Quantum treatment of Hénon–Heiles systems using oblique coordinates

Abstract

In this work we investigate the possibility of using oblique coordinates to determine the energy levels and wave functions of Hénon–Heiles coupled oscillator systems. Oblique coordinates are constructed by making a non-orthogonal linear transformation which permits the expression of the matrix representation of the second-order Hamiltonian operator of the system in a block-diagonal form. They are especially indicated for the treatment of Hénon–Heiles systems with two dissociative channels, which are representative of the stretching vibrational motions of symmetrical triatomic molecules. The ability of oblique coordinates to facilitate the solution of the vibrational Schrödinger equation versus normal coordinates and orthogonal rotated coordinates is analyzed by carrying out linear variational calculations with anharmonic basis function sets. It is shown that oblique coordinates require a considerably smaller number of basis functions than normal and rotated coordinates to variationally converge all the bound energy levels of the Hénon–Heiles potential with two dissociative channels to a given accuracy.