Synthesis, crystal and electronic structure of the new sodium chain sulfido cobaltates(II), Na₃CoS₃ and Na₅[CoS₂]₂(Br)

2.1 Preparation and phase analysis

The title compounds were synthesized in corundum crucibles sealed in steel autoclaves under an argon atmosphere starting from cobalt powder (99%, Schuchardt München) and elemental sulfur (99%, Merck) with anhydrous Na₂S (95%, ABCR GmbH Karlsruhe ) and (for the initial sample) pure sodium (Merck) as Na sources. For the synthesis of the ‘double salts’, NaSH (ABCR) and NaBr (Merck) were used as additional starting materials. The sealed steel autoclaves were heated up to maximum temperatures (T_max) between 400 and 1100°C, held there for 4 h and subsequently cooled to r. t. with rates of 20 to 30 Kh⁻¹. After the preparations, representative parts of the reguli were ground and sealed in capillaries with a diameter of 0.3 mm. X-ray powder diagrams were collected on a transmission powder diffraction system (STADI-P, Dectris Mythen 1 K detector, Stoe & Cie, Darmstadt, Mo K_α radiation, graphite monochromator). For the phase analysis, the measured powder diagrams were compared to the calculated (program Lazy-Pulverix [18]) reflections of the title compounds and other known phases in the systems Na–Co–S–(Br/H). Rietveld refinements were performed using the programs Gsas [19] and Expgui [20].

Black-metallic shiny crystals of the new phase Na₃CoS₃ were first obtained from a sulfur-rich sample of overall composition Na_2.4CoS_6.6, which consisted of 86.8 mg (2.78 mmol) of elemental sodium, 92.6 mg (1.57 mmol) of cobalt and 332.9 (10.38 mmol) of sulfur. Due to the exothermic reaction, the sample container was slowly heated up to 200°C followed by rapid heating to 760°C. It was held at this temperature for 4 h and cooled to r. t. with a rate of 10 Kh⁻¹. Most reflections of the complex powder diagram could be indexed with the calculated data of Na₃CoS₃ (Tables 1 and 2). In addition, reflections of CoS₂ (which could be optically identified as octahedral crystals with a golden metallic lustre) and Na₆[CoS₄] (light-orange transparent crystals) are also found in the diagram. A further minor phase, which was identified using single crystal methods, are the metallic-greenish needles of the known compound Na₂[CoS₂]. Attempts to synthesize Na₃CoS₃ from samples of stoichiometric composition of the elements failed due to breaking of the corundum sample tubes, even with smaller amounts of the starting materials. The choice of Na₂S as an alternative sodium source, i.e. starting from Na₂S, Co and S in a 3:2:3 ratio, resulted only in inhomogeneous products when applying the previously used maximum temperature of 760°C. At an elevated T_max of 1100°C this protocol turned out to be very successful for the synthesis of Na₃CoS₃. Samples with an overall weight of approx. 1 g (e.g. 522.4 mg/6.69 mmol Na₂S, 263.0 mg/4.46 mmol Co and 214.6 mg/6.69 mmol S) yielded homogeneous reguli of the target compound, consisting of xenomorphic aggregates with dark-gray matt metallic gloss. Very small traces of Na₆[CoS₄] can be identified in the powder pattern as well as from an optical inspection of the products.

The same sodium source Na₂S (together with Co, S and NaBr) and T_max of 1100°C were used to obtain the bromide Na₅[CoS₂]₂(Br) in X-ray pure form. Samples consisting of, e.g. 353.2 mg/4.53 mmol Na₂S, 267.5 mg/4.54 mmol Co, 145.6 mg/4.54 mmol S and 233.3 mg/2.27 mmol NaBr yielded homogeneous reguli of black-metallic shiny needles, which became dark-brown transparent when broken into thin pieces. Attempts to obtain the OH⁻ or SH⁻ derivatives of Na₅[CoS₂]₂(Br) following this protocol and adding NaOH/NaSH instead of NaBr resulted in the formation of the hydrogen-free products Na₂[CoS₂] and Na₆[CoS₄] only. At a much lower T_max of 400°C, Na₅[CoS₂]₂(SH) could finally be obtained (together with Na₆[CoS₄] and Co₉S₈) from 5:2 mixtures of NaSH and elemental cobalt. Unfortunately, the hydrogensulfide was yielded in the form of dark-gray powder only. The Rietveld refinement of the powder data resulted in the lattice parameters a=913.41(8) and c=621.94(7) pm, which agree within 0.2% with the crystal data reported by Klepp and Bronger for Na₅[CoS₂]₂(S) [11]. In an argon atmosphere Na₅[CoS₂]₂(SH) decomposes at 600°C into Na₆[CoS₄] and Co₉S₈.

