Spectroscopic study of some diatomic molecules via the proper quantization rule
aa r X i v : . [ phy s i c s . c h e m - ph ] A p r Spectroscopic study of some diatomicmolecules via the proper quantization rule
Babatunde J. Falaye a, † , , Sameer M. Ikhdair b, andMa jid Hamzavi c, a Applied Theoretical Physics Division, Department of Physics, Federal University Lafia, P. M. B.146, Lafia, Nigeria. b Department of Physics, Faculty of Science, an-Najah National University,New campus, P. O. Box 7, Nablus, West Bank, Palestine. c Department of Physics, University ofZanjan, Zanjan, Iran.
J. Math. Chem. (2015) DOI 10.1007/s10910-015-0491-9
Abstract
Spectroscopic techniques are very essential tools in studying electronic structures,spectroscopic constants and energetic properties of diatomic molecules. These tech-niques are also required for parametrization of new method based on theoretical analy-sis and computational calculations. In this research, we apply the proper quantizationrule in spectroscopic study of some diatomic molecules by solving the Schr¨odingerequation with two solvable quantum molecular systems-Tietz-Wei and shifted Deng-Fan potential models for their approximate nonrelativistic energy states via an appro-priate approximation to the centrifugal term. We show that the energy levels can bedetermined from its ground state energy. The beauty and simplicity of the methodapplied in this study is that, it can be applied to any exactly as well as approximatelysolvable models. The validity and accuracy of the method is tested with previoustechniques via numerical computation for H and CO diatomic molecules. The resultalso include energy spectrum of 5 different electronic states of NO and 2 differentelectronic state of ICl. Keywords : Proper quantization rule; formula method; Schr¨odinger equation; Tietz-Weidiatomic molecular potential; Shifted Deng-Fan diatomic molecular potential.
PACs No. : 003.65.Fd, 03.65.Ge, 03.65.Ca, 03.65-W
The exact solutions of solvable quantum potential models have received much interest sincethey provide us some insight into the physical problem under consideration. Over the E-mail: [email protected]; babatunde.falaye@fulafia.edu.ng † Corresponding author E-mail: [email protected]; [email protected]. E-mail: [email protected]
In this section, we give a brief review to this method [8, 12, 13]. The one dimensionalSchr¨odinger equation takes the form: d dx ψ ( x ) + 2 µ ¯ h [ E − V ( x )] ψ ( x ) = 0 , (1)and can be re-written as φ ′ ( x ) + φ ( x ) + k ( x ) = 0 , with k ( x ) = r µ ¯ h [ E − V ( x )] , (2)where φ ( x ) = ψ ′ ( x ) /ψ ( x ) is the logarithmic derivative of the wave function ψ ( x ). Theprime denotes the derivative with respect to the variable x . µ denotes the reduced massof the two interacting particles. k ( x ) is the momentum and V ( x ) is a piecewise continuousreal potential function of x . According to Yang [18] “For the Sturm-Liouville problem,the fundamental trick is the definition of a phase angle which is monotonic with respectto the energy”[18]. Thus, for the Schr¨odinger equation, the phase angle is the logarithmicderivative φ ( x ). From equation (2), as x increases across a node of wave function ψ ( x ), φ ( x )decreases to −∞ , jumps to + ∞ and then decreases again.In 2005, Ma and Xu [8] by carefully studying one-dimensional Schr¨odinger equation gen-eralized this exact quantization rule to the 3D radial Schr¨odinger equation with sphericallysymmetric potential by simply making the replacements x → r and V ( x ) → V eff ( r ): Z r b r a k ( r ) dr = N π + Z r b r a φ ( r ) (cid:20) dk ( r ) dr (cid:21) (cid:20) dφ ( r ) dr (cid:21) − dr, k ( r ) = r µ ¯ h [ E − V eff ( r )] , (3)where r A and r B are two turning points determined by E = V eff ( r ). The N = n + 1 isthe number of the nodes of φ ( r ) in the region E nℓ = V eff ( r ) and is larger by one thanthe number n of the nodes of wave function ψ ( r ). The first term N π is the contributionfrom the nodes of the logarithmic derivative of wave function, and the second is called thequantum correction. Ma and Xu [8] found that for all well-known exactly solvable quantumsystems, this quantum correction is independent of the number of nodes of wave function.3ccordingly, it is enough to consider the ground state in calculating the quantum correction Q c = R r B r A k ′ ( r ) φ φ ′ dr .The integrals in equations (3) and the calculation of quantum correction term are noteasy to obtain for various quantum mechanical problems. This motivated Serrano et al in2010 to propose Qiang-Dong proper quantization rule [12], so as to simplify the quantumcorrection terms. This rule can be summarized as follows: Z r b r a k ( r ) dr = Z r b r a k ( r ) dr + nπ with n = N − . (4)In the approach, it is required to first calculate the integral on the LHS of equation (4)and then replace energy levels E n in the result by the ground state energy E to obtain thesecond integral (RHS). This quantization rule has been used in many physical systems toobtain the exact solutions of many exactly solvable quantum systems [8, 12, 13, 21, 22] In this section, we apply the Qiang-Dong proper quantization to study the rotation vibra-tional of some diatomic molecular potentials. Also, where necessary, we compare our resultswith the ones obtained before in the literature.
