Expansion in Lorentzian functions of spectra of quantum autocorrelations

Abstract

We show that in a quantum mechanical many-body system of Boltzmann particles having space inversion symmetry the spectrum of the autocorrelation function of a local observable can always be given, similarly to the classical case [Phys. Rev. E 85, 022102 (2012)], in terms of a series of Lorentzian functions multiplied by the proper quantum detailed balance factor. This is done by transforming the continued fraction representation, which is derived via recurrent relations and without the use of the generalized Langevin equation hierarchy, into a series expansion. In this way characteristic frequencies can be defined, also in quantum mechanics, which refer to the particular autocorrelation. These are the frequencies of the eigenmodes of the relaxation function connected to the observable. We also show that in practical cases of interest in experimental spectroscopy, and particularly in inelastic neutron and x-ray scattering, the use of a finite number of Lorentzian shapes for an approximate description of the data is related to a reduction of the number of the relevant dynamical variables taken into account, equivalent to the lowering of the dimensionality of the orthogonalized space onto which the dynamic of the system is projected. Examples of application are given for the spectrum of the velocity autocorrelation function of liquid parahydrogen, calculated with a quantum simulation algorithm (path-integral centroid molecular dynamics), and for the molecular center-of mass dynamic structure factor in liquid carbon dioxide as computed by means of classical molecular dynamics simulation.