2.2 Crystal structure determination

For the single crystal structure analyses, dark-metallic crystals of Na₃CoS₃ and tiny needles of Na₅[CoS₂]₂(Br) were selected using a stereo microscope. The crystals, which are very sensitive to moisture and air, were fixed in glass capillaries (diameter <0.1 mm) under dried paraffine oil and centered on a diffractometer equipped with a microsource and a CCD area detector (Na₃CoS₃) or an image plate detector [Na₅[CoS₂]₂(Br)].

The reflections of the crystals of Na₃CoS₃ could be indexed using an orthorhombic C-centered lattice. The extra extinction conditions ‘reflections h0l only present for h and l=2 n’ and ‘reflections 00l only present for l=2 n’ restricted the possible space groups to Cmcm and Cmc2₁. The structure solution using Direct Methods (program Shelxs-2013 [23]) was successful in the noncentrosymmetric space group Cmc2₁ only. The solution already yielded the four cobalt, all sulfur and four of the seven sodium atomic sites. The positions of the three missing Na⁺ cations could be easily determined from the difference Fourier maps calculated after the first structure refinements (program Shelxl-2013 [23]). After standardization (program Structure Tidy [24]) the atomic parameters and anisotropic displacement parameters (ADP) could be refined to a final R1 value of 0.0205.

The diffraction images of Na₅[CoS₂]₂(Br) show a tetragonal pattern of high Laue symmetry. The reflections could be indexed by the expected tetragonal I-centered lattice and do not show any extra extinction conditions. The lattice parameter a virtually coincides with the value observed for Na₅[CoS₂]₂(S) (915.0 pm, [11]); the c parameter is only slightly (0.5%) increased compared to the sulfide (622.2 pm, [11]). Using the atomic parameters of the I4mm structure model of Na₅[CoS₂]₂(S) as starting values [11] and substituting the isolated sulfur position by bromine, the atomic and ADP parameters of Na₅[CoS₂]₂(Br) could be directly refined to a conclusively low R1 value of 0.0410 with the methods and programs described above. Whereas the absolute structure of Na₃CoS₃ could be reliably determined, all examined crystals of Na₅[CoS₂]₂(Br) needed to be refined as inversion twins.

The crystal data and the results of the anisotropic structure refinement of both title compounds are collected in the Tables 1 and 2 (see also [25]). Selected interatomic distances can be found in Tables 4 and 5 (for Na₃CoS₃) and in Table 6 [for Na₅[CoS₂]₂(Br)].

2.3 Band structure calculations

DFT calculations of the electronic band structure were performed for the title compounds Na₃CoS₃ and Na₅ [CoS₂]₂(Br) and the reference phase Na₂[CoS₂] (K₂ZnO₂-type structure, crystal data taken from [5]) using the FP-LAPW method (program Wien2k [26]). The exchange-correlation contribution was described by the generalized gradient approximation (GGA) of Perdew, Burke and Ernzerhof [27]. Muffin-tin radii were chosen as 111 pm (2.1 a.u.) for Na, Co and S, and 103 pm (1.95 a.u.) for the sulfur atoms of the disulfide groups in Na₃CoS₃. Cutoff energies used are Emaxpot=190 eV (potential) and Emaxwf=116 eV (interstitial PW). In the spin-polarized GGA+U calculation the additional Coulomb potential (Hubbard parameter U_eff) used for the strongly correlated Co d electrons was 2 eV [28], [29]. This value was chosen following the proven parameters, which were used for the calculation of other Fe and Co chalcogenides [28], [29]. In addition, the SIC (self-interaction correction) [30], [31] double counting correction method has been applied. Due to the magnetic ordering experimentally observed in several reference compounds, e.g. K₂[CoS₂] [5], an antiferromagnetic spin ordering was applied for the cobalt cations in the dimer and along the chains. For this AFM treatment of the Co spins, the crystal structures of Na₅[CoS₂]₂(Br) and Na₂[CoS₂] (space group Ibam) were transformed into the appropriate subgroups P4mm and I222 (cf. Section 3.4 for details). A Bader analysis of the electron density maps was performed to evaluate the charge distribution between the atoms [32]. Electron and spin densities were calculated and visualized using the programs XCrysDen [33] and DRAWxtl [34]. Further parameters and selected results of the calculations are summarized in Table 3. The total (tDOS) and selected partial Co and S density of states (pDOS) are depicted in Fig. 5.

