The calculations of vibration-rotation bound states and new metastable states of a diatomic beryllium molecule important for laser spectroscopy are presented. The problem is solved using the potential curve and the authors' software package that implements the iteration Newton method and the high-accuracy finite element method. The efficiency of the proposed approach is demonstrated by calculating vibration-rotation bound states and, for the first time, rotation-vibration metastable states with complex- valued energy eigenvalues (with negative imaginary parts of the order of (10-20 ÷ 6) cm-1) in a diatomic beryllium molecule. The existence of these metastable states is confirmed by calculating the corresponding scattering states with real-values resonance energies.
The computational scheme and calculation results of bound, metastable and Rydberg states of atomic and molecular systems important for laser spectroscopy are presented. The solution to the problem is performed using the authors' software package (see program libraries of the Computer Physics Communications journal and of the Joint Institute for Nuclear Research) that implement the high-accuracy finite element method. The FORTRAN procedure of matching tabulated potential functions with van der Waals asymptotic potential using interpolation Hermite polynomials which provides continuity of both the function itself and its derivative is presented. The efficiency of the proposed approach is demonstrated by calculated for the first time sharp metastable states with complex eigen-energies in a diatomic beryllium molecule and weakly bound Rydberg states of antiprotonic helium atom.
The eigenvalue problem for second-order ordinary differential equation (SOODE) in a finite interval with the boundary conditions of the first, second and third kind is formulated. A computational scheme of the finite element method (FEM) is presented that allows the solution of the eigenvalue problem for a SOODE with the known potential function using the programs ODPEVP and KANTBP 4M that implement FEM in the Fortran and Maple, respectively. Numerical analysis of the solution using the KANTBP 4M program is performed for the SOODE exactly solvable eigenvalue problem. The discrete energy eigenvalues and eigenfunctions are analyzed for vibrational-rotational states of the diatomic beryllium molecule solving the eigenvalue problem for the SOODE numerically with the table-valued potential function approximated by interpolation Lagrange and Hermite polynomials and its asymptotic expansion for large values of the independent variable specified as Fortran function. The efficacy of the programs is demonstrated by the calculations of twelve eigenenergies of vibrational bound states with the required accuracy, in comparison with those known from literature, and the vibrational-rotational spectrum of the diatomic beryllium molecule.