On the numerical solution of the multidimensional vibrational time‐independent Schroedinger equation

Abstract

A preliminary study of the capability of the finite difference and finite element methods (FDM, FEM) to evaluate eigenvalues of one-, two-, and three-dimensional self-adjoint operators is reported with reference to applications dealing with the description of vibrational levels. Results of harmonic oscillator model potentials and ab initio PES for the water molecule are obtained by using the FDM. In spite of the large matrices used, low accuracy, nonvariational results are found. A different method, based on FEM and normal coordinates, is therefore proposed. Two nearly harmonic cases are studied and it is shown that variational results with higher accuracy can be obtained with a moderate cost. The vibrational levels of the water molecule are also calculated in order to compare the results with those of the FDM treatment.

Solving the vibrational Schrödinger equation on an arbitrary multidimensional potential energy surface by the finite element method

“…Our results are shown in Table 1 along with previous results calculated by FDM [60] and FEM [61] methods. The element volumes are specified by the shape type and number of nodes; for example, HEX27 is a hexahedral basis function with 27 nodes.…”

The vibrational structure of H+4 and D+4

“…I͒ a pure stretching 1D harmonic model is a very poor representation of the kinetic energy operator T d , since large 3D movements of the H d nucleus are involved. An internal coordinate [33][34][35][36][37][38][39][40][41][42][43][44][45][46][47] 3D model seems to be a better representation of T d .…”

“…Also, it may be observed that the distances R ab , R ac , and R ad are maintained to first order and that the two bending movements become equivalent when h ϭ0, the triatomic case. 45 The Jacobian matrix J r ͓Eq. ͑4͔͒ can be easily obtained from Eqs.…”