3.1 Syntheses

The new chain sulfido cobaltate(II) Na₃CoS₃ was obtained in the course of a systematic synthetic study on the phase formation of sodium sulfido cobaltates, starting from mixtures of elemental Co and S together with elemental sodium or Na₂S as sodium sources. For the first time, the phase appeared in samples composed of the three pure elements in the sulfur-rich composition range of the ternary system Na–Co–S. Later, it could be obtained as a nearly X-ray pure phase in targeted syntheses from stoichiometric mixtures of Na₂S, cobalt and sulfur at a maximum temperature of 1100°C (cf. Experimental Section 2.1).

Due to the doubt about the exact chemical composition of the mixed valent Co(II/III) chain compound Na₅[CoS₂]₂(S) [11] and its potential SH⁻ content, the isotypic bromine Na₅[CoS₂]₂(Br) was successfully synthesized using a similar temperature program and the same sodium source (together with NaBr) to obtain crystals suitable for a single crystal structure analysis. At a decreased maximum temperature of 400°C the hydrogen sulfide Na₅[CoS₂]₂(SH) could also be obtained when substituting NaBr by NaSH, unfortunately only in the form of powdered material. In contrast to the hydrogen-free sulfido cobaltate, Na₅[CoS₂]₂(SH) decomposes at 600°C under an argon atmosphere, which is not inconsistent with the original synthesis of ‘Na₅[CoS₂]₂(S)’ at 730°C in a stream of hydrogen sulfide. The lattice parameters of Na₅[CoS₂]₂(SH) refined by the Rietveld method from powder data are in perfect agreement with those reported in the single crystal study of ‘Na₅[CoS₂]₂(S)’ by Klepp and Bronger [11].

3.2 Crystal structure description of Na₃CoS₃

According to Na₃CoS₃=Na₁₂[Co₂S₅(S₂)][Co₂S₃(S₂)] the orthorhombic structure of the first title compound contains two different types of sulfido/disulfido cobaltate anions (Fig. 1), which both exhibit m symmetry in occupying the mirror plane of the space group Cmc2₁ (Fig. 2b). The cobalt atoms Co(1) and Co(2) together with the sulfido and disulfido ligands S(1X) and S(2X) form chains [Co₂S₃(S₂)]⁴⁻ (Fig. 1b), whereas Co(3), Co(4), S(3 X) and S(4 X) form the dinuclear units [Co₂S₅(S₂)]⁸⁻ depicted in Fig. 1a. The sulfur atom labels are chosen to indicate the actually connected cobalt atom(s). The interatomic distances within the two anions are labeled using small bold characters and are collected in Table 4.

Fig. 1:

Ortep representations (90% probability ellipsoids [35]) of the two sulfido cobaltate anions in the crystal structure of Na₃CoS₃. (a) Dimers [Co₂^IIS₅ (η¹−S₂)]⁸⁻; (b) chains [Co₂^IIS₃ (µ−1,2−S₂)]⁴⁻ (cf. Table 4 for the labeled interatomic distances).

Fig. 2:

Different views of the crystal structure of Na₃CoS₃. (a) ‘Layers’ around the mirror plane (x≈0) perpendicular to the a axis of the space group Cmc2₁; (b) hexagonal rod packing along the c axis; (c) perspective view of the overall structure; (S: green balls; Co: red balls; Na: yellow balls; [CoS₄]: light-gray (anion a) and dark-gray (anion b) polyhedra, [34]).

1. In the anions [Co₂S₅(S₂)]⁸⁻ (Fig. 1a), two [Co(3,4)S₄] tetrahedra are connected via a common [S(34)₂] edge. Whereas Co(3) is additionally coordinated by the two terminating S(31) and S(32) ligands (Co–S distances g, i), Co(4) exhibits only one terminal sulfido ligand [S(41)] and the fourth tetrahedron corner is occupied by the η¹-disulfido ligand [S(42)–S(43)]²⁻. The S–S distance of dS(42) − S(43)s=213.4 pm within this S₂ dumbbell compares to other disulfido ligands (s. further discussion below). All Co(3)–S distances (g, h, i) are found in the expected narrow range 232.1 to 234.3 pm. In contrast, the terminal Co(4)–S(41) distance j is considerably decreased to 226.9 pm, an effect which is also observed in the chain anion for the Co(II) ions carrying disulfido ligands in addition (see below). As expected, the Co–S(42) bond l towards the η¹-S₂ dumbbell is significantly larger (239.5 pm) than the Co–S²⁻ bond lengths. Due to the edge-sharing and the kink in the double tetrahedra, the Co–Co distance w (282.2 pm) is somewhat shorter than the value in the simple linear chains of Na₂[CoS₂] (292.5 pm, [5]). To the best of our knowledge, such double tetrahedra with one η¹-disulfido ligand represent a new anion among sulfido metalates.

