2ħ2α2+ll+14Bħ28μB/ħ2+4ll+1+1+n2+n+1/24Bħ28μB/ħ2+4ll+1+1+n+1/24Bħ2×1+2n+8μBħ2+4ll+1+12-1×2μB/ħ2-μA/ħ2α-μV0/2ħ2α2+ll+1+n2+n+1/2+n+1/28μB/ħ2+4ll+1+11+2n+8μB/ħ2+4ll+1+1-2ħ2α2μ2μB/ħ2-μA/ħ2α-μV0/2ħ2α2+ll+1+n2+n+1/2+n+1/28μB/ħ2+4ll+1+11+2n+8μB/ħ2+4ll+1+12.Also, equating (62) to (65) gives 〈p2〉. Thus,(67)p2=8ħ2α2μ1+2n+8μBħ2+4ll+1+12Bħ2-Aħ2α-v02ħ2α2+n+1/24Bħ28μB/ħ2+4ll+1+1-2μB/ħ2-μA/ħ2α-μV0/2ħ2α2+ll+14Bħ28μB/ħ2+4ll+1+1+n2+n+1/24Bħ28μB/ħ2+4ll+1+1+n+1/24Bħ2×1+2n+8μBħ2+4ll+1+12-1×2μBħ2-μAħ2α-μV02ħ2α2+ll+1+n2+n+12+n+128μBħ2+4ll+1+1×1+2n+8μBħ2+4ll+1+1-1-4ħ2α22μBħ2-μAħ2α-μV02ħ2α2+ll+1+n2+n+12+n+128μBħ2+4ll+1+1×1+2n+8μBħ2+4ll+1+1-12.


## 6. Numerical Solutions for the Different Expectation Values

MATLAB algorithm is developed to solve numerically equations (58), (61), (66), and (67) for expectation values of 〈r-2〉,〈r-1〉,〈T〉, and 〈p2〉, respectively, for the different diatomic molecules. The diatomic molecules considered in this work are hydrogen, lithium hydride, hydrogen chloride, and carbon (ii) oxide. Table 1 is the spectroscopic constant used in the numerical computation of the expectation values. Tables 2–17 show the numerical expectation values of 〈r-2〉,〈r-1〉,〈T〉, and 〈p2〉, respectively, for the four different diatomic molecules.. These tables show that some of the expectation values increase with an increase in quantum state while some decrease with an increase in quantum state. Tables 2–5 show the expectation values of 〈r-2〉 for the diatomic molecules which decrease with an increase in quantum state for LiH but increase with an increase in quantum state for H2, HCl, and CO, respectively. The expectation value 〈r-1〉 decreases with an increase in quantum state for all the diatomic molecules as shown in Tables 6–9. The expectation value 〈T〉 decreases with an increase in quantum state for H2 and LiH, respectively, but increases with an increase in quantum state for HCl and CO as shown in Tables 10–13. Finally, the expectation value 〈p2〉 increases with an increase in quantum state for H2 and CO but decreases with an increase in quantum state for LiH and HCl as shown in Tables 14–17, respectively.Table 1
Molecular parameter using Nikiforov-Uvarov method.
Molecule
AÅ
B1/Å
CÅ
α
μ
a
m
u
Hydrogen(H2)
0.7416
1.9426
1.440558
0.20990
0.5039100
Lithium hydrideLiH
1.5956
1.1280
1.7998368
1.55000
0.8801221
Hydrogen chlorideHCl
1.2746
1.8677
2.38057
0.20039
0.9801045
Carbon (II) oxideCO
1.1283
2.2994
2.59441
0.39000
6.8606719Table 2
Expectation values ofr-2 for hydrogen molecule.
