2.1 Synthesis

Single crystals of Nd_12.4Mg_0.6Mo₁₃O₃₆ were obtained in an attempt to synthesize a phase MgNd₁₆Mo₂₀O₅₆ isostructural with Nd₁₆Mo₂₁O₅₆ [24]. Starting reagents were La₂O₃ (Rhône-Poulenc, 99.999%), MoO₃ (Strem Chemicals, 99.9%) and Mo (Cimebocuze, 99.9%), all in powder form. The rare-earth oxide was prefired at 1000 °C before use and the Mo powder was heated under a hydrogen flow at 1000 °C for 6 h. The stoichiometric mixture was pressed into a ca. 5 g pellet and loaded into a molybdenum crucible (depth: 2.5 cm; diam: 1.5 cm) which was previously cleaned by heating at 1500 °C for 15 min under a dynamic vacuum of about 10^–5 Torr and then sealed under a low argon pressure using an arc welding system. The crucible was heated at a rate of 300 °C h^–1 to 1600 °C and held there for 3 days, then cooled at 100 °C h^–1 to 1100 °C and finally furnace cooled to room temperature

2.2 Single crystal structure determination

A black crystal of dimensions 0.26 × 0.060 × 0.026 mm³ was selected for data collection. Intensity data were collected on a Nonius Kappa CCD diffractometer using a graphite-monochromatized Mo Kα radiation (λ = 0.71073 Å) at room temperature. The COLLECT program package [25] was used to establish the angular scan conditions (φ and ω scans) used in the data collection. The data set was processed using EvalCCD [26] for the integration procedure. An absorption correction (T_min = 0.01131, T_max = 0.2315) was applied using the description of the crystal faces and the analytical method described by de Meulenaar and Tompa [27]. The structure was solved by direct methods using Sir97 [28] and subsequent difference Fourier syntheses in the space group R3 m. All structure refinements and Fourier syntheses were carried out using SHELXL-97 [29]. Refinement of the occupancy factors of the Nd sites showed that the Nd2, Nd3 and the Nd5 sites are slightly deficient and occupy at 98.6 (4)%, 0.972 (4)% and 0.98 (5)%, respectively. As qualitative microanalyses using a JEOL JSM-35 CF scanning electron microscope equipped with a Tracor energy-dispersive-type X-ray spectrometer indicated the presence of magnesium in the crystals, we could expect that the deficiencies observed on some of the Nd sites results from the presence of magnesium since in the final difference Fourier map we did not observed peaks compatible with Mg positions. Refinements taking into account an occupation of the deficient Nd sites simultaneously by Nd and Mg atoms were unsuccessful. However, if assume a complete occupation of the Nd2, Nd3 and Nd5 sites by Nd and Mg atoms, we get the composition Nd_12.4(1)Mg_0.6(1)Mo₁₃O₃₆ which is in good agreement with that found by I.C.P. measurement, Nd_12.5(3)Mg_0.51(5)Mo_13.11(4)O₃₆. The positional and anisotropic displacement parameters for all atoms as well as the occupancy factors for the Nd atoms were refined to the values R₁ = 0.0311, wR₂ = 0.0782 for 126 parameters and 3184 reflections with I > 2 σ(I) and the residual electron densities were 3.310 and –3.517 e Å^–3. A summary of the X-ray crystallographic and experimental data are presented in Table 1, and selected interatomic distances are reported in Table 2.

2.3 Theoretical calculations

Extended Hückel [30] molecular calculations were carried out using the program CACAO [31]. The exponents (ζ) and the valence shell ionization potentials (H_ii in eV) used were, respectively: 2.275, –32.3 for O 2s; 2.275, –14.8 for O 2p; 1.956, –8.34 for Mo 5s; 1.921, –5.24 for Mo 5p; 2.14. The H_ii value for Mo 4d was set equal to –10.50. A linear combination of two Slater-type orbitals of exponents ζ₁ = 4.542 and ζ₂ = 1.901 with equal weighting coefficients was used to represent the Mo 4d atomic orbitals.

