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On the bound-state solutions of the Manning–Rosen potential including an improved

approximation to the orbital centrifugal term

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2011 Phys. Scr. 83 015010

(http://iopscience.iop.org/1402-4896/83/1/015010)

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Phys. Scr. 83 (2011) 015010 (10pp) doi:10.1088/0031-8949/83/01/015010

On the bound-state solutions of theManning–Rosen potential including animproved approximation to the orbitalcentrifugal termSameer M Ikhdair

Physics Department, Near East University, Nicosia, Mersin 10, Turkey

E-mail: sikhdair@neu.edu.tr

Received 15 January 2010Accepted for publication 24 November 2010Published 10 January 2011Online at stacks.iop.org/PhysScr/83/015010

AbstractThe approximate analytical bound-state solution of the Schrödinger equation for theManning–Rosen (MR) potential is found by taking a new approximation scheme to the orbitalcentrifugal term. The Nikiforov–Uvarov method is used in the calculations. We obtain analyticforms for the energy eigenvalues and the corresponding normalized wave functionsin terms of Jacobi polynomials or hypergeometric functions for different screening parameters1/b. The rotational–vibrational energy states for a few diatomic molecules are calculated forarbitrary quantum numbers n and l with different values of the potential parameter α. Thepresent numerical results agree within five decimal digits with the previously reported resultsfor different 1/b values. A few special cases of the s-wave (l = 0) MR potential and theHulthén potential are also studied.

PACS numbers: 03.65.−w, 02.30.Gp, 03.65.Ge, 34.20.Cf

1. Introduction

The exact analytic solutions of the wave equations(nonrelativistic and relativistic) are only possible for certainpotentials of physical interest under consideration since theycontain all the necessary information on the quantum system.It is well known that the exact solutions of these waveequations are only possible in a few simple cases such asfor the Coulomb potential, the harmonic oscillator potential,the pseudoharmonic potentials, etc [1–5]. The analytic exactsolutions of the wave equation with some exponential-typepotentials are impossible for l 6= 0 states. Therefore,approximation schemes have to be used to deal with the orbitalcentrifugal term like the Pekeris approximation [6–8] and theapproximation scheme suggested by Greene and Aldrich [9].Some of these exponential-type potentials include the Morsepotential [10], the Hulthén potential [11], the Pöschl–Tellerpotential [12], the Woods–Saxon potential [13], Kratzer-typeand pseudoharmonic potentials [14], Rosen–Morse-typepotentials [15], the Manning–Rosen (MR) potential [16–19]and other multiparameter exponential-type potentials [20, 21].

The MR potential has been one of the most useful andconvenient models for studying the energy eigenvalues ofdiatomic molecules [16]. As an empirical potential, the MRpotential gives an excellent description of the interactionbetween two atoms in a diatomic molecule, and it is verygood for describing such interactions close to the surface. Theshort-range MR potential is defined by [16–19]

V (r)=h2

2µb2

[α(α− 1)

(er/b − 1)2−

A

er/b − 1

], (1)

where A and α are two constants and the parameter bcharacterizes the range of the potential [22]. The abovepotential may be also put in the following simple form:

V (r)= −Cer/b + D

(er/b − 1)2, C = A, D = −A −α(α− 1),

(2)which is usually used for the description of diatomicmolecular vibrations and rotations [23, 24]. It is alsoused in several branches of physics for their boundstates and scattering properties. This potential remains

0031-8949/11/015010+10$33.00 Printed in the UK & the USA 1 © 2011 The Royal Swedish Academy of Sciences

Phys. Scr. 83 (2011) 015010 S M Ikhdair

invariant by mapping α → 1 −α and has a relativeminimum at r0 = b ln[1 + 2α(α− 1)/A] with valueV (r0)= −h2 A2/8µb2α(α− 1) for α < 0 or α > 1 andA > 0. Moreover, the second derivative determines the forceconstants at r = r0, which is given by

d2V

dr2

∣∣∣∣r=r0

=A2 [A + 2α(α− 1)]2

8b4α3(α− 1)3. (3)

If α = 0 or α = 1, the potential (1) reduces to the Hulthénpotential [11]. For the potential in equation (1) [16–19],the Schrödinger equation (SE) can be easily solved forthe s-wave, angular momentum quantum number l = 0.However, for the general solution, one needs to includesome approximations to obtain analytical or semi-analyticalsolutions to the SE. Also, it is often necessary to determinethe l-wave (l 6= 0 states), so an analytic procedure would beadvantageous [25–27]. Hence, in the previous papers, severalapproximations have been developed to find better analyticalformulae for the energy bound states and wave functions. Forinstance, in the l = 0 case, the bound-state energy spectrafor the MR potential have already been calculated by usingthe path-integral approach [17] and the function analysismethod [18]. For the l 6= 0 case, the potential cannot be solvedexactly without using the approximation scheme. Recently,Qiang and Dong [19] approximated the centrifugal term

1

r2≈

1

b2

[1

er/b − 1+

1

(er/b − 1)2

]=

1

b2

er/b

(er/b − 1)2

and studied l-wave bound-state solutions of the SE for MRpotential. Further, the scattering state solutions for the samepotential and approximation have also been investigated [25].The above approximation has also been applied to obtainthe l-wave solutions of SE with the MR potential in threedimensions and D dimensions and also with the Hulthénpotential using the Nikiforov and Uvarov (NU) method[11, 19, 26, 27]. The present approximations provide goodresults, which are in agreement within five decimal digitswith the previously reported numerical integration method ofLucha and Schöberl [28] for short-range potential (large b andsmall l) but not for long-range potential (small b and large l).

The main purpose of this paper is to improve the accuracyof the previous approximations introduced in [26, 29], so thatwe apply a different approximation scheme proposed recentlyin [27] for the centrifugal term l(l + 1)r−2, to make the resultsagree better with [28]. Thus, with this new approximationscheme, we calculate the l 6= 0 energy levels and wavefunctions for the MR potential using the NU method [30],which has shown its power in calculating the exact energylevels for some solvable quantum systems. For this, the resultsare in better agreement with those obtained by means ofthe numerical integration method [28]. As an illustration,the method is applied to find the rotational–vibrational(ro-vibrational) energy states for a few diatomic molecules:HCl, CH, LiH, CO, NO, O2, I2, N2, H2 and Ar2.

