Pseudospectral method of solution of the Schrödinger equation for the Kratzer and pseudoharmonic potentials with nonclassical polynomials and applications to realistic diatom potentials

Abstract

The pseudoharmonic and Kratzer potentials have been extensively employed by numerous workers to model the vibrational states of diatomic molecules. These potentials belong to SUperSYmmetric (SUSY) quantum mechanics and the eigenvalues of the Schrödinger equation are known. The energy eigenvalues of the Schrödinger equation for the pseudoharmonic and Kratzer potentials have been determined with different numerical methods as a benchmark of the numerical schemes. We employ a pseudospectral method based on a quadrature grid defined with a nonclassical polynomial basis set. This basis set is defined orthogonal with respect to the square of ground state wavefunction as the weight function. A discrete matrix representation of the Hamiltonian of dimension N is constructed and vibrational energies are calculated with the numerical diagonalization of this matrix. This pseudospectral method is employed to calculate the vibrational energy levels of H2, CO and NO modelled with the pseudoharmonic and Kratzer potentials in comparison with the more realistic Morse potential. The vibrational energy eigenstates for H2 with a realistic quantum mechanical potential and an approximate Morse potential are also calculated. The oversimplified pseudoharmonic and Kratzer potentials are useful for benchmarking numerical methods. However, they represent potential energy curves that can deviate drastically from more exact potentials such as the Morse potential.