The Tietz-Wei diatomic molecular potential we examine in this section is defined as [11, 23,24] V ( r ) = D (cid:20) − e − b h ( r − r e ) − c h e − b h ( r − r e ) (cid:21) , (5)with b h = δ (1 − c h ), r e is the molecular bond length, δ is the Morse constant (denoted as β in some other research papers), D is the potential well depth and c h is an optimizationparameter obtained from ab initio or Rydberg-Klein-Rees (RKR) intramolecular potentials. r is the internuclear distance. When the potential constant approaches zero, i.e. c h → .1 0.2 0.3 0.4012345678 r ( f m ) V ( r )( e V ) IIIIIIIVV 0.2 0.4051015 r ( f m ) V ( r )( e V ) VIVIIVIIIIXX
Figure 1:
Shape of Tietz-Wei diatomic molecular potential for different diatomic molecules: (I) NO (cid:0) a Π i (cid:1) (II) NO (cid:0) B Π r (cid:1) (III) NO (cid:0) L ′ φ (cid:1) (IV) NO (cid:0) b Σ − (cid:1) (V) NO( X Π r ) (VI) H (cid:0) X Σ + g (cid:1) (VII) CO (cid:0) X Σ + (cid:1) (VIII)ICl (cid:0) X Σ + g (cid:1) (IX) ICl (cid:0) A Π (cid:1) (X) ICl (cid:0) A ′ Π (cid:1) . (cid:18) P m + V ( r ) − E nℓ (cid:19) ψ n,ℓ,m ( r, θ, φ ) = 0 . (6)In this section, we take the V ( r ) as the Tietz-Wei potential. Now we begin by applyingthe method of variable separation so as to split equation (6) into radial and angular part.Thus, by taking the wavefunction ψ n,ℓ,m ( r, θ, φ ) as r − R nℓ ( r ) Y ℓm ( θ, φ ) the radial part canbe found as d R nℓ ( r ) dr + 2 µ ¯ h " E nℓ − D (cid:20) − e − b h ( r − r e ) − c h e − b h ( r − r e ) (cid:21) − ℓ ( ℓ + 1)¯ h µr R nℓ ( r ) = 0 , (7)where n , ℓ and E nℓ denote the principal quantum numbers, orbital angular momentumnumbers and the bound state energy eigenvalues of the system under consideration (i.e., E nℓ < ℓ = 0, problem (7) is exactly solvablebut for ℓ = 0, it isn’t. Therefore, in order to solve the above equation for ℓ = 0 states,Hamzavi et al [24] found that the following formula1 r ≈ r e D + D e − b h ( r − r e ) − c h e − b h ( r − r e ) + D e − b h ( r − r e ) (1 − c h e − b h ( r − r e ) ) ! , (8)5 h (Dimensionless Unit) E n ℓ ( e V ) (a) −4 −2 0 2 4−70−60−50−40−30−20−1001020 c h (Dimensionless Unit) E n ℓ ( e V ) (b) ℓ = 0 ℓ = 1 ℓ = 2 Figure 2: (a) The variation of the ground state energy spectrum for various values of ℓ as afunction of the potential constant c h . We choose µ = 1, b h = 5, r e = 0 . D = 15. (b) Thevariation of the first excited energy state for various ℓ as a function of the potential constant c h . E n ℓ ( e V ) µ (a.u.)(a) 0 0.5 1−250−200−150−100−50050 E n ℓ ( e V ) µ (a.u.)(b) ℓ = 0 ℓ = 1 ℓ = 2 Figure 3: (a) The variation of the ground state energy state for various values of ℓ as a functionof the particle mass µ . We choose c h = 0 . b h = 5, r e = 0 . D = 15. (b) The variation ofthe first excited energy state for various ℓ as a function of the particle mass µ . with D = 1 − α (1 − c h )(3 + c h ) + 3 α (1 − c h ) , lim c h → D = 1 − α + 3 α (9a) D = 2 α (1 − c h ) (2 + c h ) − α (1 − c h ) , lim c h → D = 4 α − α (9b) D = − α (1 − c h ) (1 + c h ) + 3 α (1 − c h ) , lim c h → D = − α + 3 α , (9c)6 b h ( f m − ) E n ℓ ( e V ) (a) 0 2 4 6024681012141618 b h ( f m − ) E n ℓ ( e V ) (b) ℓ = 0 ℓ = 1 ℓ = 2 Figure 4: (a) The variation of the ground state energy state for various values of ℓ as a functionof the parameter b h . We choose c h = 0 . µ = 1, r e = 0 . D = 15. (b) The variation of thefirst excited energy state for various ℓ as a function of the parameter b h . E n ℓ ( e V ) (a)r e (fm) 0.2 0.4 0.6 0.8102030405060708090 E n ℓ ( e V ) r e (fm)(b) ℓ = 0 ℓ = 1 ℓ = 2 Figure 5: (a) The variation of the ground state energy state for various values of ℓ as a function ofthe molecular bond length r e . We choose µ = 1, b h = 5, c h = 0 .
03 and D = 15. (b) The variationof the first excited energy state for various ℓ as a function of the molecular bond length r e is a good approximation scheme to deal with the centrifugal potential term. Constant α = b h r e has been introduced for the sake of simplicity. Now, by inserting this approximation7
10 20−15−10−505101520 E n ℓ ( e V ) D ( eV ) (a) 0 10 20−40−30−20−100102030 E n ℓ ( e V ) D ( eV ) (b) ℓ = 0 ℓ = 1 ℓ = 2 Figure 6: (a) The variation of the ground state energy state for various values of ℓ as a functionof the potential well depth D . We choose µ = 1, b h = 5, c h = 0 .
