Advances in High Energy Physics
Volume 2017 (2017), Article ID 9671816, 24 pages
https://doi.org/10.1155/2017/9671816

Research Article

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

Ituen B. Okon,¹ Oyebola Popoola,² and Cecilia N. Isonguyo¹

¹Theoretical Physics Group, Department of Physics, University of Uyo, Uyo, Nigeria
²Theoretical Physics Group, Department of Physics, University of Ibadan, Ibadan, Nigeria

Correspondence should be addressed to Ituen B. Okon; ituenokon@uniuyo.edu.ng

Received 7 February 2017; Revised 24 February 2017; Accepted 19 March 2017; Published 30 May 2017

Academic Editor: Saber Zarrinkamar

Copyright © 2017 Ituen B. Okon et al. This is an open access article distributed under the Creative Commons Attribution License, which permits unrestricted use, distribution, and reproduction in any medium, provided the original work is properly cited. The publication of this article was funded by SCOAP³.

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 values , , , and 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.

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

where is the potential depth, is the adjustable parameter known as the screening parameter. , , and are all spectroscopic constants which are molecular bond length, molecular constant, and potential range, respectively.

Figure 1 shows the graph of quantum interaction potential for various values of the screening parameter 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 , 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 of .

Figure 2: The graph of the HYIQP as compared to that of individual potentials. = HYIQP potential, 1 = Hulthen potential, 2 = Yukawa potential, and 3 = inversely quadratic potential.

Figure 3: Wave function plot for the interaction potential with orbital angular quantum number .

Figure 4: Wave function plot for the interaction potential with orbital angular quantum number .

Figure 5: Wave function plot for the interaction potential with orbital angular quantum number .

Figure 6: Wave function plot for the interaction potential with orbital angular quantum number .

Figure 7: Wave function plot for the interaction potential with orbital angular quantum number .

Figure 8: Wave function plot for the interaction potential with orbital angular quantum number .

Figure 9: Probability density plots for the interaction potential with orbital angular quantum number , 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 the equation can be written as

is a polynomial of degree one while and are polynomials of at most degree two.

In order to find the exact solution to (2), we set the wave function as

and substituting (3) into (2) reduces (2) into hypergeometric type.

where the wave function is defined as the logarithmic derivative

where is at most polynomial of degree one.

Likewise, the hypergeometric type function in (4) for a fixed is given by the Rodrigue relation

is the normalization constant and the weight function ρ(s) satisfies the condition

such that

In order to accomplish the conditions imposed on the weight function , it is necessary that the classical orthogonal polynomials be equal to zero and its derivative be less than zero; that is,

Therefore, the function and the parameter required for the NU method are defined as follows:

The -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

where 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

Substituting (1) into (13) gives

Equation (14) can only be solved analytically to obtain exact solution if the angular orbital momentum number . However, for (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

Substituting (15) into (14) with the transformation gives

where

Simplifying (16) further reduced to

Comparing (18) to (2) we obtained the following:

Then, using (10), the polynomial equation then becomes

To find the value of we consider the discriminant such that

Hence,

Substituting the values of and into (20), then the four values of are given below.

has four solutions and one of the solutions satisfied bound state condition which is

is the condition for bound state solution. Using (8) we have it that

such that

which satisfies the bound state condition. However using (11)

Using (12)

Equating (25) and (26) gives the result

Substituting parameters of (17) into (27) gives

Equation (28) is the energy for the combined potential.

3.1. Calculation of the Wave Function

By using (5), but

then

Taking integral of (32) gives

which then leads to

Equation (34) can further be reduced to

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

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)

Substituting the parameters gives

Integrating (38) gives

Rewriting (39) in its Rodrigue form by making use of (6) gives

Let us define standard associated Laguerre polynomial as

Then rewriting (40) in terms of (41) gives the second part of the wave function as

Hence the total wave function is given by

Substituting (17) into (43) reduced it to

Equation is the total wave function for the proposed potential. Equation can further be reduced to

The normalized wave function of (44b) is given as

where the normalization constant is given as

 , where

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

Setting in (1) results in Hulthen potential given as

The energy of this potential is given as

However , and then (46) becomes

According to Arda and Sever, 2012, the standard Hulthen potential is given by

Setting and 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 and . 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

Setting in (1) then the potential reduced to Yukawa potential.

The corresponding energy equation for this potential is given as

Equation (49a) can further be simplified as

Hamzavi et al. [33] present the eigenenergy equation for Yukawa potential as

Equations (49b) and (49c) are the same with just variation in parameter where

4.3. Inversely Quadratic Potential

Setting in (1) then the potential reduced to inversely quadratic potential.

The corresponding energy for this potential is given as

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 for a particular quantum mechanical system is given as a function of some parameters , then let and be the eigenvalues and the eigenfunctions of Hamiltonian , respectively; then

However, the Hamiltonian which contains the effective potential can be expressed as

Let us recall the energy equation as given in (28).