2. The chain anions [Co₂S₃(S₂)]⁴⁻ in the structure of Na₃CoS₃ (Fig. 1b) are formed by [Co(1,2)S₄] tetrahedra, which are alternately connected via common sulfido edges [S(12)₂] and corners [S(21)]. At these common corners, the tetrahedra are additionally bridged by the μ-1,2-disulfido ligands [S(22)−S(23)]²⁻. In an equivalent description, the [CoS₄] tetrahedra are alternatingly connected via sulfido [(μ-S)₂] and mixed sulfido/disulfido [(μ-S)(μ-S₂)] edges forming slightly folded Co₂S₂ four-membered and planar Co₂S₃ five-membered rings. Thus, the formula of the chain can be summarized after Niggli as ∞1[Co(S)3/2(S2)1/2]2−=  ∞1[Co2S3(S2)]4− and belongs to a general series of chain metalates [M(μ-S)_2−x(μ-S₂)_x]ⁿ⁻. Similar chains of [MS₄] tetrahedra, which are alternatingly bridged by [(S²⁻)₂] and [(S²⁻)(S₂²⁻)] edges, are also found in the gallates Cs₂Ga₂S₅ [36] and Cs₂Ga₂Se₅ [37] and in a series of chalcogenido gallates and indates with organic ammonium cations and metal complexes as countercations ([38] and references therein). The Co–(S²⁻) distances (a, b, d, e) in the chain anion [Co₂S₃(S₂)]⁴⁻ of Na₃CoS₃ are in the narrow range from 226.9 to 229.5 pm (Table 4). As already detailed above for the dinuclear anion, these values are considerably shorter than Co–S contacts in simple chain sulfido cobaltates(II), evidently due to the further coordination by disulfido ligands. The Co–(S₂)²⁻ distances c and f of 229.9 and 233.2 pm are again somewhat larger than those involving simple sulfide ions. The Co–Co distances dCo(1) − Co(2)u=266.3 pm between the centers of tetrahedra sharing sulfido edges are much shorter than in the simple chain compounds A₂[CoS₂] and in Na₅[CoS₂]₂(Br) or Na₅[CoS₂]₂(SH) (Section 3.3). Compared to layered cobaltate anions like those found in the structure of KCo₂S₂ (d_Co−Co=263.7 pm [6], [9]) the observed Co^II–Co^II distances are again within the already known limits.

Compared to free disulfide ions (e.g. in Na₂S₂), where d_S−S amounts to 215.8 pm [39], the S–S bond lengths decrease on increasing bonding of the S₂²⁻ ligands: In the η¹-ligand of the dinuclear anion (a) the bond length (s) is reduced to 213.4 pm. In the bridging disulfide ligand of the chain anion (b) the S–S distance (r) is further decreased to 210.8 pm. This characteristic feature of dichalcogenide ligands is sufficiently documented in the literature [38], [40]. It indicates an increased π donation of S₂²⁻, which leads to a slight depopulation of the π^* states of the ligand and additionally causes the – compared to monosulfido metalates – decreased band gaps of corresponding disulfido metalate salts (cf. Section 3.4).

