A Coloring Study of the Ga Richest Alkali Gallides: New In- and Hg-Containing Gallides with the RbGa₇- and the K₃Ga₁₃-Type Structure

Abstract

The Ga-rich gallides of the alkali metals present an interesting, yet still scarcely investigated case of polyanionic cluster compounds with subtle variations in the character of their chemical bonding. In the present work, the Ga richest phases K${}_3$Ga${}_{13}$, RbGa${}_7$, and CsGa${}_7$, which are formally electron-precise Zintl/Wade cluster compounds, are systematically studied with respect to a partial substitution of Ga by In and Hg. The pure hepta-gallides AGa${}_7$ (A = Rb/Cs; $R\overline{3}m$), which were formerly obtained from Ga-rich melts in powder form only, were crystallized from Hg-rich melts. Herein, up to 9.9/13.6% (Rb/Cs) of Ga could be substituted by In, which partly takes the four-bonded [M${}_2$] dumbbells connecting layers of Ga-icosahedra. Even though the structures are electron precise, the pseudo band gap does not coincide with the Fermi level. In the most Ga-rich potassium compound K${}_3$Ga${}_{13}$ ($Cmcm$) only 1.2% of In and 2.7% of Hg could be incorporated. Although Rb${}_3$Ga${}_{13}$ remains unknown, ternary variants containing 5.2 to 8.2% In could be obtained; this structure is also stabilized by a small Hg-proportion. The likewise closed-shell 3D polyanion consists of all-exo-bonded Ga-icosahedra and closo [Ga${}_{11}$] clusters, which are connected by two tetrahedrally four-bonded Ga${}^{-}$ and a trigonal-planar three-bonded Ga${}^0$. The aspects of the electronic structures and the site-specific Ga↦Hg/In substitution in the polyanion (“coloring”) are discussed for the title compounds and other mixed Ga/In trielides.

1. Introduction

Alkali (A^I) gallides form a large variety of fascinating complex structures associated with diverse chemical bonding features [1,2,3,4]. The Ga-rich phases are mostly electron-precise Zintl/Wade compounds with a covalently bonded electron-poor (v.e./Ga only slightly above three) gallide polyanion containing different types of closo clusters. For example, the structure of KGa₃ according to KGa₃ $\stackrel{\times 3}{\to }$ 3 K⁺ + [Ga₈]²⁻ + Ga⁻ contains dodecahedral clusters [Ga₈]²⁻ connected via tetrahedrally bonded Ga⁻ atoms [5,6]. Occasionally, the polyanions of those gallides even resemble the structures of elemental boron or B-rich borides (e.g., in the case of the title compound RbGa₇ [7]). The large number of chemically different triel (M) positions in their structures predestines this class of compounds for studies on the site preferences (“coloring”) of the different atom sites within the polyanions with element atoms differing in size, electronegativity, and even v.e. numbers. In addition to these closed-shell cluster compounds, simple intermetallics like, e.g., NaGa₄ (BaAl₄-type [8]) appear among the alkali gallides as well. These compounds show no (pseudo) band gaps, as are typically associated with a larger variety of elements that may (partly) substitute gallium [9]. A puzzling bonding situation is also observed for the less triel-rich trielides like, e.g., Cs₈Ga₁₁ [10,11,12], which contain isolated clusters as anionic structure motifs.

The Ga richest gallides in the systems A–Ga of the three heavier alkali elements (A = K, Rb, Cs) are the starting point of the “coloring” study presented herein: RbGa₇ und CsGa₇ are the most electron-poor alkali metal trielides known (v.e./Ga=3.143). They are still incompletely characterized to date, as the structure of RbGa₇ was solved and refined by Belin in 1981 [7] in the monoclinic subgroup $C2/m$ of the correct trigonal space group $R\overline{3}m$. This symmetry mistake was detected by Marsh [13], who suggested the transformation to the trigonal rhombohedral structure. These retroactively transformed data are recorded in the inorganic structure databases as final crystal data for RbGa₇ until today. In 1985, van Vucht [14] reported a refinement of the $R\overline{3}m$ structure model of the isotypic Cs compound CsGa₇ based on X-ray powder data. Unfortunately, this work lacks details of the parameter refinement procedure, which furthermore yielded implausible atomic displacement parameters. More recent structure refinements are not known, possibly due to the incongruent melting behavior of both compounds [15,16,17,18,19]. This behavior requires a huge excess of gallium for the crystal growth, the separation of which from the desired product is not straightforward (cf. the procedure described in [20]). The most Ga-rich potassium gallide is the yet singular orthorhombic phase K₃Ga₁₃ (v.e./Ga = 3.23). Its complex structure was conclusively solved by Belin forty years ago using single crystal four-circle diffractometer intensity data [20]. Both types of Ga-rich gallides belong to the family of electron-precise cluster compounds: In the polyanion of RbGa₇, [Ga₁₂]⁰ icosahedra are connected via [Ga₃] triangles (2e3c bonds) to form layers that resemble the structure of $\alpha$-rhombohedral boron. The icosahedra layers are further connected via the dumbbells of four-bonded Ga⁻ atoms. K₃Ga₁₃ exhibits similar building blocks and is likewise a closed-shell compound: All-exo-connected icosahedra [Ga₁₂]²⁻ and closo clusters [Ga₁₁]²⁻ (in a 1:1 ratio) are connected by two four-bonded Ga⁻ and one trigonal-planar bonded Ga⁰ atom.

