Energy States of Some Diatomaic Molecules: The Exact Quantisation Rule Approach

1 Introduction

According to the Schrödinger formulation of quantum mechanics, a total eigenfunction provides implicitly all relevant information about the behaviour of a physical system. Thus, if it is exactly solvable for a given potential, an obtained eigenfunction can be used to describe such a system completely. This has made the exact solutions of quantum systems to be an important subject and also to attract much attention to the development of quantum mechanics [1–25].

Up to now, there have been several efficient methodologies developed to find the exact solutions of quantum systems within the framework of the non-relativistic and relativistic quantum mechanical wave equations. A few of these methods include the recently proposed formula method [1], the Feynman integral formalism [5, 16], asymptotic iteration method [2–5, 17–21], functional analysis approach [22–24], exact quantisation rule method [25–33], proper quantisation rule [27, 34], Nikiforov–Uvarov method [35–38], supersymetric quantum mechanics [6, 39–44], etc.

Very recently, Ma and Xu [25, 26] have proposed an exact (improved) quantisation rule and applied it to calculate the energy levels of some exactly solvable quantum mechanical problems. These include the finite square well, the Morse, the symmetric and asymmetric Rosen–Morse, the harmonic oscillator, and the first and second Pöschl–Teller potentials [25]. By using this same rule, Dong and Morales [44] obtained the analytical solutions of the Schrödinger equation for the deformed harmonic oscillator in one dimension, the Kratzer potential, and pseudoharmonic oscillator in three dimensions. The energy levels of all the bound states are easily calculated from this quantisation rule.

The main purpose of the present work was to obtain the bound-state solution of the Deng–Fan diatomic potential model via this quantisation rule. The Deng–Fan diatomic potential model is also known as the generalised Morse potential, which was proposed some decades ago by Deng and Fan [45]. This was done in an attempt to find a more suitable diatomic potential to describe the vibrational spectrum, qualitatively similar to the Morse potential with the correct asymptotic behaviour as the inter-nuclear distance approaches zero. This potential model can be used to describe the motion of the nucleons in the mean field produced by the interactions between nuclei [46]. This potential has been used to describe diatomic molecular energy spectra and electromagnetic transitions, and it is an ideal inter-nuclear potential in diatomic molecules with the same behaviour for r→0. Because of its importance in chemical physics, molecular spectroscopy, molecular physics, and related fields, the bound-state solutions of the relativistic and non-relativistic wave equations have been studied by several authors [7, 29, 45, 46]. The shape of this potential with respect to some diatomic molecules is shown in Figure 1.

Figure 1

Shape of the Deng–Fan diatomic molecular potential for different diatomic molecules.

The organisation of this work is as follows. In the next section, we briefly introduced the exact quantisation rule. In Section 3, we apply this quantisation rule to obtain the bound-state solutions of the Deng–Fan molecular potential. Section 4 presents the numerical results, and, finally, a brief conclusion is presented.

2 Exact Quantisation Rule

In this section, a brief review of this quantisation rule is given. The details can be found in [25, 26]. It is well known that, in one dimension, the Schrödinger equation

(1)d2dx2ψ(x) + 2μℏ2[E − V(x)]ψ(x) = 0, (1)

can be written in the following form:

(2)ϕ′(x) + ϕ(x)2 + k(x)2 = 0,  with  k(x) = 2μℏ2[E − V(x)], (2)

where ϕ(x)=ψ′(x)/ψ(x) is the logarithmic derivative of the wave function ψ(x), the prime denotes the derivative with respect to the variable x, μ denotes the reduced mass of the two interacting particles, k(x) is the momentum, and V(x) is a piecewise continuous real potential function of x. For the Schrödinger equation, the phase angle is the logarithmic derivative ϕ(x). From (2), as x increases across a node of wave function ψ(x), ϕ(x) decreases to –∞, jumps to +∞, and then decreases again.

In the recent years, Ma and Xu [25, 26] generalised this exact quantisation rule to the three-dimensional radial Schrödinger equation with spherically symmetric potential by simply making the replacements x→r and V(x)→V_eff(r):

(3)∫rarbk(r)dr = Nπ + ∫rarbϕ(r)[dk(r)dr][dϕ(r)dr]−1,  k(r) = 2μℏ2[E−Veff(r)]. (3)

where r_a and r_b are two turning points determined by E=V_eff(r). N=n+1 is the number of the nodes of ϕ(r) in the region E_nℓ=V_eff(r) and is larger by 1 than the n of the nodes of wave function ψ(r). The first term Nπ is the contribution from the nodes of the logarithmic derivative of the wave function, and the second one is called the quantum correction. It was found that, for all well-known exactly solvable quantum systems, this quantum correction is independent of the number of nodes of the wave function. This means that it is enough to consider the ground state in calculating the quantum correction, i.e.,

(4)Qc = ∫rarbk′0(r)ϕ0ϕ′0dr (4)

In the recent years, this quantisation rule has been used in many physical systems to obtain the exact solutions of many exactly solvable quantum systems [25–29, 34].

