Lu₅Pd₄Ge₈ and Lu₃Pd₄Ge₄: Two More Germanides among Polar Intermetallics

Abstract

In this study, two novel Lu₅Pd₄Ge₈ and Lu₃Pd₄Ge₄ polar intermetallics were prepared by direct synthesis of pure constituents. Their crystal structures were determined by single crystal X-ray diffraction analysis: Lu₅Pd₄Ge₈ is monoclinic, P2₁/m, mP34, a = 5.7406(3), b = 13.7087(7), c = 8.3423(4) Å, β = 107.8(1), Z = 2; Lu₃Pd₄Ge₄ is orthorhombic, Immm, oI22, a = 4.1368(3), b = 6.9192(5), c = 13.8229(9) Å, Z = 2. The Lu₅Pd₄Ge₈ analysed crystal is one more example of non-merohedral twinning among the rare earth containing germanides. Chemical bonding DFT studies were conducted for these polar intermetallics with a metallic-like behavior. Gathered results for Lu₅Pd₄Ge₈ and Lu₃Pd₄Ge₄ permit to described both of them as composed by [Pd–Ge]^δ– three dimensional networks bonded to positively charged lutetium species. From the structural chemical point of view, the studied compounds manifest some similarities to the Zintl phases, containing well-known covalent fragments i.e., Ge dumbbells as well as unique cis-Ge₄ units. A comparative analysis of molecular orbital diagrams for Ge₂^6– and cis-Ge^10– anions with COHP results supports the idea of the existence of complex Pd–Ge polyanions hosting covalently bonded partially polarised Ge units. The palladium atoms have an anion like behaviour and being the most electronegative cause the noticeable variation of Ge species charges from site to site. Lutetium charges oscillate around +1.5 for all crystallographic positions. Obtained results explained why the classical Zintl-Klemm concept can’t be applied for the studied polar intermetallics.

1. Introduction

In RE–Pd–Ge systems (RE = rare earth metal) more than one hundred ternary compounds have already been discovered [1], which have been extensively studied with respect to crystal structure, chemical bonding and physical properties [2,3,4,5,6].

The structures of Ge-rich compounds are characterized by a variety of Ge covalent fragments, with topologies depending both on global stoichiometry and on the nature of the RE component. These units are often joined together through Pd atoms, meanwhile the RE species are located in bigger channels inside the structure [2,3,7]. The frameworks formed by Pd and Ge atoms have been interpreted as polyanions of general formula [Pd_xGe_y]^δ– counterbalanced by the rare earth cations, coherently with the definition of these compounds as polar intermetallics [4].

It is interesting to remark that the ternary RE–Pd–Ge compounds manifest a tendency to be stoichiometric with ordered distributions of constituents through distinct Wyckoff sites. Moreover, within Pd–Ge fragments, both species have small coordination numbers (usually four or five) with very similar topological distributions of neighbours (tetrahedral coordination or its derivatives). These features may be considered as geometrical traces of a similar chemical role of Pd and Ge. That is why symmetry reduction from certain aristotypes can conveniently depict the distortions related with an ordered distribution of atom sorts. Such analysis has been conducted in the literature for AlB₂ derivative polymorphs of REPdGe [8] and BaAl₄ derivatives of the RE₂Pd₃Ge₅ [7,9] family of compounds. In systems where such types of relationships exist, the geometric factor is surely of great importance. Thus, varying RE, different polymorphs [8] or even novel compounds may form. As an example, heavy rare earth containing RE₅Pd₄Ge₈ (RE = Er, Tm) [4] and RE₃Pd₄Ge₄ (RE = Ho, Tm, Yb) [3] series of compounds may be cited.

During exploratory syntheses conducted in the Lu–Pd–Ge system in the framework of our ongoing studies on Ge-rich ternary compounds, the Lu representatives of the abovementioned 5:4:8 and 3:4:4 stoichiometries were detected for the first time. In this paper, results on the synthesis and structural characterization/analysis of these new germanides are reported, together with an extensive study of their chemical bonding, including Bader charges, Density of States (DOS) and Crystal Orbital Hamilton Population (COHP) curves as well as Molecular Orbitals (MO) diagrams for Zintl anions composed by Ge.

2. Experimental

2.1. Synthesis and SEM-EDXS Characterization

The Lu–Pd–Ge alloys were synthesized from elements with nominal purities >99.9% mass. Lutetium was supplied by Newmet Koch, Waltham Abbey, England, and palladium and germanium by MaTecK, Jülich, Germany.

Different synthetic routes were followed, including arc melting and direct synthesis in resistance furnace. In the latter case, proper amounts of components were placed in an alumina crucible, which was closed in an evacuated quartz ampoule to prevent oxidation at high temperatures, and submitted to one of the following thermal cycles in a resistance furnace:

(1)

25 °C → (10 °C/min) → 950 °C (1 h) → (−0.2 °C/min) → 600 °C (168 h) → (−0.5 °C/min) → 300 °C → furnace switched off

(2)

25 °C → (10 °C/min) → 1150 °C (1 h) → (−0.2 °C/min) → 300 °C → furnace switched off

A continuous rotation of the quartz ampoule during the thermal cycle was applied. In some cases, the thermal treatment followed arc melting. A scanning electron microscope (SEM) Zeiss Evo 40 (Carl Zeiss SMT Ltd., Cambridge, UK) coupled with a Pentafet Link Energy Dispersive X-ray Spectroscopy (EDXS) system managed by INCA Energy software (Oxford Instruments, Analytical Ltd., Bucks, UK) was used for microstructure observation and phase analysis. For this last purpose, calibration was performed with a cobalt standard. Samples to be analyzed were embedded in a phenolic resin with carbon filler, by using the automatic hot compression mounting press, Opal 410 (ATM GmbH, Mammelzen, Germany), and smooth surfaces for microscopic examinations were obtained with the aid of the automatic grinding and polishing machine, Saphir 520 (ATM GmbH, Mammelzen, Germany). SiC papers with grain sizes decreasing from 600 to 1200 mesh and diamond pastes with particle sizes decreasing from 6 to 1 µm were employed for grinding and polishing, respectively.