n
l
r
-
2
n
l
Å
-
2
0
0
−2.0357926925
1
0
−0.3696297856
2
0
−0.0592081872
3
0
−0.0012360898
4
0
−0.0104609211
5
0
−0.0408431125
6
0
−0.0781701991
7
0
−0.1174681786
8
0
−0.1568879315
0
1
−0.5500657326
1
1
−0.1011084016
2
1
−0.0093831431
3
1
−0.0017047903
4
1
−0.0190389837
5
1
−0.0445644291
6
1
−0.0726573984
7
1
−0.1012658894
8
1
−0.1296232040
0
2
−0.1008800199
1
2
−0.0120707336
2
2
−0.0002713032
3
2
−0.0106774421
4
2
−0.0282925702
5
2
−0.0482198274
6
2
−0.0686930232
7
2
−0.0890515939
8
2
−0.1090597335
0
3
−0.0119105671
1
3
−0.0000320975
2
3
−0.0070284422
3
3
−0.0201151142
4
3
−0.0351718504
5
3
−0.0507225834
6
3
−0.0662155593
7
3
−0.0814518703
8
3
−0.0963742259Table 3
Expectation values ofr-2 for lithium hydride molecule.
n
l
r-2nlÅ-2
0
0
−2.5067716077
1
0
−4.3311151186
2
0
−6.0543340693
3
0
−7.7327455626
4
0
−9.3878653299
5
0
−11.0294166762
6
0
−12.6623937821
7
0
−14.2896150748
8
0
−15.9127886181
0
1
−2.5804875135
1
1
−3.8567374113
2
1
−5.0875482240
3
1
−6.2962032770
4
1
−7.4925183938
5
1
−8.6812844732
6
1
−9.8651049570
7
1
−11.0455127066
8
1
−12.2234677505
0
2
−2.5968121549
1
2
−3.4838614479
2
2
−4.3529311128
3
2
−5.2122003606
4
2
−6.0655656768
5
2
−6.9151070951
6
2
−7.7620329080
7
2
−8.6070919545
8
2
−9.4507726300
0
3
−2.5853004074
1
3
−3.2472911138
2
3
−3.9014155691
3
3
−4.5508403180
4
3
−5.1972396752
5
3
−5.8415792817
6
3
−6.4844542567
7
3
−7.1262510136
8
3
−7.7672312288Table 4
Expectation values ofr-2 for hydrogen chloride molecule.
n
l
r
-
2
n
l
(
Å
-
2
)
0
0
−3.3684130731
1
0
−0.9463978738
2
0
−0.2983665406
3
0
−0.0858762881
4
0
−0.0155115416
5
0
−0.0000100220
6
0
−0.0080075039
7
0
−0.0266967106
8
0
−0.0503597253
0
1
−1.4968915458
1
1
−0.4550672530
2
1
−0.1414988910
3
1
−0.0355399041
4
1
−0.0034391544
5
1
−0.0014140351
6
1
−0.0126923268
7
1
−0.0301145909
8
1
−0.0503830973
0
2
−0.4870936532
1
2
−0.1534956468
2
2
−0.0423781577
3
2
−0.0064304364
4
2
−0.0001287810
5
2
−0.0066922782
6
2
−0.0191475545
7
2
−0.0343426512
8
2
−0.0507669460
0
3
−0.1460080210
1
3
−0.0419218285
2
3
−0.0074968794
3
3
−0.0000019647
4
3
−0.0040520781
5
3
−0.0133574528
6
3
−0.0251076232
7
3
−0.0379661955
8
3
−0.0512710657Table 5
Expectation values ofr-2 for carbon (11) oxide molecule.
n
l
r-2nl(Å-2)
0
0
−5.4124527704
1
0
−3.0306606674
2
0
−1.7478304736
3
0
−1.0168948372
4
0
−0.5850753187
5
0
−0.3251871304
6
0
−0.1688134583
7
0
−0.07711780105
8
0
−0.02706175743
0
1
−4.7293036688
1
1
−2.6662325623
2
1
−1.5422086726
3
1
−0.8968395046
4
1
−0.5138531529
5
1
−0.2830295670
6
1
−0.1444688867
7
1
−0.0638852202
8
1
−0.0207992373
0
2
−3.6765597149
1
2
−2.0956762231
2
2
−1.2170016580
3
2
−0.7058809007
4
2
−0.4003901614
5
2
−0.2160772200
6
2
−0.0106187105
7
2
−0.0435491137
8
2
−0.0117284027
0
3
−2.6130004021
1
3
−1.5063610220
2
3
−0.8764213217
3
3
−0.5044821669
4
3
−0.2806791571
5
3
−0.1459868053
6
3
−0.0669337108
7
3
−0.2368085249
8
3
0.0040288991Table 6
Expectation values ofr-1 for hydrogen molecule.