The paper is organized as follows. In section 2, weapply the new approximation scheme to calculate the l-wavebound-state eigensolutions of the SE for MR potentialby using the NU method. In section 3, we present ourro-vibrational energy levels for a few diatomic molecules.

Section 4, is devoted for two special cases, namely s-wave(l = 0) and the Hulthén potential. Finally, we make a fewconcluding remarks in section 5.

2. Bound-state solutions

To study any quantum physical system, we solve the originalSE that is given in well-known textbooks [1, 2]:(

p2

2m+ V (r)

)ψnlm(r)= Enlψnlm(r), (4)

where the potential V (r) is taken as the MR form in (1).Further, we set the wave functions ψnlm(r)=

unl (r)r Ylm(θ, φ)

to obtain the following radial SE:

d2unl(r)

dr2+

[2µEnl

h2 − Veff(r)

]unl(r)= 0, (5a)

Veff(r)=1

b2

[α(α− 1)

(er/b − 1)2−

A

er/b − 1

]+

l(l + 1)

r2, (5b)

in which unl(0)= 0 and limr→∞

unl(r)= 0. To solve the above

equation for l 6= 0 states, we need to apply the approximatescheme to the centrifugal term given by

1

r2≈

1

b2

[D0 + D1

1

er/b − 1+ D2

1

(er/b − 1)2

], (6a)

and the higher-order terms are neglected. These solutionsare valid for r/b � 1; that is, the solutions obtained arevalid for α(α− 1)/A � 1 but positive. Obviously, the aboveapproximation to the centrifugal term turns to r−2 whenthe parameter b goes to infinity (small screening parameterδ = 1/b) as

limb→∞

[1

b2

(D0 +

1

er/b − 1+

1

(er/b − 1)2

)]=

1

r2, (6b)

which shows that the usual approximation is the limit ofour approximation (cf e.g. [31] and the references therein).The values of the parameters Di (i = 0, 1 and 2) are givenby [27, 31]

D0 '1

12, D1 = D2 = 1. (7)

However, the values of the parameters Di (i = 0, 1 and 2)used by Wei and Dong [32] are given by

D0 =12ε2

1 − 4ε1 (2A + 3ε1) log(ε2)+ ε23 log(ε2)

2

ε24 log(ε2)4

, (8a)

D1 =8ε2

1 [−6ε1 + (3A + 4ε1) log(ε2)]

Aε24 log(ε2)4

, (8b)

D2 = −16ε3

1 [−3ε1 + ε3 log(ε2)]

A2ε24 log(ε2)4

, (8c)

where ε1 = α(α− 1), ε2 = 1 + 2α(α− 1)/A, ε3 = Aε2 andε4 = bε3.

Now, we need to recast differential equations (5) andthe approximation (6a) into the form of equation (1) of [33]

2

Phys. Scr. 83 (2011) 015010 S M Ikhdair

Table 1. Bound-state energy spectrum (−Enl ) (in atomic units) for the MR potential as a function of 1/b for 2p, 3p, 3d, 4p, 4d, 4f, 5p, 5d,5f, 5g, 6p, 6d, 6f and 6g states with α = 0.75, α = 1.5 and A = 2b.

States 1/b α = 0.75 α = 1.5

Present Previous [26] Lucha and Schöberl [28] Present Previous [26] Lucha and Schöberl [28]

2p 0.025 0.120 5279 0.120 5793 0.120 5271 0.089 9715 0.090 0229 0.089 97080.050 0.108 2170 0.108 4228 0.108 2151 0.080 0414 0.080 2472 0.080 04000.075 0.096 4490 0.096 9120 0.096 4469 0.070 5703 0.071 0332 0.070 57010.100 0.085 2240 0.086 0740 0.061 5579 0.057 7157

3p 0.025 0.045 8783 0.045 9297 0.045 8779 0.036 9137 0.036 9651 0.036 91340.050 0.035 0614 0.035 2672 0.035 0633 0.027 2662 0.027 4719 0.027 26960.075 0.025 5480 0.026 0110 0.025 5654 0.018 9220 0.0193 850 0.018 94740.100 0.017 3379 0.018 1609 0.011 8813 0.012 7043

3d 0.025 0.044 7756 0.044 9299 0.044 7743 0.039 4801 0.039 6345 0.039 47890.050 0.033 6909 0.034 3082 0.033 6930 0.029 4456 0.030 0629 0.029 44960.075 0.023 7279 0.025 1168 0.023 7621 0.020 4232 0.021 8121 0.020 4663

4p 0.025 0.020 8094 0.020 8608 0.020 8097 0.017 1735 0.017 2249 0.017 17400.050 0.011 7234 0.011 9292 0.011 7365 0.008 8961 0.009 1019 0.008 91340.075 0.005 0143 0.005 4773 0.005 0945 0.003 0849 0.003 5478 0.003 1884

4d 0.025 0.020 3012 0.020 4555 0.020 3017 0.018 2106 0.018 3649 0.018 21150.050 0.010 9569 0.011 5742 0.010 9904 0.009 4775 0.010 0947 0.009 51670.075 0.003 8158 0.005 2047 0.004 0331 0.002 8919 0.004 2808 0.003 1399

4f 0.025 0.019 9801 0.020 2887 0.019 9797 0.018 6136 0.018 9223 0.018 61370.050 0.010 1938 0.011 4284 0.010 2393 0.009 3507 0.010 5852 0.009 40150.075 0.002 3157 0.005 0935 0.002 6443 0.001 8749 0.004 6527 0.002 2307