2.2. X-ray Diffraction (XRD) Measurements on Single Crystals and Powder Samples

Single crystals of Lu₅Pd₄Ge₈ and Lu₃Pd₄Ge₄ were selected from suitable samples with the aid of a light optical microscope operated in dark field mode. A full-sphere dataset was obtained in a routine fashion at ambient conditions on a four-circle Bruker Kappa APEXII CCD area-detector diffractometer equipped by the graphite monochromatized Mo Kα (λ = 0.71073 Å) radiation, operating in ω-scan mode. Crystals exhibiting metallic luster and glued on glass fibers were mounted in a goniometric head and then placed in a goniostat inside a diffractometer camera. Intensity data were collected over the reciprocal space up to ~30° in θ with exposures of 20 s per frame. Semi-empirical absorption corrections based on a multipolar spherical harmonic expansion of equivalent intensities were applied to all data by the SADABS/TWINABS (2008) software [10].

The corresponding CIF files are available in the supporting information material and they have also been deposited at Fachinformationszentrum Karlsruhe, 76344 Eggenstein-Leopoldshafen, Germany, with the following depository numbers: CSD-434226 (Lu₅Pd₄Ge₈) and CSD-434225 (Lu₃Pd₄Ge₄). Selected crystallographic data and structure refinement parameters for the studied single crystals are listed in

Table 1. Crystallographic data for Lu₅Pd₄Ge₈ and Lu₃Pd₄Ge₄ single crystals together with some experimental details of their structure determination.

Empirical Formula	Lu5Pd4Ge8	Lu3Pd4Ge4
EDXS data	Lu28.6Pd24.9Ge46.5	Lu25.7Pd35.0Ge39.5
Space group (No.)	P21/m (11)	Immm (71)
Pearson symbol-prototype, Z	mP34-Tm5Pd4Ge8, 2	oI22-Gd3Cu4Ge4, 2
а [Å]	5.7406(3)	4.1368(3)
b [Å]	13.7087(7)	6.9192(5)
c [Å]	8.3423(4)	13.8229(9)
β (°)	107.8(1)	–
V [Å3]	625.20(5)	395.66(5)
Abs. coeff. (μ), mm−1	63.5	60.7
Twin law	[$-\frac{1}{2}0\frac{1}{2};0-10;\frac{3}{2}0\frac{1}{2}$]	–
k (BASF)	0.49(1)	–
Unique reflections	2105	404
Reflections I > 2σ(I)/parameters	1877/87	398/23
GOF on F2 (S)	1.17	1.17
R indices [I > 2σ(I)]	R1 = 0.0190;wR2 = 0.0371	R1 = 0.0238;wR2 = 0.0869
R indices [all data]	R1 = 0.0247;wR2 = 0.0384	R1 = 0.0242;wR2 = 0.0871
Δρfin (max/min), [e/Å3]	2.00/−2.83	2.87/−3.33

. Details regarding the structure solution are discussed in Section 3.2. X-ray powder diffraction (XRPD) measurements were performed on all samples, using a Philips X’Pert MPD vertical diffractometer (Cu Kα radiation, λ = 1.5406 Å, graphite crystal monochromator, scintillation detector, step mode of scanning). Phase identification was performed with the help of the PowderCell software, version 2.4 [11].

2.3. Computational Details

A charge analysis based on Bader’s Quantum Theory of Atoms In Molecules (QTAIM) [12], coded in the Vienna Ab-initio Simulation Package (VASP) [13], was used to evaluate the atomic charge populations in the title compounds. Projector augmented waves (PAW) formalism was used, together with Perdew–Berke–Erzenhof parametrization of the exchange-correlation interaction. The recommended PAW sets were used, considering nine valence electrons for Lu (6s²5p⁶5d¹), ten for Pd (5s¹4d⁹), and fourteen for Ge (4s²3d¹⁰4p²). An energy cut-off of 600 eV was set for all calculations presented and the default value (10⁻⁵ eV) of the energy convergence was used.

The electronic band structures of Lu₅Pd₄Ge₈ and Lu₃Pd₄Ge₄ were calculated by means of the self-consistent, tight-binding, linear-muffin-tin-orbital, atomic-spheres approximation method using the Stuttgart TB-LMTO-ASA 4.7 program [14], within the local density approximation (LDA) [15] of DFT. The radii of the Wigner–Seitz spheres were assigned automatically so that the overlapping potentials would be the best possible approximations to the full potential, and no empty spheres were needed to meet the minimum overlapping criterion.

The basis sets included 6s/(6p)/5d orbitals for Lu with Lu 4f¹⁴ treated as core, 5s/5p/4d/(4f) for palladium and 4s/4p/(4d)/(4f) orbitals for germanium with orbitals in parentheses being downfolded.

The Brillouin zone integrations were performed by an improved tetrahedron method using a 20 × 8 × 12 k-mesh for Lu₅Pd₄Ge₈ and 16 × 16 × 16 for Lu₃Pd₄Ge₄.

Crystal Orbital Hamilton populations (COHPs) [16] were used to analyze chemical bonding. The integrated COHP values (iCOHPs) were calculated in order to evaluate the strengths of different interactions. Plots of DOS and COHP curves were generated using wxDragon [17], setting the Fermi energy at 0 eV as a reference point.

