Vibrational properties and bonding nature of Sb2Se3 and their implications for chalcogenide materials† †Electronic supplementary information (ESI) available: Additional computational data and discussion. See DOI: 10.1039/c5sc00825e Click here for additional data file.

There is more to chemical bonding in chalcogenides than the shortest, strongest bonds, as revealed by microscopic quantum-chemical descriptors.


Supplementary methods
The data reported in the main text have been obtained in the local density approximation (LDA), not only due to its conceptual simplicity but also because the LDA has previously shown excellent performance in reproducing experimentally determined on-site force constants in the chemically related Sb2Te3. S1 Nonetheless, the results reported in the paper must be independent of one particular DFT method and be reproducible at other, higher levels of theory. We show in this ESI document that this is indeed the case.
For this purpose, supplementary computations were performed using the VASP package as described in the main text, with the following methods (and otherwise comparable computational parameters):  The generalised gradient approximation (GGA)-that is, the second rung of Perdew's "ladder" of DFT functionals S2 -using, in particular, o the Perdew-Burke-Ernzerhof (PBE) functional; S3 o the PBE functional revised for solids (PBEsol); S4 o the Armiento-Mattsson (AM05) functional. S5  Dispersion corrections to GGA, which have been seen to be important for layered tellurides and might also improve the description of Sb2Se3 (see also the discussion in

Ref. S6):
o the "D3" scheme of Grimme and co-workers, S7 which is a pairwise a posteriori correction added to energies (and forces), both in the initially proposed zerodamping scheme (D3 in the following) and o using Becke-Johnson damping (BJ in the following); S8 o furthermore, the vdW-DF2 method of Langreth, Lundqvist, and co-workers (abbreviated as "DF2" in the following). S9  Finally, the meta-GGA functional after Tao, Perdew, Staroverov, and Scuseria (TPSS) was employed to explore the effect of a "third-rung" functional (that is, one exceeding LDA and GGA in formal and methodological scope). S10 ESI for Chemical Science S3 DFT methods beyond the LDA Optimised lattice parameters. The structure of Sb2Se3 contains a range of "weak" interatomic contacts whose description is notoriously nontrivial for traditional DFT methods. S11 We thus start by assessing the computed lattice parameters at different levels of theory; the results (for cells fully optimised at the respective computational level) are provided in Figure S1 below.
Not surprisingly, the cell volume in particular is underestimated by LDA, and overestimated by the PBE functional; some of the methods can alleviate this, but there is no clear "failure" of the LDA among its competitors. It is also interesting to note that along the b-axis, the deviations are mainly very small-this is the direction in which the covalently bonded chains extend.
Along the a-axis, where only weak stacking interactions occur, the deviations are generally more pronounced. Finally, we stress that the aim of these data is not to benchmark DFT methods or judge their merit (note, e.g., that several other formulations of "vdW-DF"-type functionals exist, S9 some of which will likely provide better lattice parameters). Figure S1. Quality of structural descriptions as judged from the computed lattice parameters and their deviation from experiment (single-crystal X-ray diffraction; data from Ref. S12).

S4
Atomic positions. In addition to the computed lattice parameters, it is worthwhile to quantify the description of the individual atoms' positions; this gives more detailed insight into how well interatomic distances are reproduced at the different levels of theory. For this purpose, it has been suggested to inspect the root mean square (rms) deviation of experimental and computed Cartesian coordinates; S13 later, this definition has been extended by George et al. S14 and decomposed according to the different spatial directions, which seems particularly useful for anisotropic structures. By definition, the y component (rmsy) equals zero because all atoms reside on Wyckoff sites with y = ¼ or y = ¾. The results for the other two spatial directions (which are most important as the "weak" contacts occur in these) are provided in Figure S2. Again, certain improvements are possible, but there is not a clear failure of LDA, and we note that a minor deviation remains even with the best methods. Figure S2. Quality of structural descriptions as measured by the directionally resolved and overall root mean square deviation of Cartesian coordinates. S14 Bond-projected force constants with different DFT methods. Among the key results of this study is a significant contrast in bond-projected force constants: they are large for the "classically" covalent bonds, but very low for the medium-range contacts. One might argue that more sophisticated formulations are required to capture properties of the latter, too, and hence we have repeated the phonon computations described in the main text at all levels of theory. The results-separated, for clarity, into two panels-are given below (Figures S3 and S4), and they unambigously show that the conclusions obtained in the LDA hold also at the other DFT levels investigated.

