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The Manning–Rosen potential using J-matrix approachI. Nasser a , M.S. Abdelmonem a & Afaf Abdel-Hady ba Department of Physics, King Fahd University of Petroleum & Minerals, Dhahran 31261,Saudi Arabiab Faculty of Engineering, El-Asher University, El-Asher City, EgyptAccepted author version posted online: 28 May 2012.Version of record first published: 25Jun 2012.

To cite this article: I. Nasser , M.S. Abdelmonem & Afaf Abdel-Hady (2013): The Manning–Rosen potential using J-matrixapproach, Molecular Physics: An International Journal at the Interface Between Chemistry and Physics, 111:1, 1-8

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Molecular Physics, 2013Vol. 111, No. 1, 1–8, http://dx.doi.org/10.1080/00268976.2012.698026

RESEARCH ARTICLE

The Manning–Rosen potential using J-matrix approach

I. Nassera, M.S. Abdelmonema* and Afaf Abdel-Hadyb

aDepartment of Physics, King Fahd University of Petroleum & Minerals, Dhahran 31261, Saudi Arabia;bFaculty of Engineering, El-Asher University, El-Asher City, Egypt

(Received 28 February 2012; final version received 16 May 2012)

Close to the bound-resonance crossover region, state energies for the Manning–Rosen (MR) potential have beencalculated. New data, for both bound and resonance state energies have been reported nearby the crossoverregion, using the J-matrix approach, for LiH and CO molecules. Furthermore, the average oscillator strength,multipole moments and transition probabilities for certain states have been reported for future comparison.

Keywords: Manning–Rosen; critical screening parameter; singular short-range potential; J-matrix method;S-matrix; Complex rotation; oscillator strength; transition probabilities

PACS numbers: 03.65.Ge; 03.65.-w; 34.20.Cf; 03.65.Nk; 34.20.Gj; 32.70.Cs; 32.10.Dk

The MR potential [1] has been proven to be one of the

most useful and convenient empirical mathematical

models that has relevant applications for the calcula-

tion of the energy eigenvalues of diatomic molecules

[2–7]. The short-range MR potential is defined by:

VðrÞ ¼ �h2�2

2m

�ð�� 1Þðe�r � 1Þ2

� A

e�r � 1

� �ð1Þ

where m is the reduced mass, the strength parameter A

and the screening parameter � characterize the range

of the potential. The parameter � is a constant and for

� ¼ 0 or 1, Equation (1) reduces to Hulthen potential

[8]. To facilitate comparison with the available data [7],

we take A ¼ 2�. The atomic units (�h ¼ a0 ¼ m ¼ e ¼ 1)

are used throughout this work unless otherwise noted.The effect of the parameter � could be understood

with the help of Figure 1. The effective potential

VeffðrÞ, where VeffðrÞ ¼ VðrÞ þ ‘ ð‘þ1Þ2r2

, for ‘ ¼ 1 and

� ¼ 0:615, is displayed in Figure 1 at different values

of �, � ¼ 0:75, 1 and 1.5. For the used value �, onefinds that the well depth decreases with inceasing �.Consequently, the number of bound states decreases.

Similar behaviour has been shown for ‘4 1 and at

different values of �. In this work we restrict our

calculation for the case of � ¼ 0:75 and range of �close to the critical value �c, which we call the critical

screening parameter. �cð�‘Þ, which is a crucial param-

eter in our calculations, is defined as the screening

parameter beyond which the state ð�‘Þ is no longer a

bound state. We will identify our bound states with the

abbreviation n‘, where n is the principle quantumnumber in atomic physics, or the vibrational quantum

number in molecular physics, and ‘ is the rotational

quantum number. The resonance state that is trans-formed from the bound state n‘ is identified with the

prefix R, where R refers to the resonance, so we call

it R n‘.Using Equation (1), the one particle Schrodinger

wave equation (in atomic units) could be written as:

ðH� EÞ�ðrÞ ¼ ðHo þ VðrÞ � EÞ�ðrÞ

¼ � 1

2

d2

dr2þ Veff � E

� ��ðrÞ ¼ 0 ð2Þ

where the symbol ‘ is used for the angular momentumquantum numbers, E is used for the total energy and

Ho is called the reference Hamiltonian. For ‘ ¼ 0,

Equation (2) has been solved exactly for bound stateenergies using different methods, e.g. standard method

[2], Feynman path integral formalism [3], theNikiforov-Uvarov (NU) method [4], and tridiagonal

matrix [5]. Unfortunately, in the case of ‘ 6¼ 0,

Equation (2) can not be solved analytically andmany approximations have been proposed. For

example: Feynman path integral formalism [6], the

NU method [7], diagonalization of the tridiagonal fullHamiltonian with oscillator basis [9] and numerical

integrating procedure using the MATHEMATICA

program [10].

*Corresponding author. Email: msmonem@kfupm.edu.sa

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2 I. Nasser et al.

