Approximate analytical solutions and mean energies of stationary Schr¨odinger equation for general molecular potential

Eyube E S Rawen B O and Ibrahim N

1Department of Physics,School of Physical Sciences,Modibbo Adama University of Technology,P M B 2076 Yola,Adamawa State,Nigeria

2Directorate of Basic and Remedial Studies,Abubakar Tafawa Balewa University(ATBU),P M B 750001,Bauchi,Bauchi State,Nigeria

3Department of Physics,Faculty of Science,P M B 1069,Maiduguri,Borno State,Nigeria

Keywords: general molecular potential,Schr¨odinger equation,improved quantization rule

1. Introduction

The wave equations are of tremendous importance in the field of quantum mechanics due to the fact that they can provide the information about the quantum mechanical system.[1,2]However,the wave functions are obtained by solving Schr¨odinger equation with potential energy function describing the system. For a given quantum state defined by the vibrational and rotational quantum numbersνandJrespectively, the exact solution of the Schr¨odinger equation is limited to the harmonic oscillator and Coulombic potential.[3,4]For many other potential energy functions, the exact solution can be obtained only for the stateJ=0 (also known as the pure vibrational state).[5]For majority of the known potential energy functions,exact solutions of the Schr¨odinger equation are not feasible as a result of the spin-orbit or centrifugal term coupled to the effective potential.[1,5]For such potentials,approximate numerical or analytical solutions are necessary.[6,7]

Analytical solutions have the advantage of being obtained in close form, a form that makes them useful in exploring the physical properties of the system which they represent.[8-10]It must be mentioned here that not all analytical solutions are reliable,the most accurate analytical solution is one that approximates more closely to its corresponding numerical solution.[11,12]Approximate analytical solution of the Schr¨odinger equation involves the application of a suitable approximation model to the spin-orbit term[13-15]and an appropriate solution method. Researchers have proposed an almost endless solution methods to solve the Schr¨odinger equation. Some of the solution methods include ansatz method,[16]asymptotic iteration method,[17]exact quantization rule,[18]Nikiforov-Uvarov method,[19]improved or proper quantization rule,[20]and so,on.

Various models of potential energy functions have been used by researchers to solve the Schr¨odinger equation.[21-23]In their quest for constructing a potential energy function suitable for explaining molecular interactions,Yanaret al.have suggested the general molecular potential (GMP).[24]It is given by the following expression:

whereA,B,C,D, ˜q, andqare adjustable real potential parameters: ˜qandqare dimensionless parameters,reis the equilibrium bond length,andr ∈(0,∞)is the internuclear separation of the interacting atoms.

The Schr¨odinger equation has been solved in the presence of GMP, and the expression for the rovibrational energy was used to model the interaction among six diatomic molecules.[24]Ikotet al. have employed the asymptotic iteration method to solve the radial Schr¨odinger equation with GMP and used the obtained energy spectra to study the thermodynamic properties of potassium dimer.[17]Finally, to the best of our knowledge,these are the only researches reported in the literature about the(GMP).

The researches about the GMP, which were earlier carried out are as follows. (i)In Ref.[24],the ansatz solution approach was used to solve the Schr¨odinger equation. However, explicit expression for energy eigenvalues was not given,which now entails obtaining separate explicit expression for energy eigenvalues for each of the potentials considered as special case.(ii)In Ref.[17],the expression for eigen energies of the GMP was obtained via asymptotic iteration method. Nonetheless,the result was neither subjected to a numerical test nor reduced into energy eigenvalues of other potential energy functions derivable from the GMP.(iii)The energy eigenvalues obtained for the GMP in previous researches are not valid forq=0. (iv) Even though the GMP remains finite asq →0,the form of the Pekeris approximation model used in Refs.[17,24]is infinite asq →0.

