Maximum entropy solution to the Stieltjes moment problem in chemical physics

Abstract

In the inverse Stieltjes moment problem, one seeks to reconstruct a non-negative distribution from its spectral moments defined on an unbounded interval 〈0,∞. In chemical physics, this problem arises when computing continuous quantities such as photoionization cross sections or electronic decay widths using discretized approximations to the electronic continuum. While Stieltjes imaging (SI) is the established method in this context, it provides only sparse, discrete sampling of the distribution. Here, we develop a maximum entropy (ME) approach to the solution of the inverse Stieltjes moment problem in the context of Fano theory of resonances, where the sought-after quantity is the decay width function. We implement two ME variants—with polynomial and exponential asymptotic damping—and introduce an averaging procedure over spectral moment orders that addresses convergence issues and provides reliable error estimates. Benchmarking against ab initio Fano-ADC data for molecular Auger decay and interatomic Coulombic decay, we show that ME achieves comparable accuracy to SI while providing a continuous representation of the function. Our results establish ME as a valuable alternative to SI, particularly when analytical continuation or additional verification is required.