03 and r e = 0 .
8. (b) The variationof the first excited energy state for various ℓ as a function of the potential well depth D Table 1: Model parameters of the diatomic molecules studied in the present work.
Molecules(states) c h µ/ − ( g ) b h (˚ A − ) r e (˚ A ) D ( cm − ) β (˚ A − ) RefsNO (cid:0) a Π i (cid:1) (cid:0) B Π r (cid:1) -0.482743 1.249 3.42650 1.428 22722 2.310923 [29]NO (cid:0) L ′ φ (cid:1) -0.073021 1.249 2.73796 1.451 14501 2.551645 [29]NO (cid:0) b Σ − (cid:1) -0.085078 1.249 3.01538 1.318 21183 2.778957 [29]NO( X Π r ) 0.013727 1.249 2.71559 1.151 53341 2.7534 [29]H (cid:16) X Σ + g (cid:17) (cid:0) X Σ + (cid:1) (cid:16) X Σ + g (cid:17) -0.086212 4.55237 2.008578 2.3209 17557 1.849159 [29]ICl (cid:0) A Π (cid:1) -0.167208 4.55237 2.542557 2.6850 3814.7 2.178324 [29]ICl (cid:0) A ′ Π (cid:1) -0.157361 4.55237 2.373450 2.6650 4875 2.050745 [29] into equation (7) and then introducing a new transformation of the form r → ̺ = r − r e r e through the mapping function ̺ = f ( r ) with r in the domain [ 0 , ∞ ) or ̺ in the domain[ − , ∞ ], we obtain the following second order differential equation:1 r e d R nℓ ( ̺ ) d̺ + 2 µ ¯ h [ E nℓ − V eff ( ̺ )] R nℓ ( ̺ ) = 0 , with (10) V eff ( ̺ ) = " ℓ ( ℓ + 1) D r e + 2 µD ¯ h + ℓ ( ℓ +1) D r e + µD ¯ h ( c h − e α̺ − c h + ℓ ( ℓ +1) D r e + µD ¯ h ( c h − ( e α̺ − c h ) ¯ h µ . Comparison of the bound-state energy eigenvalues − ( E nℓ − D )( eV ) of H and COmolecules for various n and rotational ℓ quantum numbers in Tietz-Wei potential. H CO n ℓ PQR (Present) NU (Present) NU [24] GPS [30] PQR (Present) NU (Present) NU[24]0 0 4.48160205 4.48157183 4.4815718267 4.4815797825 11.07378271 11.07370964 11.073709645 4.26706288 4.26703273 4.2658220403 11.06668238 11.06660933 11.0665960610 3.73741847 3.73738873 3.7336304360 11.04775337 11.04768033 11.047631735 0 2.28155518 2.28153387 2.2669650930 2.2815913849 9.63231601 9.63224796 9.6298689855 2.12184477 2.12182368 2.1070072990 9.62560750 9.62553946 9.62314891310 1.72567170 1.72565129 1.7105026080 9.60772327 9.60765524 9.0849200847 0 1.63001750 1.62999964 1.6130911000 9.08816748 9.08810144 9.0849200845 1.48916801 1.48915041 1.4722799590 9.08161259 9.08154656 9.07835452510 1.13910018 1.13908334 1.1225356530 9.06413799 9.06407196 9.060851440
Table 3: Bound-state energy eigenvalues for NO ( a Π i ), NO ( B Π r ), NO ( L ′ φ ), NO ( b Σ − )and NO ( X Π r ) molecules for various n and rotational ℓ quantum numbers in Tietz-Weidiatomic molecular potential. n ℓ NO (cid:0) a Π i (cid:1) NO (cid:0) B Π r (cid:1) NO (cid:0) L ′ φ (cid:1) NO (cid:0) b Σ − (cid:1) NO ( X Π r )0 0 0.05724382 -0.06511639 -0.05748847 -0.07561426 -0.047759361 0 0.16924769 -0.19664818 -0.17491875 -0.22970681 -0.143640661 0.16950472 -0.19637388 -0.17464813 -0.22937970 -0.143525092 0 0.27795077 -0.33036890 -0.29569047 -0.38771825 -0.240067571 0.27820318 -0.33009351 -0.29541545 -0.38738640 -0.239951692 0.27870799 -0.32954274 -0.29486542 -0.38672273 -0.239719933 0 0.38335537 -0.46625489 -0.41978250 -0.54962300 -0.337037991 0.38360316 -0.46597847 -0.41950311 -0.54928644 -0.336921802 0.38409872 -0.46542563 -0.41894434 -0.54861334 -0.336689423 0.38484202 -0.46459641 -0.41810624 -0.54760374 -0.336340864 0 0.48546381 -0.60428372 -0.54717398 -0.71539576 -0.434549811 0.48570698 -0.60400632 -0.54689024 -0.71505453 -0.434433322 0.48619332 -0.60345153 -0.54632279 -0.71437207 -0.434200333 0.48692277 -0.60261938 -0.54547165 -0.71334844 -0.433850854 0.48789530 -0.60150990 -0.54433688 -0.71198366 -0.43338490 The two turning points are obtained by solving V eff ( ̺ ) − E nℓ = 0 or V eff ( ρ ) − E nℓ = 0 with ρ = ( e α̺ − c h ) − . Thus, it is easy to show that the turning points ρ a and ρ b are ρ a = − ℓ ( ℓ +1) D r e + µD ¯ h ( c h −