5p 0.025 0.009 8062 0.009 8576 0.009 8079 0.008 0793 0.008 1308 0.008 08165d 0.025 0.009 5094 0.009 6637 0.009 5141 0.008 5359 0.008 6902 0.008 54155f 0.025 0.009 2751 0.009 5837 0.009 2825 0.008 6536 0.008 9622 0.008 66195g 0.025 0.009 0254 0.009 5398 0.009 0330 0.008 6066 0.009 1210 0.008 61506p 0.025 0.004 3537 0.004 4051 0.004 3583 0.003 4820 0.003 5334 0.003 48766d 0.025 0.004 1518 0.004 3061 0.004 1650 0.003 6666 0.003 8209 0.003 68136f 0.025 0.003 9566 0.004 2652 0.003 9803 0.003 6520 0.003 9606 0.003 67746g 0.025 0.003 7284 0.004 2428 0.003 7611 0.003 5278 0.004 0422 0.003 5623

by introducing the change in variables r → z through themapping function z = e−r/b and defining

εnl =

√−

2µb2 Enl

h2 +1El > 0, Enl <h2

2µb21El ,

1El = l(l + 1)D0,

(9a)

β1 = A − l(l + 1)D1, (9b)

β2 = α(α− 1)+ l(l + 1)D2, (9c)

to obtain the following compact hypergeometric equation:

d2unl(z)

dz2+(1 − z)

z(1 − z)

dunl(z)

dz+

1

[z(1 − z)]2

×{−ε2

nl +(2ε2

nl +β1)

z − (ε2nl +β1 +β2)z

2}

unl(z)= 0.

(10)

We note that for the presence of bound-state (real) solutions,εnl must be a positive real parameter and we require that

z =

{0, when r → ∞,

1, when r → 0,(11)

for the radial wave functions to fulfill the boundary conditions,i.e. unl(0)→ 0 and unl(1)→ 0. Let us begin by comparing

equation (10) with equation (1) of [33]; then we obtain thefollowing definitions:

τ (z)= 1 − z, σ (z)= z − z2,

σ (z)= −ε2nl +

(2ε2

nl +β1)

z − (ε2nl +β1 +β2)z2.

(12)

After applying relations (A.1)–(A.4) of [33], the followinguseful functions usually defined by the NU method [30] areachieved:

k = β1 − aεnl , a =

√(1 − 2α)2 + 4l(l + 1)D2, (13)

π(z)= −z

2−

1

2[(a + 2εnl) z − a] (14)

and

τ(z)= 1 + 2εnl − (2 + 2εnl + a) z, τ ′(z)= − (2 + 2εnl + a).(15)

We can also write the values of λ= k +π ′(z) andλn = −nτ ′(z)− n(n−1)

2 σ ′′(z), n = 0, 1, 2, . . ., to obtain

λ= β1 − (1 + a)( 12 + εnl) (16)

andλn = n(1 + n + a + 2εnl), n = 0, 1, 2, . . . , (17)

respectively. Furthermore, using the relation, λ= λn, oralternatively the energy equation (A.5) of [33] allows one toobtain

εnl =A +α(α− 1)+ l(l + 1) (D2 − D1)

2n + 1 + a−

2n + 1 + a

4. (18)

3

Phys. Scr. 83 (2011) 015010 S M Ikhdair

Table 2. The ro-vibrational energy spectra (−Enl ) (in eV) for HCl and CH for 2p, 3p, 3d, 4p, 4d, 4f, 5p, 5d, 5f, 5g, 6p, 6d, 6f and 6g stateswith hc = 1973.29 eV Å, µHCl = 0.980 1045 amu, µCH = 0.929 931 amu and A = 2b.

States 1/b HCl CH

α = 0, 1 α = 0.75 α = 1.5 α = 0, 1 α = 0.75 α = 1.5

2p 0.025 4.809 33 5.140 59 3.837 34 5.068 82 5.417 95 4.044 380.050 4.309 60 4.615 53 3.413 82 4.542 12 4.864 55 3.598 010.075 3.832 14 4.113 62 3.009 87 4.038 90 4.335 56 3.172 260.100 3.376 95 3.634 86 2.425 49 3.559 15 3.830 97 2.767 14

3p 0.025 1.864 14 1.956 74 1.574 39 1.964 72 2.062 31 1.659 340.050 1.414 39 1.495 39 1.162 92 1.490 71 1.576 08 1.225 660.075 1.020 23 1.089 64 0.807 04 1.075 28 1.148 43 0.850 580.100 0.681 66 0.739 47 0.506 74 0.718 44 0.779 37 0.534 09

3d 0.025 1.859 75 1.909 71 1.683 85 1.960 10 2.012 75 1.774 700.050 1.396 84 1.436 94 1.255 88 1.472 21 1.514 47 1.323 630.075 0.980 74 1.012 01 0.871 06 1.033 66 1.066 61 0.918 060.100 0.611 46 0.634 92 0.529 41 0.644 45 0.669 17 0.557 98

4p 0.025 0.850 82 0.887 53 0.732 46 0.896 72 0.935 42 0.771 980.050 0.471 04 0.500 01 0.379 42 0.496 459 0.526 989 0.399 8960.075 0.193 51 0.213 87 0.131 57 0.203 948 0.225 404 0.138 671

4d 0.025 0.846 43 0.865 86 0.776 69 0.892 099 0.912 577 0.818 5990.050 0.453 49 0.467 32 0.404 22 0.477 960 0.492 531 0.426 0290.075 0.154 02 0.162 75 0.123 34 0.162 325 0.171 527 0.129 997

4f 0.025 0.839 85 0.852 16 0.793 88 0.885 162 0.898 138 0.836 7160.050 0.427 16 0.434 77 0.398 81 0.450 211 0.458 228 0.420 3290.075 0.094 777 0.098 765 0.079 967 0.099 891 0.104 094 0.084 281

5p 0.025 0.400 99 0.418 24 0.344 59 0.422 623 0.440 805 0.363 1815d 0.025 0.396 60 0.405 58 0.364 06 0.417 998 0.427 463 0.383 7055f 0.025 0.390 018 0.395 586 0.369 08 0.411 061 0.416 929 0.388 9935g 0.025 0.381 242 0.384 94 0.367 077 0.401 811 0.405 709 0.386 8826p 0.025 0.176 998 0.185 69 0.148 51 0.186 548 0.195 706 0.156 5216d 0.025 0.172 610 0.177 08 0.156 38 0.181 923 0.186 631 0.164 8206f 0.025 0.166 028 0.168 752 0.155 759 0.174 986 0.177 856 0.164 1636g 0.025 0.157 252 0.159 02 0.150 462 0.165 736 0.167 600 0.158 580

b is in pm.