Qualitative MO arguments based on extended Hückel theory (EHT) have been developed with the CACAO package [18,19] and its graphic interface. Even if the EHT model tends to involve the most drastic approximations in MO theory, this one electron effective Hamiltonian method tends to be used to generate qualitatively correct molecular and crystal orbitals [20]. EHT is best used to provide models for understanding both molecular and solid state chemistry, as shown with great success by Roald Hoffmann and others [21].

3. Results and Discussion

3.1. Results of SEM-EDXS Characterization

An explorative study of the Ge-rich region of the Lu–Pd–Ge system was conducted by synthesis of some ternary samples with a Ge content >40 at %. The prepared samples are listed in

Table 2. Results of SEM/EDXS characterization of the Lu–Pd–Ge samples (> 40 at % Ge) obtained with different synthesis methods/thermal treatments. The highest yield phase in each sample is the first in the list.

No. Overall Composition [at %] Synthesis/Thermal Treatment	Phases	Phase Composition [at %] Lu; Pd; Ge	Crystal Structure
1 Lu21.4Pd11.2Ge67.4 Arc melting followed by thermal treatment (1)	Lu2PdGe6 Lu5Pd4Ge8 LuPd0.16Ge2 Ge	21.5; 12.1; 66.4 28.6; 25.1; 46.3 31.1; 5.4; 63.5 –; –; –;	oS72–Ce2(Ga0.1Ge0.9)7 mP34-Tm5Pd4Ge8 oS16-CeNiSi2 cF8-C
2 Lu28.9Pd24.1Ge47.0 Arc melting	Lu5Pd4Ge8 new phase Lu2PdGe6 LuPd0.16Ge2	28.8; 24.8; 46.4 32.4; 28.5; 39.1 21.7; 11.8; 66.5 30.1; 6.9; 63.0	mP34-Tm5Pd4Ge8 AlB2 related oS72–Ce2(Ga0.1Ge0.9)7 oS16-CeNiSi2
3 * Lu30.8Pd25.5Ge43.7 Direct synthesis with thermal treatment (2)	Lu5Pd4Ge8 new phase Ge	28.6; 24.9; 46.5 33.0; 26.8; 40.2 –; –; –;	mP34-Tm5Pd4Ge8 AlB2 related cF8-C
4 * Lu33.0Pd26.0Ge41.0 Arc melting followed by thermal treatment (2)	Lu3Pd4Ge4 Lu5Pd4Ge8 LuPdGe PdGe Ge	25.7; 35.0; 39.5 28.4; 25.1; 46.5 31.9; 34.5; 33.6 0.0; 53.4; 47.6 0.0; 0.0; 100.0	oI22-Gd3Cu4Ge4 mP34-Tm5Pd4Ge8 oI36-AuYbSn oP8-FeAs cF8-C
5 Lu17.9Pd29.0Ge53.1 Arc melting	Lu3Pd4Ge4 LuPdGe PdGe Ge	26.1; 34.2; 39.7 32.0; 33.5; 34.5 0.0; 52.4; 47.8 0.0; 0.0; 100.0	oI22-Gd3Cu4Ge4 oI36-AuYbSn oP8-FeAs cF8-C

* Samples from which single crystals were taken.

, together with an indication of the followed synthetic route, as well as the results of SEM/EDXS characterization. Information on phase crystal structure was obtained from X-ray diffraction results.

All samples are multiphase, as it is common for non-annealed alloys belonging to complex ternary systems, Ge is always present, in some cases in small amount. SEM images using the Back-Scattered Electron (BSE) mode are well contrasted, helping to distinguish different compounds, whose compositions are highly reproducible.

Several ternary compounds already known from the literature were detected in the samples, namely Lu₂PdGe₆, LuPd_0.16Ge₂ and LuPdGe [1,2]. For the latter, the oI36-AuYbSn structure was confirmed, in agreement with previous single crystal data [8].

A new phase of composition ~Lu₃₃Pd₂₇Ge₄₀ was detected in samples 2 and 3; the corresponding X-ray powder patterns could be acceptably indexed assuming a simple AlB₂-like structure, with a ≈ 4.28 and c ≈ 3.54 Å. Nevertheless, a deeper structural investigation would be necessary to ensure its crystal structure.

Crystal structures of the new Lu₅Pd₄Ge₈ and Lu₃Pd₄Ge₄ compounds were solved by analysing single crystals extracted from samples 3 and 4, respectively. The obtained structural models, discussed in the following section, were consistent with the measured powder patterns.

3.2. Crystal Structures of Lu₅Pd₄Ge₈ and Lu₃Pd₄Ge₄

3.2.1. Structural Determination

The Lu₅Pd₄Ge₈ crystal selected for X-ray analysis is one more example of non-merohedral twins among germanides. Previously, similar twins were found for Tb₃Ge₅ [22], Eu₃Ge₅ [23], Pr₄Ge₇ [24], La₂PdGe₆ and Pr₂PdGe₆ [2]. Based on the preliminary indexing results, the unit cell of the measured crystal might be considered as a base centered orthorhombic one with a = 8.55, b = 21.29 and c = 13.70 Å. The analysis of systematic extinctions suggested the following space groups: Cmc2₁ (No. 36), C2cm (No. 40) and Cmcm (No. 63). It should be mentioned that the average value of ⎸E²–1⎹ = 1.33, characterizing the distribution of peak intensities, deviates noticeably from the ideal value (0.968) for centrosymmetric space groups. Frequently, this is an indication of a twinned dataset [25,26]. A charge-flipping algorithm implemented in JANA2006 [27] was used, giving a preliminary structural model with 36 Lu atoms and 96 Ge atoms in the unit cell (Cmcm space group). Usually, when scatterers have such remarkable differences in electrons, the charge-flipping algorithm is quick and very efficient in discriminating them. Considering the interatomic distances criterion and U_eq values, in the successive iteration cycles, Pd atoms were introduced manually by substituting those of Ge, but no improvements were observed. There was no chance to improve this model further because the isotropic thermal displacement parameters showed meaningless values; several additional strong peaks were present at difference Fourier maps located too close to the accepted atom positions; and the R1 value stuck at ca. 10%. Looking for a correct structure solution in other space groups gave no reasonable results.