1) + rh ℓ ( ℓ +1) D r e + µD ¯ h ( c h − i − T nℓ h ℓ ( ℓ +1) D r e + µD ¯ h ( c h − i T nℓ (11a)9able 4: Bound-state energy eigenvalues for H (cid:0) X Σ + g (cid:1) , ICl (cid:0) X Σ + g (cid:1) , ICl ( A Π ), ICl( A ′ Π ) and ICl ( B ′ O + ) molecules for various n and rotational ℓ quantum numbers inTietz-Wei diatomic molecular potential. n ℓ H (cid:16) X Σ + g (cid:17) CO (cid:0) X Σ + (cid:1) ICl (cid:16) X Σ + g (cid:17) ICl (cid:0) A Π (cid:1) ICl (cid:0) A ′ Π (cid:1) ρ b = − ℓ ( ℓ +1) D r e + µD ¯ h ( c h − − rh ℓ ( ℓ +1) D r e + µD ¯ h ( c h − i − T nℓ h ℓ ( ℓ +1) D r e + µD ¯ h ( c h − i T nℓ (11b)with the following sum and product properties: ρ a + ρ b = − ℓ ( ℓ + 1) D ¯ h + 4 µr e D ( c h − T nℓ r e ¯ h and ρ a ρ b = ℓ ( ℓ + 1) D ¯ h + 2 µr e D ( c h − T nℓ r e ¯ h . (12)Furthermore, the momentum k ( ρ ) between two turning points can be found as: k ( ρ ) = s(cid:20) ℓ ( ℓ + 1) D r e + 2 µD ¯ h ( c h − (cid:21) [( ρ b − ρ )( ρ − ρ a )] . (13)The Riccati relation given by equation (2) can be re-written for the ground state as − αρ (1 + c h ρ ) r e φ ′ ( ρ ) + φ ( ρ ) = − µ ¯ h [ E ℓ − V eff ( ρ )] . (14)Since the logarithmic derivative φ ( ρ ) for the ground state has one zero and no pole, it hasto take the linear form in ρ . The only possible solution satisfying equation (14) is of theform φ ( ρ ) = A + B ρ . The substitution of this expression into equation (14), one has the10round state energy eigenvalue E ℓ = (cid:20) ℓ ( ℓ + 1) D r e − A (cid:21) ¯ h µ + D with A = 12 B (cid:20) ℓ ( ℓ + 1) r e D + 4 µD ¯ h ( c h − (cid:21) + α r e and B = c h α r e − c h α r e s r e α c h (cid:20) ℓ ( ℓ + 1) D r e + 2 µD ¯ h ( c h − (cid:21) (15)We have now reached a position of calculating the integrals given by equation (4). The LHSintegral can be calculated as follows: Z r B r A k ( r ) dr = r e Z ̺ b ̺ a k ( ̺ ) d̺ = − r e α Z ρ a ρ b k ( ρ ) ρ (1 + c h ρ ) dρ = − r e α s ℓ ( ℓ + 1) D r e + 2 µD ¯ h ( c h − Z ρ b ρ a p ( ρ b − ρ )( ρ − ρ a ) ρ (1 + c h ρ ) dρ (16)= − πr e α s ℓ ( ℓ + 1) D r e + 2 µD ¯ h ( c h − " p (1 + c h ρ a )(1 + c h ρ b ) c h − c h − √ ρ a ρ b = − πr e α "s T nℓ − c h (cid:20) ℓ ( ℓ + 1) D r e + 4 µD ¯ h ( c h − (cid:21) + R T c h − √ R T c h − p T nℓ with T nℓ = (cid:20) ℓ ( ℓ + 1) D r e + 2 µ ¯ h ( D − E nℓ ) (cid:21) and R T = ℓ ( ℓ + 1) D ¯ h µr e + D ( c h − , where we have utilized the properties (12) and the integral relation given by Z x b x a p ( x b − x )( x − x a ) x (1 + Qx ) dx = π " p ( Qx a + 1)( Qx b + 1) Q − Q − √ x a x b . (17)Now, simply by replacing E nℓ in the above equation (16) by E ℓ given by equation (15), and T nℓ as T ℓ = h ℓ ( ℓ +1) D r e + µ ¯ h ( D − E ℓ ) i , we obtain the integral in the RHS of equation (4) as Z r b r a k ( r ) dr = − πr e α "s T ℓ − c h (cid:20) ℓ ( ℓ + 1) D r e + 4 µD ¯ h ( c h − (cid:21) + R T c h − √ R T c h − p T ℓ = πr e αc h hp R T + B i . (18)With equations (16), (18) and (4), we can deduce the following relation − πr e α "s T nℓ − c h (cid:20) ℓ ( ℓ + 1) D r e + 4 µD ¯ h ( c h − (cid:21) + R T c h − p T nℓ − πr e αc h B = nπ. s T nℓ − c h (cid:20) ℓ ( ℓ + 1) D r e + 4 µD ¯ h ( c h − (cid:21) + R T c h = − (cid:18) n + r e αc h B (cid:19) αr e + p T nℓ (19)11n squaring up both sides of equation (19), it is straightforward to show that the energyeigenvalues equation can be found as E nℓ = ¯ h ℓ ( ℓ + 1) D µr e + D − α ¯ h µr e η + ℓ ( ℓ +1) α c h ( D c h − D ) + µDr e α ¯ h (cid:16) − c h (cid:17) η (20)with η = n + 12 + 12 s c h (cid:18) D ℓ ( ℓ + 1) α + 2 µDr e α ¯ h (1 − c h ) (cid:19) . The shifted Deng-Fan molecular potential model we examine in this section is