Dealing With the Shifted and Inverted Tietz–Hua Oscillator Potential Using the J-Matrix Method Mohamed S. Abdelmonem, [a] Afaf Abdel-Hady, [b] and Ibraheem Nasser* [c] The tridiagonal J-matrix approach has been used to calculate the low and moderately high-lying eigenvalues of the rotating shifted Tietz–Hua (RSTH) oscillator potential. The radial Schr€ odinger equation is solved efficiently by means of the diagonalization of the full Hamiltonian matrix, with the Laguerre or oscillator basis. Ro–vibrational bound state ener- gies for 11 diatomic systems, namely H 2 , HF, N 2 , NO, CO, O 2 , O 1 2 , Cl 2 ,N 1 2 ,I 2 , and NO 1 , are calculated with high accuracy. Some of the energy states for molecules are reported here for the first time. The results of the last four molecules have been introduced for the first time using the oscillator basis. Higher accuracy is achieved by calculating the energy corresponding to the poles of the S-matrix in the complex energy plane using the J-matrix method. Furthermore, the bound states and the resonance energies for the newly proposed inverted Tietz–Hua IRSTH-potential are calculated for the H 2 -molecule with scaled depth. A detailed analysis of variation of eigenval- ues with n, ‘ quantum numbers is made. Results are com- pared with literature data, wherever possible. V C 2015 Wiley Periodicals, Inc. DOI: 10.1002/qua.24968 Introduction Quantum mechanically, the radial Schr€ odinger equation for diatomic systems [1] is written as: ðH2EÞvðrÞ¼ 2  h 2 2l d 2 dr 2 1  h 2 ‘ð‘11Þ 2lr 2 1V ðrÞ2E   vðrÞ ¼ H o 1V ðrÞ2E ½ vðrÞ¼ 0; (1) where  h is the rationalized Plank’s constant, l is the effective mass of the two nuclei, r denotes the internuclear distance, and the symbol ‘ is used for the angular momentum quantum numbers. E is the total energy, vðrÞ is the wave function of the full Hamiltonian H, H o is called the reference Hamiltonian, and V ðrÞ is the vibrational potential of the molecule. The potential V ðrÞ in Eq. (1), and its parameters, plays a major role in studying the molecular behavior, reaction, and interac- tion. Many empirical potentials have been proposed to compre- hend the physical behavior of molecules. For example, the Morse potential (MP) [2] and a class of potentials that modified it, namely Manning–Rosen, [3] Hellmann [4] ; P€ oschl–Teller, [5] and shifted Tietz–Hua (STH) [6,7] describe the diatomic molecule vibra- tions very well. [1] This is a result of the fact that the electron cloud moves in the Coulomb potential that is produced by two positive charges (the two atomic nuclei). There are well-defined centers here, and the sum of interaction between the electrons and atoms creates the potential in which the atoms move. In the language of energy of diatomic molecules, the electronic excita- tion and the vibration and rotational molecular energies are widely separated in energy scales, creating the complete uncou- pling of the three degrees of freedom. Due to many different advantages, in molecular dynamics and modeling of diatomic molecules, the Tietz–Hua-potential (TH) has attracted recent attention. [8–12] Furthermore, the TH oscillator potential has been found to be a more realistic analyt- ical potential than the familiar MP in describing molecular dynamics at low, as well as moderately high, rotational, and vibrational quantum numbers. [10] The TH-potential has the form: V TH ¼ D 12e 2b h r2re ð Þ 12c h e 2b h r2re ð Þ   2 ; b h ¼ bð12c h Þ (2) where b is the Morse constant, r e is the molecular bond length, and D signifies the potential well depth (defined rela- tive to the dissociated atoms). c h is an optimization parameter obtained from ab initio or Rydberg–Klein–Rees intramolecular potentials, respectively. For positive values of D, V TH has a min- imum of 0 at r ¼ r e , and for negative values of D, V TH has a maximum of 0 at r ¼ r e . [a] M. S. Abdelmonem Department of Physics, Faculty of Sciences and Supporting Studies, University of Hafr Al-Batin, Hafr Al-Batin, Saudi Arabia [b] A. Abdel-Hady Department of Physics, Faculty of Engineering, El-Asher University, El-Asher City, Egypt [c] I. Nasser Department of Physics, King Fahd University of Petroleum and Minerals, Dhahran 31261, Saudi Arabia E-mail: imnasser@kfupm.edu.sa or ibraheemmnasser@gmail.com Authors’ contributions: This work was performed in equal collaboration between all authors. Author Mohammed S. Abdelmonem suggested the study, and all others (Mohammed S. Abdelmonem, Afaf Abdel-Hady, and Ibraheem Nasser) shared the responsibilities for the calculation and analy- sis, and discussed the results. Ibraheem Nasser wrote the first draft of the manuscript. Authors Mohammed S. Abdelmonem and Afaf Abdel-Hady reviewed the existing literature, managed the literature searches, and placed the research objectives of the article in perspective. All authors managed the analysis of the study, read, edited, and approved the final revised manuscript. Contract grant sponsor: KFUPM. V C 2015 Wiley Periodicals, Inc. International Journal of Quantum Chemistry 2016, 116, 897–907 897 FULL PAPER WWW.Q-CHEM.ORG