Vibrational eigenstates of NO2 by a Chebyshev-MINRES spectral filtering procedure

Abstract

It is shown that the minimum residual algorithm (MINRES) is able to generate spectral filters sharp enough to obtain bound vibrational eigenstates of NO2 (J=0) by direct access in the most dense part of the spectrum even for the worst near-degeneracy cases. The same is not true for filters constructed as an expansion of the spectral density operator via Chebyshev polynomials. The best performance is obtained in a progressively restarted scheme in which the sharpness of the filter is increased between subsequent restarts to accompany the refinement of the state. Best efficiency and most convenient handling is obtained with early restarts using the Chebyshev-filter, whereas the MINRES-filter is more efficient later in the filtering procedure, where sharp peaking of the filter action is necessary.

Investigation of Ergodic Character of Quantized Vibrational Motion

“…Quantum Mechanical Representation. After discussing two-dimensional model systems in the previous section, we now turn to a more realistic case in three dimensions, namely the high lying vibrational states of the NO 2 molecule in its adiabatic electronic groundstate as given by a slightly modified − version of the ab initio surface of Leonardi, Petrongolo, Hirsch, and Buenker . We restrict ourselves to the case of zero total angular momentum, J = 0, for which the Hamiltonian of a triatomic molecule in hyperspherical coordinates takes the following form 47,48 where L̂ 2 (θ,φ) is the grand angular operator μ is the reduced mass and M is the total mass of the molecule The volume element is given by These hyperspherical coordinates 47,48 are such that 0 ≤ θ ≤ π, 0 ≤ φ ≤ 2π, and for the present potential the ρ range 3.0 bohr ≤ ρ ≤ 6.8 bohr is suitable when we limit our calculations to bound states.…”

“…The basis used is N ρ × N θ × N φ = 168 × 115 × 105 = 2 028 600, which means that a direct diagonalization is out of the question. Since we will only need a few states near dissociation, we use instead the same method as in ref , i.e., a restarted spectral filtering method, where repeated application of the Hamiltonian is used to generate a sharp energy envelope (the spectral filter). By restarting the sequence with sharper and sharper filters, a pure state may be obtained.…”