Although bound state energies of the MR potential

have been reported previously in many articles [1–7,

9,10], the results of resonance state energies have not

been found. So, we are devoting this paper to calculate

a few bound and resonance state energies close to the

crossover region using the J-matrix method [11]. As an

application, we will consider the bound and resonance

state energies of LiH and CO molecules. The usefulness

of the method is further illustrated by calculating the

expectation values of the multipoles hrni, with

n ¼ �1, 1, 2, and 3, and the average oscillator strength

of the dipole transition of the MR potential.The main idea of the J-matrix method is based on

expanding the wave function �ðrÞ, in Equation (2), in

terms of a complete square integrable Laguerre basis

set ’nð�rÞ� �

, where:

’nðxÞ ¼ anx‘þ1e�x=2L2‘þ1

n ðxÞ; n ¼ 0, 1, 2, . . . ð3Þ

with x ¼ �r and � is the positive length scale

parameter. L2‘þ1n ðxÞ is the associate Laguerre poly-

nomial, and an ¼ffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffi�� nþ 1ð Þ=� nþ 2‘þ 2ð Þ

pis the

normalization constant. The basis set ’nð�rÞ� �

satisfies

the boundary conditions ’nð�rÞ� �

¼ 0 at r ¼ 0 and as

r ! 1. Base (3) is chosen to make the matrix

representation of the reference Hamiltonian Ho tridia-

gonal. The matrix element ðHoÞnm ¼ h’njHoj’mi in the

Laguerre basis can be written in the following

tridiagonal form [11]:

Hoð Þnm ¼ �2

82nþ �þ 1ð Þ

� ��n,m þ �2

8

ffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffinðnþ �Þ

p� ��n,mþ1

þ �2

8

ffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiðnþ 1Þðnþ �þ 1Þ

p� ��n,m�1: ð4Þ

where � ¼ 2‘þ 1. The full Hamiltonian H, in Equation(2), in the J-matrix representation is approximatelywritten as:

Hnm ffiHoð ÞnmþVnm; n,m � N� 1

Hoð Þnm; n,m4N� 1

�: ð5Þ

where N is an adequately large number.The calculation of the matrix elements of the

potential V(r), Equation (1), is usually obtained byevaluating the integral:

Vnm ¼Z 1

0

’n �rð ÞVðrÞ’m �rð Þdr ð6Þ

The evaluation of such an integral is usually donenumerically using the Gauss quadrature approxima-tion [12].

Our approach in finding resonance and bound stateenergies makes use of the ‘‘direct method’’ based on theJ-matrix calculation of the scattering S-matrix, SðEÞ, inthe complex energy plane. Physically, the bound andresonance state energies are the roots of S�1ðEÞ ¼ 0,where SðEÞ is given by [13]

SðEÞ ¼ TN�1ðEÞ1þ gN�1,N�1ðEÞJN�1,NðEÞR�

NðEÞ1þ gN�1,N�1ðEÞJN�1,NðEÞRþ

NðEÞ, ð7Þ

Table 1 [14], lists all elements that are needed tocalculate the S-matrix in Equation (7), such as thekinematics’ quantities: TnðEÞ, R�

n ðEÞ, the Green’sfunction gN�1,N�1 zð Þ ¼ H� zð Þ�1 and the matrix ele-ments JN�1,N Eð Þ of the reference wave operatorJ ¼ Ho � Eð Þ in the Laguerre basis. The coefficients

snf g1n¼0 and cnf g1n¼0 of the asymptotically sine-like andcosine-like eigenfunction of the reference Hamiltonian,

Figure 1. VeffðrÞ as a function of � and r at fixed screening parameter � ¼ 0:615 and ‘ ¼ 1 for different �.

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Molecular Physics 3

Although bound state energies of the MR potential

have been reported previously in many articles [1–7,

9,10], the results of resonance state energies have not

been found. So, we are devoting this paper to calculate

a few bound and resonance state energies close to the

crossover region using the J-matrix method [11]. As an

application, we will consider the bound and resonance

state energies of LiH and CO molecules. The usefulness

of the method is further illustrated by calculating the

expectation values of the multipoles hrni, with

n ¼ �1, 1, 2, and 3, and the average oscillator strength

of the dipole transition of the MR potential.The main idea of the J-matrix method is based on

expanding the wave function �ðrÞ, in Equation (2), in

terms of a complete square integrable Laguerre basis

set ’nð�rÞ� �

, where:

’nðxÞ ¼ anx‘þ1e�x=2L2‘þ1

n ðxÞ; n ¼ 0, 1, 2, . . . ð3Þ

with x ¼ �r and � is the positive length scale

parameter. L2‘þ1n ðxÞ is the associate Laguerre poly-

nomial, and an ¼ffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffi�� nþ 1ð Þ=� nþ 2‘þ 2ð Þ

pis the

normalization constant. The basis set ’nð�rÞ� �

satisfies

the boundary conditions ’nð�rÞ� �

¼ 0 at r ¼ 0 and as

r ! 1. Base (3) is chosen to make the matrix

representation of the reference Hamiltonian Ho tridia-

gonal. The matrix element ðHoÞnm ¼ h’njHoj’mi in the

Laguerre basis can be written in the following

tridiagonal form [11]:

Hoð Þnm ¼ �2

82nþ �þ 1ð Þ

� ��n,m þ �2

8

ffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffinðnþ �Þ

p� ��n,mþ1

þ �2

8

ffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiðnþ 1Þðnþ �þ 1Þ

p� ��n,m�1: ð4Þ

where � ¼ 2‘þ 1. The full Hamiltonian H, in Equation(2), in the J-matrix representation is approximatelywritten as:

Hnm ffiHoð ÞnmþVnm; n,m � N� 1

Hoð Þnm; n,m4N� 1

�: ð5Þ

where N is an adequately large number.The calculation of the matrix elements of the

potential V(r), Equation (1), is usually obtained byevaluating the integral:

Vnm ¼Z 1

0

’n �rð ÞVðrÞ’m �rð Þdr ð6Þ

The evaluation of such an integral is usually donenumerically using the Gauss quadrature approxima-tion [12].

Our approach in finding resonance and bound stateenergies makes use of the ‘‘direct method’’ based on theJ-matrix calculation of the scattering S-matrix, SðEÞ, inthe complex energy plane. Physically, the bound andresonance state energies are the roots of S�1ðEÞ ¼ 0,where SðEÞ is given by [13]

SðEÞ ¼ TN�1ðEÞ1þ gN�1,N�1ðEÞJN�1,NðEÞR�

NðEÞ1þ gN�1,N�1ðEÞJN�1,NðEÞRþ

NðEÞ, ð7Þ

Table 1 [14], lists all elements that are needed tocalculate the S-matrix in Equation (7), such as thekinematics’ quantities: TnðEÞ, R�

n ðEÞ, the Green’sfunction gN�1,N�1 zð Þ ¼ H� zð Þ�1 and the matrix ele-ments JN�1,N Eð Þ of the reference wave operatorJ ¼ Ho � Eð Þ in the Laguerre basis. The coefficients

snf g1n¼0 and cnf g1n¼0 of the asymptotically sine-like andcosine-like eigenfunction of the reference Hamiltonian,

Figure 1. VeffðrÞ as a function of � and r at fixed screening parameter � ¼ 0:615 and ‘ ¼ 1 for different �.

Ho, in the Laguerre basis have three-term recursionrelations. The three-term recursion relations for sn andcn (collectively shown in the last row of Table 1 as fn).The eigenvalues of the N�N Hamiltonian H are"nf gN�1

n¼0 and the eigenvalues of the truncated Hobtained by removing the last row and last columnare ~"nf gN�2

n¼0 .Furthermore, this method has been applied in the

calculation of the expectation values hrni forn ¼ �1, 1, 2, and 3 of the MR potential, and theaverage oscillator strength of the dipole transitionfrom an initial state jn‘i to a final state jn0‘0i which isdefined as [15]:

�fn0n ¼2

3

maxð‘, ‘0Þ2‘þ 1

�n0‘0

n‘ �n0‘0

n‘

� �2,

�n0‘0

n‘ ¼Z 1

0

�n‘ðrÞ�n0‘0 ðrÞr dr, ‘0 ¼ ‘� 1

ð8Þ

where �n0‘0

n‘ ¼ En‘ � En0‘0 in a.u. and maxð‘, ‘0Þ, is thelarger angular momentum of ‘ and ‘0. The transitionprobability is defined as [15]:

An0n ¼ 32:0� 109 �n0‘0

n‘

� �2 �fn0n sec�1� �

ð9Þ

The expression, Equation (9), has been usedextensively in calculating the life time, �, of thetransition, where � ¼ 1=An0n, and the dielectronicrecombination coefficients in plasma modeling [16].

The accuracy and stability of the theoretical schemethat we used are mainly dependent on two parameters,

the Laguerre length scaling � and the number ofabscissas N. It is important to choose the proper valuefor the domain of � where the calculations of thebound state energies are stable. We have determined astable domain [17–19] in which the targeting bound, orresonance, state energies are the same up to at leastfour to six decimal places, depending on how far theparameter � is far from the crossover region, i.e. from�c. We usually take the optimum value of � in themiddle of the stable domain.

Our calculation of the bound and resonance stateenergies started with estimating the values of thecritical screening parameter �c of the MR potential.The values of �c are shown in Table (2) for differentn‘� states with � ¼ 0:75. It was found that �c

decreases with increasing n and ‘. The parametersused to calculate �c are: N ¼ 70 and � ¼ 0:02 up to 2.Usually, the smaller value of � has been used forsmaller ‘-values, and vice versa for larger ‘-values. Thestate with a larger quantum number n and angularmomentum ‘ has a smaller �c due to the reduction ofthe attractive potential well, as well as the enhancementof the centrifugal potential by the screening effect.