defined as[17, 27] V ( r ) = D (cid:18) − be βr − (cid:19) − ¯ D, b = e ar e − , (21)where ( D, ¯ D ), b and β are three parameters representing the dissociation energy, the po-sition of the minimum r e and the range of the potential respectively. Very recently, Wangand co-workers found that the Manning-Rosen, Deng-Fan and Schi¨oberg potential are notbetter than the traditional Morse potential in simulating the atomic interaction for diatomicmolecules [28]. In order to overcome this problem, Hamzavi et al. suggested a modificationto the Deng-Fan potential, which they referred to as the shifted Deng-Fan potential (sDF)[17]. This modification is simply a Deng-Fan potential [19, 20] shifted by dissociation energy D [17]. The researchers [17] examined the Schr¨odinger equation with this potential and ap-plied their results to some diatomic molecules [17]. From their plot for the shifted Deng-Fanpotential and the Morse potential using the parameters set for H diatomic molecule, it wasshown that the two potentials are very close to each other for large values of r in the regions r ≈ r e and r > r e , but they are very different at r ≈
0. Also, if both the Deng-Fan and theshifted Deng-Fan potentials are deep (that is,
D >>
1) they could be well approximated bya harmonic oscillator in the region r ≈ r e [17].In Figure (7), we study the variation of this potential with respect to some diatomicmolecules of interest given in table 1. Now inserting this potential into the Schr¨odingerequation, and then use the approximation of the form [27]:1 r = (cid:20) d + e − βr (1 − e − βr ) (cid:21) , (22)12 r ( f m ) V ( r )( e V ) IIIIIIIVV 1 2 3 4 5−10−50510 r ( f m ) V ( r )( e V ) VIVIIVIIIIXX
Figure 7:
Shape of shifted Deng-Fan diatomic molecular potential for different diatomic molecules:(I) NO (cid:0) a Π i (cid:1) (II) NO (cid:0) B Π r (cid:1) (III) NO (cid:0) L ′ φ (cid:1) (IV) NO (cid:0) b Σ − (cid:1) (V) NO( X Π r ) (VI) H (cid:0) X Σ + g (cid:1) (VII)CO (cid:0) X Σ + (cid:1) (VIII) ICl (cid:0) X Σ + g (cid:1) (IX) ICl (cid:0) A Π (cid:1) (X) ICl (cid:0) A ′ Π (cid:1) . β (fm) E n ℓ ( e V ) (a) 0 5 10 15−160−140−120−100−80−60−40−20020 β (fm) E n ℓ ( e V ) (b) ℓ = 0 ℓ = 1 ℓ = 2 Figure 8: (a) The variation of the ground state energy state for various values of ℓ as a functionof the potential range β . We choose µ = 1, r e = 0 . D = 15. (b) The variation of the firstexcited energy state for various ℓ as a function of the potential range β . E n ℓ ( e V ) µ (a.u.)(a) 0 1 2−100−90−80−70−60−50−40−30−20−10010 E n ℓ ( e V ) µ (a.u.)(b) ℓ = 0 ℓ = 1 ℓ = 2 Figure 9: (a) The variation of the ground state energy state for various values of ℓ as a functionof the particle mass µ . We choose β = 5, r e = 0 . D = 15. (b) The variation of the firstexcited energy state for various ℓ as a function of the particle mass µ . e (fm) E n ℓ ( e V ) (a) 0 0.5 1 1.5−45−40−35−30−25−20−15−10−505 r e (fm) E n ℓ ( e V ) (b) ℓ = 0 ℓ = 1 ℓ = 2 Figure 10: (a) The variation of the ground state energy state for various values of ℓ as a functionof the molecular bond length r e . We choose µ = 1, β = 5 and D = 15. (b) The variation of thefirst excited energy state for various ℓ as a function of the molecular bond length r e the effective potential takes the following form: V eff ( y ) = P + Qy + Ry with P = D − ¯ D + ℓ ( ℓ + 1) β d ¯ h µ (23) Q = ℓ ( ℓ + 1) β ¯ h µ − Db and R = ℓ ( ℓ + 1) β d ¯ h µ + Db ,
10 20 30−20−15−10−50510 E n ℓ ( e V ) (a) D ( eV ) E n ℓ ( e V ) D ( eV ) (b) ℓ = 0 ℓ = 1 ℓ = 2 Figure 11: (a) The variation of the ground state energy state for various values of ℓ as a functionof the dissociation energy D e . We choose µ = 1, β = 5 and r e = 0 .
