Ro-vibrational energies and expectation values of selected diatomic molecules via Varshni plus modified Kratzer potential model

Abstract

The analytical solutions of the Klein–Gordon equation of diatomic molecules with an approximation to the centrifugal term for the Varshni plus modified Kratzer potential model are obtained approximately within the framework of the Nikiforov–Uvarov (NU) method. The expression for ro-vibrational energy, normalized wave function, and expectation values for the diatomic molecules ${\text{N}}_{2} {\text{, CO, NO, I}}_{2} {\text{ and H}}_{2}$ have all been obtained. The ro-vibrational energies for the dimer molecules were computed using their separate spectroscopic parameters. Utilizing Hellman–Feynman theorem, the expectation values of the inverse square position $\left\langle {r^{ - 2} } \right\rangle$, kinetic energy $\left\langle T \right\rangle$, square of momentum $\left\langle {P^{2} } \right\rangle$, and their separate numerical values for the selected diatomic molecules were obtained explicitly. Two special cases of the potential are also studied, and their numerical energy eigenvalues obtained are reliable and very consistent with the existing literature.

Appendices

Appendix

Appendix methodology: review of Nikiforov-Uvarov (NU) method

The NU method was proposed by Nikiforov and Uvarov [70] to transform Schrödinger-like equations into a second-order differential equation of the form

$$\psi^{\prime\prime}\left( s \right) + \frac{{\tilde{\tau }\left( s \right)}}{\sigma \left( s \right)}\psi^{\prime}\left( s \right) + \frac{{\tilde{\sigma }\left( s \right)}}{{\sigma^{2} \left( s \right)}}\psi \left( s \right) = 0,$$

(60)

where $\tilde{\sigma }\left( s \right),{\text{ and }}\sigma \left( s \right)$ are polynomials, at most second degree and $\tilde{\tau }\left( s \right)$ is a first-degree polynomial. The exact solution of Eq. (60) can be obtain by using the transformation.

$$\psi \left( s \right) = \phi \left( s \right)y\left( s \right)$$

(61)

This transformation reduces Eq. (60) into a hypergeometric-type equation of the form

$$\sigma \left( s \right)y^{\prime\prime}\left( s \right) + \tau \left( s \right)y^{\prime}\left( s \right) + \lambda y\left( s \right) = 0$$

(62)

The function $\phi (x)$ can be defined as the logarithm derivative

$$\frac{{\phi^{\prime}\left( s \right)}}{\phi \left( s \right)} = \frac{\pi \left( s \right)}{{\sigma \left( s \right)}},$$

(63)

with $\pi (s)$ being at most a first-degree polynomial. The second part of $\psi \left( s \right)$ being $y(s)$ in Eq. (61) is the hypergeometric function with its polynomial solution given by Rodrigues relation as

$$y\left( s \right) = \frac{{B_{nl} }}{\rho \left( s \right)}\frac{{d^{n} }}{{ds^{n} }}\left[ {\sigma^{n} \left( s \right)\rho \left( s \right)} \right]$$

(64)

where $B_{nl}$ is the normalization constant, and $\rho \left( s \right)$is the weight function which satisfies the condition as follows:

$$\left( {\sigma \left( s \right)\rho \left( s \right)} \right)^{\prime } = \tau \left( s \right)\rho \left( s \right)$$

(65)

where also

$$\tau \left( s \right) = \tilde{\tau }\left( s \right) + 2\pi \left( s \right)$$

(66)

For bound solutions, it is required that

$$\tau^{/} (s) < 0$$

(67)

The eigenfunctions and eigenvalues can be obtained using the definition of the following function $\pi \left( s \right)$ and parameter λ, respectively:

$$\pi \left( s \right) = \frac{{\sigma^{\prime}\left( s \right) - \tilde{\tau }\left( s \right)}}{2} \pm \sqrt {\left( {\frac{{\sigma^{\prime}\left( s \right) - \tilde{\tau }\left( s \right)}}{2}} \right)^{2} - \tilde{\sigma }\left( s \right) + k\sigma \left( s \right)}$$

(68)

and

$$\lambda = k_{ - } + \pi^{\prime}_{ - } \left( s \right)$$

(69)

The value of $k$ casn be obtained by setting the discriminant in the square root in Eq. (68) equal to zero. As such, the new eigenvalues equation can be given as

$$\lambda_n + n\tau^{^{\prime}} \left( s \right) + \frac{n(n - 1)}{2}\sigma^{^{\prime\prime}} \left( s \right) = 0,(n = 0,1,2,...)$$

(70)