Circumventing the Heisenberg principle: A rigorous demonstration of filter‐diagonalization on a LiCN model

In a previous paper [J. Chem. Phys. 93, 2611 (1990)] a new method, filter diagonalization, was introduced for extracting highly excited rovibrational states from an arbitrary Hamiltonian, in any desired energy range. In the method, an arbitrary initial wave packet is propagated for a short time and during the propagation a ‘‘short time filter’’ of the wave packet is accumulated at various energies in any desired ‘‘window,’’ yielding a small set of functions which span the eigenfunctions of the Hamiltonian in the desired range. A small Hamiltonian matrix is then evaluated in the filtered‐functions basis, to yield the eigenvalues in the desired range. The combination of the time‐dependent (TD) propagation with the small matrix diagonalization eliminates the uncertainty‐relation limitation associated with a pure TD approach and the large‐matrix diagonalization necessary in a purely time‐independent approach. In this paper we give the first demonstration of the power of filter diagonalization for a molecular Hamiltonian exhibiting accidental near degeneracies, thereby supplying a stringent test of our approach. A two‐dimensional model of LiCN (J=0) is used. Good agreement is established with previous results for high‐energy states. For a further check of consistency, we perform a large‐scale direct diagonalization of the Hamiltonian, and verify a very‐high accuracy even for nearly degenerate levels. Extraction of these levels by a purely TD approach would have necessitated ≊700‐fold increase in propagation time.