Research Article: A method to estimate statistical errors of properties derived from charge-density modelling

Date Published: May 01, 2018

Publisher: International Union of Crystallography

Author(s): Bertrand Fournier, Benoît Guillot, Claude Lecomte, Eduardo C. Escudero-Adán, Christian Jelsch.


Errors on molecular properties including the topology of electron density and electrostatics are estimated from a sample of deviating models generated using the variance–covariance matrix issued at the end of the charge-density refinement.

Partial Text

Errors on electron-density-derived properties, such as topological characteristics or electrostatic potential, are generally poorly addressed in the relevant literature. To the best of our knowledge, no available computer software designed for charge-density analysis on the basis of multipolar modelling computes properly analytical standard deviations on electron-density-derived properties. For instance, in the XD2006 program (Volkov et al., 2006 ▸), there is a feature that allows one to compute estimated uncertainties of the electron density ρ(r), of the Laplacian ∇2ρ and of dipole moment values using the variance–covariance matrix, but it only accounts for the contributions of some of the parameters used in the Hansen & Coppens (1978 ▸) model, i.e. monopole and multipole populations. It implies that the propagation of errors due to the contributions of the atomic coordinates and of the contraction/expansion coefficients κ and κ′ is not taken into account. This could lead, consequently, to an overall underestimation of standard deviations on electron-density-derived properties.

At the convergence of a least-squares crystallographic refinement against diffraction data, the e.s.d.’s of the parameters used to model the molecular structure and electron density can be directly retrieved. However, the uncertainties on derived molecular properties are not readily available. To estimate the errors of properties, series of models at ‘standard deviation’ from the final refined model can be easily generated by using vectors of random numbers and a square root of the inverted normal matrix. The SSDs obtained for the properties derived from a sample of such deviating structures can be used as estimated values of their uncertainties. For instance, samples of 20 perturbed structures yield SSD values with an expected relative precision of 16%. The average value of properties P in the perturbed models appears to be generally within one SSD from the final refined value; in the case of topological integrated charges and electrostatic energies, it was, for instance, found that .




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