Maximum entropy solution to the Stieltjes moment problem in chemical physics

Area of Science:

• Chemical Physics
• Computational Chemistry
• Quantum Mechanics

Background:

• The inverse Stieltjes moment problem involves reconstructing distributions from spectral moments.
• Discretized approximations in chemical physics require methods to compute continuous quantities.
• Stieltjes imaging (SI) is a standard method but yields discrete data.

Purpose of the Study:

• Develop a maximum entropy (ME) approach for the inverse Stieltjes moment problem.
• Apply ME to reconstruct decay width functions within Fano theory.
• Provide a continuous alternative to SI for spectral distribution reconstruction.

Main Methods:

• Implemented two variants of maximum entropy (ME) with polynomial and exponential damping.
• Introduced an averaging procedure over spectral moment orders to improve convergence and error estimation.
• Benchmarked ME against ab initio Fano-ADC data for molecular Auger and interatomic Coulombic decay.

Main Results:

• The ME method achieves accuracy comparable to Stieltjes imaging (SI).
• ME provides a continuous representation of the spectral distribution, unlike SI's sparse sampling.
• Averaging procedures enhance convergence and yield reliable error estimates for ME.

Conclusions:

• Maximum entropy (ME) is a valuable alternative to Stieltjes imaging (SI) for the inverse Stieltjes moment problem.
• ME offers a continuous representation of spectral distributions, crucial for analytical continuation and verification.
• The developed ME approach is effective for computing quantities like photoionization cross sections and electronic decay widths.