A NONDIRECT PRODUCT DISCRETE VARIABLE REPRESENTATION-LIKE METHOD FOR CALCULATING VIBRATIONAL SPECTRA OF POLYATOMIC MOLECULES

We present a new method for solving the vibrational Schroedinger equation for polyatomic molecules. It has the following advantages: 1) the size of the matrix eigenvalue problem is the size of the required pruned (nondirect product) polynomial-type basis; 2) it requires solving a regular, and not a generalized, symmetric matrix eigenvalue problem; 3) accurate results are obtained even if quadrature points and weights are not good enough to yield a nearly exact overlap matrix; 4) the potential matrix is diagonal; 5) the matrix-vector products required to compute eigenvalues and eigenvectors can be evaluated by doing sums sequentially, despite the fact that the basis is pruned. To achieve these advantages we use sets of nested Leja points and appropriate Leja quadrature weights and special hierarchical basis functions. Matrix-vector products are inexpensive because transformation matrices between the basis and the grid, and their inverses, are lower triangular. Vibrational energy levels of CH$_2$NH are calculated with the new method. For this purpose a simple harmonic oscillator kinetic energy operator and a quartic force field are used