At this point, a more careful analysis of diffraction spots in reciprocal space was performed using RLATT [10] software. It was noticed that a remarkable number of peaks distributed in a regular way had a small intensity and might be considered as super reflections. Therefore, they were ignored during the indexing procedure, and a four times smaller primitive monoclinic unit cell with a = 5.73, b= 13.70, c = 8.34 Å and β = 107.8° was derived. The dataset was newly integrated and semi-empirical absorption corrections were applied by SADABS [10] software. This time, an mP34 structural model, containing all the atomic species, was proposed by the charge-flipping algorithm. Even so, the refinement was not satisfactory because some Wyckoff sites manifested partial occupancy and it was not possible to refine the structure anisotropically. It was decided to test the ROTAX [28] algorithm implemented in WinGx [29] and check the possibility of interpreting our crystal as a non-merohedral twin. In fact, a two-fold rotation along the [101] direction $\left(\begin{array}{ccc}\raisebox{1ex}{$-1$}\!\left/ \!\raisebox{-1ex}{$2$}\right.& 0& \raisebox{1ex}{$1$}\!\left/ \!\raisebox{-1ex}{$2$}\right.\\ 0& -1& 0\\ \raisebox{1ex}{$3$}\!\left/ \!\raisebox{-1ex}{$2$}\right.& 0& \raisebox{1ex}{$1$}\!\left/ \!\raisebox{-1ex}{$2$}\right.\end{array}\right)$ was proposed as a twin law obtaining a good figure of merit.

To check this hypothesis and refine the collected data as accurately as possible, the initially selected batch of ca. 1000 reflections (comprising those of weak intensity considered as super reflections) was separated into two groups with the help of the CELL_NOW [10] program, suggesting the same twin law for the two monoclinic domains. Successively, the information on the reciprocal domain orientation stored in the .p4p file was used to integrate the dataset considering the simultaneous presence of both domains. After that, the resulting intensities set was scaled, corrected for absorption and merged with the help of the TWINABS [10] program. As a result, the output in HKLF5 format with a flag indicating the original domains, was generated. Using the latter and testing one more time the charge flipping procedure, the structural model was immediately found and element species were correctly assigned. The Lu₅Pd₄Ge₈ was of monoclinic symmetry (space group P2₁/m, mP34-Tm₅Pd₄Ge₈) and contained 3 Lu, 2 Pd and 6 Ge crystallographic sites. All the atom positions were completely occupied and did not manifest any considerable amount of statistical mixture. The anisotropically refined Lu₅Pd₄Ge₈ showed excellent residuals and flat difference Fourier maps (see Table 1). The refined volume ratio of twinned domains was 0.49/0.51.

The RLATT program was used to generate a picture showing the distribution of X-ray diffraction spots originating from the two domains, differentiated by color, in Figure 1 (upper part). The distribution of the non-overlapped peaks of the second domain was also easily visible on the precession photo of the h3l zone, demonstrated in Figure 1 (lower part). In the same figure, a schematic real space representation of the mutual orientation of the twinned-crystal components is shown.

Indexing of the diffraction dataset of the Lu₃Pd₄Ge₄ single crystal gave an orthorhombic base centered unit cell with a = 4.137, b = 6.919, c = 13.823 Å. Systematic extinction conditions related to the presence of symmetry elements were not found for this dataset. The structure solution was found in Immm with the aid of the charge flipping algorithm implemented in JANA2006 [27]. The proposed preliminary structural model contained five crystallographic sites, giving the Lu₃Pd₄Ge₄ formula and corresponding to the oI22-Gd₃Cu₄Ge₄ prototype. Partial site occupation (due to a possible statistical mixture of the species) was checked in separate cycles of least-squares refinement, but no significant deviation from full occupation was detected. The final structure model was refined as stoichiometric with the anisotropic displacement parameters for all crystallographic sites, giving small residual factors and a flat difference Fourier map (see Table 1). The standardized atomic coordinates for Lu₅Pd₄Ge₈ and Lu₃Pd₄Ge₄ are given in

Table 3. Atomic coordinates standardized by Structure Tidy [30] and equivalent isotropic displacement parameters for Lu₅Pd₄Ge₈ and Lu₃Pd₄Ge₄.