3 Application to Deng–Fan Molecular Potential

In this section, we apply the exact quantisation rule to study the ro-vibrational energy states of some diatomic molecules. To begin, we write the Schrodinger equation with the Deng–Fan diatomic molecular potential as

(5)d2Rnℓ(r)dr2 + 2μℏ2[Enℓ − D(1 − bear − 1)2 − ℓ(ℓ + 1)ℏ22μr2]Rnℓ(r) = 0, (5)

where n and ℓ denote the radial and orbital angular momentum quantum numbers, respectively; r is the internuclear separation of the diatomic molecules; and E_nℓ is the bound-state energy eigenvalues. The μ and V(r) represent the reduced mass and interaction potential, respectively. It is well known that the equation of form (5) is an exactly solvable problem for the S-wave. But for the ℓ -wave states, the problem is not analytically solvable. To obtain the bound-state solutions, we must therefore resort to using an approximation similar to those used in other works [47–50] to deal with the centrifugal term or, alternatively, we must solve numerically. It is noted that, for short potential range, the following formula is a good approximation to the centrifugal term:

(6)1r2 ≈ α2[d0 + eαr(eαr − 1)2], (6)

where d0 = 112. It should be noted that this approximation reduces to the one used by Dong and Gu [29], Qiang and Dong [23], and Dong and Qiang [24] when d₀=0. Now, if we consider the approximation (6) and (5), we can find the effective potential as

(7)Veff(ϱ) = D + ℓ(ℓ + 1)α2d0ℏ22μ + (ℓ(ℓ + 1)α2ℏ22μ − 2Db)ϱ + (ℓ(ℓ + 1)α2d0ℏ22μ + Db2)ϱ2, (7)

where we have introduced a new transformation of the form ϱ = e−αr1 − e−αr. Now, let us begin the application of the quantisation rule to study the potential V_eff(ϱ). To perform this task, we have to first calculate the turning points ϱ_a and ϱ_b determined by solving V(ϱ)=E_nℓ. Thus we have

(8)ϱa = − Q2R − Q2 − 4R(P − Enℓ)2R,ϱb = − Q2R + Q2 − 4R(P − Enℓ)2R with k(ϱ) = 2μRℏ[(ϱa − ϱ)(ϱ − ϱb)]1/2 (8)

where k(ϱ) is the momentum between the two turning points ϱ_a and ϱ_b. We have also introduced three parameters, P, Q, and R, for mathematical simplicity. These parameters are as follows:

(9)P = D + ℓ(ℓ + 1)α2d0ℏ22μ, Q = ℓ(ℓ + 1)α2ℏ22μ − 2Db   and  R = ℓ(ℓ + 1)α2d0ℏ22μ + Db2. (9)

The non-linear Riccati equation for the ground state is written in terms of the new variable ϱ as

(10)− aϱ(1 + ϱ)ϕ′0(ϱ) + ϕ02(ϱ) + 2μℏ2[E0ℓ − Veff(ϱ)]ϕ0(ϱ) = 0. (10)

Thus, since the logarithmic derivative ϕ₀(ϱ) for the ground state has one zero and no pole, therefore we assume the following solution for the ground states

(11)ϕ0(ϱ) = A + ℬϱ. (11)

On substituting (11) into (10) and then solving the non-linear Riccati equation, we obtain the ground state energy as

(12)E0ℓ = P − ℏ2A22μ. (12)

Also, A and ℬ are as follows:

(13)A = μℏ2Q − RB + B2  and ℬ = a2 + 12a2 + 8μRℏ2. (13)

It is worth noting that we only chose the positive sign in front of the square root for ℬ. This is because the logarithmic derivative ϕ₀(ϱ) will decrease exponentially, which is required physically. Let us now calculate the quantum correction. For this purpose, we utilised the integrals given by Appendix A, and we obtain

(14)∫rarbk′0(r)ϕ0(r)ϕ′0(r)dr = − ∫ϱaϱbk′0(ϱ)αρ(1 + ϱ)ϕ0(ϱ)ϕ′0(ϱ)dϱ = 1α2μRℏ2∫ϱaϱb[A + ℬ]ϱ[ϱ − (ϱa + ϱb2)]dϱℬϱ(1 + ϱ)(ϱa − ϱ)(ϱ − ϱb) = 1α2μRℏ2∫ϱaϱbdϱ(ϱa − ϱ)(ϱ − ϱb) [(Aℬ − 1)(ϱa + ϱb2 + 1)1 + ϱ − Aℬ(ϱa + ϱb2)ϱ + 1] = πα2μRℏ2[1 + 2μRℬℏ]. (14)