Fig. 2a shows a projection of the orthorhombic unit cell of Na₃CoS₃ along the a axis with both anions located at the mirror plane at x=0. The chains [Co₂S₃(S₂)] (b) are aligned along [001], i.e. along the direction of the 2₁ screw axis, at y≈0. The double tetrahedra (a) are lined up in the same direction, whereby their two [CoS₄] tetrahedra match four [CoS₄] tetrahedra of the chain. Along [010], two such columns (at y≈13 and 23) are arranged between the chains, which finally leads to an overall 1:1 ratio of dinuclear [Co₂^IIS₅ (S₂)]⁸⁻ and chain segments [Co₂^IIS₃(S₂)]⁴⁻ (=Na₁₂Co₄S₁₂=4×Na₃CoS₃) in each layer. According to the C-centered lattice, the adjacent layers (at x≈±12) are shifted by 12 in the a and b direction, which becomes obvious in the view of the structure along the chain direction [001] shown in Fig. 2b. This projection also illustrates the pseudo-hexagonal rod packing, which is a characteristic feature of all sulfido metalates containing chains of edge-sharing tetrahedra [41], [42], and also for cutouts of such chains, e.g. the tetraferrates like (Rb/Cs)₈ [Fe₄S₁₀] [12], [13]. Each chain anion (dark-gray polyhedra in Fig. 2) is surrounded by six rods of aligned double tetrahedra (light-gray polyhedra). In contrast to the distribution of the countercations in the simple chain metalates, the Na⁺ ions in Na₃CoS₃ are not homogeneously interspersed among the chains/aligned dimers: With the only exception of Na(1) and Na(2), which occupy m/4 a positions, the sodium cations are arranged in puckered layers (x=0.22 – 0.30) between the Co/S ‘layers’ around the mirror planes (cf. projection of the structure in Fig. 2b, Na(1) and Na(2) are hidden by the anions). The seven crystallographically distinct cations in the structure of Na₃CoS₃ are coordinated by five to eight sulfur atoms (Table 5); the belonging arrangements are depicted in Fig. 3. The shortest Na–S distance of 269.6 pm is slightly shorter than the sum of Shannon’s ionic radii of 286 pm (for CN=6, [43]). The effective coordination numbers (ECoN) of the sodium cations are within the typical range observed in sodium salts: For Na(1) and Na(2) located at the mirror plane the ECoN values are with 4.31/4.88 somewhat reduced compared to the coordination numbers of the remaining Na⁺ ions (5.09 to 6.95, Table 5).

Fig. 3:

Sodium coordination spheres in the crystal structure of Na₃CoS₃ (thick/thin red lines: distances below/above 310 pm, [34]). (a) Na(1) and Na(2); (b) Na(3) to Na(5); (c) Na(6) and Na(7).

3.3 Crystal structure description of Na₅[CoS₂]₂ (X) (X=Br⁻, SH⁻)

The second title compound, Na₅[CoS₂]₂(Br), is a ‘double salt’ containing linear SiS₂-analogous chains and isolated bromide ions. Figure 4 shows the two anions (a, b) and a perspective view of the unit cell (c). In the tetragonal acentric structure, sulfido cobaltate chains [CoS_4/2]²⁻ of edge-sharing tetrahedra, comparable to those in the common chain compounds of the K₂ZnO₂-type structure, like e.g. Na₂[CoS₂], are running along the tetragonal c axis at 0, 12,z. The Br⁻ ions are located in between these chains. They are surrounded by five Na⁺ cations forming square pyramids (point group symmetry 4mm, blue polyhedra in Fig. 4c). The pyramids are equally oriented along [001], with one [BrNa₅] pyramid per two [CoS₄] tetrahedra of the cobaltate chains. The structure of Na₅[CoS₂]₂(Br) is isotypic to the reported mixed-valent phase ‘Na₅[CoS₂](S)’ ([11], see Discussion below). Both compounds formally belong to the Na₆PbO₅-type structure, with the unusual assignment (Na₆PbO₅≡Na₅Co₂S₄Br) according to Pb≡Br (2 a) and O(1)≡Na(2) (2 a), but O(2)≡S (8 d), Na(2)≡Co (4 b) and Na(1)≡Na(1) (8 c).

Fig. 4:

Crystal structure of Na₅[CoS₂]₂(Br). (a, b: Ortep representations with 90% probability ellipsoids [35]; cf. Table 6 for the interatomic distances; c: unit cell: green balls: S; red balls: Co; yellow balls: Na, [34]).