n
l
r-1nl(Å-1)
0
0
1.2603680546
1
0
0.4379795401
2
0
0.1517172611
3
0
0.1960018427
4
0
−0.0520387537
5
0
−0.0951819716
6
0
−0.0123158672
7
0
−0.0142326424
8
0
−0.1560297050
0
1
0.6791360070
1
1
0.2467913485
2
1
0.0664104304
3
1
−0.0256311394
4
1
−0.0788474859
5
1
−0.1123603234
6
1
−0.1348197402
7
1
−0.1506017288
8
1
−0.1621130507
0
2
0.2992866512
1
2
0.9089728082
2
2
−0.1229187417
3
2
−0.7079611201
4
2
−0.1071332977
5
2
−0.1312363775
6
2
−0.1480391542
7
2
−0.1602175908
8
2
−0.1693251067
0
3
0.1047211078
1
3
−0.0489334886
2
3
−0.0663845048
3
3
−0.1042945872
4
3
−0.1293034551
5
3
−0.1466642182
6
3
−0.1592049675
7
3
−0.1685578998
8
3
−0.1757185637Table 7
Expectation values ofr-1 for lithium hydride molecule.
n
l
r-1nlÅ-1
0
0
−1.6004003285
1
0
−1.7169221172
2
0
−1.7576288115
3
0
−1.7764517192
4
0
−1.7866702782
5
0
−1.7928291806
6
0
−1.7968253335
7
0
−1.7995644507
8
0
−1.8015233696
0
1
−1.6824412202
1
1
−1.7440039428
2
1
−1.7697345550
3
1
−1.7828775641
4
1
−1.7904817568
5
1
−1.7952728019
6
1
−1.7984847995
7
1
−1.8007424714
8
1
−1.8023895794
0
2
−1.7364379965
1
2
−1.7662014135
2
2
−1.7809515122
3
2
−1.7893186518
4
2
−1.7945174323
5
2
−1.7979668174
6
2
−1.8003719661
7
2
−1.8021154796
8
2
−1.8034195262
0
3
−1.7642012758
1
3
−1.7798804427
2
3
−1.7886797501
3
3
−1.7941062087
4
3
−1.7976867501
5
3
−1.8001727144
6
3
−1.8019687138
7
3
−1.8033083180
8
3
−1.8043340174Table 8
Expectation values ofr-1 for hydrogen chloride molecule.
n
l
r-1nl(Å-1)
0
0
1.6729006170
1
0
0.7484646279
2
0
0.3703627204
3
0
0.1796456609
4
0
0.0702095120
5
0
0.0166105989
6
0
−0.4409486494
7
0
−0.0761475106
8
0
−0.0994688026
0
1
1.1368069123
1
1
0.5412411880
2
1
0.2694738513
3
1
0.1231413170
4
1
0.0354355317
5
1
−0.0212411621
6
1
−0.0599699313
7
1
−0.0876003075
8
1
−0.1080007236
0
2
0.6614557236
1
2
0.3289571474
2
2
0.1568012770
3
2
0.0562991328
4
2
−0.0074281360
5
2
−0.0503567338
6
2
−0.0806430409
7
2
−0.1028045106
8
2
−0.1195073036
0
3
0.3674917640
1
3
0.1780764140
2
3
0.0692601602
3
3
0.0010437052
4
3
−0.0445186237
5
3
−0.0764508536
6
3
−0.0996933551
7
3
−0.1171352594
8
3
−0.1305576611Table 9
Expectation values ofr-1 for carbon (11) oxide molecule.