The present work on the “coloring” [21] of the polyanions in the two Ga-rich gallides K₃Ga₁₃ and (Rb/Cs)Ga₇ with the larger In atoms extends our recent studies on mixed Ga/In trielides of the RbGa₃, Cs₂In₃ [22], BaAl₄ [23], and Cs₈Ga₁₁ [12] structure types. Due to the closed-shell character of the two gallides, the partial substitution of Ga by Hg, which differs not only in size and electronegativity, but in v.e. numbers as well, is initially not expected. The only hint towards this possibility is the Au-containing derivative of RbGa₃ reported by Henning and Corbett [23].

2. Experimental Section

2.1. Preparation and Phase Analysis

The synthesis of the title compounds was performed starting from the pure alkali metals, gallium and indium or mercury, as obtained from commercial sources (K, Rb, Cs: Metallhandelsgesellschaft Maassen, Bonn, 99%; Ga: ABCR Karlsruhe, 99.999%; In: rods, ABCR Karlsruhe, 99.99%; Hg: 99.9%, Merck KGaA). The elements were filled into tantalum crucibles in a glovebox under an argon atmosphere, and the welded containers were heated up with a rate of 200 ${}^{\circ }$Ch${}^{-1}$ to maximum temperatures between 370 and 700 ${}^{\circ }$C. For all samples, the crystallization was performed applying a slow cooling rate of 2 to 10 ${}^{\circ }$Ch${}^{-1}$. The temperature programs applied can be found in

Table 1. Details of the synthesis of Ga-rich alkali gallides (tr: traces, s.c.: single crystals, ${}^{*}$ full s.c. dataset; cf. Table 2 and Table 6).

