Vibrational energies of some diatomic molecules for a modified and deformed potential

A molecular potential model is proposed and the solutions of the radial Schrӧdinger equation in the presence of the proposed potential is obtained. The energy equation and its corresponding radial wave function are calculated using the powerful parametric Nikiforov–Uvarov method. The energies of cesium dimer for different quantum states were numerically obtained for both negative and positive values of the deformed and adjustable parameters. The results for sodium dimer and lithium dimer were calculated numerically using their respective spectroscopic parameters. The calculated values for the three molecules are in excellent agreement with the observed values. Finally, we calculated different expectation values and examined the effects of the deformed and adjustable parameters on the expectation values.

www.nature.com/scientificreports/ possible variation of the electron-to-proton mass ratio and of the fine-structure constant 48 . It is noted that 3 3 + g state of cesium dimer has a strong Fermi contact interaction with the nuclei, and possesses a large hyperfine splitting 49 . The potential energy curve of the cesium dimer for 3 3 + g and a 3 + u states has been reported in ref. 49,50 . The modified and deformed exponential-type molecular potential model under consideration, is given as where C is a modified parameter, q 0 is a deformed parameter and q 1 is an adjustable parameters whose value can be taken as ±1. When the value of the adjustable parameter equals the value of the deformed parameter within ±1, the results of potential (1) gives other useful results. D e is the dissociation energy r e is the equilibrium bond separation and α is the screening parameter. Its numerical value can be obtain using the formula where W is the Lambert function, µ is the reduced mass, c is the speed of light and ω e is the vibrational frequency.

Parametric Nikiforov-Uvarov method
The parametric Nikiforov-Uvarov method is one of the shortest and accurate traditional techniques to solve bound state problems. This method was derived from the conventional Nikiforov-Uvarov method by Tezcan and Sever 17 . According to the authors, the reference equation for the parametric Nikiforov-Uvarov is given as Following the work of these authors, the condition for eigenvalues equation and wave function are respectively given by 17,51 The parametric constants in Eqs. (3) and (4) are deduced as follows

The radial Schrӧdinger equation and the interacting potential
To obtain the energy eigenvalues of the Schrödinger equation with potential (1), we consider the original Schrödinger equation given by Setting the wave function ψ(r) = U n,ℓ (r)Y m,ℓ (θ, φ)r −1 , and consider the radial part of the Schrӧdinger equation, Eq. (7) becomes where V (r) is the interacting potential given in Eq. (1), E nℓ is the non-relativistic energy of the system, is the reduced Planck's constant, µ is the reduced mass, n is the quantum number, U nℓ (r) is the wave function. Substituting Eq. (1) into (8), and by defining y = 1 e −αr , the radial Schrӧdinger equation with the deformed exponentialtype potential turns to be where (1) V (r) = D e − D e Ce −αr + q 0 e αr e + q 1 − (e αr e + q 1 ) 2 Ce −αr + q 0 ,

Expectation values
In this section, we calculated some expectation values using Hellmann-Faynman Theorey (HFT) [52][53][54][55][56] . When a Hamiltonian H for a given quantum system is a function of some parameter v, the energy-eigenvalue E n and the eigenfunction U n (v) of H are given by with the effective Hamiltonian as Setting v = µ and v = D e, , we have the expectation values of p 2 and V respectively are The average deviation of the calculated results from the experimental results is obtained using the formula E n,ℓ q 0 D e − q 0 + e αr e + q 1 , D e e −αr + q 0 e αr e + q 1 − (e αr e + q 1 ) 2 e −αr + q 0 .

Discussion of result
The comparison of the observed values of RKR and calculated values for 3 3 + g state of cesium dimer with q 0 = q 1 = 1, q 0 = q 1 = −1, D e = 2722.28 cm −1 , r e = 5.3474208 Å, and ω e = 28.891 cm −1 is reported in Table 1. The results for two values for each of the deformed parameter and adjustable parameter agreed with the observed values of the cesium dimer. However, the results obtained with q 0 = q 1 = −1 are higher than their counterpart obtained with q 0 = q 1 = 1. In Table 2, the comparison of vibrational energies of sodium dimer and lithium dimer respectively are reported. When the deformed parameter and the adjustable parameter are taken as one with D e = 79885 cm −1 , r e = 1.097 Å, ω e = 2358.6 cm −1 , the results agreed with the observed values of 5 1 g state of sodium dimer. Taken the deformed parameter and adjustable parameter respectively as minus one, with D e = 2722.28 cm −1 , r e = 4.173 Å and ω e = 65.130 cm −1 , the results obtained correspond to the observed values of lithium dimer.
To deduce the effect of the deformed and adjustable parameters on the numerical values and discrepancy of the calculated results from the experimental data, we used the formula given in Eq. (28). For cesium dimer, the average percentage deviation for q 0 = q 1 = 1 is 0.0038% while the average percentage deviation for q 0 = q 1 = −1 is 0.0002%. For sodium dimer with q 0 = q 1 = 1, the average percentage deviation is 0.0342% while the average percentage deviation for lithium dimer with q 0 = q 1 = −1, is 0.0016%. In Table 3, we presented the numerical results for the two different expectation values calculated in Eq. (20) and Eq. (21). The effect of the deformed and adjustable parameters on the expectation values can be seen in Table 3. For p 2 , the values obtained with q 0 = q 1 = 1 are higher than their counterpart obtained with q 0 = q 1 = −1. However, for V n the values obtained with q 0 = q 1 = −1 are higher than their counterpart obtained with q 0 = q 1 = 1.
The effect of the screening parameter on the energy eigenvalues with two values each of the deformed parameter and adjustable parameter are shown in Fig. 1. In each case, the energy of the system varies inversely with Table 1. Comparison of theoretical values with experimental values for the vibrational energy levels of the modified deformed exponential-type molecular potential for 3 3 + g state of cesium dimer. n RKR 1 cm −1 q 0 = q 1 = 1 cm −1 RKR 49 cm −1 q 0 = q 1 = − www.nature.com/scientificreports/ the screening parameter. The energy of the system for q 0 = q 1 = 1 are lower than the energy of the system for q 0 = q 1 = −1.

Conclusion
The solutions of a one-dimensional Schrӧdinger equation is obtained for a molecular potential model using parametric Nikiforov-Uvarov method. By changing the numerical values of the deformed parameter and adjustable parameter, the results obtained for different molecules agreed with experimental values. However, the results obtained with q 0 = q 1 = −1 are closer to the experimental values compared with the results obtained with q 0 = q 1 = 1. The results for lithium dimer are more closer to the experimental values followed by the results for cesium dimer obtained with q 0 = q 1 = −1. Table 3. Expectation values at various quantum states with = µ = 1,r e = 0.4,α = 0.25 and D e = 5. n q 0 = q 1 = −1. q 0 = q 1 = 1.