Refinement Statistics and R Factors
The table of crystal data at the end of a paper, read one number at a time: what each quantity is a statement about, which of them are not comparable with each other, and what the set as a whole cannot tell you. Type in the numbers a paper printed and leave the rest empty.
- You supply
- Any subset of the numbers a structure determination reports — wavelength and the highest θ measured, completeness, Rint, R1, wR2, the goodness of fit, the reflection, parameter and restraint counts, the difference-map extremes, and the Flack parameter with its standard uncertainty. Every box is optional and nothing is assumed for an empty one.
- Reading it
- This page does not grade a structure. It prints the arithmetic that follows from the numbers — the resolution from Bragg’s law, reflections per parameter, wR2 / R1 — and says what each quantity measures. It quotes no acceptable range for any of them, because what counts as good depends on the compound, the data and the discipline, not on a number this site could check.
Worked examples: a Flack parameter that decides nothing · data that stop early · a refinement held together by restraints
See also: Friedif Calculator · Difference Map · CIF Parser · Displacement Parameters and NPD Atoms
Input
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
| Resolution dmin | 0.7696 Å |
|---|---|
| sin θ / λ | 0.6497 Å−1 |
| Completeness | 99.6 % |
| Rint | 0.0412 |
The highest angle measured sets the finest detail the data can carry: dmin = λ / (2 sin θmax), which is Bragg’s law rearranged. Two features closer together than that are not resolved by the measurement, however well the model fits — so a structure refined against data stopping early is a coarser statement about the atoms, whatever its R factor says.
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.
Rint compares reflections that symmetry says must be equal, after merging. 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. A structure cannot fit its data better than the data agree with themselves.
How well the model fits
| R1 | 0.0389 |
|---|---|
| wR2 | 0.0951 |
| wR2 / R1 | 2.44 |
| Goodness of fit S | 1.043 |
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 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
| Reflections per parameter | 15.9 |
|---|---|
| Restraints | 0 |
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 is left over
| Largest peak | 0.31 e Å−3 |
|---|---|
| Deepest hole | -0.28 e Å−3 |
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.
Where this comes from
- Reporting and evaluating absolute-structure and absolute-configuration determinations
H. D. Flack, G. Bernardinelli, J. Appl. Cryst. 2000, 33, 1143–1148 · doi:10.1107/S0021889800007184
The criteria for when a Flack parameter has actually determined an absolute structure, and how the answer differs for a material already known to be enantiopure. This page deliberately quotes no threshold of its own: the condition it depends on is not something a table of numbers can be asked, and a figure this site cannot check is one a reader would quote anyway.