Self-consistent ab initio band structure calculations were performed on the model compound Nd₁₃Mo₁₃O₃₆ with the scalar relativistic tight-binding linear muffin-tin orbital (LMTO) method in the atomic spheres approximation including the combined correction [32–36]. Exchange and correlation were treated in the local density approximation using the von Barth and Hedin [37] local exchange correlation potential. Within the LMTO formalism interatomic spaces are filled with interstitial spheres. The optimal positions and radii of these additional ‘empty spheres’ were determined by the procedure described in Ref. [38]. Twenty-four non-symmetry-related ‘empty spheres’ with 0.62 Å ≤ r_ES ≤ 1.60 Å were introduced for the calculations. The full LMTO basis set consisted of 6s, 6p, 5d and 4f functions for Nd spheres, 5s, 5p, 4d and 4f functions for Mo spheres, 2s, 2p and 3d functions for O spheres, and s, p and d functions for ‘empty spheres’. The eigenvalue problem was solved using the following minimal basis set obtained from the Löwdin downfolding technique: Nd 6s, 5d, 4f, Mo 5s, 5p, 4d, O 2s, 2p and interstitial 1s LMTOs. The k space integration was performed using the tetrahedron method [39]. Charge self-consistency and the average properties were obtained from 46 irreducible k points. A measure of the magnitude of the bonding was obtained by computing the crystal orbital Hamiltonian populations (COHP) which are the Hamiltonian population weighted density of states (DOS) [40]. As recommended [41], a reduced basis set (in which all ES LMTO's have been downfolded) was used for the COHP calculations. DOS and COHP curves are shifted so that ε_F lies at 0 eV.

3.1 Structure description

A perspective view of the crystal structure of Nd_12.4Mg_0.6Mo₁₃O₃₆ along the b axis is represented in Fig. 1. It shows that this structure can be viewed as a stacking along the c-axis of two different slabs interconnected through oxygen atoms. The A slab is made up of molybdenum forming Mo₁₃ clusters, neodymium and oxygen atoms, and the B slab of neodymium and oxygen atoms.

In the A slab, the Mo₁₃ cluster is similar to that encountered in the compound Pr₄Mo₉O₁₈. It has 3 m (C_3v) symmetry and may be regarded as resulting from the edge-sharing-condensation of three Mo₆ octahedra having the same apex in common, as depicted in Fig. 2. The Mo–Mo bond distances within the Mo₁₃ cluster range from 2.692(1) to 2.848(1) Å with an average value of 2.74 Å and are thus quite comparable to those found in one-dimensional condensed Mo clusters. The molybdenum atoms are linked to three (Mo1), four (Mo4) or five (Mo2, Mo3) oxygen atoms. These bridge the Mo-Mo edges (O2, O7, O8), cap the triangular Mo faces (O1, O4, O9) or are bonded exo to the Mo apices of the Mo₁₃ polyhedron (O5, O6) leading to the Mo₁₃O₃₁ unit shown in Fig. 3. The Mo–O bond distances lie between 1.969(5) and 2.135(5) Å with an average value of 2.08 Å. In the A slab, we also find the neodymium atoms Nd1 and Nd2 that are surrounded by 10 and 8 oxygen atoms, respectively. The coordination polyhedra around both neodymium atoms are shown in Fig. 4. The Nd1O₁₀ polyhedron may be described as the stacking of a triangle formed by the O5 on a hexagon of O2 atoms which is capped by the O4 atom. The environment of the Nd2 atoms may be viewed as a distorted bicapped trigonal prism. The Nd–O distances range from 2.339(11) to 2.832(5) for the Nd1 site and from 2.148(2) to 2.774(4) for the Nd2 site. The arrangement of the Mo₁₃O₃₁ units and the Nd1O₁₀ and Nd2O₈ polyhedra in the A slab is shown in Fig. 5.

