Maximum Entropy charge density studies: Bayesian viewpoint and test applications

The Maximum Entropy (abbreviated MaxEnt) method has been used in the field of accurate charge density studies for some time now (see Section 2.2): it has the potential to overcome some of the limitations of traditional multipolar modelling, but great care must be taken not to apply it outside the range of validity of its own foundations. In this paper, after a brief discussion of the main sources of error affecting the present day implementation of multipolar and MaxEnt charge density studies (Sections 1.1 and 2.2), we present a rationale for the well-known drawbacks of the MaxEnt method as applied to charge density studies. In particular, we will show that the use of a uniform prior-prejudice distribution gives rise to artefacts when the dynamic range of the electron density to be reconstructed is large enough that the exponential modelling of the density requires non-negligible Lagrange multipliers past the resolution limit of the available diffraction data. The artefacts are not due to insufficient numerical precision, but to series termination effects in the Fourier series with Lagrange multipliers as coefficients. In the last section of the paper, we discuss a Bayesian approach to the treatment of experimental error variances, and its first limited implementation to obtain MaxEnt distributions from a fit to noisy data.