5.1. Expectation Value of

Substituting into (53) then

Taking the partial derivative of (28) with respect to gives

Taking the partial derivative of (54) with respect to gives

Equating (56) to (57) gives the expectation values of for different orbital quantum number. Hence,

5.2. Expectation Value of

Taking the partial derivative of (28) with respect to gives

Also, taking the partial derivative of (54) with respect to gives

Equating (59) to (60) then gives

5.3. Expectation Values for and

Taking the partial derivative of (28) with respect to , this then gives

However, taking the partial derivative of (54) with respect to gives

Hence,

From the relation , substituting for in (64) gives

Equating (62) to (64) gives the expectation value of . Therefore,

Also, equating (62) to (65) gives . Thus,

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 , and , 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 , and , 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 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 decreases with an increase in quantum state for all the diatomic molecules as shown in Tables 6–9. The expectation value 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 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

Hydrogen
0.7416
1.9426
1.440558
0.20990
0.5039100

Lithium hydride
1.5956
1.1280
1.7998368
1.55000
0.8801221

Hydrogen chloride
1.2746
1.8677
2.38057
0.20039
0.9801045

Carbon (II) oxide
1.1283
2.2994
2.59441
0.39000
6.8606719

Table 2: Expectation values of for hydrogen molecule.

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
−0.0963742259

Table 3: Expectation values of for lithium hydride molecule.

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
−7.7672312288

Table 4: Expectation values of for hydrogen chloride molecule.

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
−0.0512710657

Table 5: Expectation values of for carbon () oxide molecule.

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
0.0040288991

Table 6: Expectation values of for hydrogen molecule.

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
−0.1757185637

Table 7: Expectation values of for lithium hydride molecule.

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
−1.8043340174

Table 8: Expectation values of for hydrogen chloride molecule.

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
−0.1305576611

Table 9: Expectation values of for carbon () oxide molecule.

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
0.0436667613

Table 10: Expectation values of for hydrogen molecule.

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
−5.8478274746

Table 11: Expectation values of for lithium hydride molecule.

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
−197.3506175640

Table 12: Expectation values of for hydrogen chloride molecule.

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
0.1879629685

Table 13: Expectation values of for carbon () oxide molecule.

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
−0.6270405479

Table 14: Expectation values of for hydrogen molecule.

0
0

1
0

2
0

3
0

4
0

5
0

6
0

7
0

8
0

0
1

1
1

2
1

3
1

4
1

5
1

6
1

7
1

8
1

0
2

1
2

2
2

3
2

4
2

5
2

6
2

7
2

8
2

0
3

1
3

2
3

3
3

4
3

5
3

6
3

7
3

8
3

Table 15: Expectation values of for lithium hydride molecule.

0
0

1
0

2
0

3
0

4
0

5
0

6
0

7
0

8
0

0
1

1
1

2
1

3
1

4
1

5
1

6
1

7
1

8
1

0
2

1
2

2
2

3
2

4
2

5
2

6
2

7
2

8
2

0
3

1
3

2
3

3
3

4
3

5
3

6
3

7
3

8
3

Table 16: Expectation values of for hydrogen chloride molecule.

0
0

1
0

2
0

3
0

4
0

5
0

6
0

7
0

8
0

0
1

1
1

2
1

3
1

4
1

5
1

6
1

7
1

8
1

0
2

1
2

2
2

3
2

4
2

5
2

6
2

7
2

8
2

0
3

1
3

2
3

3
3

4
3

5
3

6
3

7
3

8
3

Table 17: Expectation values of for carbon () oxide molecule.

0
0

1
0

2
0

3
0

4
0

5
0

6
0

7
0

8
0

0
1

1
1

2
1

3
1

4
1

5
1

6
1

7
1

8
1

0
2

1
2

2
2

3
2

4
2

5
2

6
2

7
2

8
2

0
3

1
3

2
3

3
3

4
3

5
3

6
3

7
3

8
3

Table 18: Bound state energy of Hulthen potential in comparison to other works.

Present work
Arda and Sever, 2012

0
1
0.025

0
1
0.050

0
1
0.075

0
1
0.100

0
1
0.150

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

The wave function is given in

Basically to normalize a wave function we get the integral of wave function and its complex conjugate to be equal to one. That is,

In a situation where and its complex conjugate are real function, then (68) can be expressed as

Considering the fact that then when ,   and when ,  ,

Hence the wave function will be physically valid for and .

However from let

Equation can then be expressed as

Substituting (71) into (69) gives

Jacobi polynomial can be expressed in two different hypergeometric functions by

where

Equations (73) and (74) are used simultaneously in evaluating the Jacobi polynomial.

Considering the Jacobi polynomial of (72)

Using (73) then the Jacobi polynomial becomes

The summation sign in (73) can be evaluated simultaneously for and as a partial sum.

Evaluating it for

For

Substituting (78) and (79) into (73) gives

Using the second expression for the Jacobi polynomial that is (74) then

Then the square of the Jacobi polynomial in (72) then becomes

  (80) multiplied by (81).

Equation (82) can be further simplified to

Substituting (83) into (72) gives

Confluent hypergeometric function can be defined as follows:

assuming that

However,

Considering

therefore

Substituting (89) into (84) gives

Let

However,

Hence, the normalized wave function then becomes

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. Figure 1 shows the graph of the quantum interaction potential with different values of the screening parameter. However, Figure 2 shows the graph of the potential () plotted against the individual potentials (1, 2, and 3). Some of the numerical expectation values either increase or decrease with an increase in quantum state.

Conflicts of Interest

The authors declare that there are no conflicts of interest regarding the publication of this paper.

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