The sulfido cobaltate chain in the tetragonal structure of Na₅[CoS₂]₂(Br) is fully comparable to the similar chain anion present in the orthorhombic phase Na₂[CoS₂] [5]: The Co–S distances of 233.6 (a) and 234.3 pm (b, Table 6) (Co site symmetry: 2mm/C_2v) are well comparable to the values found for Na₂[CoS₂] (d_Co−S=232.8 pm; 222/D_2h). In contrast, Na₅[CoS₂]₂(SH) exhibits two significantly different Co–S distances along the chains (d_Co−S=226.3 and 240.4 pm); the mean value of 233.4 pm nevertheless coincides with the Co–S distances in the common chain sulfido cobaltates(II). Apparently, the underlying shift of the S²⁻ ligand enables the formation of S–H···S contacts between the SH⁻ groups and the surrounding sulfido ligands of the chain through the trianglar faces of the square pyramid (d_S−S=418.3 pm). Due to the incorporation of the extra Br⁻/SH⁻ ions in between the cobaltate anions, the Co–Co distances of 312.8/311.1 pm in the linear chain of Na₅[CoS₂]₂(Br) are significantly enlarged compared to the values in Na₂[CoS₂] (d_Co−Co=292.5 pm, [5]). Characteristic for trans edge-connected tetrahedra, the ∠_S−Co−S of the two connecting edges (95.9/96.3° (X=Br⁻) and 92.6/100.3° (SH⁻) are much smaller than the four of the non-connecting tetrahedra edges (≈116°, 4×). In agreement with the increased Co–Co distances, this stretching of the tetrahedra along the 4̅ axis is even more pronounced in the ‘double salts’ than in the reference structure of Na₂[CoS₂] [102.2° (2×) and 110.1/116.4° (4×)].

The extra anions X⁻ are coordinated by a square pyramid of sodium cations with X–Na distances of 291.1 and 300.6 pm for the bromide and the hydrogensulfide, respectively. A further Na⁺ ion is located above the square face of the pyramid at a distance of 334.5 pm (Fig. 4b). The Br⁻–Na⁺ and SH⁻–Na⁺ distances as well as the coordination numbers of X⁻ are thus in good agreement with the values found in the (for X=SH⁻: distorted) rocksalt structures of NaSH (299.2 pm) and NaBr (298.1 pm). The sum of the Shannon radii [43] of 299 pm (for Na⁺/Br⁻) are also similar.

The sodium coordination numbers in both X-containing compounds of 5 (4 S²⁻+1 Br⁻) for Na(1) and 5+2 (4 S²⁻+(1+1) Br⁻) for Na(2) (Table 6) also fit the values in simple binary sodium salts, as well as in Na₂[CoS₂] and Na₃CoS₃. The Na–X distances (d_Na−S=289−333 pm, average value: 301 pm) perfectly compare to the distances observed in the structure of NaBr (298.1 pm) and NaSH (299.2 pm). In contrast (and despite the enlarged coordination number of 8) the Na⁺–S²⁻ distance in the sulfide Na₂S (anti-CaF₂-type structure) is much smaller (283 pm). This provides a further argument for the presence of SH⁻ in ‘Na₅[CoS₂]₂(S)’. Unfortunately, the detection of the SH⁻ vibrational mode by means of IR or Raman spectroscopy failed due to metallic lustre of the compound.

The molar volumes of Na₅[CoS₂]₂(SH) and Na₅[CoS₂]₂(Br) are practically equal, which agrees with the likewise similar molar volumes of NaSH (52.47×10⁶ pm³) und NaBr (52.98×10⁶pm³). For both compounds, the ‘double salt’ character leads to a nearly perfect match of the observed molar volumes (V_obs) with the values calculated from the doubled volume of Na₂[CoS₂] and the volume of NaBr/NaSH (X=SH⁻: 261.5/260.5; X=Br⁻: 261.6/262.0×10⁶ pm³ for V_calc/ V_obs).

3.4 Electronic structure and chemical bonding

The band structures of the title compounds and of the more simple reference chain sulfido cobaltate Na₂[CoS₂] were calculated using FP-LAPW DFT methods (GGA+ U) including spin-polarization and a suitable Hubbard U parameter. Computational details can be found in the Experimental Section 2.3 and in Table 3. For the orthorhombic structure of Na₂[CoS₂] a symmetry reduction from space group Ibam to I222 mimics the magnetic space group Ibam’, which was experimentally verified for the potassium compound K₂[CoS₂] by a Rietveld refinement of neutron powder diffraction data [5]. For the tetragonal structure of Na₅[CoS₂]₂(Br) a symmetry lowering from space group I4mm to P4mm was applied to enable the antiferromagnetic ordering of the Co spins along the chains. In the case of Na₃CoS₃ the implementation of the AFM ordering of Co spins within the dinuclear unit as well as along the chain (which are not yet experimentally verified) is possible without a symmetry reduction [Co(1)/Co(3): ↑ and Co(2)/Co(4): ↓].

