Refinement Statistics and R Factors

These numbers are illustrative

Nothing was submitted, so the boxes hold a made-up set that hangs together — a Mo Kα data set, an anisotropic model, no restraints. They are not a real structure and no compound is being described. Paste your own numbers over them.

What was measured

The highest angle measured sets the finest detail the data can carry: d_min = λ / (2 sin θ_max), which is Bragg’s law rearranged. It is the finest detail a Fourier sum over those reflections can put into the electron density — a limit on the MAP. It is not a minimum distance between atoms. A refined coordinate is routinely located to a small fraction of d_min, because the refinement adds atom shapes and chemical knowledge the raw Fourier sum does not have. What data stopping early costs is detail in the map, and with it the number of parameters the refinement can support — not the precision of any one of them, and not the R factor, which can look excellent on coarse data. Where 0.6 Å⁻¹ comes from: that is the sin θ / λ a full data set is expected to reach for publication — d = 0.83 Å, which is θ = 25.24° with Mo Kα and 67.68° with Cu Kα, which is why a copper diffractometer has to swing so much further for the same data.

Completeness is what fraction of the reflections that COULD have been measured to that angle actually were. It is the other half of the resolution statement: data taken to a fine d but missing a wedge of reciprocal space carry systematic gaps rather than a uniform loss of detail.

R_int compares reflections that symmetry says must be equal, and it is computed from those equivalents as they are merged: once they have been averaged into one value the differences it measures are no longer there to compare. It is therefore a statement about the MEASUREMENT — counting statistics, absorption, crystal quality, how well the correction worked — and not about the model, which has not been built yet when it is computed. Read it beside the R factor rather than as a bound on it: the two are computed over different observations, on different quantities, with different weights, so neither is a floor or a ceiling for the other.

How well the model fits

R₁ and wR₂ are not the same quantity measured two ways. R₁ is computed on F, unweighted, and quoted for the reflections a paper calls observed, usually I > 2σ(I). wR₂ is computed on F², weighted, over ALL data including the weak reflections that carry the least information and the largest relative error. Squaring the quantity and keeping the weak data both push it up, which is why the two are quoted together and why wR₂ is the larger. The definitions: R₁ = Σ‖F_o| − |F_c‖ / Σ|F_o|; wR₂ = $\sqrt{\sum w{({F_{\text{o}}}^2-{F_{\text{c}}}^2)}^2/\sum w{\left({F_{\text{o}}}^2\right)}^2}$; S = $\sqrt{\sum w{({F_{\text{o}}}^2-{F_{\text{c}}}^2)}^2/(n-p)}$. The weight w is the refinement’s own, and SHELXL’s usual form is 1 / [σ²(F_o²) + (aP)² + bP] with P = [max(F_o², 0) + 2F_c²] / 3 — the scheme a CIF prints in _refine_ls_weighting_details, and the one the CIF report reads back.

The goodness of fit S compares the residuals against the uncertainties the weighting scheme assigned them, over the n − p degrees of freedom. Its expected value is 1 when those uncertainties are right, so a value far from 1 is first of all a statement about the WEIGHTS rather than about the structure — which is why refinement programs that optimise the weighting scheme drive it towards 1 without any atom moving.

What stands behind it

Every refined parameter has to be paid for with observations. The ratio of independent reflections to parameters is how much data stands behind each number in the model, and it is the reason a structure refined anisotropically needs data a coarser one does not: six displacement parameters per atom instead of one. Restraints are counted separately here because they are not observations — they are chemical knowledge added to the refinement, which is a different kind of support and is worth seeing separately. What counts as enough: checkCIF asks for at least 10 reflections per parameter for a centrosymmetric structure and 8 for a non-centrosymmetric one (PLAT088), so a ratio below those is a question to answer rather than a fault. A paper’s “data / restraints / parameters” line is three counts, and the ratio is its first over its third, as here. Restraints join the observations in one place only: SHELXL’s restrained goodness of fit, whose denominator is n − p + r for r restraints.

What is left over

The difference map is what the model does not account for. Its extremes are quoted as a peak and a hole, in electrons per cubic ångström, and WHERE they sit decides what they mean: features at bonding distances from a heavy atom are usually series termination and absorption, a peak at a chemically sensible position is a missing atom or a disorder component, and a peak on top of an atom is a scattering power the model has got wrong. A number alone cannot tell those apart, which is why this page prints it and does not grade it. The site’s difference map page shows what each of those looks like.

What these numbers do not say

These numbers describe how well a model accounts for a measurement. None of them says the chemistry is right: a structure with an excellent R factor can still have the wrong element at a site, a mis-assigned hydrogen, or a molecule that is chemically impossible. That is what the geometry is for — see distances and angles and displacement parameters.