Error calculation

Once a best-fit structure has been determined, it is useful to see how strongly small changes to specific parameters affect the \(R\) factor. This can be obtained via a so-called “error calculation”. In the error calculation, displacements are applied to one parameter at a time, and the \(R\) factor is determined for each step of the variation range, resulting in an “error curve”.

When using the Pendry \(R\) factor, its standard error, historically called \(\mathrm{var}(R_\mathrm{P})\),[1] provides an estimate of the uncertainty of a given parameter: The point of intersection between the error curve for one parameter and \(R_{\mathrm{P,min}} + \mathrm{var}(R_\mathrm{P,min})\) can be taken as a measure of the statistical error of the parameter itself [5]. This gives the “error calculation” its name. \(\mathrm{var}(R_\mathrm{P})\) is defined as

\[\mathrm{var}(R_\mathrm{P}) = R_\mathrm{P} \sqrt{ \frac{8 |V_{0\mathrm{i}}| }{ \delta E} },\]

where \(R_\mathrm{P}\) is the value of the Pendry \(R\) factor, \(V_{0\mathrm{i}}\) the imaginary part of the inner potential, and \(\delta E\) the total energy range of the beam set (i.e., the sum of the energy ranges for all the symmetry-inequivalent beams). For more information, see Ref. 21.

Note

Note that, when multiple parameters are varied separately, they are assumed to be statistically independent from one another. This is not always the case. For example, if a site can be occupied by different chemical elements, the site occupation (i.e., the element concentrations) and vibration amplitude can be strongly correlated [22]. In such a case, the increase of the \(R\) factor when changing only one of these parameters does not provide a good estimate for the accuracy of that parameter.

Aside from the estimate of the accuracy of a displacement parameter, error curves can be used to judge the impact of a certain displacement on the overall \(R\) factor. While estimates on the parameter accuracy only apply to the Pendry \(R\) factor, the impact of a parameter on the \(R\) factor is also meaningful for other \(R\)-factor definitions.

• You may find the \(R\) factor to be very insensitive to the displacement of some atoms (i.e., much less sensitive than for other atoms with a similar depth and similar scattering properties). This is an indication that the atom is absent, or that its position is far away from the correct one. In this case, consider increasing the displacement range for this atom. Note also that what is “far away” depends on how strongly the atom scatters (i.e., on the chemical species and on the depth within the solid), and in some cases may be as small as 0.1 Å, e.g., for displacements along the surface-normal, \(z\) direction of a surface atom.

• Hydrogen is a very weak electron scatterer: the \(R\) factor depends only weakly on its position.

Attention

The \(R\) factor values obtained in the error calculation contain errors from the tensor-LEED approximation.

Error calculation in ViPErLEED

To run the error calculation, set RUN = 5 in the PARAMETERS file. It is recommended to first run a reference calculation (e.g., via RUN = 1 5), as the error curves may not be centered otherwise.

Aside from the PARAMETERS file, the required input is the same as for running a Delta-amplitudes calculation and a Structure search: structural input files (POSCAR, and, optionally, VIBROCC), experimental beams, a set of Tensors from a reference calculation, and a DISPLACEMENTS file defining which parameters should be varied. If multiple parameters are linked (e.g., by symmetry), they are treated as one and varied together.

Note that defining multiple sections in the DISPLACEMENTS file, as is possible for the search, is not allowed here. Only one section of the DISPLACEMENTS file is read: the last one if the error calculation is run following a Structure search, the first one otherwise. Defining geometric, vibrational, and occupation variations all in the same DISPLACEMENTS file is allowed, but the different variations are split up, so the result is the same as executing multiple error calculations. This means that you cannot have simultaneous error calculations for multiple geometric-displacement directions (e.g., \(x\) and \(z\)) for the same atom, since this would require multiple consecutive blocks in the DISPLACEMENTS file.

Tip

Simultaneous geometric or vibrational variations of multiple chemical elements occupying the same site is possible, and the displacement values for the different elements may differ. However, all displacement ranges must have the same number of steps.

The error calculation does not require a set of Delta files, since the normal delta-calculation routines mix geometric and vibrational displacements. Instead, the error calculation runs the required delta calculations automatically, splitting the geometric and vibrational variations into separate delta files to reduce computational cost.

The results of the error calculation consists of the Errors_summary.csv, Errors.zip, and Errors.pdf files, stored in the OUT directory.

If the Pendry \(R\) factor is used, the value of \(R_{\mathrm{P,min}} + \mathrm{var}(R_\mathrm{P,min})\) is calculated for each error type and drawn as a horizontal line in Errors.pdf. If ViPErLEED finds an intersection between the error curve for any parameter \(p\) and the line \(R_{\mathrm{P,min}} + \mathrm{var}(R_\mathrm{P,min})\), the intersection points are used to calculate the (estimate of the) statistical uncertainty for \(p\). This is written to Errors_summary.csv and plotted in Errors.pdf.

[1]

It is a common mistake to refer to \(\mathrm{var}(R_\mathrm{P})\) as the “variance” of the \(R\) factor. This is wrong, as, from a dimensional point of view, it does not correspond to a variance (it is proportional to \(R_\mathrm{P}\), not to its square). If it were a variance, it would not make sense to calculate \(R_{\mathrm{P,min}} + \mathrm{var}(R_\mathrm{P,min})\). The original “\(\mathrm{var}\)” abbreviation was probably intended as a contraction for “variation” [21].