M. S. Abdelmonem

The rotating Morse potential model for diatomic molecules in the tridiagonal J-matrix representation: I. Bound states

This is the first in a series of articles in which we study the rotating Morse potential model for diatomic molecules in the tridiagonal J-matrix representation. Here, we compute the bound-state energy spectrum by diagonalizing the finite-dimensional Hamiltonian matrix of H2, LiH, HCl and CO molecules for arbitrary angular momentum. The calculation was performed using the J-matrix basis that supports a tridiagonal matrix representation for the reference Hamiltonian. Our results for these diatomic molecules have been compared with available numerical data satisfactorily. The proposed method is handy, very efficient, and it enhances accuracy by combining analytic power with a convergent and stable numerical technique.

The analytic inversion of any finite symmetric tridiagonal matrix

We use the theory of orthogonal polynomials to write down explicit expressions for the polynomials of the first and second kind associated with a given infinite symmetric tridagonal matrix H. The Greens function is the inverse of the infinite symmetric tridiagonal matrix (H-zI). By calculating the inverse of the finite symmetric tridiagonal matrix we can find the analytical form of the inverse of the finite symmetric tridiagonal matrix, .

Taming the Yukawa potential singularity: improved evaluation of bound states and resonance energies

Using the tools of the J-matrix method, we absorb the 1/r singularity of the Yukawa potential in the reference Hamiltonian, which is handled analytically. The remaining part, which is bound and regular everywhere, is treated by an efficient numerical scheme in a suitable basis using Gauss quadrature approximation. Analysis of resonance energies and bound states spectrum is performed using the complex scaling method, where we show their trajectories in the complex energy plane and demonstrate the remarkable fact that bound states cross over into resonance states by varying the potential parameters.

Relativistic scattering with a spatially dependent effective mass in the Dirac equation

We formulate a relativistic algebraic method of scattering for systems with spatially dependent mass based on the J-matrix method. The reference Hamiltonian is the three-dimensional Dirac Hamiltonian but with a mass that is position-dependent with a constant asymptotic limit. Additionally, this effective mass distribution is locally represented in a finite dimensional function subspace. The spinor couples to spherically symmetric vector and pseudo scalar potentials that are short-range such that they are accurately represented by their matrix elements in the same finite dimensional subspace. We calculate the relativistic phase shift as a function of energy for a given configuration and study the effect of spatial variation of the mass on the energy resonance structure.

Analytical treatment of the Yukawa potential

Using a suitable Laguerre basis set that ensures a tridiagonal matrix representation of the reference Hamiltonian, we were able to evaluate exactly the matrix elements of the Yukawa potential in this representation. This enabled us to compute the bound state spectrum as the eigenvalues of the associated analytical matrix representing the full Hamiltonian. We also used the complex scaling method to evaluate the resonance energies and compared our results with those obtained using the Gauss quadrature approach and the corresponding results from the literature.

J-Matrix approach for the exponential-cosine-screened Coulomb potential

The bound state energies for the exponential-cosine-screened Coulomb potential were calculated by using the Gauss quadrature method. The resonance energies were calculated using the complex rotation method and were then used as a seed for our J-matrix approach in order to achieve improved results. The calculated bound and resonance state energies are compared with the available numerical data in this paper. The trajectories of the resonance states in the complex E-plane are also shown. New data, for both bound and resonance state energies close to the crossover region of the transition from bound to resonance, have been reported for guiding future studies.

Regularization in the J-matrix method of scattering revisited

In three dimensional scattering, the energy continuum wavefunction is obtained by utilizing two independent solutions of the reference wave equation. One of them is typically singular (usually, near the origin of configuration space). Both are asymptotically regular and sinusoidal with a phase difference (shift) that contains information about the scattering potential. Therefore, both solutions are essential for scattering calculations. Various regularization techniques were developed to handle the singular solution leading to different well-established scattering methods. To simplify the calculation the regularized solutions are usually constructed in a space that diagonalizes the reference Hamiltonian. In this work, we start by proposing solutions that are already regular. We write them as infinite series of square integrable basis functions that are compatible with the domain of the reference Hamiltonian. However, we relax the diagonal constraint on the representation by requiring that the basis supports an infinite tridiagonal matrix representation of the wave operator. The hope is that by relaxing this constraint on the solution space a larger freedom is achieved in regularization such that a natural choice emerges as a result. We find that one of the resulting two independent wavefunctions is, in fact, the regular solution of the reference problem. The other is uniquely regularized in the sense that it solves the reference wave equation only outside a dense region covering the singularity in configuration space. However, asymptotically it is identical to the irregular solution. We show that this natural and special regularization is equivalent to that already used in the J-matrix method of scattering.

The Manning–Rosen potential using J-matrix approach

Close to the bound-resonance crossover region, state energies for the Manning–Rosen (MR) potential have been calculated. New data, for both bound and resonance state energies have been reported nearby the crossover region, using the J-matrix approach, for LiH and CO molecules. Furthermore, the average oscillator strength, multipole moments and transition probabilities for certain states have been reported for future comparison.

Hellmann potential in the J-matrix approach: I. Eigenvalues

The trajectories of the poles of the S-matrix for Hellmann potential, in the complex energy plane, have been studied near the critical screening parameter. The calculation has been performed using the J-matrix approach, which uses a suitable L2 basis to tridiagonalize the reference Hamiltonian matrix. The calculated bound and resonance state energies have been compared with available numerical data.

The rotating Morse potential model for diatomic molecules in the J-matrix representation: II. The S-matrix approach

This is the second article in which we study the rotating Morse potential model for diatomic molecules using the tridiagonal J-matrix approach. Here, we further improve the accuracy of computing the bound states and resonance energies for this potential model from the poles of the S-matrix for arbitrary angular momentum. The calculation is performed using an infinite square integrable basis that supports a tridiagonal matrix representation for the reference Hamiltonian, which is included in the computations analytically without truncation. Our method has been applied to both the regular and inverted Morse potential with favourable results in comparison with available numerical data. We have also shown that the present method adds a few significant digits to the accuracy obtained from the finite dimensional approach (e.g. the complex rotation method). Moreover, it allows us to easily handle both analytic and non-analytic potentials as well as 1/r singular potentials.