Next, we proceed to find the roots of S�1ðEÞ, usingthe values generated by the complex rotation method[20] as a seed for different values of potentialparameters and ‘. This strategy increases the speed ofconvergence of the root finding algorithm. Thecalculated eigenvalues using the J-matrix approachwill be more accurate than those calculated using the

Table 1. The explicit form of the kinematics quantities: nðEÞ, R�n ðEÞ, the Green’s function gN�1,N�1ðzÞ and JN�1,NðEÞ in the

Laguerre basis. The three-term recursion relations for sn and cn (collectively shown as fn) are also given.

gN�1,N�1ðzÞ 1

Nþ 2‘þ 1

YN�2

m¼0~"m � zð Þ=

YN�1

n¼0"n � zð Þ

h i

Tn Eð Þ cn � isncn þ isn

T0ðEÞ ¼ c0�is0c0þis0

e2i�2F1ð�‘, 1; ‘þ 2; e2i�Þ2F1ð�‘, 1; ‘þ 2; e�2i�Þ ; cos � ¼ 8E� �2

8Eþ �2

R�n Eð Þ

cn � isncn�1 � isn�1

R�1 ðEÞ ¼ c1�is1

c0�is0

1

‘þ 2ð Þ e�i� 2F1 �‘, 2; ‘þ 3; e�i�

� �

2F1 �‘, 1; ‘þ 2; e�i�ð Þ

JN�1,NðEÞ ¼ ’N�1

� ��ðH0 � EÞ ’N�� �

Eþ �2=8� � ffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffi

NðNþ 2‘þ 1Þp

Recursion Relation of both sn and cn (shown as fn) 2ðcos �Þðnþ ‘þ 1Þ fn �ffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffinðnþ 2‘þ 1Þ

pfn�1

�ffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiðnþ 1Þðnþ 2‘þ 2Þ

pfnþ1 ¼ 0; n � 1

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complex rotation method. The reason for this advan-tage, of the J-matrix method, is mainly because thereference Hamiltonian Ho is included fully (seeEquation (5)) in the calculation. In the complexrotation, the Ho Is truncated to be an N�N matrix.The results reported here are correct up to theprecision that maintained calculation stability, and allour results are truncated rather than rounded-off. Thus,all results may be considered accurate up to thereported decimal place.

From our calculations it was found that the s-stateof the MR potential does not exhibit a resonancebehavior. Such behavior has been noticed in theHulthen potential [14]. For ‘ ¼ 1, the calculatedenergies for the bound states, np, and resonancestates Rnp, with n ¼ 2, 5 and 8 are given in Table 3.The parameters used in this step are: N ¼ 70 and �ranging from 0.1 to 3.0. Away from the crossoverregion for each state, our results and others [5,7,9,10]are in excellent agreement up to the given seven digitfor all values of ‘ and n. As � increases and comescloser to �c, for which no literature results could befound, the number of digits of bound and resonancestate energies decreases. In general, the propagation ofthe resonance states, for example R2p, in the complexplane is developed by increasing the parameter � from0.385 to 0.845. The same behaviour is shown for theother resonance states R5p and R8p.

From our calculations we noticed that the energyeigenvalues with �� �c have a large number ofsignificant figures, which will be reduced as �approaches �c. In order to understand the reason forthis reduction, we plotted in Figure (2) the radialprobability distribution function, PðrÞ ¼ j�ðrÞj2, for the2 p-state, where �cð2pÞ ¼ 0:38428, for different �. Wenoticed that for � ¼ 0:075 ð�� �cð2pÞÞ the wavefunction �ðrÞ in configuration space is very muchlocalized and can hence be described by a few elementsof the configuration basis set ’nð�rÞ

� �to reach the

desired accuracy. On the other hand, for � ¼ 0:38

Table 3. Roots of S�1ðEÞ-matrix calculation of the boundnp-states and resonance energies of MR potential fordifferent values of �. The numerical parameters werechosen as � ¼ 0:75, � ¼ 0:2 to 2 and N¼ 70. The superscriptis the reference number.

State � Energy (a.u.)

2p 0.075 �0.09644690�0.09644699, �0.096446910,�0.09644907

0.1 �0.08522531�0.08522539, �0.08522407

0.2 �0.045913400.3 �0.016069510.35 �0.005243300.37 �0.00188750.375 �0.0011630.38 �4.99�10�4

R2p 0.385 7.15� 10�5�4.59� 10�6 i0.39 5.618� 10�4�1.038� 10�4 i0.61 0.0110035–0.0265617 i0.7 0.0094904–0.04376136 i0.8 0.0040076–0.0656974 i0.845 2.499� 10–4–0.076400 i

5p 0.025 �0.0098079�0.00980799, �0.009807910,�0.00980627, �0.00980806

0.05 �0.0028947�0.00289479

0.07 �2.029� 10�4

0.072 �8.54� 10�5

R5p 0.075 1.88� 10�5–6.7� 10�6 i0.078 6.37� 10�5–6.07(�5) i0.082 7.38� 10�5–1.609� 10�4 i0.086 2.41� 10�5–2.64� 10�4 i

8p 0.025 �0.00031940.029 �0.00031949

�2.80� 10�5

R8p 0.0305 4.4� 10�6–2.6� 10�6 i0.031 7.1� 10–6–7.9� 10� 6 i0.032 6.7� 10�6–2.08� 10�5 i

Table 2. Estimated values of the critical screening parameter �c of Manning-Rosen potential for different n‘-states for� ¼ 0:75. Our parameters are N ¼ 200, and � ¼ 0:1 to 2.