Atom	Site	x/a	y/b	z/c	Ueq (Å2)
Lu5Pd4Ge8
Lu1	2e	0.71858(8)	1/4	0.93028(6)	0.0047(1)
Lu2	4f	0.13606(7)	0.11370(2)	0.78913(7)	0.0051(1)
Lu3	4f	0.62176(8)	0.11902(2)	0.28943(7)	0.0056(1)
Pd1	4f	0.07436(13)	0.08476(3)	0.14089(12)	0.0072(1)
Pd2	4f	0.42601(13)	0.58211(3)	0.35985(12)	0.0075(1)
Ge1	2e	0.0515(2)	1/4	0.28977(15)	0.0081(2)
Ge2	2e	0.3343(2)	1/4	0.58221(15)	0.0078(2)
Ge3	2e	0.7797(2)	1/4	0.5814(2)	0.0063(2)
Ge4	4f	0.15453(17)	0.04252(5)	0.44776(16)	0.0071(1)
Ge5	2e	0.2797(2)	1/4	0.0606(2)	0.0048(2)
Ge6	4f	0.34622(17)	0.54443(4)	0.05049(16)	0.0060(1)
Lu3Pd4Ge4
Lu1	2a	0	0	0	0.0110(2)
Lu2	4j	1/2	0	0.37347(4)	0.0081(2)
Pd	8l	0	0.30094(10)	0.32738(5)	0.0155(3)
Ge1	4h	0	0.18745(17)	1/2	0.0084(3)
Ge2	4i	0	0	0.21754(10)	0.0132(3)

.

Similar to (Tm/Er)₅Pd₄Ge₈ [4], the presence of Ge covalent fragments in Lu₅Pd₄Ge₈ is obvious. Among these, there were two almost identical Ge–Ge dumbbells distanced at 2.49 Å and one more finite fragment composed of four germanium atoms having a cis-configuration (Figure 2,

Table 4. Interatomic distances and integrated crystal orbital Hamilton populations (-iCOHP, eV/cell) at E_F for the strongest contacts within the first coordination spheres in Lu₅Pd₄Ge₈. Symbols (2b) and (1b) indicate the number of homocontacts for corresponding Ge species.

Central Atom	Adjacent Atoms	d (Å)	-iCOHP	Central Atom	Adjacent Atoms	d (Å)	-iCOHP	Central Atom	Adjacent Atoms	d (Å)	-iCOHP
Lu1	Ge6 (×2)	2.853	1.26	Lu3	Ge5	2.904	1.25	(1b)Ge6	Ge6	2.494	2.39
	Ge1	3.025	0.81		Ge3	2.938	1.25		Pd1	2.516	2.16
	Ge5	3.033	1.02		Pd1	3.042	0.71		Pd2	2.533	2.12
	Ge3	3.036	0.85		Ge1	3.051	0.92		Pd1	2.619	1.80
	Ge2	3.064	0.79		Pd2	3.063	0.68		Lu1	2.853	1.26
	Ge5	3.069	0.81		Ge6	3.072	0.90		Lu2	3.016	0.96
	Pd1 (×2)	3.194	0.52		Ge6	3.094	0.75		Lu2	3.050	0.80
	Pd2 (×2)	3.258	0.46		Pd2	3.100	0.63		Lu3	3.072	0.90
Lu2	Ge5	2.857	1.29		Ge4	3.105	0.71		Lu3	3.094	0.75
	Ge3	2.918	1.20		Ge4	3.120	0.84	Pd1	Ge6	2.516	2.16
	Ge2	2.994	0.99		Pd1	3.236	0.52		Ge4	2.526	2.11
	Ge6	3.016	0.95		Ge4	3.493	0.27		Ge1	2.606	1.66
	Ge4	3.043	0.91	(1b)Ge3	Ge2	2.559	1.92		Ge6	2.619	1.80
	Ge4	3.043	0.79		Pd2 (×2)	2.699	1.46		Ge5	2.730	1.35
	Ge6	3.050	0.80		Lu2 (×2)	2.918	1.20		Lu3	3.042	0.71
	Pd1	3.087	0.68		Lu3 (×2)	2.938	1.25		Lu2	3.087	0.68
	Pd1	3.104	0.63		Lu1	3.036	0.86		Lu2	3.104	0.63
	Pd2	3.114	0.64	(1b)Ge4	Ge4	2.492	2.48		Lu1	3.194	0.52
	Pd2	3.156	0.57		Pd2	2.512	2.14	Pd2	Ge4	2.512	2.14
(2b)Ge1	Ge2	2.484	2.98		Pd1	2.526	2.11		Ge6	2.533	2.12
	Pd1 (×2)	2.606	1.66		Pd2	2.566	1.94		Ge4	2.566	1.94
	Ge5	2.627	1.69		Lu2	3.043	0.91		Ge2	2.649	1.53
	Lu1	3.025	0.81		Lu2	3.043	0.79		Ge3	2.699	1.46
	Lu3 (×2)	3.051	0.92		Lu3	3.105	0.71		Lu3	3.063	0.68
(2b)Ge2	Ge1	2.484	2.98	(1b)Ge5	Ge1	2.627	1.69		Lu3	3.100	0.63
	Ge3	2.559	1.92		Pd1 (×2)	2.730	1.35		Lu2	3.114	0.65
	Pd2 (×2)	2.649	1.53		Lu2 (×2)	2.857	1.29		Lu2	3.156	0.58
	Lu2 (×2)	2.994	1.00		Lu3 (×2)	2.904	1.26
	Lu1	3.064	0.79		Lu1	3.033	1.02
					Lu1	3.069	0.81

). The latter manifests a small geometrical distortion from the ideal conformation due to slightly different chemical arrangements around terminal Ge atoms (terminal atoms are located at 2.56 and 2.63 Å far from central dumbbell; the internal obtuse angles are ca. 111° and 113°, respectively). The cis unit is planar and lays at the mirror plane of the P2₁/m space group. The cited covalent fragments are joined together through Pd–Ge contacts shortened with respect to metallic radii sum (ranging from 2.51 to 2.73 Å) in a complex network hosting Lu atoms in the biggest cavities (see Figure 2). The shortest Lu–Pd and Lu–Ge contacts do not manifest noticeable deviations from the expected values and are ca. 3.0 Å.