8. (b) The variation of the firstexcited energy state for various ℓ as a function of the dissociation energy D Table 5:
Comparison of the bound-state energy eigenvalues − E nℓ ( eV ) of H and CO moleculesfor various n and rotational ℓ quantum numbers in sDF diatomic molecular potential. H Diatomic Molecule CO Diatomic Molecule n ℓ
Present GPS [26] AIM [27] N-U [17] Present GPS [26] AIM [27] N-U[24] (CO)0 0 4.39461978 4.39462330967 4.394619779 4.39444 11.08074990 11.0807513815 11.08075178 11.080685 4.17661316 4.176618048 4.17644 11.07253746 11.07253985 11.0724710 3.62182049 3.621838424 3.62165 11.05064208 11.05064581 11.050575 0 1.75845157 1.758451567 1.75835 9.68814442 9.688146187 9.688095 1.61740572 1.617410615 1.61731 9.68022402 9.680226284 9.6801710 1.26043371 1.260451640 1.26034 9.65910731 9.659110919 9.659057 0 1.07763699 1.077636993 1.07756 9.15916229 9.159164003 9.159115 0.96180989 0.961814782 0.96174 9.15135744 9.151359661 9.1513110 0.66982613 0.669844065 0.66976 9.13054886 9.130552425 9.13050 after an appropriate coordinate transformation of the form y = (cid:0) e βr − (cid:1) − has been intro-duced. Now, we can write the non-linear Riccati equation for the ground state as − ay (1 + y ) φ ′ ( y ) + φ ( y ) = − µ ¯ h [ E ℓ − V eff ( y )] (24)Since the logarithmic derivative φ ( y ) for the ground state has one zero and no pole, it has15able 6: Bound-state energy eigenvalues for NO ( a Π i ), N O ( B Π r ), NO ( L ′ φ ), NO ( b Σ − )and NO ( X Π r ) molecules for various n and rotational ℓ quantum numbers in sDF diatomicmolecular potential. n ℓ NO (cid:0) a Π i (cid:1) NO (cid:0) B Π r (cid:1) NO (cid:0) L ′ φ (cid:1) NO (cid:0) b Σ − (cid:1) NO ( X Π r )0 0 -1.96954814 -2.75045354 -1.73990613 -2.54988834 -6.490843201 0 -1.85431702 -2.61950265 -1.62684906 -2.40042016 -6.249198501 -1.85394591 -2.61914169 -1.62645839 -2.39995170 -6.248660002 0 -1.74265661 -2.49186027 -1.51767759 -2.25557591 -6.012328671 -1.74228862 -2.49150219 -1.51729002 -2.25511093 -6.011793902 -1.74155263 -2.49078605 -1.51651490 -2.25418099 -6.010724383 0 -1.63455757 -2.36751730 -1.41238208 -2.11534494 -5.780222071 -1.63419268 -2.36716213 -1.41199768 -2.11488343 -5.779691032 -1.63346278 -2.36645179 -1.41122876 -2.11396043 -5.778628963 -1.63236803 -2.36538629 -1.41007542 -2.11257593 -5.777035884 0 -1.53001045 -2.24646485 -1.31095323 -1.97971663 -5.552867131 -1.52964860 -2.24611256 -1.31057185 -1.97925858 -5.552339812 -1.52892485 -2.24540801 -1.30980918 -1.97834250 -5.551285173 -1.52783925 -2.24435119 -1.30866514 -1.97696839 -5.549703244 -1.52639185 -2.24294214 -1.30713977 -1.97513628 -5.54759404 to take the linear form in y . Thus, we assume the following solution for the ground state φ ( y ) = A + By. (25)By putting equation (25) into (35) and then solve the non-linear Riccati equation, it isstraightforward to obtain the ground state energy and values of A and B as E ℓ = P − ¯ h A µ with A = µ ¯ h Q − RB + B B = β r β + 8 µR ¯ h . (26)Furthermore, in a similar fashion to the previous problem, the two turning points as well astheir sum and product properties are given by y a = − Q R − R p Q − R ( P − E nℓ ) , and y b = − Q R + 12 R p Q − R ( P − E nℓ ) y a + y b = − QR , y a y b = P − E nℓ R and k ( y ) = r µR ¯ h [ − ( y − y a ) ( y − y b )] / . (27)Now, we have all necessary tools required to perform our calculations. Therefore, we proceed16able 7: Bound-state energy eigenvalues for H (cid:0) X Σ + g (cid:1) , ICl (cid:0) X Σ + g (cid:1) , ICl ( A Π ), ICl( A ′ Π ) and ICl ( B ′ O + ) molecules for various n and rotational ℓ quantum numbers in sDFdiatomic molecular potential. n ℓ H (cid:16) X Σ + g (cid:17) CO (cid:0) X Σ + (cid:1) ICl (cid:16) X Σ + g (cid:17) ICl (cid:0) A Π (cid:1) ICl (cid:0) A ′ Π (cid:1) to calculate integral (4) Z r b r a k ( r ) dr = − Z y b y a k ( y ) βy (1 + y ) dy = − Z y b y a s µRβ ¯ h [( y − y a ) ( y b − y )] / y (1 + y ) dy = − πβ r µR ¯ h hp (1 + y a )(1 + y b ) − − √ y a y b i (28)= − πβ r µR ¯ h " √ R − Q + P − E nℓ R − − r P − E nℓ R , where we have used the following standard integral Z y b y a [ − ( y − y a ) ( y − y b )] / y (1 + y ) = π hp ( y a + 1)( y b + 1) − − √ y a y b i . (29)17urthermore, we can find Z r b r a k ( r ) dr = − Z y b y a k ( y ) βy (1 + y ) dy = − Z y b y a r µRa ¯ h [ − ( y − y a ) ( y − y b )] / y (1 + y )= − πβ r µR ¯ h hp (1 + y a )(1 + y b ) − − √ y a y b i (30)= − πβ r µR ¯ h " √ R − Q + P − E ℓ R − − r P − E ℓ R = − πβ r µR ¯ h " ¯ h µR ( A − B ) − r P − E ℓ R − = πβ r µR ¯ h B s ¯ h µR + 1 . From equations (4), (29) and (31), we can find the energy spectrum for the sDF as E nℓ = D ( b + 1) + ℓ ( ℓ + 1) β ¯ h d µ − ¯ h β µ (cid:16) Bβ + n (cid:17) µDb ( b + 2)2¯ h β (cid:16) Bβ + n (cid:17) − ¯ D. (31) Eigenfunctions-eigenvalue relation is very important in quantum mechanics because of itsprominence in the equations which relate the mathematical formalism of the theory withphysical results. eigenfunctions could be considered as trial functions in variational-typeprocedures for deriving energy levels anl also for computing line intensities. Since properquantization rule cannot be used to obtain these eigenfunctions, we therefore resort to usingthe recently proposed formula method [1]. This method is very easy to use in obtainingnot only the eigenfunctions but also energy eigenvalues. In the approach, it is requiredto transform the Schr¨odinger equation with two solvable quantum molecular systems-Tietz-Wei and shifted Deng-Fan potential models into the form given by equation (1) of ref. [1] viaan apprropriate coordinate transformation of the form τ = e β̺ (for TH) and t = e − αr (forsDF), which maintained the finiteness of the transformed wave functions on the boundaryconditions to have d R nℓ ( τ ) dτ + 1 τ dR nℓ ( τ ) dτ + 1 τ (1 − c h τ ) (cid:26)(cid:20) µr e ¯ h α ( E nℓ − D ) − ℓ ( ℓ + 1) α D (cid:21) + (cid:20) − c h (cid:18) µr e E nℓ α ¯ h − ℓ ( ℓ + 1) α D (cid:19) + 4 µr e D ¯ h α − ℓ ( ℓ + 1) α D (cid:21) τ + (cid:20) c h (cid:18) µr e E nℓ α ¯ h − ℓ ( ℓ + 1) α D (cid:19) + ℓ ( ℓ + 1) α ( D c h − D ) − µr e D ¯ h α (cid:21) τ (cid:27) R nℓ ( τ ) = 0 , (32a)18 R nℓ ( t ) dt + 1 t dR nℓ ( t ) dt + 1 t (1 − t ) (cid:20) µ ¯ h ( E nℓ − D ) − µDbβ ¯ h ( b + 2) − ℓ ( ℓ + 1) d + t (cid:18) µbDβ ¯ h − µβ ¯ h ( E nℓ − D ) − ℓ ( ℓ + 1)(1 − d ) (cid:19) + t (cid:18) µβ ¯ h ( E nℓ − D ) − ℓ ( ℓ + 1) d (cid:19)(cid:21) R nℓ ( t ) = 0 . (32b)Considering equation (32a) with reference to [1], k , k , k , A T − H , B T − H and C T − H can befound. Then, parameters k and k can be obtained as k = s(cid:20) µr e ¯ h α ( D − E nℓ ) + ℓ ( ℓ + 1) α D (cid:21) and k = 12 ( s c h (cid:20) ℓ ( ℓ + 1) α D + 2 µr e D ¯ h α (1 − c h ) (cid:21)) , (33)Hence, the eigenfunctions for TW can be found as R nℓ ( ̺ ) = N nℓ e − k α̺ (1 − c h e − α̺ ) k F (cid:0) − n, n + 2( k + k ); 2 k + 1 , c h e − α̺ (cid:1) . (34)Similarly, the eigenfunctions for sDF can be found as R nℓ ( z ) = N nℓ t w (1 − t ) v F ( − n, w + v ) ; 2 w + 1; t ) , (35)where w = s − (cid:18) µ ¯ h ( E nℓ − D ) − µDbβ ¯ h ( b + 2) + ℓ ( ℓ + 1) d (cid:19) and v = 12 + s(cid:18) ℓ + 12 (cid:19) + 2 µDb β ¯ h . (36) In Figure 1, we plotted the Tietz-Wei (TW) potential for different diatomic molecules. Inwhat follows, to see the behavior of the ground n = 0 and first excited n = 1 states, weplotted the energy for these states with potential parameters for three different orbital states ℓ = 0 , ,
2. In Figure 2, we show the variation of E n,ℓ with the potential constant c h . Itshows that for c h <
0, the energy is negative whereas when c h >
0, the energy is positivefor n = 0. On the other hand, for n = 1, the energy becomes strongly bound for c h < c h >
