Fig. 3: 2D and 3D probability density distributions and the Helmholtz free energy surface of the chlorides of the [TaCl₆]/[NbCl₆] polyanions.

a, b Shown for (a) NaTaCl₆ and (b) NaNbCl₆ are schematic representations of M-Cl bond orientations and probability distributions of [MCl₆] octahedra. The central blue and green spheres in the left panel are Ta and Nb atoms, respectively, while the bonded light gray spheres denote Cl atoms. The angle between the M-Cl bond and the z axis is defined as θ, and φ is the angle between the x axis and the projection of the M-Cl vector in the xy plane. The probability density isosurfaces depict the spatial distributions of Ta/Nb (blue/green) and Cl (light yellow) atoms within a [MCl₆] octahedron. c, d Shown for (c) NaTaCl₆ and (d) NaNbCl₆ are the 2D probability distributions of Cl atoms coordinated to Ta and Nb, respectively. The color bar on the right indicates the probability density values. e, f Shown in (e) NaTaCl₆ and (f) NaNbCl₆ are the Helmholtz free energy surfaces as a function of angle θ and φ. The Helmholtz free energy A was computed as A(θ, φ) = –k_BTln[ρ(θ, φ)], where k_B is the Boltzmann constant, T is temperature, ρ(θ, φ) is the probability density distribution of the Cl ligands of [TaCl₆]/[NbCl₆] anions from the AIMD simulations. Source data for figure are provided as a Source Data file.