The present work aims at obtaining approximate analytical solutions and mean energies of stationary Schr¨odinger equation with the general molecular potential. The objectives to be studied include: i) deriving the eigen energies of the GMP via the improved quantization rule and eigenfunctions by ansatz solution,with consideration for the cases ofq/=0 andq=0;ii)using a Pekeris-like approximation recipe to model the pseudospin orbit term of the effective potential; iii) deriving the formulas of expectation values of kinetic energy,potential energy, and total energy of the GMP, iv) computing the mean energies of the GMP for four diatomic molecules: H2, CO, HF, and O2; v) comparing the obtained results with those available from the literature.

The rest of this paper is organized as follows.In Section 2,a brief review of the improved quantization rule is presented. In Section 3,we use ideas of improved quantization rule to derive the expression for bound state energy eigenvalues of the GMP,and obtain the corresponding eigenfunctions via ansatz solution. Mean kinetic energy and potential energy for the system are discussed in Section 4. Section 5 is devoted to the special cases of the GMP and numerical computations. Finally, some conclusions are drawn from the present work in Section 6.

2. Review of improved quantization rule

In this section,a brief outline of the rudiments of improved quantization rule applicable to the present work is given. A detailed narration of this concept is presented elsewhere.[25]The improved quantization rule comes into being due to the need to overcome some of the complicated integrals encountered in exact quantization rule.At the beginning, the exact quantization rule was used to solve the one-dimensional Schr¨odinger equation given by[25]

where double prime denotes the second derivative ofψ(x) with respect to argumentx,ψ(x),x ∈(-∞,∞) is the one-dimensional wave function, andk(x) is the momentum of the system,and given in terms of the energyEand effective potentialVeff(x)by[18,25,26]

In the usual notation,equation(2)expressed as a nonlinear Riccati equation is given as[18,25,26]

The logarithmic derivativeφ(x) =ψ′(x)/ψ(x) of the wave functionψ(x) is the phase angle.[18,26]For the Sturm-Liouville problem, the phase angle must be monotonic with respect to the energy.[18,26]Therefore,from Eq. (2),ψ(x) must decreases monotonically with respect toxbetween two turning points whereE ≥Veff(x). Specifically, asxincreases across a node of the wave functionψ(x),φ(x) decreases to-∞, jumps to+∞,and then decreases again.[26]Having carefully studied the one-dimensional Schr¨odinger equation,the exact quantization rule is given as[18,25,26]

where the turning pointsxAandxBare the roots of the equationVeff(x)=E. The number of nodesNofφ(x)in the regionE ≥Veff(x) is by 1 greater than the number of nodesνof the wave functionψ(x). That is,N=ν+1.[18,26]In Eq.(5),N πis the contribution from the nodes of the phase angle while the second term on the right-hand side is the quantum correction. The quantum correction is independent ofνfor all exactly solvable quantum systems. Accordingly, it is evaluated for the ground state (ν=0). For a spherically symmetric potential,equations(2)-(4)are expressed as[18,20,25,26]

withL=J(J+1) being the angular momentum of the system. In a spherical coordinate system, equation (5)takes the form[18]

where

Settingν=0 in Eq. (11) and eliminatingQc, the improved quantization rule is written in compact form as follows:[18]

3. Solution of stationary Schr¨odinger equation with GMP

3.1. Solution of stationary Schr¨odinger equation with GMP corresponding to q/=0

Substituting Eq.(1)into Eq.(10),we have

Equation (6) has an exact solution in the presence of Eq.(15)only for the pure vibrational state(J=0). However,approximate analytical solutions are achievable if a suitable approximation scheme is employed to model the centrifugal term of the effective potential. In the earlier studies, the Pekeris-like approximation scheme used is given as[17,24]

where the constant coefficientsc-1+j(j=1,2,3)are given by[17,24]

Evidently,asq →0,equation(1)is finite. Also,from Eq.(19)we have

which is also finite. Therefore, equation (18) holds for bothq/=0 andq=0. The “M” appearing in Eq. (20) is designated as “expression corresponding toq=0”. In the remainder part of this work, This notation will be used unless otherwise stated. Substituting Eq.(18)into Eq.(15)we have

Using Eq. (22) to transform Eq. (8) into one involving variableτ,the Riccati equation leads to

