Validation of experimental charge-density refinement strategies: when do we overfit?

Abstract

A cross-validation method is supplied to judge between various strategies in multipole refinement procedures. Its application enables straightforward detection of whether the refinement of additional parameters leads to an improvement in the model or an overfitting of the given data. For all tested data sets it was possible to prove that the multipole parameters of atoms in comparable chemical environments should be constrained to be identical. In an automated approach, this method additionally delivers parameter distributions of k different refinements. These distributions can be used for further error diagnostics, e.g. to detect erroneously defined parameters or incorrectly determined reflections. Visualization tools show the variation in the parameters. These different refinements also provide rough estimates for the standard deviation of topological parameters.

New Insights into the Catalytic Activity of Cobalt Orthophosphate Co3(PO4)2from Charge Density Analysis

“…Even with full refinement of all feasible multipole parameters from theoretical data, it was impossible to resolve them. We refrained from that endeavor because it clearly indicated overfitting . Local mirror symmetry had to be taken into account.…”

“…We refrained from that endeavor because it clearly indicated overfitting. [22] Local mirror symmetry had to be taken into account. Additionally,t he multipole population at all oxygen atoms had to be set to be the same.…”

Is Hexachloro‐cyclo‐triphosphazene Aromatic? Evidence from Experimental Charge Density Analysis

“…[30] Some of the atoms showed anharmonic motion [31] and were refined using the Gram-Charlier coefficients up to third order.F inally,t he resulting models were validated to exclude overfitting. [32] Crystal data for 1:C l 6 N 3 P 3 ,M= 347.64 gmol À1 ,o rthorhombic space group Pnma, a = 13.8710(1), b = 12.8320(1), c = 6.0870(1) , V = 1083.44(2) 3 , Z = 4, (sinq max /l) = 1.163 À1 ,c ompleteness = 99.9 %, 1 calcd = 2.131 Mg·m À3 , m = 1.978 mm À1 ,1 44 450 reflections measured, 7304 unique, R 1 (I > 0) = 0.0088, wR 2 (I > 0) = 0.0189, GoF = 1.718, wGoF = 1.018, residual density from À0.139 to 0.182 e· À1 after multipole refinement. Crystal data for 2a:C l 8 N 4 P 4 ,M = 463.52 gmol À1 ,t etragonal space group P4 2 /n, a = b = 15.1621(6), c = 5.8946(2) , V = 1355.11(12) 3 , Z = 4, (sinq max /l) = 1.111 À1 ,c ompleteness = 99.9 %, 1 calcd = 2.272 Mg·m À3 , m = 2.108 mm À1 ,3 7307 reflections measured, 3803 unique, R 1 (I > 0) = 0.0114, wR 2 (I > 0) = 0.0237, GoF = 1.429, wGoF = 0.975, residual density from À0.188 to 0.188 e· À3 after multipole refinement.…”

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

“…The uncertainties obtained may also not account totally for all systematic errors present in the data measurements. Krause et al (2017) recently presented a method based on R free calculations. Sample standard deviations computed on the relevant models refined on subsets of the measured reflections (for example, 20 subsets of 95% reflections) can yield a rough estimate of the standard deviation on topological properties of the electron density.…”