Plugging the parameters given in equation (9) into (18),we finally obtain the following discrete bound-state energyeigenvalues:

Enl = E (approx)nl =

h2l(l + 1)D0

2µb2

−h2

2µb2

[A +α(α− 1)+ l(l + 1) (D2 − D1)

2n + 1 +√(1 − 2α)2 + 4l(l + 1)D2

−2n + 1 +

√(1 − 2α)2 + 4l(l + 1)D2

4

]2

, (19)

where 06 n 6 nmax and l = 0, 1, 2, . . . signify the usualvibrational and rotational quantum numbers, respectively. It isfound that the parameter a in equation (13) remains invariantby mapping α → 1 −α, and so do the bound-state energiesEnl . An important quantity of interest for the MR potentialis the critical coupling constant Ac, which is that value of Afor which the binding energy of the level in question becomeszero. Hence, using equation (19), in atomic units h2

= µ=

Z = e = 1, we find the following critical coupling constant:

Ac =14

(2n + 1 +

√(1 − 2α)2 + 4l(l + 1)D2

)2

−α(α− 1)− l(l + 1) (D2 − D1). (20)

Let us now turn to the calculation of the radial partof the normalized wave functions. After applying relations

(A.6)–(A.10) of [33], we obtain

φ(z)= zεnl (1 − z)(1+a)/2, (21)

ρ(z)= z2εnl (1 − z)a, (22)

ynl(z)= Cnz−2εnl (1 − z)−a dn

dzn[zn+2εnl (1 − z)n+a]. (23)

The functions ynl(z), up to a numerical factor, are in theform of Jacobi polynomials, i.e. ynl(z)' P (2εnl ,a)

n (1 − 2z)(the physical interval (0,∞) for variable r is mapped tothe interval (0, 1) for variable z) [13, 14]. Hence, theapproximated radial wave functions satisfying equations (5)are given by

unl(r)= u(approx)nl (r)=Nnle

−εnlr/b(1 − e−r/b)νl

×2 F1(−n, n + 2 (εnl + νl) ; 2εnl + 1; e−r/b),

εnl > 0, νl = (1 + a) /2> 1, (24)

where a and εnl are given in equations (13) and (18),respectively, and Nnl is a normalization constant determinedin the appendix.

When l = 0, we deal with the s-wave case; the possibleenergies for the bound states and the corresponding wave

4

Phys. Scr. 83 (2011) 015010 S M Ikhdair

Table 3. The ro-vibrational energy spectra (−Enl) (in eV) for LiH and CO for 2p, 3p, 3d, 4p, 4d, 4f, 5p, 5d, 5f, 5g, 6p, 6d, 6f and 6g stateswith µLiH = 0.880 1221 amu, µCO = 6.860 6719 amu and A = 2b.

States 1/b LiH CO

α = 0, 1 α = 0.75 α = 1.5 α = 0, 1 α = 0.75 α = 1.5

2p 0.025 5.355 68 5.724 57 4.273 26 0.687 053 0.734 377 0.548 1960.050 4.799 18 5.139 85 3.801 63 0.615 663 0.659 367 0.487 6930.075 4.267 47 4.580 92 3.351 79 0.547 453 0.587 664 0.429 9850.100 3.760 57 4.047 78 2.923 74 0.482 425 0.519 270 0.375 073

3p 0.025 2.075 91 2.179 02 1.753 24 0.266 308 0.279 536 0.224 9150.050 1.575 07 1.665 27 1.295 03 0.202 058 0.213 629 0.166 1330.075 1.136 13 1.213 42 0.898 72 0.145 749 0.155 664 0.115 2920.100 0.759 101 0.823 478 0.564 311 0.097 381 0.105 640 0.072 393

3d 0.025 2.071 02 2.126 65 1.875 14 0.265 681 0.272 818 0.240 5530.050 1.555 52 1.600 18 1.398 54 0.199 550 0.205 279 0.179 4120.075 1.092 15 1.126 98 0.970 015 0.140 107 0.144 574 0.124 4390.100 0.680 918 0.707 045 0.589 556 0.087 352 0.090 703 0.075 631

4p 0.025 0.947 473 0.988 358 0.815 668 0.121 547 0.126 792 0.104 6380.050 0.524 555 0.556 813 0.422 528 0.067 293 0.071 431 0.054 2040.075 0.215 490 0.238 160 0.146 518 0.027 644 0.030 552 0.018 796

4d 0.025 0.942 586 0.964 223 0.864 926 0.120 920 0.123 695 0.110 9570.050 0.505 009 0.520 405 0.450 139 0.064 785 0.066 760 0.057 7460.075 0.171 512 0.181 234 0.137 354 0.022 002 0.023 250 0.017 620

4f 0.025 0.935 256 0.948 967 0.884 069 0.119 979 0.121 738 0.113 4130.050 0.475 690 0.484 161 0.444 117 0.061 024 0.062 111 0.056 9740.075 0.105 544 0.109 984 0.089 051 0.013 540 0.014 109 0.011 424

5p 0.025 0.446 540 0.465 751 0.383 735 0.057 284 0.059 749 0.049 2275d 0.025 0.441 654 0.451 655 0.405 420 0.056 658 0.057 941 0.052 0095f 0.025 0.434 324 0.440 525 0.411 008 0.055 717 0.056 513 0.052 7265g 0.025 0.424 551 0.428 669 0.408 777 0.054 464 0.054 992 0.052 4406p 0.025 0.197 105 0.206 782 0.165 379 0.025 286 0.026 527 0.021 2166d 0.025 0.192 219 0.197 193 0.174 148 0.024 659 0.025 297 0.022 3416f 0.025 0.184 889 0.187 922 0.173 454 0.023 718 0.024 108 0.022 2526g 0.025 0.175 116 0.177 085 0.167 554 0.022 465 0.022 717 0.021 495

b is in pm.

functions are written explicitly as follows:

for α < 1/2

En = −h2

8µb2

[A +α(α− 1)

n −α + 1− (n −α + 1)