n
l
r-1nl(Å-1)
0
0
2.3183952626
1
0
1.6087565101
2
0
1.1442089011
3
0
0.8236269179
4
0
0.5931311104
5
0
0.4218775175
6
0
0.2911757802
7
0
0.1891645008
8
0
0.1080220439
0
1
2.1698058118
1
1
1.5137194441
2
1
1.0797914397
3
1
0.7779687438
4
1
0.5596010388
5
1
0.3965336986
6
1
0.2715561482
7
1
0.1736671893
8
1
0.0955685247
0
2
1.9172571191
1
2
1.3495663209
2
2
0.9671517266
3
2
0.6973560297
4
2
0.4999374576
5
2
0.3511456004
6
2
0.2362292864
7
2
0.1456344871
8
2
0.0729524039
0
3
1.6206607845
1
3
1.1522380728
2
3
0.8292959293
3
3
0.5972813496
4
3
0.4250064850
5
3
0.2935928640
6
3
0.1910702660
7
3
0.1095511163
8
3
0.0436667613Table 10
Expectation values ofT for hydrogen molecule.
n
l
T
n
l
(
e
V
)
0
0
−5.7775010957
1
0
−0.1575248128
2
0
0.3833647660
3
0
0.1004688440
4
0
−0.4051760492
5
0
−1.0142757581
6
0
−1.7009007969
7
0
−2.4609241866
8
0
−3.2955155097
0
1
−1.1305296358
1
1
0.4062543383
2
1
0.2903023018
3
1
−0.1806436110
4
1
−0.7793821663
5
1
−1.4584654527
6
1
−2.2092740425
7
1
−3.0323596290
8
1
−3.9301708526
0
2
0.3945367735
1
2
0.3765550363
2
2
−0.0842946033
3
2
−0.6872416982
4
2
−1.3716032980
5
2
−2.1263145693
6
2
−2.9517276048
7
2
−3.8505394856
8
2
−4.8255106670
0
3
0.4231977151
1
3
−0.0331938070
2
3
−0.6403179766
3
3
−1.3293878653
4
3
−2.0879210976
5
3
−2.9161860233
6
3
−3.8170461261
7
3
−4.7934410194
8
3
−5.8478274746Table 11
Expectation values ofT for lithium hydride molecule.
n
l
T
n
l
(
e
V
)
0
0
−4.2305585674
1
0
−8.5398534703
2
0
−15.8689882670
3
0
−26.0744836125
4
0
−39.0894871134
5
0
−54.8817584651
6
0
−73.4342035101
7
0
−94.7369941855
8
0
−118.7841018590
0
1
−7.8088669292
1
1
−14.2714091244
2
1
−23.6854511210
3
1
−35.9457058061
4
1
−51.0026938513
5
1
−68.8311306737
6
1
−89.4169299119
7
1
−112.7516689680
8
1
−138.8300177540
0
2
−14.2391535700
1
2
−23.4467711704
2
2
−35.5261470800
3
2
−50.4157170230
4
2
−68.0844826720
5
2
−88.5154126054
6
2
−111.6984397090
7
2
−137.6272527440
8
2
−166.2977020400
0
3
−23.4729539098
1
3
−35.4898681236
2
3
−50.3254081811
3
3
−67.9449624090
4
3
−88.3296507196
5
3
−111.4683804780
6
3
−137.3542318380
7
3
−165.9826734470
8
3
−197.3506175640Table 12
Expectation values ofT for hydrogen chloride molecule.
n
l
T
n
l
(
e
V
)
0
0
−16.8602470534
1
0
−3.8173740938
2
0
−0.7264968799
3
0
0.8170812272
4
0
0.1808796145
5
0
0.0076433944
6
0
−0.2925912249
7
0
−0.6654441904
8
0
−1.0891026610
0
1
−1.8259881179
1
1
−0.1652596937
2
1
0.2050704178
3
1
0.1331442742
4
1
−0.1233945608
5
1
−0.4738642417
6
1
−0.8832796542
7
1
−1.3374731433
8
1
−2.7518917695
0
2
−0.3749865298
1
2
0.1982728862
2
2
0.1926154142
3
2
−0.0409008669
4
2
−0.3834013618
5
2
−0.7902268789
6
2
−1.2436082882
7
2
−1.7363979570
8
2
−0.5275303052
0
3
0.1879629685
1
3
0.2257156907
2
3
0.0056050075
3
3
−0.3335743383
4
3
−0.7405492881
5
3
−1.1951594384
6
3
−1.6893574408
7
3
−2.2201023828
8
3
0.1879629685Table 13
Expectation values ofT for carbon (11) oxide molecule.