In the B slab, we find the remaining Nd atoms, i.e. Nd3, Nd4, Nd5 and Nd5′ (Fig. 6). The Nd3 and Nd4 atoms are eight-coordinate in oxygen but present different environments. Nd3has a bicapped trigonal prismatic environment and Nd4 is at the center of a tricapped square-based pyramid. The Nd3–O bond distances lie between 2.331(6) and 2.904(6) Å and the Nd4–O between 2.345(5) and 2.883(5) Å. The last Nd atom is delocalized over two the two positions Nd5 and Nd5′ in a monocapped square-based pyramid. The Nd5–O and Nd5′–O distances vary between 2.313(11) and 2.927(7) Å and between 2.210(17) and 2.750(10) Å, respectively. The Fig. 7 shows the arrangement of the Nd3O₈, Nd4O₈ and (Nd5, Nd5′) O₆ polyhedra within the B slab.

3.2 Bond-length bond-strength formula

The estimation of the oxidation states of molybdenum and, consequently, of the number of electrons per Mo₁₃ cluster could be performed on this compound using the empirical bond length-bond strength relationship developed by Brown and Wu [42] for Mo–O bonds:

In the above formula, s(Mo–O) is the bond-strength in valence units, d(Mo–O) is the observed Mo–O bond distance in Å, 1.882 Å corresponds to a Mo–O single bond distance, and the exponential parameter-6 is characteristic of the Mo atom. Thus, the sum of the Mo–O bond strengths s (in valence units) about a particular Mo atom is equal to the oxidation state of that Mo atom. For the four crystallographically independent Mo atoms, these calculations lead to the number oxidation states of +1.44(2), 2.67(4), 3.19(5) and +2.06(3) for Mo1, Mo2, Mo3 and Mo4, respectively. From these values, we could estimate the metallic electron (ME) count per Mo₁₃ cluster to 44.7(4), which is in good agreement with the value of 44.4(3) e^– based on the stoichiometry.

3.3 Electronic structure

Extended Hückel calculations were performed on the experimental Mo₁₃O₃₁ cluster of C_3v symmetry. The corresponding molecular orbital (MO) diagram is shown in Fig. 8. Because of the electronegativity difference of molybdenum and oxygen atoms, MOs that lie in the HOMO/LUMO part of MO diagram shows a major metallic d character. The 48e, 49e and 33a₁ MOs show an overall Mo–Mo non-bonding character as revealed by the metal–metal overlap populations examination, whereas lower and higher metallic MOs are Mo–Mo bonding and antibonding, respectively. The 48e, 49e and 33a₁ MOs become the HOMO for the ME count of 42, 46, and 48, respectively. No well-defined HOMO/LUMO gaps are observed for those ME counts. Such a situation has been frequently encountered for metal clusters with a high metal–metal connectivity [43]. Spin-polarized periodic density functional (DF) calculations were carried out on the model compound Nd₁₃Mo₁₃O₃₆ where no Mg atoms are considered and all Nd5 and Nd5′ positions are fully occupied and vacant, respectively. Since no close metal–metal contacts are observed between the Mo₁₃ clusters in the crystal, no significant perturbation of the Mo₁₃ clusters' MO pattern is expected in the three-dimensional solid. Spin-polarized total, Nd and Mo projected DOS and Mo–Mo COHP are sketched in Fig. 9. The Fermi level cuts a spin-up DOS peak mainly centered on Nd 4f. Since three electrons occupy this DOS peak inducing a + 3 oxidation state for the Nd atoms, the ME count of the Mo₁₃ cluster is equal to 45. The Fermi level cuts also a rather large Mo projected spin-up DOS peak and a narrow Mo projected spin-down DOS peak that are both weakly Mo–Mo antibonding. Theses bands are separated by an energy of more than half an eV from lower Mo–Mo bonding bands. Assuming a rigid band model, this gap becomes the band gap for the ME count of 42. Although some Mo–Mo antibonding bands are occupied, the C_3v symmetry architecture is preserved for this cluster because of the high connectivity of the different atoms constituting the cluster. Periodic DF calculations suggest that an open-shell configuration is encountered for the Mo₁₃O₃₁ unit in this model compound. Because of the absence of significant HOMO/LUMO gaps, strong Jahn–Teller distortions of the cluster could occur for the low spin state configuration that is less stable than the high spin state. Magnetic properties consistent with the presence of Nd³⁺ and some polarized electrons on Mo atoms are expected.