]2

,

n = 0, 1, 2, . . . ,

nmax = [√

A +α(α− 1)+α− 1] and

un0(r)=Nne−(εn/b)r (1 − e−r/b)(1−α)

×2 F1(−n, n + 2 (εn −α + 1) ; 2εn + 1; e−r/b),

(25)

where εn =12

[ A+α(α−1)n−α+1 − (n −α + 1)

],

and for α > 1/2

En = −h2

8µb2

[A +α(α− 1)

n +α− (n +α)

]2

,

n = 0, 1, 2, . . . ,

nmax = [√

A +α(α− 1)−α] and

un0(r)= Nne−(ε′n/b)r (1 − e−r/b)α

×2 F1(−n, n + 2(ε′n +α); 2ε′

n + 1; e−r/b),

(26)

where ε′n =

12

[ A+α(α−1)n+α − (n +α)

]. The normalization

constantsNn and Nn are calculated explicitly in the appendix.Note that nmax is the number of bound states for the wholebound spectrum near the continuous zone. nmax is the largestinteger which is less than or equal to the value of n that makes

the right-hand side of equations (25) and (26) vanish. Theabove results are identical to equations (12) and (13) givenin [34].

3. Applications to diatomic molecules

To show the accuracy of the new approximation scheme,we have calculated the ro-vibrational energy spectra forvarious n and l quantum numbers with two different valuesof the parameters α. The results obtained by means ofequation (19) are compared with those obtained by aMathematica package programmed by Lucha andSchöberl [28] as listed in table 1 for short-range (large b) andlong-range (small b) potentials. Table 1 can be used to assessthe validity and usefulness of our present approximations. Theresults of the energy spectrum for the p-state show that thepercentage accuracy decreases as either n or 1/b increases,for example, when 1/b = 0.025; then the range of accuraciescan be 0.000 75, 0.000 87, 0.0014, 0.017 and 0.11% forn = 0, 1, 2, 3 and 4, respectively. However, when 1/b variesbetween 0.025 and 0.075, the range of accuracies can be0.000 75–0.0022%, 0.000 87–0.068% and 0.0014–1.57%for n = 0, 1 and 2, respectively. In addition, we present thero-vibrational energy states for a few diatomic moleculesHCl, CH, LiH, CO, NO, O2, I2, N2, H2 and Ar2 in tables 2–6.The lowest eigenvalues of l = 0, 1, 2 and 3 are given at fourvalues of 1/b in the range 0.025–0.1 covering both weakerand stronger interactions to demonstrate the generality of

5

Phys. Scr. 83 (2011) 015010 S M Ikhdair

Table 4. The ro-vibrational energy spectra (−Enl) (in eV) for NO and O2 for 2p, 3p, 3d, 4p, 4d, 4f, 5p, 5d, 5f, 5g, 6p, 6d, 6f and 6g stateswith µNO = 7.468 441 amu, µO2 = 7.997 457 504 amu and A = 2b.

States 1/b NO O2

α = 0, 1 α = 0.75 α = 1.5 α = 0, 1 α = 0.75 α = 1.5

2p 0.025 0.631 142 0.674 615 0.503 585 0.589 393 0.629 990 0.470 2740.050 0.565 561 0.605 709 0.448 005 0.528 150 0.565 642 0.418 3700.075 0.502 903 0.539 841 0.394 993 0.469 637 0.504 132 0.368 8650.100 0.443 166 0.477 013 0.344 550 0.413 852 0.445 459 0.321 759

3p 0.025 0.244 637 0.256 788 0.206 612 0.228 454 0.239 802 0.192 9450.050 0.185 615 0.196 245 0.152 613 0.173 337 0.183 263 0.142 5180.075 0.133 888 0.142 996 0.105 910 0.125 032 0.133 537 0.098 9040.100 0.089 457 0.097 043 0.066 502 0.083 539 0.090 624 0.062 103

3d 0.025 0.244 061 0.250 617 0.220 977 0.227 917 0.234 039 0.206 3600.050 0.183 311 0.188 574 0.164 812 0.171 186 0.176 100 0.153 9100.075 0.128 706 0.132 809 0.114 312 0.120 192 0.124 024 0.106 7500.100 0.080 243 0.083 322 0.069 477 0.074 935 0.077 810 0.064 881

4p 0.025 0.111 655 0.116 474 0.096 123 0.104 270 0.108 769 0.089 7640.050 0.061 816 0.065 618 0.049 793 0.057 727 0.061 277 0.046 4990.075 0.025 395 0.028 066 0.017 267 0.023 715 0.026 210 0.016 124

4d 0.025 0.111 080 0.113 629 0.101 928 0.103 732 0.106 113 0.095 1850.050 0.059 513 0.061 327 0.053 047 0.055 576 0.057 271 0.049 5380.075 0.020 212 0.021 358 0.016 187 0.018 875 0.019 945 0.015 116

4f 0.025 0.110 216 0.111 831 0.104 184 0.102 925 0.104 434 0.097 2920.050 0.056 058 0.057 056 0.052 337 0.052 350 0.053 282 0.048 8750.075 0.012 438 0.012 961 0.010 494 0.011 615 0.012 104 0.009 800

5p 0.025 0.052 623 0.054 887 0.045 221 0.049 142 0.051 256 0.042 2305d 0.025 0.052 047 0.053 225 0.047 777 0.048 604 0.049 705 0.044 6175f 0.025 0.051 183 0.051 914 0.048 435 0.047 797 0.048 480 0.045 2315g 0.025 0.050 031 0.050 517 0.048 173 0.046 722 0.047 175 0.044 9866p 0.025 0.023 228 0.024 368 0.019 489 0.021 691 0.022 756 0.018 2006d 0.025 0.022 652 0.023 238 0.020 523 0.021 154 0.021 701 0.019 1656f 0.025 0.021 788 0.022 146 0.020 441 0.020 347 0.020 681 0.019 0896g 0.025 0.020 637 0.020 869 0.019 746 0.019 272 0.019 488 0.018 439

b is in pm.

our results. The formalism is quite simple, computationallyefficient, reliable and illustrated very accurately.