0. The c h = 0 represents the Morse energy. Thebest choice c h = 0 .
03 restores the results of Morse potential. At this value the energy curvescoincide and have same behavior for ℓ = 0 , ,
2. Figure 3 shows the variation of E n,ℓ withthe reduced mass µ for three orbital states. The energy is very similar for 0 . < µ < . n = 1 but different when n = 0. Its seen that when µ increases for more than 0.3, theenergy spectrum becoming positive for ground state while for excited state, it is positive forany value of µ In Figure 4, we plotted the variation of E n,ℓ with the potential parameter b h . It isincreasing in the positive direction within the interval 0 < b h < n = 0. However,when n = 1, the energy increases in positive side for 0 < b h < < b h <
8. In Figure 5 we show the variation of the energy states E n,ℓ as a functionof molecular bond length r e . The ground state energy drops with nearly 2eV for all orbitalstates at r e = 0 . f m and r e = 0 . f m whereas the first excited state has drop of about0.45eV and coincide at 0.56fm. Finally, Figure 6 demonstrates the energy versus the welldepth D . Its seen that the ground state energy span from negative to positive spectrum at D = 2 eV . for orbital states ℓ = 0 , ,
2. However the first excited energy state span fromnegative to positive spectrum at D = 8 eV for all orbital states.A very similar behavior to TW potential model (sDF shape) for various moleculesis shown in Figure 7. In addition we have obtained the energy spectrum for differentdiatomic molecules with the help of TW molecular model for various states using themodel potential parameters in Table 1. This spectroscopic parameter are taken from Refs.[27], [29] and [31] and the conversion factors used are taken from NIST database [32]:1 cm − = 1 . hc = 1973 . eV ˚ A and 1 amu = 931 . M ev/c . In Table 2,we test the accuracy of the method utilized in this study by finding the energy spectra of H and CO diatomic molecules. We found that the spectrum obtained by NU method in [24]have some error in the Maple codes. We therefore re-compute these spectrum in the presentwork for the sake of comparison. As it can been seen from the table, our results are veryclose to the ones of the Nikiforov-Uvarov method. Tables 3 and 4 present the spectrum forH (cid:0) X Σ + g (cid:1) , CO ( X Σ + ) and various electronic states of NO and ICl diatomic molecules.Considering sDF molecular potential, Figure 8 shows the variation of E n,ℓ as a functionof β , in the ground state. The restriction on choice of the parameter β of sDF molecularpotential can be observed. The energy for ℓ = 1 , ℓ = 0the energy increases in the negative side. On the other hand, in the first excited state, theenergy increases in the negative side in the interval 0 < β <
15 for ℓ = 0 , , E n,ℓ as a function of reduced mass µ . Small values of20article mass µ result into a sharp change in energy values for ground and first excited statesfor the orbital states. The energy becomes stable when µ >
1. This plot indicates how tochoose or read the most reasonable values of µ which provides the most appropriate andnot overlapping spectrum amongst orbital states. Figure 10 is a plot of energy versus bondlength r e . The spectrum E , > E , > E , when r e > . E , > E , > E , when r e > .
4. Figure 10 set restrictions on the most suitable values of r e . At r e < . − . f m ,the energy of different orbital states overlap and deteriorate sharply.The variation of the energy versus parameter D is shown in Figure 11. For n = 0 theenergy increases and then decreases in the given range 0 < D <
25 whereas for n = 1, it isincreasing in the same interval. Furthermore we generated the spectrum of several diatomicmolecules using the sDF molecular potential for various states. The behavior of the plotenergy against each potential parameter for various states provides us the most appropriatephysical values for each parameter. Figures 6 and 10 have different behaviors since theyare a plots of energy against r e for two potentials. However, if one has set to choose largevalues for r e (say r e > .
6) then the two curves will be similar.Table 5 compares our results for H and CO with those of the GPS method, Nikiforov-Uvarov method and AIM methods. Our currently found energy states are reasonably com-pared with the other findings. The vibrational energy is close to 7 digits with AIM [27]but found to agree with GPS [26] up to 4 digits. However, the rotational-vibrational en-ergy states are close close to 5 digits with AIM. This is due to the approximation madeto the centrifugal restorsion term. Also, it should be noted that the model is a parameterdependent which may result into slight variation in energy spectrum if parameters are notconverted/adjusted properly. Table 6 displays the energy spectrum for different speciesof N O diatomic molecules. However, Table 7 present ones for H (cid:0) X Σ + g (cid:1) , CO ( X Σ + ),various electronic states of ICl diatomic molecules
In this research work, we applied proper quantization rule in a spectroscopic study of somediatomic molecules. This task is made possible by solving the Schr¨odinger equation with twomolecular models; namely, Tietz-Wei and shifted Deng-Fan potential models. This solution21erves as the basis for the description of the quantum aspects of diatomic molecules. Weobtained the energy spectra of different diatomic molecules. The validity and accuracyof the method is tested with previous techniques via numerical computation for H andCO molecules. Our reasonable results show the efficiency and simplicity of the presentcalculations. The approximation to the centrifugal restorsion is valid for the lowest orbitalquantum number ℓ . As ℓ increases, the accuracy of the energy states reduces and vice-versa. The present research work represents a new procedure in dealing with the diatomicmolecules. Our results are reasonable and credible in generating the spectrum as the othercommonly known methods. Acknowledgments
We thank the kind referees for the positive enlightening comments and suggestions, whichhave greatly helped us in making improvements to this paper. In addition, BJF acknowl-edges eJDS (ICTP).
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