The Lu₃Pd₄Ge₄ contains less germanium with respect to Lu₅Pd₄Ge₈ and, consequently, only a simple Ge–Ge dumbbell forms being, however, more stretched (2.59 Å,

Table 5. Interatomic distances and integrated crystal orbital Hamilton populations (-iCOHP, eV/cell) at E_F for the strongest contacts within the first coordination spheres in Lu₃Pd₄Ge₄. Symbols (1b) and (0b) indicate the number of homocontacts for corresponding Ge species.

Central Atom	Adjacent Atoms	d (Å)	-iCOHP	Central Atom	Adjacent Atoms	d (Å)	-iCOHP
Lu1	Ge4 (×4)	2.992	1.21	(0b)Ge2	Pd (×4)	2.562	1.88
	Ge5 (×2)	3.006	1.05		Pd (×2)	2.577	1.86
	Pd (×8)	3.445	0.41		Lu2 (×2)	2.988	0.83
Lu2	Ge5 (×2)	2.988	0.83		Lu1	3.006	1.05
	Ge4 (×4)	3.003	0.99	Pd	Ge4	2.512	2.23
	Pd (×4)	3.003	0.79		Ge5 (×2)	2.562	1.88
	Pd (×2)	3.100	0.58		Pd	2.755	0.97
(1b)Ge1	Pd (×2)	2.512	2.23		Lu2 (×2)	3.003	0.79
	Ge1	2.595	1.82		Pd (×2)	3.058	0.46
	Lu1 (×2)	2.992	1.22		Lu2	3.100	0.58
	Lu2 (×4)	3.003	0.99		Lu1 (×2)	3.445	0.41

). The trend of other interactions is similar as for Lu₅Pd₄Ge₈; Pd and Ge construct an extended network with infinite channels of hexagonal and pentagonal forms hosting Lu atoms.

One more structural relation can be proposed for the title compounds: both compounds contain common structural ThCr₂Si₂-like building blocks [31] (highlighted by blue lines in Figure 2) defined in many related compounds as “linkers” within various polyanionic fragments [32].

3.2.2. Lu₅Pd₄Ge₈: Structural Relationships

Looking for structural relationships is not an easy task, since this process is often strongly affected by human factors and is based on sometimes arbitrary criteria. From this point of view, one of the most rigorous approaches is based on the symmetry principle within the group-subgroup theory [33]. The most frequent chemical reason causing the reduction of symmetry is so-called “coloring”, which can be interpreted as an ordered distribution of different chemical elements within distinct Wyckoff sites. Müller [34] and Pöttgen [35] depict numerous examples of these.

Structural relationships between Tm₅Pd₄Ge₈ (isostructural with Lu₅Pd₄Ge₈) and RE₃T₂Ge₃ (T = late transitional element) were proposed in the literature [2] based on topological similarities between polyanionic fragments and the spatial distribution of cations. An alternative description of relationships between the abovementioned structures in terms of symmetry reduction is proposed here. The stoichiometries of these compounds are related as follows:

4 RE₃T₂Ge₃ − 2 RE + 4 Ge = 2 RE₅T₄Ge₈

(1)

This relation, even if purely numerical, finds support when comparing the crystal structures of the two chemically affine representatives Lu₃Fe₂Ge₃ (oS32) and Lu₅Pd₄Ge₈ (mP34). As is evidenced in Figure 3, one of the Lu sites in the former is substituted by a Ge dumbbell in the latter.

From the chemical interaction point of view, this should be a drastic change; instead, the remaining atoms apparently do not suffer noticeable displacements. This is why it was checked whether a Bärnighausen tree might be constructed relating the oS32 and mP34 models. In fact, only two reduction steps were needed:

-

a traslationengleiche (t2) decentering leading to a monoclinic Niggli cell (mP16-P2₁/m).

-

a klassengleiche transformation (k2) giving a monoclinic model with doubled cell volume (mP32-P2₁/m). As a result, all the independent sites split in two (see Figure 4).

The Lu2′ site (2e: 0.211 1/4 0.430) was further substituted by two germanium atoms (positions Ge1 and Ge2 in the final mP34-P2₁/m structural model). As a result, the already cited cis-Ge4 unit forms (see Figure 2), whose chemical role is discussed in the next section. The presence of the cis-Ge₄ units is quite intriguing, since the trans conformation is more favorable in numerous molecular chemistry examples. Therefore, it was decided to generate a structural model of Lu₅Pd₄Ge₈ composition hosting the trans-Ge unit and optimize it (see Figure S1 and Table S1). The relaxed structure perfectly coincided with the experimental results, confirming that minimal energy is associated with the cis conformation. More details on this, including an animation showing the evolution of the structural model after each relaxation step, are available in the Supplementary Material.

3.3. Chemical Bonding Analysis

Frequently, chemical bonding in polar intermetallics is preliminary addressed using the Zintl-Klemm concept. Taking into account the interatomic distances between Ge atoms, the presence of [(1b)Ge³⁻] with [(2b)Ge²⁻] Zintl species in Lu₅Pd₄Ge₈ and [(1b)Ge³⁻] with [(0b)Ge⁴⁻] ones in Lu₃Pd₄Ge₄ could be guessed. In order to guarantee the precise electron count, the average number of valence electrons per Ge atom [VEC(Ge)] should amount to 6.75 for Lu₅Pd₄Ge₈ and to 7.50 for Lu₃Pd₄Ge₄. Although it is reasonable to hypothesize a formal charge transfer of 3 valence electrons per Lu atom (Lu³⁺), as a first approximation, the Pd could be considered as a divalent cation (Pd²⁺) or a neutral species (Pd⁰). However, none of the possible electron distribution formulae listed below are suitable for the studied compounds, giving VEC(Ge) values that deviate somewhat from ideal values.