4. Some special cases

Let us study a few special cases. In the case where α=0or α = 1, the MR potential (1) reduces to the Hulthénpotential [9, 11]

V (H)(r)= −V0e−δr

1 − e−δr, V0 = Ze2δ, δ = b−1, (27)

where Ze2 is the strength, δ is the screening parameter and b isthe range of the potential. If the potential is used for atoms, theZ is identified with the atomic number. Furthermore, takingb = 1/δ and identifying (Ah2/2µb2) as Ze2δ, we are able toobtain the ro-vibrating energy states and the normalized wavefunctions deduced from equations (19) and (24), respectively:

Enl = −h2δ2

2µ

[(2µZe2/h2δ

)+ l(l + 1) (D2 − D1)

2n + 1 +√

1 + 4l(l + 1)D2

−2n + 1 +

√1 + 4l(l + 1)D2

4

]2

+h2δ2l(l + 1)D0

2µ, 06 n, l <∞, (28)

and

unl(r)=Nnle−(εnl/b)r (1 − e−r/b)νl

×2 F1(−n, n + 2(εnl + νl); 2εnl + 1; e−r/b),

εnl =

√−

2µEn,l

h2δ2+ l(l + 1)D0 > 0,

(29)

νl =1

2(1 +

√1 + 4l(l + 1)D2)> 1,

whereNnl is as given in the appendix. Also, for s-wave (l = 0)states, we obtain

En = −µ

(Ze2

)2

2h2

[1

(n + 1)−

h2δ

2Ze2µ(n + 1)

]2

, 06 n <∞

(30)and

un(r)=Nne−(εn/b)r (1 − e−r/b)

×2 F1(−n, n + 2(εn + 1); 2εn + 1; e−r/b),

εn =

√−

2µEn0

h2δ2> 0, (31)

where Nn can be easily found from either relation (A.7)or (A.9) after setting α = 0 or α = 1 in the appendix,

6

Phys. Scr. 83 (2011) 015010 S M Ikhdair

Table 5. The ro-vibrational energy spectra (−Enl) (in eV) for I2 and N2 for 2p, 3p, 3d, 4p, 4d, 4f, 5p, 5d, 5f, 5g, 6p, 6d, 6f and 6g states withµI2 = 63.452 235 02 amu, µN2 = 7.003 35 amu and A = 2b.

States 1/b I2 N2

α = 0, 1 α = 0.75 α = 1.5 α = 0, 1 α = 0.75 α = 1.5

2p 0.025 0.074 2866 0.079 4033 0.059 2729 0.673 056 0.719 416 0.537 0280.050 0.066 5676 0.071 2930 0.052 7310 0.603 120 0.645 934 0.477 7570.075 0.059 1925 0.063 5403 0.046 4914 0.536 300 0.575 692 0.421 2250.100 0.052 1615 0.056 1452 0.040 5541 0.472 597 0.508 691 0.367 431

3p 0.025 0.028 7942 0.030 2244 0.024 3186 0.260 883 0.273 841 0.220 3330.050 0.021 8472 0.023 0983 0.017 9628 0.197 941 0.209 277 0.162 7480.075 0.015 7589 0.016 8309 0.012 4658 0.142 780 0.152 493 0.112 9430.100 0.010 5292 0.011 4221 0.007 8274 0.095 397 0.103 488 0.070 918

3d 0.025 0.028 7264 0.029 4980 0.026 0094 0.260 269 0.267 260 0.235 6520.050 0.021 5761 0.022 1955 0.019 3987 0.195 485 0.201 097 0.175 7570.075 0.015 1489 0.015 6319 0.013 4547 0.137 253 0.141 629 0.121 9030.100 0.009 4448 0.009 8072 0.008 1775 0.085 572 0.088 855 0.074 090

4p 0.025 0.013 1420 0.013 7091 0.011 3138 0.119 070 0.124 209 0.102 5060.050 0.007 2759 0.007 2330 0.005 8607 0.065 922 0.069 976 0.053 1000.075 0.002 9890 0.003 3034 0.002 0323 0.027 081 0.029 930 0.018 413

4d 0.025 0.013 0743 0.013 3744 0.011 9971 0.118 456 0.121 175 0.108 6970.050 0.007 0048 0.007 2183 0.006 2437 0.063 465 0.065 400 0.056 5700.075 0.002 3790 0.002 5138 0.001 9052 0.021 554 0.022 776 0.017 261

4f 0.025 0.012 9726 0.013 1628 0.012 2626 0.117 535 0.119 258 0.111 1020.050 0.006 5981 0.006 7156 0.006 1602 0.059 781 0.060 845 0.055 8130.075 0.001 4640 0.001 5256 0.001 2352 0.013 264 0.013 822 0.011 191

5p 0.025 0.006 1938 0.006 4603 0.005 3226 0.056 117 0.058 532 0.048 2255d 0.025 0.006 1260 0.006 2647 0.005 6234 0.055 503 0.056 760 0.050 9505f 0.025 0.006 0243 0.006 1104 0.005 7009 0.054 582 0.055 361 0.051 6525g 0.025 0.005 8888 0.005 9459 0.005 6700 0.053 354 0.053 872 0.051 3726p 0.025 0.002 7340 0.002 8682 0.002 2939 0.024 771 0.025 987 0.020 7836d 0.025 0.002 6662 0.002 7352 0.002 4155 0.024 156 0.024 782 0.021 8856f 0.025 0.002 5645 0.002 6066 0.002 4059 0.023 235 0.023 616 0.021 7986g 0.025 0.002 4290 0.002 4563 0.002 3241 0.022 007 0.022 255 0.021 057

b is in pm.

respectively. Here, in this case εn = ε′n and the number of

bound states nmax is also the same in both relations (A.8) and(A.10). In the usual approximation [19] where D0 = 0 andD1 = D2 = 1, equations (28) and (29) become

Enl = −µ

(Ze2

)2

2h2

[1

(n + l + 1)−

h2δ

2µZe2(n + l + 1)

