Relaxed structure containing the muon (shown in Fig. S1(a)) can be found in 'Re_octahedral.cif'. Here the muon is represented by a 
hydrogen atom.

Displacements of the Re ions as a function of distance from the muon site (shown in Fig. S1(b)) can be found in 'displacements.dat'.
First column is distance of the ion from the muon site, second column is radial displacement. Both are measured in angstroms.

Density of states for the Re atoms closest to the muon site without including the muon can be found in Re_nomuon.pdos.dat. The first column
gives the energy and the subsequent columns give the density of states for each of the four nearest-neighbour Re atoms in states per eV. 
This data was used to create Fig. S1(c).

The corresponding density of states obtained when including the muon (Fig. S1(d)) can be found in Re_oct.pdos.dat. Columns 2-7 contain the DOS
for the six Re atoms forming the coordination octhedron of the muon and column 8 contains the muon DOS.