n
l
T
n
l
(
e
V
)
0
0
−129.9291130830
1
0
−75.6124342843
2
0
−45.8529997998
3
0
−28.4567951051
4
0
−17.7933612549
5
0
−11.0290843042
6
0
−6.6378456500
7
0
−3.7514369504
8
0
−1.8531332454
0
1
−117.6340205260
1
1
−69.0411092188
2
1
−42.0768329533
3
1
−26.1683200430
4
1
−16.3516907261
5
1
−10.0962337215
6
1
−6.0247765063
7
1
−3.3471874404
8
1
−1.5898907876
0
2
−97.7238969338
1
2
−58.1864276360
2
2
−35.7490746782
3
2
−22.2936665438
4
2
−13.8933817128
5
2
−8.4988638639
6
2
−4.9737074744
7
2
−2.6557694881
8
2
−1.1430395374
0
3
−76.0158584874
1
3
−46.0107012922
2
3
−28.5036356479
3
3
−17.7913843239
4
3
−11.0086725080
5
3
−6.6145424645
6
3
−3.7334128723
7
3
−1.8447253427
8
3
−0.6270405479Table 14
Expectation values ofP2 for hydrogen molecule.
n
l
p2nleV/c2
0
0
-
5.8226811543
1
0
-
0.1587566569
2
0
0.3863626785
3
0
0.1012545103
4
0
-
0.4083445259
5
0
-
1.0222073945
6
0
-
1.7142018411
7
0
-
2.4801686138
8
0
-
3.3212864410
0
1
-
1.1393703776
1
1
0.4094312472
2
1
0.2925724658
3
1
-
0.1806436110
4
1
-
0.7854769348
5
1
-
1.4698706526
6
1
-
2.2265505655
7
1
-
3.0560726813
8
1
-
3.9609047886
0
2
0.3945367735
1
2
0.3794996967
2
2
-
0.8495378710
3
2
-
0.6926159282
4
2
-
1.3823292358
5
2
-
2.1429423493
6
2
-
2.9748101146
7
2
-
3.8806507044
8
2
-
4.8632461604
0
3
0.4265071213
1
3
-
0.0334533826
2
3
-
0.6453252632
3
3
-
1.3397836784
4
3
-
2.1042486406
5
3
-
2.9389905980
6
3
-
3.8468954268
7
3
-
4.8309257282
8
3
-
5.8935574854Table 15
Expectation values ofp2 for lithium hydride molecule.
n
l
p2nleV/c2
0
0
-
7.4468161811
1
0
-
15.0322275400
2
0
-
27.9332945570
3
0
-
45.8974585468
4
0
-
68.8070429723
5
0
-
96.6052970237
6
0
-
129.2621308100
7
0
-
166.7602445400
8
0
-
209.0890263500
0
1
13.7455127207
1
1
-
25.1211651371
2
1
-
41.6921779602
3
1
-
35.9457058061
4
1
-
89.7771960362
5
1
-
121.1595985480
6
1
-
157.3956322590
7
1
-
198.4704713410
8
1
-
244.3747335380
0
2
-
14.2391535700
1
2
-
41.2720429614
2
2
-
62.5346943459
3
2
-
88.7439734787
4
2
-
119.8453157330
5
2
-
155.8087416490
6
2
-
196.6165306470
7
2
-
242.2575734060
8
2
-
292.7245654890
0
3
-
41.3181309765
1
3
-
62.4708345234
2
3
-
88.5850078633
3
3
-
119.5997260000
4
3
-
155.4817553670
5
3
-
196.2115702200
6
3
-
241.7769899380
7
3
-
292.1700382350
8
3
-
347.3852799350Table 16
Expectation values ofp2 for hydrogen chloride molecule.