]2

,

06 n, l <∞, (32)

and

unl(r)=Nnle−(εnl/b)r (1 − e−r/b)l+1

×2 F1(−n, n + 2(εnl + l + 1); 2εnl + 1; e−r/b),

εnl =

√−

2µEnl

h2δ2> 0, (33)

where Nnl can be found from relation (A.6) by settingνl = l + 1. Essentially, these results coincide with thoseobtained by the Feynman integral method [17] and thestandard way [18, 19]. Following [27] by taking D1 = D2 = 1and D0 = 1/12, equations (28) and (29) become

Enl = −µ

(Ze2

)2

2h2

[1

(n + l + 1)−

h2δ

2µZe2(n + l + 1)

]2

+l(l + 1)h2δ2

24µ, 06 n, l <∞, (34)

and

unl(r)=Nnle−(εnl/b)r (1 − e−r/b)l+1

×2 F1(−n, n + 2 (εnl + l + 1) ; 2εnl + 1; e−r/b),

εnl =

√−

2µEnl

h2δ2+

l(l + 1)

12> 0, (35)

which coincide, for the ground state, with equation (6) inGönül and Zorba [9]. The Hulthén potential behaves likethe Coulomb potential near the origin (r → 0), but in theasymptotic region (r � 1) the Hulthén potential decreasesexponentially, so its capacity for bound states is smallerthan the Coulomb potential. However, for small values ofthe screening parameter or for δr � 1 (i.e. r/b � 1), theHulthén potential becomes the Coulomb potential given byVC(r)= −

Ze2

r with energy levels and wave functions

Enl = −ε0

(n + l + 1)2, n, l = 0, 1, 2, . . . ,

ε0 =Z2h2

2µa20

, a0 =h2

µe2, (36)

7

Phys. Scr. 83 (2011) 015010 S M Ikhdair

Table 6. The ro-vibrational energy spectra (−Enl) (in eV) for H2 and Ar2 for 2p, 3p, 3d, 4p, 4d, 4f, 5p, 5d, 5f, 5g, 6p, 6d, 6f and 6g stateswith µH2 = 0.504 07 amu, µAr2 = 19.9812 amu and A = 2b.

States 1/b H2 Ar2

α = 0, 1 α = 0.75 α = 1.5 α = 0, 1 α = 0.75 α = 1.5

2p 0.025 9.351 18 9.995 28 7.461 26 0.235 904 0.252 153 0.188 2270.050 8.379 51 8.974 35 6.637 77 0.211 392 0.226 398 0.167 4520.075 7.451 14 7.998 44 5.852 33 0.187 972 0.201 778 0.147 6380.100 6.566 08 7.067 55 5.104 95 0.165 644 0.178 295 0.128 784

3p 0.025 3.624 60 3.804 64 3.061 22 0.091 439 0.095 981 0.077 2260.050 2.750 12 2.907 61 2.261 16 0.069 378 0.073 351 0.057 0430.075 1.983 72 2.118 67 1.569 19 0.050 044 0.053 448 0.039 5860.100 1.325 41 1.437 82 0.985 31 0.033 437 0.036 272 0.024 857

3d 0.025 3.616 07 3.713 20 3.274 05 0.091 2234 0.093 6738 0.082 59530.050 2.715 99 2.793 96 2.441 90 0.068 5169 0.070 4839 0.061 60240.075 1.906 94 1.967 73 1.693 68 0.048 1067 0.049 6405 0.042 72680.100 1.188 90 1.234 52 102 938 0.029 9927 0.031 1436 0.025 9685

4p 0.025 1.654 32 1.725 70 1.424 18 0.041 734 0.043 535 0.035 9280.050 0.915 89 0.972 21 0.737 75 0.023 105 0.024 526 0.018 6110.075 0.376 25 0.415 84 0.255 83 0.009 4918 0.010 4904 0.006 4538

4d 0.025 1.645 78 1.683 56 1.510 19 0.041 5186 0.042 4716 0.038 09780.050 0.881 76 0.908 64 0.785 96 0.022 2444 0.022 9225 0.019 82750.075 0.299 46 0.316 44 0.239 82 0.007 5547 0.007 9829 0.006 0501

4f 0.025 1.632 99 1.656 93 1.543 61 0.041 1957 0.041 7996 0.038 94100.050 0.830 57 0.845 36 0.775 44 0.020 9530 0.021 3261 0.019 56230.075 0.184 28 0.192 04 0.155 49 0.004 6490 0.004 8445 0.003 9225

5p 0.025 0.779 67 0.813 22 0.670 01 0.019 6690 0.020 5152 0.016 90265d 0.025 0.771 14 0.788 60 0.707 88 0.019 4538 0.019 8943 0.017 85785f 0.025 0.758 34 0.769 17 0.717 63 0.019 1309 0.019 4040 0.018 10395g 0.025 0.741 28 0.748 47 0.713 74 0.018 7004 0.018 8818 0.018 00566p 0.025 0.344 15 0.361 05 0.288 76 0.008 6820 0.009 1082 0.007 28456d 0.025 0.335 62 0.344 30 0.304 07 0.008 4667 0.008 6859 0.007 67086f 0.025 0.322 82 0.328 12 0.302 86 0.008 1439 0.082 7750 0.007 64026g 0.025 0.305 76 0.309 20 0.292 56 0.007 7134 0.007 8001 0.007 3804

b is in pm.

where ε0 = 13.6 eV and a0 is the Bohr radius for the Hydrogenatom [3]. The wave functions also take the form

unl(r)= Nnl exp

[−µZe2

h2

r

(n + l + 1)

]× r l+1 P(

(2µZe2/h2δ(n+l+1)),2l+1)n (r), (37)

which is found to be identical to [11, 13].