State

n

2 3 4 5 6 7 8

p 0.38428 0.19043 0.11264 0.07413 0.05238 0.03893 0.03005d 0.15866 0.09828 0.06659 0.04799 0.03618 0.02823f 0.08667 0.06019 0.04414 0.03370 0.02655g 0.05460 0.04067 0.03142 0.02498h 0.03754 0.02932 0.02351i 0.02740 0.02214k 0.02087

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Molecular Physics 5

complex rotation method. The reason for this advan-tage, of the J-matrix method, is mainly because thereference Hamiltonian Ho is included fully (seeEquation (5)) in the calculation. In the complexrotation, the Ho Is truncated to be an N�N matrix.The results reported here are correct up to theprecision that maintained calculation stability, and allour results are truncated rather than rounded-off. Thus,all results may be considered accurate up to thereported decimal place.

From our calculations it was found that the s-stateof the MR potential does not exhibit a resonancebehavior. Such behavior has been noticed in theHulthen potential [14]. For ‘ ¼ 1, the calculatedenergies for the bound states, np, and resonancestates Rnp, with n ¼ 2, 5 and 8 are given in Table 3.The parameters used in this step are: N ¼ 70 and �ranging from 0.1 to 3.0. Away from the crossoverregion for each state, our results and others [5,7,9,10]are in excellent agreement up to the given seven digitfor all values of ‘ and n. As � increases and comescloser to �c, for which no literature results could befound, the number of digits of bound and resonancestate energies decreases. In general, the propagation ofthe resonance states, for example R2p, in the complexplane is developed by increasing the parameter � from0.385 to 0.845. The same behaviour is shown for theother resonance states R5p and R8p.

From our calculations we noticed that the energyeigenvalues with �� �c have a large number ofsignificant figures, which will be reduced as �approaches �c. In order to understand the reason forthis reduction, we plotted in Figure (2) the radialprobability distribution function, PðrÞ ¼ j�ðrÞj2, for the2 p-state, where �cð2pÞ ¼ 0:38428, for different �. Wenoticed that for � ¼ 0:075 ð�� �cð2pÞÞ the wavefunction �ðrÞ in configuration space is very muchlocalized and can hence be described by a few elementsof the configuration basis set ’nð�rÞ

� �to reach the

desired accuracy. On the other hand, for � ¼ 0:38

Table 3. Roots of S�1ðEÞ-matrix calculation of the boundnp-states and resonance energies of MR potential fordifferent values of �. The numerical parameters werechosen as � ¼ 0:75, � ¼ 0:2 to 2 and N¼ 70. The superscriptis the reference number.

State � Energy (a.u.)

2p 0.075 �0.09644690�0.09644699, �0.096446910,�0.09644907

0.1 �0.08522531�0.08522539, �0.08522407

0.2 �0.045913400.3 �0.016069510.35 �0.005243300.37 �0.00188750.375 �0.0011630.38 �4.99�10�4

R2p 0.385 7.15� 10�5�4.59� 10�6 i0.39 5.618� 10�4�1.038� 10�4 i0.61 0.0110035–0.0265617 i0.7 0.0094904–0.04376136 i0.8 0.0040076–0.0656974 i0.845 2.499� 10–4–0.076400 i

5p 0.025 �0.0098079�0.00980799, �0.009807910,�0.00980627, �0.00980806

0.05 �0.0028947�0.00289479

0.07 �2.029� 10�4

0.072 �8.54� 10�5

R5p 0.075 1.88� 10�5–6.7� 10�6 i0.078 6.37� 10�5–6.07(�5) i0.082 7.38� 10�5–1.609� 10�4 i0.086 2.41� 10�5–2.64� 10�4 i

8p 0.025 �0.00031940.029 �0.00031949

�2.80� 10�5

R8p 0.0305 4.4� 10�6–2.6� 10�6 i0.031 7.1� 10–6–7.9� 10� 6 i0.032 6.7� 10�6–2.08� 10�5 i

Table 2. Estimated values of the critical screening parameter �c of Manning-Rosen potential for different n‘-states for� ¼ 0:75. Our parameters are N ¼ 200, and � ¼ 0:1 to 2.