n
l
p2nleV/c2
0
0
-
7.446816181
1
0
-
15.0322275400
2
0
-
27.9332945570
3
0
-
45.8974585468
4
0
-
68.8070429723
5
0
-
96.6052970237
6
0
-
129.2621308100
7
0
-
166.7602445400
8
0
-
209.0890263500
0
1
-
13.7455127207
1
1
-
25.1211651371
2
1
-
41.6921779602
3
1
-
35.9457058061
4
1
-
89.7771960362
5
1
-
121.1595985480
6
1
-
157.3956322590
7
1
-
198.4704713410
8
1
-
244.3747335380
0
2
-
14.2391535700
1
2
-
41.2720429614
2
2
-
62.5346943459
3
2
-
88.7439734787
4
2
-
119.8453157330
5
2
-
155.8087416490
6
2
-
196.6165306470
7
2
-
242.2575734060
8
2
-
292.7245654890
0
3
-
41.3181309765
1
3
-
62.4708345234
2
3
-
88.5850078633
3
3
-
119.5997260000
4
3
-
155.481755367
5
3
-
196.2115702200
6
3
-
241.7769899380
7
3
-
292.1700382350
8
3
-
347.3852799350Table 17
Expectation values ofp2 for carbon (11) oxide molecule.
n
l
p2nleV/c2
0
0
-
1782.8020302400
1
0
-
1037.5042063700
2
0
-
629.1647745140
3
0
-
390.4654690820
4
0
-
244.1488271370
5
0
-
151.333857537
6
0
-
91.0801622550
7
0
-
51.474756141
8
0
-
25.4274783679
0
1
-
1614.0968382200
1
1
-
947.3367959250
2
1
-
577.3506909670
3
1
-
26.1683200430
4
1
-
224.3671701640
5
1
-
138.5338939770
6
1
-
82.6680297618
7
1
-
45.9279096337
8
1
-
21.8154381022
0
2
-
97.7238969338
1
2
-
798.3959780860
2
2
-
490.5253441900
3
2
-
305.8990632100
4
2
-
190.6358670260
5
2
-
116.6158329860
6
2
-
68.2459502172
7
2
-
36.4407262008
8
2
-
15.6840384698
0
3
-
1043.0397285600
1
3
-
631.3286509110
2
3
-
391.1081842740
3
3
-
244.1217009860
4
3
-
151.0537802640
5
3
-
90.7604112360
6
3
-
51.2274415685
7
3
-
25.3121106450
8
3
-
8.6038389354Table 18
Bound state energy of Hulthen potential in comparison to other works.
n
l
δ
Present work
Arda and Sever, 2012
0
1
0.025
-
0.1128125
-
0.1128130
0
1
0.050
-
0.1012500
-
0.1012500
0
1
0.075
-
0.0903125
-
0.0903120
0
1
0.100
-
0.0800000
-
0.0800000
0
1
0.150
-
0.0612500
-
0.0612500


## 7. Normalizing the Wave Function of Quantum Interaction Potential Using Confluent Hypergeometric Functions

The wave function is given in44Ψns=BnsPn2μc/2ħ2α2-μE/2ħ2α2-48μB/ħ2+4ll+1+1,-2μc/2ħ2α2-μE/2ħ2α2-48μB/ħ2+4ll+1+11-s×1-s1/2+1/28μB/ħ2+4ll+1+1sμc/2ħ2α2-μE/2ħ2α2.Basically to normalize a wave function we get the integral of wave function and its complex conjugate to be equal to one. That is,(68)∫0∞ΨrΨ∗rdr=1.In a situation where Ψ(r) and its complex conjugate are real