The total (tDOS, gray shaded) and selected partial (pDOS) Co and S density of states of Na₂[CoS₂], Na₅[CoS₂]₂(Br) and Na₃CoS₃ are depicted in Fig. 5. Consistently, the tDOS of the three sodium salts show definite band gaps and a valence band region of approx. 5 eV formed by Co- d and S- p states. The conduction band is equally determined by the single-occupied Co-d_xy/d_yz/d_xz states of the upper Hubbard band. Thus, in the ZSA scheme [44] all compounds belong to the class 2B of ‘charge transfer semiconductors’. In all compounds, the Bader charges (q) and volumes (V_BB) as well as the heights of the bond critical points at the Co–S bonds (ρ_BCP) are similar: The sodium and cobalt cations exhibit the expected positive charges (Na⁺: q=+0.82 – +0.86; V_BB=9.7 – 10.4 e⁻×10⁻⁶ pm⁻³; Co²⁺: q=+0.72 – +0.79; V_BB=12.8 – 14.3 e⁻×10⁻⁶ pm⁻³, Table 3). The sulfur charges differ only for Na₃CoS₃ with its mixed sulfide/disulfide ligands (see below). For all compounds, the values of ρ_BCP for the Co–S bonds are found in a narrow range (ρ_BCP=0.41 – 0.53 e⁻×10⁻⁶ pm⁻³) and steadily decrease with increasing bond lengths (Table 3). They are somewhat smaller than the values calculated for sulfido ferrates(III) (e.g. Cs[FeS₂]: 0.57 – 0.61 e⁻×10⁻⁶ pm⁻³), which is evidently only an effect of the reduced ionic radius of Fe(III) compared to Co(II). These values also agree with those found for pyrite in an experimental electron density study (ρ_BCP=0.53 e⁻×10⁻⁶ pm⁻³ [45]). The Laplacian of the electron density at the BCPs (∇²ρ_BCP), which is discussed as an additional indicator for the character of the bonding, is also very comparable in the cobaltates (Na₂[CoS₂]: +3.63 e⁻×10⁻¹⁰ pm⁻⁵) and the ferrates(III) (Cs[FeS₂]: +3.55–3.80 e⁻×10⁻¹⁰ pm⁻⁵). They are significantly smaller than the experimental as well as the calculated values of ∇²ρ_BCP in pyrite (Fe(II) in an octahedral coordination, exp.: 6.20 e⁻×10⁻¹⁰ pm⁻⁵ [45]) indicating an increased covalency for the tetrahedral coordination.

Fig. 5:

Calculated total density of states (tDOS, gray) together with the Co and S partial DOS (colored lines) of Na₂[CoS₂] (a), Na₅[CoS₂]₂(Br) (b) and Na₃CoS₃ (d) in the range between −4.9 and 3.8 eV relative to E_F. (c): schematic representation of the different Co d state energies; possible superexchange path; calculated spin density map in Na₂[CoS₂] at a ±0.01 e⁻×10⁻⁶pm⁻³ level.

Na₂[CoS₂] contains 19 valence electrons per formula unit (v.e./f.u.), excluding the sulfur 2 s states [2 (2×Na) +9 (Co) +8 (2×S)]. Due to the AFM ordering, the total DOS of the α (^↑) and β (^↓) spins are equal, but the splitting of the Co sites allows to assign the individual d states of the Co^IId⁷ ions. For Na₂[CoS₂], the z direction of the local cartesian coordinate system of Co points along the chain direction, x and y are also oriented between the ligands (cf. scheme in Fig. 5, top right). In the calculation of Na₅[CoS₂]₂(Br) x and y are rotated by 45 °. This difference was taken into account for by the coloring of the different d states in the pDOS plots at the left hand side of Fig. 5. As expected, Na₂[CoS₂] exhibits a distinct band gap of 1.4 eV and the 19 occupied valence states (Z=2), which spread from −4.7 eV to the Fermi level E_F, are formed by S²⁻- p and Co^II- d⁷ orbitals. The position of the individual d states of Co was used to extract the simplified scheme of Fig. 5 (top right). The ^↑t₂ Co- d orbitals (Mulliken symbols as for an ideal tetrahedron, green/blue/cyan) together with the two a₁ S- p ligand SALCs are located at the bottom of the valence band region. This region represents the σ-bonding contribution between S p and the Co d orbitals d_xy/d_yz/d_xz pointing towards the ligands, which are also responsible for the strong antiferromagnetic coupling of the Co spins along the chain (see below). The broad region between −3.7 and −1.7 eV, where the Co- d^↑ states of e symmetry (red/orange) and six S- p orbitals coincide, represents the mainly π-bonding contribution between Co dx2 − y2/dz2 and the ligand p orbitals both of ideal symmetry t₂. The upper region of the valence band is formed by the remaining four sulfur orbitals (non/π^*-bonding) and the e set of the ^↓d-Co states of the upper Hubbard band. The relative energy of the three narrow empty ^↓t bands of the upper Hubbard band determines the band gap of the compound. Its value of 1.42 eV fits the dark-metallic color of the crystals of Na₂[CoS₂].

