High energy electron diffraction of molecules in the gas measures what? For diffraction of an ensemble of molecules under suitable conditions, one can often read, that what one measures is a ensemble/thermal average of inverse distances, between scattering centers (atoms) of the molecules. Using the distances $r$ between these scattering centres the resulting experimentally measured parameters can then be expressed as $<r^{-1}>^{-1}$ when $<>$ denotes the averaging.
I would be interested in a concise explanation/derivation or at least some reference where I can check that.
 A: The question is in the context of Gas Electron Diffraction. The energy of the electrons is chosen such that each molecule of the gas diffracts the electrons. Since the molecules are randomly oriented, the situation is similar to that of powder crystallography: one measures only one angle of diffraction $\theta$. Precisely, the experimental output is the intensity $I(s)$ for a range of $s=4\pi(\sin\theta/2)/\lambda$, where $\lambda$ is the wavelength of the electrons. This is then compared to a theoretical prediction [2] (my go-to reference for gas electron diffraction: it is old but as far as the theory goes, it is still relevant),
$$I(s)\propto \sum_{ij}f_i(s)f_j(s)\frac{\sin sr_{ij}}{sr_{ij}},$$
where the indices $i$ and $j$ label the atoms of a molecule: $r_{ij}$ is then the distance between atom $i$ and $j$ whereas $f_i(s)$ models the diffraction by atom $i$. 
The curve of $I(s)$ will have peaks at the values of $s$ for which $\sin(sr_{ij})/sr_{ij}$ is maximum. It is a classic result that those are $r_{ij}^{-1}$ times the solutions of $\tan x = x$. Thus, it can be said that by locating one peak in the measured curve $I(s)$, one effectively measures one $r_{ij}^{-1}$. In practice, peaks may overlap for large-ish molecules and one resorts to least-squares fit of the model to the measured values to find the $r_{ij}$'s but since the data point near the maxima discussed above will dominate, we reach the same conclusion, blurred by the statistical machinery, but the same nevertheless: we effectively measure the inverse of the intra-molecular atomic distances.
Final point: the above model is too crude as it neglects intra-molecular vibrations: the electrons take a frozen snapshot of the $r_{ij}$'s for each molecule and they differ slightly from one molecule to the next. Thus we can only determine an average of the values of $1/r_{ij}^{-1}$, which is what the equation (1) in the paper you cited is about [1].
[1] Derek A. Wann, Alexander V. Zakharov, Anthony M. Reilly, Philip D. McCaffrey, and David W. H. Rankin. Experimental equilibrium structures: Application of molecular dynamics simula- tions to vibrational corrections for gas electron diffraction. The Journal of Physical Chemistry A, 113(34):9511–9520, 2009.
[2] L. O. Brockway. Electron diffraction by gas molecules. Reviews of Modern Physics, 8(3):231–266, 1936.