function, then (68) can be expressed as (69)∫0∞Ψr2dr=1.Considering the fact that s=e-2αr then when r=0,  s=1 and when r=∞,  s=0,Hence the wave function will be physically valid fors∈0,1 and r∈(0,∞).However from44 let (70)ξ1=μc2ħ2α2-μE2ħ2α2,∧=8μBħ2+4ll+1+1.Equation 44 can then be expressed as (71)Ψns=Bnssξ11-s1/2+1/2∧×Pn2ξ1-4∧,-2ξ1-4∧1-s.Substituting (71) into (69) gives(72)-Bn22α∫01s2ξ1-11-s1+∧×Pn2ξ1-4∧,-2ξ1-4∧1-s2ds=1.Jacobi polynomial Pnρ,ν(ξ1) can be expressed in two different hypergeometric functions by (73)Pnρ,νξ1=2-n∑p=0n-1n-pn+ρpn+νp1-ξ1n-p1+ξ1p,(74)Pnρ,νξ1=Γn+ρ+1n!Γn+ρ+ν+1∑r=0nnrΓn+ρ+ν+r+1Γr+ρ+1ξ1-12r,where (75)nr=crn=n!n-r!r!=Γn+1Γn-r+1Γr+1.Equations (73) and (74) are used simultaneously in evaluating the Jacobi polynomial.Considering the Jacobi polynomial of (72)(76)Pn2ξ1-4∧,-2ξ1-4∧1-s⇓ρ=2ξ1-4∧,ν=-2ξ1-4∧,ξ=1-s.Using (73) then the Jacobi polynomial becomes(77)Pn2ξ1-4∧,-2ξ1-4∧1-s=2-n∑p=0n-1n-pn+2ξ1-4∧pn-2ξ1-4∧n-psn-p2-sp⟹Pn2ξ1-4∧,-2ξ1-4∧1-s=2-nsn-p2-sp∑p=0n-1n-pn+2ξ1-4∧pn-2ξ1-4∧n-p.The summation sign in (73) can be evaluated simultaneously for p=0 and p=0,n as a partial sum.Evaluating it forp=0(78)∑p=0n-1n-pn+2ξ1-4∧pn-2ξ1-4∧n-p=∑p=0n-1n-pn+2ξ1-4∧0n-2ξ1-4∧n=-1nn+2ξ1-4∧!n+2ξ1-4∧-0!0!n-2ξ1-4∧!-2ξ1-4∧!n!=-1nΓn-2ξ1-4∧+1Γn+1Γ1-2ξ1-4∧.For p=0,n(79)∑p=0n-1n-pn+2ξ1-4∧pn-2ξ1-4∧n-p=-1pn+2ξ1-4∧!n+2ξ1-4∧-p!p!n-2ξ1-4∧!n-2ξ1-4∧-n-p!n-p!=∑p=0n-1pΓn+2ξ1-4∧Γn-2ξ1-4∧p!n-p!Γn+2ξ1-4∧-p+1Γp-2ξ1-4∧+1.Substituting (78) and (79) into (73) gives(80)Pn2ξ1-4∧,-2ξ1-4∧1-s=-1nΓn-2ξ1-4∧+1Γn+1Γ1-2ξ1-4∧∑p=0n-1pΓn+2ξ1-4∧+1Γn-2ξ1-4∧+12-nsn-p2-spp!n-p!Γn+2ξ1-4∧-p+1Γp-2ξ1-4∧+1.Using the second expression for the Jacobi polynomial that is (74) then (81)Pn2ξ1-4∧,-2ξ1-4∧1-s=Γn+2ξ1-4∧+1Γn+1-8∧∑r=0n-12rΓn+r+1-8∧r!n-r!Γr+2ξ1-4∧+1sr.Then the square of the Jacobi polynomial in (72) then becomesPn2ξ1-4∧,-2ξ1-4∧(1-s)2= (80) multiplied by (81).(82)Pn2ξ1-4∧,-2ξ1-4∧1-s2=-1nΓn-2ξ1-4∧+1Γn+1Γ1-2ξ1-4∧×∑p=0n-1pΓn+2ξ1-4∧Γn-2ξ1-4∧2-nsn-p2-spp!n-p!Γn+2ξ1-4∧-p+1Γp-2ξ1-4∧+1×Γn+2ξ1-4∧+1Γn+1-8∧×∑r=0n-12rΓn+r+1-8∧r!n-r!Γr+2ξ1-4∧+1sr.Equation (82) can be further simplified to (83)Pn2ξ1-4∧,-2ξ1-4∧1-s2=-1nΓn-2ξ1-4∧+12Γn+2ξ1-4∧+12Γn+1Γ1-2ξ1-4∧×Γn+r+1-8∧Γn+1-8∧∑p=0n∑r=0n-1p+r2-n-rsn-p+r2-spp!n-p!r!n-r!Γn+2ξ1-4∧-p+1Γp-2ξ1-4∧+1Γr+2ξ1-4∧+1.Substituting (83) into (72) gives(84)--1nΓn-2ξ1-4∧+12Γn+2ξ1-4∧+122αΓn+1Γ1-2ξ1-4∧×Γn+r+1-8∧Γn+1-8∧∑p=0n∑r=0n-1p+r2-n-r2-spBn2p!n-p!r!n-r!