Eigensolutions and expectation values of shifted-rotating Möbius squared oscillator

Abstract

In this paper we have employed exact quantization rule and the Hellman–Feynman theorem to obtain bound states eigensolutions and expectation values of the shifted-rotating Möbius squared oscillator, and we have used the Pekeris-like approximation recipe to effect solution for bound states energy eigenspectra, and in solving the Riccati equation for the eigenfunctions, cases of Q ≠ 0 and Q = 0 were considered in our solutions. We have also applied our derived formulas to compute numerical values to the spectroscopic parameters of four diatomic molecules: CO, LiH, H₂ and ICl. Our analysis showed that numerical results we obtained for potential parameters, bound states energy eigenspectra and expectation values are consistent with available data in the literature and that molecular interactions in CO, LiH, H₂ and ICl are best described by improved Wei oscillator than by Morse oscillator.

Appendix

List of standard integrals used in the present work

$$ \int\limits_{{s_{n\,A} }}^{{s_{n\,B} }} {\frac{d\,s}{{\left( {a + b\,s} \right)\sqrt {\left( {s - s_{n\,A} } \right)\left( {s_{n\,B} - s} \right)} }} = \frac{\pi }{{\sqrt {\left( {a + b\,s_{n\,A} } \right)\left( {a + b\,s_{n\,B} } \right)} }}} \quad \left[ {14} \right] $$

(A.1)

$$ \int\limits_{{s_{n\,A} }}^{{s_{n\,B} }} {\frac{{\sqrt {\left( {s - s_{n\,A} } \right)\left( {s_{n\,B} - s} \right)} }}{{s\left( {1 + Q\,s} \right)}}d\,s = \pi \left\{ {\frac{{\sqrt {\left( {1 + Q\,s_{n\,A} } \right)\left( {1 + Q\,s_{n\,B} } \right)} }}{Q} - \frac{1}{Q} - \sqrt {s_{n\,A} \,s_{n\,B} } } \right\}} \quad \left[ {19} \right] $$

(A.2)

$$ \int\limits_{{s_{n\,A} }}^{{s_{n\,B} }} {\frac{s}{{\sqrt {\left( {s - s_{n\,A} } \right)\left( {s_{n\,B} - s} \right)} }}d\,s = \frac{\pi }{2}\left( {s_{n\,A} + s_{n\,B} } \right)} \quad \left[ {17} \right] $$

(A.3)

$$ \int\limits_{{s_{n\,A} }}^{{s_{n\,B} }} {\frac{d\,s}{{s\sqrt {\left( {s - s_{n\,A} } \right)\left( {s_{n\,B} - s} \right)} }} = \frac{\pi }{{\sqrt {s_{n\,A} \,s_{n\,B} } }}} \quad \left[ {17} \right] $$

(A.4)

$$ \int\limits_{{s_{n\,A} }}^{{s_{n\,B} }} {\frac{d\,s}{{\sqrt {\left( {s - s_{n\,A} } \right)\left( {s_{n\,B} - s} \right)} }} = \pi } \quad \left[ {17} \right] $$

(A.5)

$$ \int\limits_{{s_{n\,A} }}^{{s_{n\,B} }} {\frac{{\sqrt {\left( {s - s_{n\,A} } \right)\left( {s_{n\,B} - s} \right)} }}{s}d\,s = \pi \left\{ {\frac{1}{2}\left( {s_{n\,A} + s_{n\,B} } \right) - \sqrt {s_{n\,A} \,s_{n\,B} } } \right\}} \quad \left[ {17} \right] $$

(A.6)