5. Conclusions

We have applied an alternative improved approximationscheme for the centrifugal potential l(l + 1)r−2 to obtainthe energy levels and corresponding wave functions forthe MR potential in the framework of the NU methodfor arbitrary l-waves. We have calculated the bound-stateenergy eigenvalues for the MR potential with α = 0.75, 1.5and A = 2b and several 1/b screening parameter values.The wave functions are physical and bound-state energiesare in good agreement with the results obtained by othermethods for short-range potential, small α and l. Theprecision of the resulting approximation of wave functions(24) for Veff(r) in equation (5b) is due to the approximativecharacter of the centrifugal term 1/r2 in equation (6a)for l 6= 0 states because the wave functions are relevantto the bound-state energy approximation in equation (19).The approximation (6a) for the centrifugal potential allowsus to obtain analytic approximations (34) and (35) for the

eigenvalues and eigenfunctions for the MR potential inthe framework of the NU method for arbitrary l-waves.It is not possible to compute the residual (the error inthe solution u(approx)

nl (r) given by equation (24)) becausethe correct (exact) wave functions, u(correct)

nl (r) of equations(5), are still not found. Hence, the notation residualcan be used for R = Hu(approx)

nl (r)− E (approx)nl u(approx)

nl (r) andthe error (or deviation) for the difference u(exact)

nl (r)−

u(approx)nl (r) and E (exact)

nl − E (approx)nl . Due to the slowness of

the numerical calculation of the hypergeometric functions2F1(−n, n + 2(εnl + νl); 2εnl + 1; e−r/b) and their derivatives inMathematica, the residual R is not evaluated. This residual isexpected to be six orders of magnitude smaller than typicalvalues E (approx)

nl u(approx)nl (r). Accordingly, the error u(exact)

nl (r)−

u(approx)nl (r) is also expected to be small. Furthermore, the

error in approximation of the Hamiltonian (4) with potential(5) is already small, since the approximation used in (6a) isvalid only when r � b (small screening parameter δ = 1/b).To demonstrate this, the NU results have been comparedwith the results of numerical integration procedures usinga Mathematica program [28] and the results obtained fromusual approximation schemes of the centrifugal potential [26].For small 1/b values, the NU results are in good agreementwith the ones obtained in [28], but in the high-screeningregion (large 1/b values) the agreement is poor. It is obviousfrom table 1 that five (three) decimal digits are expectedto be correct in the present (previous) approximation. Thereason is simply that when r/b increases in the high-screening

8

Phys. Scr. 83 (2011) 015010 S M Ikhdair

region, the agreement between the approximation expressionand the centrifugal potential decreases. We have also studiedtwo special cases for l = 0, l 6= 0 and the Hulthén potential.As we have seen, the NU method puts no constraint on thepotential parameter values involved and is easy to implement.Our results are sufficiently accurate for practical purposes.Therefore, we have applied the present solution in equation(19) to obtain the ro-vibrational energies (−Enl) for the HCl,CH, LiH, CO, NO, O2, I2, N2,H2 and Ar2 diatomic molecules.

Acknowledgments

I thank the anonymous referees for their enlighteningcomments and suggestions. The support provided by theScientific and Technological Research Council of Turkey(TÜBITAK) is highly appreciated.

Appendix. Normalization for the radial wavefunctions

The normalization constant Nnl can be determined inclosed form. We start by using the relation between thehypergeometric function and the Jacobi polynomials (seeformula (8.962.1) in [35]), namely

2F1

(−n, n + ν +µ+ 1; ν + 1;

1 − x

2

)=

n!

(ν + 1)nP (ν,µ)

n (x), (ν + 1)n =0(n + ν + 1)

0(ν + 1), (A.1)

to rewrite the wave functions in (24) as

unl(r)=Nnln!0(2εnl + 1)

0(n + 2εnl + 1)e−εnlr/b

× (1 − e−r/b)νl P (2εnl ,2νl−1)n (1 − 2e−r/b). (A.2)

From the normalization condition∫

∞

0 [unl(r)]2 dr = 1

and under the coordinate change x = 1 − 2e−r/b, thenormalization constant in (A.2) is given by

N−2nl = b

[n!0(2εnl + 1)

0(n + 2εnl + 1)

]2 ∫ 1

−1

(1 − x

2

)2εnl

×

(1 + x

2

)2νl−1 (1 + x

2

) [P (2εnl ,2νl−1)

n (x)]2

dx .

(A.3)

The calculation of this integral can be performed by writing

1 + x

2= 1 −

(1 − x

2

)and by making use of the following two integrals (see formula(7.391.5) in [35]):∫ 1

−1(1 − x)ν−1 (1 + x)µ

[P (ν,µ)

n (x)]2

dx

= 2ν+µ0(n + ν + 1)0(n +µ+ 1)

n!ν0(n + ν +µ+ 1), (A.4)

which is valid for Re(ν) > 0 and Re(µ) >−1, and (seeformula (7.391.1) in [35])∫ 1

−1(1 − x)ν (1 + x)µ

[P (ν,µ)

n (x)]2

dx

= 2ν+µ+1 0(n + ν + 1)0(n +µ+ 1)

n!0(n + ν +µ+ 1)(2n + ν +µ+ 1), (A.5)

which is valid for Re(ν) >−1, Re(µ) >−1. This leads to

Nnl =1

0(2εnl + 1)

[εnl(n + εnl + νl)

2b(n + νl)

×0(n + 2εnl + 1)0(n + 2εnl + 2νl)

n!0 (n + 2νl)

]1/2

, (A.6)

where 06 n, l <∞. In the s-wave (l = 0) case, the aboveresult is written explicitly as follows:for α < 1/2

Nn =1

0(2εn + 1)

[εn(n + εn −α + 1)

2b(n −α + 1)

×0(n + 2εn + 1)0(n + 2εn − 2α + 2)

n!0 (n − 2α + 2)

]1/2

, (A.7)

where

εn =A +α(α− 1)

2 (n −α + 1)−

n −α + 1

2,

06 n < nmax = [√

A +α(α− 1)+α− 1] (A.8)

in which α = 0 is included in (−∞, 1/2),and for α > 1/2

Nn =1

0(2εn + 1)

[ε′

n(n + ε′n +α)

2b(n +α)

×0(n + 2ε′

n + 1)0(n + 2ε′n + 2α)

n!0 (n + 2α)

]1/2

, (A.9)

where

ε′n =

A+α(α−1)2(n+α) −

n+α2 ,

06 n < nmax = [√

A +α(α− 1)−α](A.10)

in which α = 1 is included in (1/2,∞).

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Phys. Scr. 83 (2011) 015010 S M Ikhdair

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