Γn+2ξ1-4∧-p+1Γp-2ξ1-4∧+1Γr+2ξ1-4∧+1×∫01s2ξ1-1+n-p+r1-s1+∧ds=1Confluent hypergeometric function can be defined as follows:(85)F12α0,β0:α0+1;1=α0∫01sα0-11-s-β0ds=1assuming that (86)γ0=α0+1,thenF12α0,β0:γ0;1=α0∫01sα0-11-s-β0ds=1.However, (87)F12α0,β0:γ0;1=Γγ0Γγ0-α0-β0Γγ0-α0Γγ0-β0.Considering(88)∫01s2ξ1-1+n-p+r1-s1+∧ds,α0=2ξ1+n-p+r,β0=-1+∧therefore(89)∫01s2ξ1-1+n-p+r1-s1+∧ds=Γα0+1Γ∧+2α0Γα0+∧+2.Substituting (89) into (84) gives(90)--1nΓn-2ξ1-4∧+12Γn+2ξ1-4∧+122αΓn+1Γ1-2ξ1-4∧×Γn+r+1-8∧Γn+1-8∧∑p=0n∑r=0n-1p+r2-n-r2-spBn2p!n-p!r!n-r!Γn+2ξ1-4∧-p+1Γp-2ξ1-4∧+1Γr+2ξ1-4∧+1×Γα0+1Γ∧+2α0Γα0+∧+2=1.Let(91)s1=--1nΓn-2ξ1-4∧+12Γn+2ξ1-4∧+122αΓn+1Γ1-2ξ1-4∧×Γn+r+1-8∧Γn+1-8∧∑p=0n∑r=0n-1p+r2-n-r2-spp!n-p!r!n-r!Γn+2ξ1-4∧-p+1Γp-2ξ1-4∧+1Γr+2ξ1-4∧+1×Γα0+1Γ∧+2α0Γα0+∧+2.However,(92)Bn2s1=1⟹Bn=1s1.Hence, the normalized wave function then becomes(93)Ψns=1s1sPn2μc/2ħ2α2-μE/2ħ2α2-48μB/ħ2+4ll+1+1,-2μc/2ħ2α2-μE/2ħ2α2-48μB/ħ2+4ll+1+11-s×1-s1/2+1/28μB/ħ2+4ll+1+1sμc/2ħ2α2-μE/2ħ2α2.


## 8. Conclusion

In this work, we obtained analytical solution of Schrodinger equation with quantum interaction potential (HYIQP) and obtained the energy eigenvalue and the wave function using conventional Nikiforov-Uvarov method. Under some special cases, the potential reduced to three well-known potentials which are as follows: Hulthen, Yukawa, and inversely quadratic potential. The result from these potentials is in agreement with that reported in existing literature. Expectation values for four diatomic molecules are computed using Hellmann-Feynman theorem. Numerical solutions of the expectation values are obtained by implementing MATLAB algorithm. However, by implementing mathematica algorithm, the wave function and probability density plots were obtained for four diatomic molecules with an increasing orbital angular momentum. The quantum interaction potential model developed is suitable in studying bound state energies of diatomic molecules. Figure1 shows the graph of the quantum interaction potential with different values of the screening parameter. However, Figure 2 shows the graph of the potential (F) plotted against the individual potentials (F1, F2, and F3). Some of the numerical expectation values either increase or decrease with an increase in quantum state.



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*Source: 9671816-2017-05-30.xml*