Stable highly excited vibrational eigenvalues without the variational principle

The energy spectrum in the stochastic region for a model Hamiltonian of two strongly coupled modes is calculated by diagonalizing small matrices which do not provide the Hylleraas–Undheim–MacDonald variational energy upper bounds. In addition, a method for selecting the most important basis functions by artificial intelligence algorithms is utilized. The energy convergence is determined by the Hazi–Taylor stabilization method and by the nearest-neighbor-spacing distribution function which measures the local fluctuations in the spectrum.