The quantum resonance spectrum of the H3+ molecular ion for J=0. An accurate calculation using filter diagonalization

Abstract

Three-dimensional variational calculations of bound and resonance spectra of the H 3 + molecular ion for the total angular momentum J = 0 are presented. All N E + N A1 + N A2 = 2 × 604 + 358 + 256 bound states of the MBB PES and some N E + N A1 + N A2 = 2 × 264 + 142 + 119 resonances with widths Γ < 100 cm −1 are computed. Such calculations are very challenging due to (i) a very deep potential-energy well which gives rise to a high density of bound and resonance states and (ii) the floppiness of the H 3 + molecular ion which leads to various numerical difficulties. The computation was done using the filter diagonalization method. Here a series of subspace diagonalizations is performed on a Hamiltonian with a complex absorbing potential, Ĥ − iŴ, represented in a relatively small adapted basis set. Each small basis set is obtained by acting with the ABC (absorbing boundary conditions) Green's function on an initial wavepacket, using the modified by ABC Chebyshev recursion polynomial expansion. The method is described in some detail. Additionally, we describe an efficient and simple procedure of using Jacobi coordinate DVR (discrete variable representation) for a floppy triatomic molecule resulting in a sparse Hamiltonian matrix with a small spectral range, which in particular, overcomes the slow convergence of iterative Krylov type methods. Comparison between the present version of the filter diagonalization and the Lanczos method is made. The level statistics of the H 3 + bound and resonance states are analysed showing the strongly chaotic nature of the spectra, starting already at very moderate energies, caused by the extreme floppiness of the system. The widths of computed resonances fluctuate by orders of magnitude. At energies just above the dissociation threshold the resonances are essentially very narrow and isolated, but start to overlap very quickly as energy increases. The number of resonance states per effective decay channel is sufficient to carry out a statistical analysis of decay rates by fitting their distribution to the χ 2 distribution with ν eff degrees of freedom, where ν eff is the effective number of decay channels. The result of this analysis is however negative, suggesting that the concept of average width on which it is partially based is not correct when resonances are overlapping, and that more reliable statistical theories of the effective Hamiltonian must be developed for the barrierless reactions.