State

n

2 3 4 5 6 7 8

p 0.38428 0.19043 0.11264 0.07413 0.05238 0.03893 0.03005d 0.15866 0.09828 0.06659 0.04799 0.03618 0.02823f 0.08667 0.06019 0.04414 0.03370 0.02655g 0.05460 0.04067 0.03142 0.02498h 0.03754 0.02932 0.02351i 0.02740 0.02214k 0.02087

ð� � �cÞ the wave function �ðrÞ starts having a longrange tail and needs many elements of ’nð�rÞ

� �to

reach the desired accuracy. So, close to the crossoverregion, i.e. � � �c, the most suitable basis set ’nð�rÞ

� �should have long extensions (small �) and/or a biggersize (large N) to ensure that the potential is sampledcorrectly in regions away from the origin. In Figure (2)our parameters were: N ¼ 20, � ¼ 0:5.

Figure 3 shows a snapshot of the dynamicalbehaviour of the crossover, in the complex energyplane, of the states nk at � ¼ 0:2, � ¼ 0:019, and� ¼ 1:2 rad. The string of points in the third quadrantis the set of energy eigenvalues that mimic thediscretized continuum line. In the complex energyplane, we have one bound state, 8 k, on the real axis at

�4.588� 10�4, and three resonance states, R9k, R10kand R11k, located in the third quadrant at the valuesof 2.59� 10�4� 8.0� 10�6 i, 3.41� 10�4 � 2.33� 10�4

i, and 2.0� 10�4 � 5.7� 10�4 i, respectively.Moreover, few of the crossover resonance statesappeared in the ‘‘unphysical’’ third quadrant, whichwe called ‘shadow’ resonances. They are well separatedfrom the continuum line. Table 4 shows the calculationof the bound state energy of 8 k at � ¼ 0:019 and theresonance state energies for nk-states, with n¼ 8, 9, 10

Figure 2. The radial probability distribution function, PðrÞ ¼ j�ðrÞj2, for the 2p-state with N ¼ 20, and � ¼ 0:5 for different �.

Table 4. Roots of S�1ðEÞ -matrix calculation of the boundstates 8 k and resonance energies of MR potential fordifferent values of �. The numerical parameters werechosen as � ¼ 0:75, � ¼ 0:2 to 2, � ¼ 1rad and N¼ 70.

� State Energy (a.u.)

0.019 8 k �4.588� 10�4

R9k 2.59� 10�4–8.0� 10�6 iR10k 3.41� 10�4–2.33� 10�4 iR11k 2.0� 10�4–5.7� 10�4 i

0.022 R8k 2.390823� 10�4–1.24 (�7) iR9k 5.2795� 10�4–2.0199� 10�4 iR10k 4.247� 10�4–6.608� 10�4 iR11k 6.6� 10�5–1.133� 10�3 i

0.027 R8k 9.7434100� 10�4–2.32916355� 10�4 iR9k 8.849828� 10�4–9.050316� 10�4 iR10k 4.3728� 10�4–1.60507� 10�3 i

0.04 R8k 2.181693947979� 10�3

–2.18508316745� 10�3 iR9k 1.07959051� 10�3

–3.6085737� 10�3 i

0.09 R8k 4.447118581� 10�4

–1.709695997285� 10�2 i

Figure 3. The bound state energy of 8 k is on the negativereal axis of the complex energy plane. The three resonancestate energies R9k, R10k and R11k are shown in the fourthquadrant. In the third quadrant we have more ‘shadow’resonances.

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6 I. Nasser et al.

Table 5. The ro-vibrational energy spectra ( n‘ in eV) of the MR potential for LiH and CO for 2p to 8p, 3d to 8d, 4f to 8f andR4p states. We used the reduced masses mLiH¼ 0.8801221 amu, mCO¼ 6.8606719 amu, �hc ¼ 1973:29 eV.A, and � is in pm�1.

� States

LiH CO

Our work Ikhdair [7] Our work Ikhdair [7]

0.025 2p �5.7245882 �5.72457 �0.7343795 �0.7343773p �2.1790248 �2.17902 �0.2795365 �0.2795364p �0.9883803 �0.988358 �0.1267945 �0.1267925p �0.4658391 �0.465751 �0.0597602 �0.0597496p �0.2070027 �0.206782 �0.0265554 �0.0265277p �0.0755389 �0.00969058p �0.01517 �0.00194613d �2.1266084 �2.12665 �0.2728122 �0.2728184d �0.9642549 �0.964223 �0.1236995 �0.1236955d �0.4518845 �0.451655 �0.0579700 �0.0579416d �0.1978204 �0.197193 �0.0253774 �0.0252977d �0.0692281 �0.00888098d �0.0112411 �0.00144214f �0.9489591 �0.948967 �0.1217373 �0.1217385f �0.4408842 �0.440525 �0.0565589 �0.0565136f �0.1890475 �0.187922 �0.0242520 �0.0241087f �0.0621670 �0.00797518f �0.006299 �0.0008081

0.075 2p �4.5808606 �4.58092 �0.5876562 �0.5876443p �1.2142604 �1.21342 �0.1557715 �0.1556644p �0.2419688 �0.238160 �0.0310410 �0.0305523d �1.1286070 �1.12698 �0.1447835 �0.1445744d �0.1915555 �0.181234 �0.0245737 �0.0232504f �0.1255960 �0.109984 �0.0161121 �0.014109