Lu₅Pd₄Ge₈ (Pd⁰)     VEC(Ge) = 5.875

Lu₅Pd₄Ge₈ (Pd²⁺)     VEC(Ge) = 6.875

Lu₃Pd₄Ge₄ (Pd⁰)     VEC(Ge) = 6.250

Lu₃Pd₄Ge₄ (Pd²⁺)     VEC(Ge) = 8.250

Even if the obtained VEC(Ge) values are closer to 6.75/7.50, in the case of Pd²⁺, this assumption is not coherent with the valence electrons flow when considering any of the known electronegativity scales. For example, taking into account the Pearson electronegativity values for Pd (4.45 eV) and Ge (4.60 eV) it is clear that a charge transfer from Pd to Ge is hardly probable. Strictly speaking, it is not possible to successfully apply the (8–N) rule to interpret the Ge–Ge covalent interactions. Thus, it becomes clear that these simplified considerations are not sufficient to account for the chemical bonding of the studied intermetallics. In particular, it is not reliable to consider covalent Ge fragments as isolated and more complex interactions should be taken into account. Therefore, a deeper chemical bonding investigation was conducted.

In

Table 6. Calculated QTAIM effective charges and atomic basin volumes for Lu, Pd and Ge in their elemental structure, in Lu₅Pd₄Ge₈ and in Lu₃Pd₄Ge₄.

Element/Compound	Atom/Site	Volume, [Å3]	QTAIM Charge, Qeff	Compound	Atom/Site	Volume, [Å3]	QTAIM Charge, Qeff
Lu (hP2)	Lu/2c	29.74 #	0	Lu5Pd4Ge8	Lu1/2e	15.88	+1.45
Pd (cF4)	Pd/4a	14.71 #	0	(mP34)	Lu2/4f	15.48	+1.48
Ge (cF8)	Ge/8a	22.66 #	0		Lu3/4f	15.85	+1.51
					Pd1/4f	19.91	−0.79
Lu3Pd4Ge4	Lu1/2a	16.90	+1.57		Pd2/4f	19.77	−0.76
(oI22)	Lu2/4j	15.05	+1.53		(2b)Ge1/2e	19.39	−0.23
	Pd/8l	19.43	−0.67		(2b)Ge2/2e	19.51	−0.30
	(1b)Ge1/4h	22.35	−0.89		(1b)Ge3/2e	22.66	−0.87
	(0b)Ge2/4i	16.42	−0.09		(1b)Ge4/4f	18.73	−0.30
					(1b)Ge5/2e	23.96	−1.14
					(1b)Ge6/4f	19.67	−0.59

^#—the QTAIM volumes of atoms in pure elements are equal to the volumes of their Wigner–Seitz polyhedra; structural data were taken from Ref [1].

, the volumes of the atomic basins and Bader effective charges for all the atoms in Lu₅Pd₄Ge₈ and Lu₃Pd₄Ge₄ are listed together with those for the same species in their pure element form. Comparing these values, one can qualitatively estimate the chemical role of constituents in binary/ternary compounds.

In both ternary germanides, the QTAIM basins of Lu were shrunk with respect to Lu-hP2, and the corresponding charges oscillated around +1.5, confirming the active metal-like role of Lu. The significant difference between Lu effective charges and the formal charges suggest that some of its valence electrons may contribute to covalent interactions.

The palladium atoms had similar volumes of atomic basins (ca. 20 ų) and are negatively charged (−0.7 ÷ −0.8), suggesting a bonding scenario coherent with the electronegativity values, i.e., with Pd taking part in a polyanionic network, as was hypothesized from the crystal structure analysis.

It is noteworthy that in the same compound, Ge atoms had pronounced differences in charge values (always negative) from site to site. More on the structural/chemical reasons for this will be discussed in the following.

The total and projected DOS for Lu, Pd and Ge for the studied intermetallics are shown in Figure 5. Orbital projected DOS can be found in the Supplementary Material (Figure S3). Focusing on the total DOS, a difference between the two compounds at the Fermi energy (E_F) is evident: for Lu₅Pd₄Ge₈ a pseudo-gap is visible just above E_F, instead for Lu₃Pd₄Ge₄ the Fermi level corresponds to a local maximum of the DOS, indicating a potential electronic instability. This might be a sign of particular physical properties (e.g., superconductivity or magnetic ordering) [36] or of small structural adjustments (e.g., off-stoichiometry due to statistical mixture or increase of vacancy concentration) [37] which, adequately modelled, would shift the E_F towards a local minimum. Even if EDXS elementary composition is compatible with a slightly off-stoichiometry, there is no strong indication of this coming from XRD data, so, the stoichiometric model was considered here. Further experimental investigations will be carried out aiming physical properties studies of this compound.

For both compounds, the valence orbital mixing of the three components over the whole energy range is noteworthy. Below E_F, both DOSs showed a gap of around −7 eV separating the two regions, with the lowest being mostly dominated by the 4s Ge states. The Pd-d states are mainly distributed in the range between −5 and −2.5 eV. Their width and energy overlapped with 4p Ge and Lu states, supporting the bonding relevance of Pd–Ge and Pd–Lu interactions. The fact that the majority of Pd 4d states are located well below the E_F indicates the electron acceptor character of this species. A significant contribution of 5d Lu states just below the E_F is a common feature of cations in polar intermetallics, characterized by an incomplete charge transfer (confirmed here also by Bader charge values).