Figure S3
(as supplement to Figure 7 of the main text). Bond-projected force constants, computed with different GGA functionals. The LDA data used in the main text are given by red triangles for comparison.
ESI for Chemical Science S6 Figure S4 (as supplement to Figure 7 of the main text). Bond-projected force constants, computed with different dispersion-corrected DFT methods and at the meta-GGA (TPSS) level. The LDA data used in the main text are given by red triangles for comparison.

Figure S5
(as supplement to Figure 5 of the main text). Computed phonon band structure along the more comprehensive reciprocal-space pathway that had also been used for the electronic bands in Figure 2. No imaginary contributions are seen over the entire range.

Figure S6
(as supplement to Figure 5 of the main text). Computed partial density of phonon states, resolved according to symmetry-inequivalent antimony (top) and selenium (bottom) atoms. Note the pronounced contribution of both Sb(2) and Se(1) at around 200 cm -1 , corresponding to a stretching vibration of the shortest covalent bond. The large partial density of states of Se(1) at ≈ 120 cm -1 also seems an interesting target for experimental investigation. The vibrations projected on the antimony atoms (top panel), which lie in the "interior" of the chains, are more evenly distributed over the range of wavenumbers.

Born effective charges.
To better understand the bonding nature of the compound, we computed Born effective charges, using density-functional perturbation theory (DFPT) S15 as implemented in VASP (settings: LEPSILON = .TRUE., LRPA = .FALSE.). S16 The analysis of Born effective charges also in comparison to experiment has recently proven a valuable tool to elucidate the nature of bulk and nanoscale oxides and heavier chalcogenides. S17 Here, doing so is especially interesting as it allows for comparison with recent results for Sb2S3. S18 The computed Born effective charges are listed in Table S1, and throughout, they are slightly larger than those of the lighter sulphide compound. S18 Table S1 (as supplement to Figure 5

S10
Force constants and Badger's relation. Here, we provide raw data for the bond-projected force constants as discussed in the main text, and also supply additional data to further justify the hypothesis of low force constants in the "weak" Sb-Se bonds. According to Badger, the force constants  and bond distances d in molecules obey S19 where d0 is an element-specific constant. Plotting the cubic root of  -1 hence yields a linear relationship if Badger's relation is satisfied. Doing so for the dataset obtained here for Sb2Se3 reveals that the short bonds within the 1D chains appear to follow a Badger-like relation, whereas the weaker bonds deviate because their force constants are lower ( Figure S7). Table S2 (as supplement to Figure 7 of the main text). Bond-projected force constants.

S12
The results provided so far have been obtained for structural fragments from the optimised crystal structure. However, subsequent optimisation of the 1D models is also an option: this alters the structural details compared to the bulk network, but it can provide additional insight regarding the dynamic stability. Results for these fully re-optimised models are given in Figure   S9: again, a dynamic instability is observed for the single chain at all three levels of theory.

Figure S9
(as supplement to Figure 6 of the main text). Phonon band structures as before, but for fully re-optimised models at the respective level of theory. The double wire bridged via longer bonds relaxed into a distinctly different, but ultimately unstable structure; it is hence not considered here. In the double wire characterised in panel (b), on the contrary, the character of the "interchain" contacts remained similar, changing only from 2.977 Å to 2.960 Å (LDA), for example.
ESI for Chemical Science S13 Structural raw data Listing S1. Optimised structure of Sb2Se3 at the LDA-DFT level. Lattice vectors and coordinates are provided in VASP POSCAR format for convenience.