0.1 2p �4.0478781 �4.04778 �0.5192825 �0.5192703p �0.8266246 �0.823478 �0.1060436 �0.1056404p �0.0469922 �0.00602843d �0.7138119 �0.707045 �0.0915714 �0.087352

0.11 2p �3.8419971 �0.4928713p �0.6896684 �0.0884744p �0.0061684 �0.000791

0.112 2p �3.8013244 �0.4876533p �0.6635367 �0.0851214p �0.001224 �0.00015

0.113 2p �3.7810512 �0.4850523p �0.6506293 �0.083466R4p 5.4� 10�4–8.1� 10�5 i 6.9� 10�5–1.0� 10�5 i

0.115 2p �3.7406308 �0.4798673p �0.6251321 �0.080195R4p 3.328� 10�3–1.382� 10�3 i 4.270� 10�4–1.773� 10�4 i

0.120 2p �3.6403168 �0.4669983p �0.5632503 �0.072256R4p 8.4013� 10�3–7.363� 10�3 i 1.0777� 10�3–9.445� 10�4 i

0.130 2p �3.4428549 �0.4416673p �0.4475453 �0.057413R4p 0.010255–0.024270 i 1.3156�10�3�3. 1135� 10�3 i

0.140 2p �3.2496312 �0.4168793p �0.3427761 �0.043973R4p 3.05� 10�4–0.041862 i 3.91�10�5–5.3703� 10�3 i

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Molecular Physics 7

Table 5. The ro-vibrational energy spectra ( n‘ in eV) of the MR potential for LiH and CO for 2p to 8p, 3d to 8d, 4f to 8f andR4p states. We used the reduced masses mLiH¼ 0.8801221 amu, mCO¼ 6.8606719 amu, �hc ¼ 1973:29 eV.A, and � is in pm�1.

� States

LiH CO

Our work Ikhdair [7] Our work Ikhdair [7]

0.025 2p �5.7245882 �5.72457 �0.7343795 �0.7343773p �2.1790248 �2.17902 �0.2795365 �0.2795364p �0.9883803 �0.988358 �0.1267945 �0.1267925p �0.4658391 �0.465751 �0.0597602 �0.0597496p �0.2070027 �0.206782 �0.0265554 �0.0265277p �0.0755389 �0.00969058p �0.01517 �0.00194613d �2.1266084 �2.12665 �0.2728122 �0.2728184d �0.9642549 �0.964223 �0.1236995 �0.1236955d �0.4518845 �0.451655 �0.0579700 �0.0579416d �0.1978204 �0.197193 �0.0253774 �0.0252977d �0.0692281 �0.00888098d �0.0112411 �0.00144214f �0.9489591 �0.948967 �0.1217373 �0.1217385f �0.4408842 �0.440525 �0.0565589 �0.0565136f �0.1890475 �0.187922 �0.0242520 �0.0241087f �0.0621670 �0.00797518f �0.006299 �0.0008081

0.075 2p �4.5808606 �4.58092 �0.5876562 �0.5876443p �1.2142604 �1.21342 �0.1557715 �0.1556644p �0.2419688 �0.238160 �0.0310410 �0.0305523d �1.1286070 �1.12698 �0.1447835 �0.1445744d �0.1915555 �0.181234 �0.0245737 �0.0232504f �0.1255960 �0.109984 �0.0161121 �0.014109

0.1 2p �4.0478781 �4.04778 �0.5192825 �0.5192703p �0.8266246 �0.823478 �0.1060436 �0.1056404p �0.0469922 �0.00602843d �0.7138119 �0.707045 �0.0915714 �0.087352

0.11 2p �3.8419971 �0.4928713p �0.6896684 �0.0884744p �0.0061684 �0.000791

0.112 2p �3.8013244 �0.4876533p �0.6635367 �0.0851214p �0.001224 �0.00015

0.113 2p �3.7810512 �0.4850523p �0.6506293 �0.083466R4p 5.4� 10�4–8.1� 10�5 i 6.9� 10�5–1.0� 10�5 i

0.115 2p �3.7406308 �0.4798673p �0.6251321 �0.080195R4p 3.328� 10�3–1.382� 10�3 i 4.270� 10�4–1.773� 10�4 i

0.120 2p �3.6403168 �0.4669983p �0.5632503 �0.072256R4p 8.4013� 10�3–7.363� 10�3 i 1.0777� 10�3–9.445� 10�4 i

0.130 2p �3.4428549 �0.4416673p �0.4475453 �0.057413R4p 0.010255–0.024270 i 1.3156�10�3�3. 1135� 10�3 i

0.140 2p �3.2496312 �0.4168793p �0.3427761 �0.043973R4p 3.05� 10�4–0.041862 i 3.91�10�5–5.3703� 10�3 i

and 11. The parameters we used are: N ¼ 70 and �ranging from 0.1 to 3.0.