Although the Zintl–Klemm (8–N) rule cannot be applied for the title compounds, it was decided to trace interaction similarities comparing the electronic structures of ideal Zintl anions Ge₂⁶⁻ and cis-Ge₄¹⁰⁻ coming from the extended Hückel calculation with those obtained by means of TB-LMTO-ASA, in terms of COHP curves. Molecular orbital diagrams (MO) for Ge₂⁶⁻ (point group D_∞h) and Ge₄¹⁰⁻ (the point symmetry of this anion was forced to C_2v fixing for all distances to 2.56 Å and obtuse internal angles to 111°) are presented in the Supplementary Materials (Figure S2) with the accordingly labeled orbitals.

In Figure 6a, the molecular orbital overlap population (MOOP) for Ge₂⁶⁻ is shown, together with COHP curves for Ge–Ge interactions (in dumbbells) existing in Lu₃Pd₄Ge₄ and Lu₅Pd₄Ge₈.

These partitioning methods could not be directly compared, since MOOP partitions the electron number, instead, COHP partitions the band structure energy. Since they both permit to easily distinguish between bonding and antibonding states, it was decided to perform a qualitative comparison targeting to figure out the similarities/differences between the isolated molecular fragments analogous with those found in the studied compounds.

The presence of the gap (at ca. −7eV) may be attributed to the energy separation of the σss and σ*ss of Ge₂ dumbbells from the σp, πp and π* orbitals. For the Lu₃Pd₄Ge₄ there are some occupied π* states close to E_F, whereas in Lu₅Pd₄Ge₈, the cited interactions are almost optimized at E_F. From these observations it derives that Ge dumbbells are not completely polarized; for Lu₅Pd₄Ge₈ the dispersion of σ and σ* states is more pronounced. One of the possible explanation of this is the existence of additional covalent interactions between germanium dumbbells and neighboring atoms.

From the structural data it is known that in Lu₃Pd₄Ge₄, Ge atoms are distanced at 2.59 Å as in diverse metal-like salts studied before [38,39,40]. Instead, in Lu₅Pd₄Ge₈ this distance is shortened to 2.49 Å. Usually, the trend of Ge–Ge dumbbell distances is related with electrostatic repulsion between atoms. This statement is coherent with integrated COHP values (–iCOHP, see Tables 4 and 5) that reflect the same trend, being of −1.82 eV/cell for Lu₃Pd₄Ge₄ and of −2.39 and −2.48 eV/cell for Lu₅Pd₄Ge₈.

Within the cis-Ge₄^10– anion the number of covalent interactions is higher, as a result the energy dispersion of its molecular states increases. For example, in the range −18 ÷ −14eV there are four MOs instead of two MOs for dumbbells. A very similar trend/type of interactions derives from COHP curves for Lu₅Pd₄Ge₈. As for the dumbbells, the interactions for the cis fragment are optimized at the E_F confirming its partial polarization.

Based on –iCOHP values listed in Tables 4 and 5 it derives that Pd–Ge interactions are very relevant, so one may assume the covalent type of bonding between them. The –COHP plots in Figure 7 confirm that they are mainly of bonding type over a large range below E_F with a weak unfavorable antibonding interaction in the vicinity of E_F, probably due to electrostatic repulsion between Ge orbitals and filled d states of Pd.

Inside Lu₃Pd₄Ge₄ the presence of a Pd–Pd short interaction can be highlighted. The –COHP plots for this are similar to those reported for Ca₂Pd₃Ge [41] showing a sharp antibonding character around −3 eV commonly attributed to enhanced repulsion between filled d states of Pd. Nevertheless, they are of bonding type in average as deducible from the –iCOHP values for this interaction (0.97 eV/cell), comparable to those reported in [41].

The remaining Lu–Pd and Ge–Lu interactions are weaker being however very similar for both germanides. All of them are of bonding type, Lu–Pd interactions are practically optimized at Fermi level. Numerous interactions between Lu and Pd (Lu and Ge) suggest that some covalent-like interaction may exist due to mixing between d states of Lu and Pd (or d states of Lu with p of Ge; similarly, as it was reported for Ca₅Ge₃ [36] and CaSi [42]). More detailed studies are needed in order to interpret these interactions.

The existence of the complex Pd–Ge polyanion (illustrated in Figure 2) and the electronegativity difference between Pd and Ge explains the trend of Ge species charges (listed in Table 6 ). The Ge dumbbell in Lu₃Pd₄Ge₄ has four neighboring Pd atoms, instead those in Lu₅Pd₄Ge₈ install six Pd–Ge polar interactions. As a result, the latter Ge species has lower negative charges. The same is true for (0b)Ge atom with six palladium atoms around in Lu₃Pd₄Ge₄: its charge approaches to zero. Within crystal structure, the number of Pd–Ge contacts is the same for terminal and central atoms of cis–Ge₄ units; thus, their charges trend is similar as for ideal cis-Ge₄^10– anion, terminal atoms being more negative.

4. Conclusions

The two new Lu₅Pd₄Ge₈ and Lu₃Pd₄Ge₄ polar intermetallics were synthesized and characterized in this work. They were found to crystallize in the mP34–Tm₅Pd₄Ge₈ and oI22–Gd₃Cu₄Ge₄ structures respectively. A detailed description of crystal structure solution in the case of the non-merohedral twinned crystal of Lu₅Pd₄Ge₈ was proposed, highlighting the difficulties/problems encountered here along with practical suggestions to manage them.

Joined crystal chemical analysis and combined DFT studies suggest the presence of [Pd₄Ge₈]^7.4– and [Pd₄Ge₄]^4.6– polyanions. The interactions of Lu with these frameworks cannot be viewed as purely ionic as derives from its states distribution, COHP analysis and Bader charges. The Lu–Pd and Lu–Ge bonding interactions are one of the most interesting aspects arisen from our study and their nature deserves further investigations.