In order to satisfy the Sturm-Liouville requirement,we choose a first-degree polynomial as a trial wave function for the ground stateviz

whereωandγare constant coefficients. Substituting Eq.(32)into Eq.(31)forν=0,we have Equation (33) is true if and only if the separate coefficients ofτ2,τ,andτ0each are equal to zero. This yields the following expressions

The parameterωis obtained from Eq.(34a)as

Withωandγgiven by Eqs. (35) and (34b) respectively,our trial solution defined by Eq.(32)is completely known. Our next task is to evaluate the momentum integral. Substituting Eq. (22) into Eq. (12) and inserting Eq.(30)into the resulting relation,we have

With the help of standard integral (A1) in Appendix A,equation(37)gives

Corresponding toν=0,equation(38)yields

Inserting Eqs. (34a)-(34c) into Eq. (39) and conducting simplification, the resulting equation is obtained as follows:

where

subject to the following constraints:

For a polynomial solution,[24]equation (46) has a solution given by

where2F1(-ν,ν+2s+2t+1;2s+1;z) is the hypergeometric function. Equations (42) and (45) give the complete solution of the Schr¨odinger equation with the GMP,and these equations remain valid for none zero values ofq.

3.2. Solution of stationary Schr¨odinger equation with GMP corresponding to q=0

To obtain the solution of the Schr¨odinger equation with the GMP forq=0,first,it must be observed that by taking limits,equation(21)remains valid since

Note that in arriving at Eq. (51a) from Eq. (34a), the negative square root is considered to meet the Sturm-Liouville requirements of a trial wave function of one node and zero-pole. Forq=0,equation(37)gives

Inserting Eqs.(53)and(54)into Eq.(14), and eliminatingη0M,η1M,η2M,andην JM,we obtain

The radial eigenfunctions are obtained by settingq=0 in Eq.(31),resulting in

In order to solve Eq.(56),we assume a trial solution in the form of(Ref.[1])

whereρandυare the parameters,Nν JMis the normalization constant, andφν JM(τ) is an unknown function ofτ.Substitution of Eq.(57)into Eq.(56)gives the Laguerre hypergeometric differential equation

on condition that

For bound state solution,equation(58)gives

where1F1(-n,υ+1;ρ τ)is the Laguerre hypergeometric function.

4. Mean values of kinetic energy and potentia energy in GMP

Now that we have obtained explicit analytical relations for bound state energy eigenvalues of the GMP for cases whereq=0 andq/=0, we are in the position to apply these expressions to driving the important formulas for mean values of kinetic energy and potential energy. The Hellmann-Feynman theorem (HFT)[1,9,10]is used, and it states that if the HamiltonianHof a quantum mechanical system is a function of some parameterζ,thenH(ζ),ψν J(ζ),andEν J(ζ)are related by[1,9,10]

4.1. Mean values of kinetic energy and potential energy of system in GMP for q/=0

Choosingζ=L,equations(62)and(63)give

ELis obtained by differentiating Eq.(42)with respect toL,and expressed as

The mean value of the kinetic energy is found by settingζ=μ,separately,in Eqs.(62),(63),and(42). Thus,we have

From Eqs. (80) and (81), we find the expectation value of the kinetic energy to be

Another important relation which is deduced by the HFT is the expectation value of the GMP.Takingζseparately as potential parametersA,B, and ˜q, we have the mean value of the GMP〈V〉as given below:

LetΩν Jbe the sum of kinetic energy and potentials,then we have

4.2. Mean values of kinetic energy and potential energy of system in GMP for q=0

In the previous section,the equations derived for expectation values of kinetic energy and potential energy are valid only for nonzero values ofq. Corresponding toq=0,equations(70)and(74)transform into

are obtained from Eq. (64) by takingζseparately asLand asμ,with the following expressions:

Similarly,the mean value of the GMP〈V〉M,forq=0 is deduced from Eq.(75)and expressed as

whereEAM,EBM,andE˜qMare obtained by differentiating Eq. (55) with respect toAM,BM, and ˜qM, their expressions are given below:

5. Results and discussion

Having obtain the explicit expressions for energy eigenvalues,eigenfunctions and expectation values of the GMP, at this point we will test the accuracy of our results. Through the suitable choice of parametersA,B,C,D,q,and ˜q,we deduce the bound state energy eigenvalues, radial wave functions and expectation values of some potential models that are special cases of the GMP.