As an application to diatomic molecules, we usedthe J-matrix method to calculate the bound andresonance state energies for CO and LiH for thestates 2p to 8p, 3d to 8d, 4f to 8f and R4p. (away andclose to the crossover region). Table 5 shows the boundstate energies, away from the crossover region, for theavailable states n‘, ‘¼ 1, 2, 3. Our results are in goodagreement, up to 3 or 4 digits, with those reported

values of [7]. In addition, new state-energies have beenreported in Table 5 for future comparison. In theneighborhood of the crossover region of the 4p-state,where �cð4pÞ ¼ 0:112, the bound and resonance stateenergies have been calculated for the states: 2p, 3p and4p. No reported data have been found for theresonances for the MR potential of these molecules.

Finally, we calculate the expectation values hri andhr�1i of the MR potential. Table 6 shows the calculatedbound state energies and the expectation values hri and

Table 6. The calculated bound state energies and the expectation values hri and hr�1i of the MRpotential for the states 2p, 3p, 4p and 3d. Our parameters are: � ¼ 0:75 and � ¼ 0:1, � ¼ 0:2 to 2, withdifferent N. The superscript is used for the reference number.

State N �Energy (a.u.) hri hr�1i

2 p 10 0.0852172 4.8513238 0.2601817915 0.0852210 4.8510993 0.2602010020 0.0852226 4.8507531 0.2602080970 0.0852253,0.08522539, 0.08522407

3 p 10 0.0174017 13.353311 0.1040554315 0.0174028 13.353985 0.1040575020 0.0174032 13.353791 0.1040598070 0.0174040,0.01740409, 0.01733797

4 p 10 0.0009817 34.505483 0.0398283715 0.0009872 34.847781 0.0396328720 0.0009881 34.851618 0.0396369570 0.0009894, 0.00098949

3 d 10 0.0150287 11.489523 0.1032236115 0.0150288 11.489476 0.1032242820 0.0150288 11.488341 0.1032242970 0.0150288,0.01502889

Table 7. Multipole and the average oscillator strengths �fn0n for the 3d-np transitions (n¼ 2 to 4) and its transition probabilitiesAn0n sec�1

� �. Our parameters are: � ¼ 0:75, � ¼ 0:1, � ¼ 0:2 to 2 with different N.

Physical quantity N 3d!2p 3d!3p 4p!3d

Dipole moment square: hri2 10 18.7286 128.39 4.0606915 18.7273 128.399 4.06820 18.7258 128.396 4.07278

Quadrupole square: hr2i2 10 1.3730� 104 7.2960� 104 1.9156� 104

15 1.3732� 104 7.2981� 104 1.9173� 104

20 1.3729� 104 7.2977� 104 1.9204� 104

Octupole square: hr3i2 10 1.3357� 105 1.9812� 107 2.0441� 107

15 1.3375� 105 1.9828� 107 2.0527� 107

20 1.3384� 105 1.9828� 107 2.0513� 107

Average oscillator strength �fn0n 10 �0.350541 �0.081246 0.0253515 �0.350537 �0.081286 0.0253820 �0.350517 �0.081299 0.02549

transition probability An0n sec�1� �

10 5.5261� 107 1.4640� 104 1.6007� 105

15 5.5266� 107 1.4660� 104 1.6017� 105

20 5.5269� 107 1.4668� 104 1.6082� 105

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hr i (in a.u.) of the MR potential for the states 2p, 3p,4p and 3d, for which no literature results could befound. Our parameters are: � ¼ 0:75 and � ¼ 0:1,� ¼ 0:2 to 2, with different N. Our calculation showsthat even with small N, N ¼ 10, we get an accuracy ofup to 4 digits. In order to calculate the radiativetransition probabilities for the state 3d-np (n¼ 2–4) wehave to choose a screening parameter away from thecross-over region. In our case, we used � ¼ 0:1.Table 7 shows the calculated squares of the dipole,quadrupole and octupole moments for the giventransitions, as well as the average oscillator strengths�fn0n, Equation (8), for the 3d-np transitions (n¼ 2–4)and its transition probabilities An0n sec�1

� �, Equation

(9). Our parameters are: � ¼ 0:75, � ¼ 0:1, � ¼ 0:2 to 2with different N.

In summary, we have reported the bound andresonance state energies associated with Manning–Rosen (MR) using the Laguerre basis. The bound stateenergies were generated and compared favorably, withthose available in the literature, for different values ofthe quantum numbers n and ‘, and with differentvalues of the screening parameter and � ¼ 0:75. Themethod has been applied to calculate the bound andresonance state energies for MR potential of LiH andCO molecules. In the neighbourhood of the crossoverregion, no available data of: resonance state energies,oscillator strengths, or transition probabilities tocompare with. We believe that our results will be agood guide for future work.

Acknowledgments

The authors would like to acknowledge the supportprovided by the Deanship of Scientific Research (DSR) atKing Fahd University of Petroleum & Minerals (KFUPM)for funding this work through project No RG1109-1 &RG1109–2.

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