Furthermore, the integral of the momentum k(r) can be found as follows:

(15)∫rarbk(r)dr = − ∫ϱaϱbk(ϱ)αϱ(1 + ϱ)dϱ = − 1α2μRℏ2∫ϱaϱb(ϱa − ϱ)(ϱ − ϱb)ϱ(1 + ϱ) = − πα2μRℏ2[(ϱa + 1)(ϱb + 1) − 1 − ϱaϱb] = − πα[R − Q + P − EnℓR − 1− P − EnℓR], (15)

where we used an appropriate standard integral in Appendix A. Now, by combining the results obtained by (14) and (15) with (3), i.e.,

(16)− πα[R − Q + P − EnℓR − 1− P − EnℓR] = Nπ + πα2μRℏ2[1 + 2μRℬℏ], (16)

the energy eigenvalues spectrum can then be found as

(17)Enℓ = D(b + 1)2 + ℓ(ℓ + 1)a2ℏ2d02μ − ℏ2a22μ[(ℬ + αn)2α + 2μDb(b + 2)2ℏ2α(ℬ + αn)]. (17)

Let us now obtain the corresponding wave function for this system. For this purpose, we introduced a new transformation of the form z=e^–αr ∈(e^α, 0) in (5), which maintained the finiteness of the transformed wave functions on the boundary conditions we have

(18)z2d2Rnℓ(z)dz2 + zdRnℓ(z)dz + (U + Vz + Wz2(1 − z)2)Rnℓ(z) = 0 with W = 2μα2ℏ2(Enℓ − D) − ℓ(ℓ + 1)d0,V = 4μbDα2ℏ2 − 4μα2ℏ2(Enℓ − D) − ℓ(ℓ + 1)(1 + 2d0)and U = 2μℏ2(Enℓ − D) − 2μDbα2ℏ2(b + 2) + ℓ(ℓ + 1)d0. (18)

Furthermore, (18) is transformed into a more convenient second-order homogeneous linear differential equation via the following transformation:

(19)Rnℓ(z) = zp(1 − z)ℬαGnℓ(z), with  p = 12 + −W. (19)

Substituting (19) into (18), we can find

(20)G″nℓ(z) + G′nℓ(z)[(2p + 1) − z(2p + ℬα + 1)z(1 − z)] + Gnℓ(z)[(ℬα + p)2 + Uz(1 − z)] = 0. (20)

The solution to the above second-order differential equation can be expressed in terms of the hypergeometric function as

(21)Gnℓ(z) = 2F1(ζ, η; γ; z) or Rnℓ(z) = zp(1 − z)ℬα 2F1(ζ, η; γ; z). (21)

The following notations

(22)ζ = p + ℬα − −U,  η = p + ℬα + −U,  γ = 2p + 1, (22)

were introduced in (21) so as to avoid mathematical complexity. By considering the finiteness of the solutions, G_nℓ(z) approaches infinity unless ζ is a negative integer. This is an indication that G_nℓ(z) will not be finite everywhere unless we take p + ℬα − −U = − n. Thus, the η given by (22) can be re-written in terms of this condition, and, finally, we can write the wave function as

(23)Rnℓ(z) = Nnℓzp(1 − z)ℬαGnℓ(z) = zp(1 − z)ℬα 2F1(−n, 2(p + ℬα); 2p + 1; z). (23)

N_nℓ is the normalisation constant.

4 Results and Conclusion

By using the known spectroscopic values in Table 1, we obtained the energy states of some selected diatomic molecules for various vibrational n and rotational ℓ angular momentum, as shown in Tables 2 and 3. These diatomic molecules are H2(X1Σg+), CO(X1Σ+), HF(X1Σ+),O2(X3Σg+), and O2+(X2Πg). The spectroscopic parameters were taken from the work of Kuncand Gordillo-Vázquez [51] and Oyewumi et al. [50]. We also applied the following conversion, μ/10^–23g=μ×6.0221415×931.494028e6/c² with ħc=1973.29 eV Å, throughout our numerical computation.

In order to test the accuracy of our results, we performed a numerical comparison of our obtained energy spectrum with those of H2(X1Σg+) and CO(X1Σ+) obtained previously in the literature, for the shifted Deng–Fan potential (i.e., the Deng–Fan potential shifted by the dissociation energy D). As shown in Table 2, our approximate results are in excellent agreement with those obtained previously by other authors and via other approach. Furthermore, we also proceeded to obtain the ro-vibrational energy spectrum (in electronvolt units) for the five selected diatomic molecules.

It is worth noting that the advantage of the method presented in this study is that, for an exactly solvable quantum system, it allows finding the energy spectrum directly in a simple way. Finally, we recommend an extension of this method to some other potentials such as the Wood–Saxon potential, Yukawa potential, Hellmann potential, etc., to obtain the bound-state solution.