5.1. Woods-Saxon potential

The Woods-Saxon potential is a useful potential energy function and used as a major part of nuclear shell model to obtain nuclear energy level spacing and properties of electron distribution in atoms, nuclei and atomic clusters.[20]By choosingδ(0)=δ(2)=0,δ(1)=-V0,and the mappings

in Eqs. (23) and (42), we obtain Eqs. (B1) and (B2)in Appendi B. These equations are identical to Eqs. (9)and (29) of Ref. [20], the effective potential and bound state energy eigenvalue of the Woods-Saxon potential inDdimensions. Here, sinceq/=0, it follows that equations(70)and(75)are applicable for the computation of expectation values of spin-orbit term and potential energy function. On the other hand, equation (74) is suitable for the mean kinetic energy of the potential.

5.2. Morse potential

The Morse potential has been described as an important empirical potential model suitable for representing vibration-rotation states of diatomic molecules.[2,24,28]Equations (B3) and (B4) shown in Appendix B are obtained for the following choice of potential parameters applied to Eqs.(23)and(42)respectively:By comparison,equarions(B3)and(B4)are identical to Eqs.(48)and(56)for the Morse potential in Ref.[1]. In this case,sinceq=0,equations(81),(80),and(86)are employed to compute mean values of kinetic energy,centrifugal term,and Morse potential function respectively.

5.3. Tietz-Hua oscillator

Equations (B5) and (B6) shown in Appendix B are obtained from Eqs.(23)and(42)respectively,subject to the following choice of potential parameters:δ(2)=de,˜qD2=De, andB+2 ˜qCD=2De. As it is evident that,equation(B5)corresponds to the expression for effective potential of the Tietz-Hua oscillator.[2,28,31]On the other hand,withq=chandbh=αM(1-ch),equation(B6)is just the Eq.(13)of Ref.[31]for the ro-vibrational energy.The Tietz-Hua oscillator is known to be more realistic than the Morse potential in the description of molecular dynamics at moderate and high rotational and vibrational quantum number,and it reduces to Morse potential whench=0.[28]Forch/=0,equations(70),(75),and(74)are appropriate for the computation of the mean value of centrifugal term,the mean value of the Tietz-Hua potential energy function,and the mean value of kinetic energy.

5.4. M¨obius squared oscillator

The M¨obius squared oscillator is a very important potential energy model that has wide applications in chemical physics, nuclear physics, molecular physics,and solid-state physics.[1,29]Bound state energy eigenvalues and some expectation values of this oscillator have been studied in the literature.[1,29]It is noted that with the following expressions

equations (23) and (46) are transformed into the forms given by Eqs. (B7) and (B8) in the Appendix B. Except for the coefficientsd1,d2, andd3, equations (B7)and (B8) are respectively equivalent to Eqs. (16) and(37)in Ref.[1]for the effective potential energy and rovibrational energy of shifted-rotating M¨obius squared oscillator. If in addition,ηis reduced to zero and the coefficientsd-1+j(j=1,2,3)are chosen asα2,2α2e-α re,andα2e-2α rein Eqs. (B7) and (B8), the first term and second term of Eq. (B7) are correspondingly equivalent to Eqs. (5) and (6) of Ref. [29] for the Mobius squared potential and the approximation to the centrifugal term respectively,where for validity,D′=-C′. Furthermore,ifLis replaced with(D+2ℓ-3)(D+2ℓ-1)/4,equation(B8)is identical to the energy eigenvalues of M¨obius squared oscillator inDdimensions given by Eq.(17b)in Ref.[29],the parameterλis related toγof Eq.(17c)byλ=γ+1/2.

The data in Table 1 show spectroscopic constants of four diatomic moleculesviz.H2,CO,HF,and O2studied in the present work.

Table 1. Spectroscopic parameters of selected diatomic molecules used in the present study.