The occupation of the different d states is reflected in the electron and spin densities in the vicinity of the Co atom. The v. e. density map exhibits the octahedral shape of the fully occupied superimposed e-type orbitals, i.e. the lobes are situated between the Co–S bonds. In contrast, the spin density map is of the ‘cubic’ shape resulting from the single-occupied superimposed t₂ atomic orbitals and it thus extends towards the ligands. In Fig. 5, top right, the spin density map is shown at the ±0.01 e⁻×10⁻⁶ pm⁻³ level. In accordance with the comparably long Co–Co distance, the distinct spin density at the ligands clearly shows that the magnetic ordering is due to a superexchange via the sulfur atoms. A simple explanation (e.g. applying the GKA rules [46], [47]) for an AFM ordering via a 90° d^t- p- d^t superexchange is not possible for tetrahedral d⁷ ions like Co(II) [48]. The scheme of possible σ bonding interactions of the relevant d_xy (cyan) and the d_xy+d_yz (blue, green) combination of Co show, that only a p_z-type ligand orbital is able to mediate the AFM exchange for this configuration. Although the pDOS exhibits a preferred admixture of S p_z into the singly occupied Co d states and the sign of the spin density fits a simple p_z orbital, the calculated spin distribution exhibits distinct lobes along the Co–S bonds (and thus looks like two p_z+p_x/y ‘orbitals’.)

The calculated magnetic moment of Co (2.33 μ_B) in Na₂[CoS₂] is in good agreement with the experimental value of the isotypic potassium compound K₂[CoS₂] (2.5 μ_B, [5]) and is somewhat smaller than the expected ‘spin only’ value of 3 μ_B. A reduced magnetic moment of the 3 d metal is observed in most of the sulfido metalates containing [MS₄] tetrahedra and was attributed to covalency effects, in comparison with oxides or compounds containing an octahedral coordination of M. In contrast to the complex situation in the d⁵ ferrates(III), where the reduction of the magnetic moment is pronounced and depends on the connectivity of the [FeS₄] tetrahedra [49], the Co^II- d⁷ ions in sulfido cobaltates(II) exhibit a very similar magnetic moment [3], [8], [50].

In accordance with the crystal-chemically shown ‘double salt’ character of Na₅[CoS₂]₂(Br), the electronic structure (Fig. 5, bottom left) is directly comparable to Na₂[CoS₂] (but note the rotation of x/y and Z=4). The additional pDOS of the Br⁻- p ions (6 v. e., magenta line) is situated between −1.5 and −3.5 eV. The slight increase of the band gap to 1.6 eV on formal addition of the salt NaBr to Na₂[CoS₂] is reasonable and also becomes apparent from the slight transpacency of the crystals (even regarding the DFT intrinsic underestimation of band gaps).

The DOS of the complex structure of Na₃CoS₃ (Fig. 5, bottom right) is by far too complicated for an in-depth interpretation. While the Bader charges and volumes of the seven Na⁺ and the four Co²⁺ sites of Na₃CoS₃ are very similar and also comparable to those of the simpler sodium cobaltates, the differences for the S atoms arise from their different bonding situation: Sulfur atoms of bridging ligands exhibit smaller Bader volumes than those of terminal ligands. As expected, the negative charge of the pure sulfido ligand’s atoms (q=−1.16 to −1.38) is larger than the charge of the disulfide atom (q=−0.68 to −0.83). The latter values again agree with the experimentally observed charge of this anion in FeS₂ (ρ_BCP=0.83 e⁻×10⁻⁶ pm⁻³, [45]). The band gap of Na₃CoS₃ is slightly reduced with respect to the pure sulfido cobaltates(II) (1.14 eV). A similar trend has been observed in several main group metalates containing dichalcogenide ligands; it was traced to Q-π contributions of these ligands to the top of the valence band [38].