Fig. 2: Comparison of Na⁺-ion transport properties in NaTaCl₆ and NaNbCl₆ derived from BVSE and AIMD simulations.

a Calculated Na⁺-ion isosurface (pink) in NaTaCl₆ by BVSE method, displayed in a 2a × 2b × 1c supercell. Blue polyhedra represent [TaCl₆] units. The green and yellow dash lines indicate the Na1-i1 and i1-i2 migration pathways, respectively. b, c Na⁺-ions migration barriers calculated using BVSE method for (b) NaTaCl₆ and (c) NaNbCl₆. The energy of the Na1 site is set as the reference (zero). The green and yellow segments correspond to the migration barriers along the Na1–i1 and i1–i2 paths, as illustrated in (a). d, e Na⁺-ion probability density depicted as pink isosurfaces, extracted from AIMD simulations conducted at 600 K for 200 ps in (d) NaTaCl₆ and (e) NaNbCl₆, demonstrating 3D Na⁺-ion diffusion pathways involving the Na1 site and two intermediate sites. Blue and green polyhedra represent [TaCl₆] and [NbCl₆] units, respectively. Red dashed circles highlight regions with notable differences in isosurface connectivity. f The lnD vs. 1000/T relation for NaTaCl₆ and NaNbCl₆ derived from AIMD simulations. Blue and yellow shaded areas correspond to high- and low-temperature regimes, respectively, indicating similar diffusivities at elevated temperatures and a pronounced divergence in Na⁺-ion mobility at lower temperatures. Error bars denote the relative standard deviation calculated from the mean squared displacement (MSD) based on statistical analysis. Source data for figure are provided as a Source Data file.