We compute the bound state energy eigenvalues,expectation values of centrifugal term, potential energy function, kinetic energy, and total energy. The calculations are carried out for arbitrary rotational and vibrational quantum numbers. The results obtained using Eqs. (42), (70), (75), (74), and (79) for the parameters defined in Subsection 5.3 of the Tietz-Hua oscillator are shown in Tables 2-5. Also reported in the tables are the corresponding results for the parameters of Morse potential given in Subsection 5.2,which are calculated through Eqs.(55),(80),(86),(81),and(90).To enable our results to be compared with the existing data in the literature,we compute the binding energyde-Eν Jandde-Eν JMfor the Tietz-Hua and Morse potentials respectively.The computed results and those extracted from the literature are shown in the tables. For the four diatomic molecules considered in the present work,our computed bound state energy values are in nearly perfect agreement with the results obtained by generalized pseudospectral method[30]and by Nikiforov-Uvarov method.[2]

Variations of expectation values of the kinetic energy, the potential energy, and the total energy with the vibrational quantum number are shown graphically in the following Figs.1-4.

Fig. 1. The variation of the mean centrifugal term with the vibrational quantum number for(a)the Tietz-Hua oscillator,and(b)the Morse oscillator.

Fig.2.The variation of the mean GMP with the vibrational quantum number for(a)the Tietz-Hua oscillator and(b)the Morse oscillator.

Figure 1 shows the plot of expectation values of the centrifugal termversus ν. As indicated by the plot in Fig. 1(a), whenνis increased from zero,〈VL〉decreases linearly to some threshold value and afterwards increases. However,the opposite is true for the variation of〈VLM〉withνas represented in Fig.1(b).

The variation of the expectation value of GMP withνis shown in Fig.2. In this case,asνis gradually raised from zero,〈V〉increases to a peak value,then decreases asνis further increased (Fig. 2(a)). Like the previous case,the variation of〈VM〉withνis opposite to the variation of〈V〉withν.

In Fig. 3, we plot the expectation values of the kinetic energyversus ν. Figure 3(a) shows that asνis varied positively,〈T〉increases to a maximum value and afterwards decreases with theνfurther increasing. In Fig.3(b),〈TM〉increases approximately linearly with theνincreasing.

Fig.3.The variation of the mean kinetic energy with the vibrational quantum number for(a)the Tietz-Hua oscillator,and(b)the Morse oscillator.

The variations of total energy of the GMP withνare shown in Figs.4(a)and 4(b)for the Tietz-Hua potential and Morse potential. The plots reveal that asνincreases from zero, the total energy for each of the two oscillators increases slowly,then attains a maximum value,and finally decreases.

Fig.4. The variation of the total energy with the vibrational quantum number for(a)the Tietz-Hua oscillator and(b)the Morse oscillator.

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6. Conclusions

In the present work, bound state energy eigenvalues and radial eigenfunctions of the general molecular potential are obtained within the frameworks of improved quantization rule and ansatz solution method respectively. A Pekeris-like approximation model is used to model the pseudo-spin orbit term of the Schr¨odinger equation. With the help of Hellmann-Feynman theorem, closed form expressions for mean kinetic energy and mean potential energy are derived for the system.Equations for ro-vibrational energy, wave functions and mean energy of Woods-Saxon potential, Morse potential, M¨obius squared, and Tietz-Hua oscillators are deduced from the general molecular potential energy. Furthermore,we numerically compute the values for bound state energy eigenvalues and mean energy values for four diatomic moleculesviz.H2, CO, HF, and O2. The results obtained are in perfect agreement with existing data in the literature for the potentials and molecules. Our studies also reveal that as the vibrational quantum number increases, the mean kinetic energy of the system in Tietz-Hua potential increases slowly to a threshold value then decreases. On the other hand,in a Morse potential,the mean kinetic energy increases approximately linearly with the vibrational quantum number increasing.

Appendix A:List of standard definite integral used in this work

Appendix B:List of derived expressions of effective potential and bound state ro-vibrational energy for the special cases of general molecular potentials considered in this work

B4. M¨obius squared oscillator