Sample		Weighed Elements						Temperature Program						Products, as Identified from Powder
No.	Composition	K/Rb/Cs		Ga		In/Hg		${\mathit{T}}_{\mathbf{max}}$	${\dot{\mathit{T}}}^{\downarrow }$	${\mathit{T}}_1$	${\dot{\mathit{T}}}^{\downarrow }$	${\mathit{T}}_2$	${\dot{\mathit{T}}}^{\downarrow }$	and Single Crystal (${}^{*}$) Data
		/mg	/mmol	/mg	/mmol	/mg	/mmol	(T/${}^{\circ }$C; $\dot{\mathit{T}}$/${}^{\circ }$Ch${}^{-1}$)						(Exact Compositions for s.c. Data Only)
K1a	K3Ga6.5In6.5	89	2.27	345	4.94	567	4.94	500	10					K3Ga9.6In1.4 [25], KGa1.12In2.88, In
K1b	K3Ga6.5In6.5	90	2.28	344	4.94	567	4.94	700	50	600	8	400	50	KGa1.9In2.1 [9], KGa3-type [22], In
K2	K3Ga10In3	101	2.59	602	8.63	297	2.59	500	10					K3Ga9.6In1.4 [25], In, KGa1.12In2.88 (tr.)
K3	K3Ga11In2	105	2.69	689	9.88	207	1.80	700	50	600	8	400	50	K3Ga12.85In0.15, In
K4	K3Ga11.5In1.5	108	2.75	735	10.54	157	1.37	500	10					K3Ga9.8In1.2, Cs2In3-type, In (tr.)
K5	K3Ga42In5	32	0.83	809	11.60	159	1.38	700	5	350	50			KGa3-type, K3Ga12.85In0.15${}^{*}$
K6	KGa3Hg3	91	2.33	484	6.94	1392	6.94	700	20	450	2			K3Ga12.65Hg0.35, KHg6 [26]
K7	KGaHg3	114	2.92	201	2.88	1726	8.60	700	20	450	2			K3Ga12.65Hg0.35${}^{*}$, K2Hg7 [27]
Rb1	RbGa4.5In2.5	127	1.48	458	6.56	419	3.65	650	5	300	50			RbGa2.67In0.33 (KGa3-type, [22]), In (tr.)
Rb2	RbGa6In	141	1.64	676	9.70	186	1.62	650	5	300	50			KGa3-type [22], (+Ga)
Rb3	Rb3Ga32In15	62	0.73	530	7.60	409	3.56	550	3	200	10			RbGa2.78In0.22 [22], In
Rb4	Rb3Ga37In10	67	0.76	647	9.28	229	2.51	370	3					RbGa6.45In0.55, In, (s.c. of Rb3Ga11.93In1.07${}^{*}$)
Rb5a	Rb3Ga42In5	71	0.83	779	11.17	153	1.33	550	3	200	10			Rb3Ga12.31In0.69${}^{*}$, KGa3-type [22], In
Rb5b	Rb3Ga42In5	71	0.83	779	11.17	153	1.33	550	5	200	20			RbGa6.57In0.43 (s.c. of Rb3Ga12.6In0.4)
Rb5c	Rb3Ga42In5	69	0.81	779	11.17	151	1.32	370	5					RbGa6.57In0.43, (+Ga)
Rb6	RbGa1.5In4.5	123	1.44	148	2.12	731	6.37	370	3					RbGa6.31In0.69${}^{*}$, RbIn4, In
Rb7	RbGa3.5Hg3.5	170	1.99	497	7.13	1427	7.11	700	20	450	2			RbGa7${}^{*}$ (+Hg)
Rb8	RbGaHg3	246	2.88	187	2.68	1608	8.02	650	50	400	2			Rb3Ga12.58Hg0.42${}^{*}$ + Rb5Hg19 [28]
Cs1	Cs3Ga42In5	104	0.77	751	10.8	148	1.28	550	5	200	20			CsGa6.39In0.61, (+Ga)
Cs2	Cs3Ga37In10	97	0.73	624	8.97	278	2.42	550	5	200	20			CsGa6.45In0.55, In
Cs3	CsGa4.5In4.5	137	1.04	326	4.67	536	4.67	550	5	200	20			CsGa6.05In0.95${}^{*}$, CsGa1.57In1.43, In
Cs4	CsGaHg3	345	2.60	173	2.48	1500	7.48	700	20	450	2			CsGa7${}^{*}$, Cs5Hg19 [28]

, together with the weighed sample compositions and the results of the X-ray powder diffraction data. For the latter, 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 transmission powder diffraction systems (STADI-P, Dectris Mythen 1K detector, Stoe & Cie, Darmstadt, Mo$K_{\alpha 1}$ radiation). For the phase analysis, the measured diffraction data were compared to the calculated (program Lazy-Pulverix [24]) reflections of the title compound and other known phases in the ternary system K/Rb/Cs–Ga–In/Hg.

In the ternary system K–Ga–In, indium richer samples at the pseudo-binary section A₃M₁₃ (e.g., samples K1 and K2; Table 1) yielded members of the recently newly obtained orthorhombic A₃M₁₁ phases K₃Ga_11-xIn_x ($x=1.16-1.36$, [25]) if the maximum temperature was kept at 500 ${}^{\circ }$C. At a reduced In content, i.e., below a Ga:In ratio of 11:2 (samples K3 and K4), variable amounts (depending on the temperature program) of the A₃M₁₁ trielides (for $T_{\max }$ = 500 ${}^{\circ }$C) and ternary In-substituted derivative of the K₃Ga₁₃-type ($T_{\max }$ = 700 ${}^{\circ }$C) are formed. As the binary gallide K₃Ga₁₃ [20] is known to melt incongruently and the peritectic line at 510 ${}^{\circ }$C meets the liquidus curve at approximately 6% K [18,29], well-developed needle-shaped crystals of the most In-rich K₃Ga₁₃-type phase were obtained from the triel-rich sample K₃Ga₄₂In₅ (K5), again at a maximum temperature of 700 ${}^{\circ }$C. By-products were In-substituted KGa₃-type phases [5,22] and elemental gallium. Attempts to crystallize the incongruently melting Ga-rich gallides from liquid mercury in samples of overall composition KGa₃Hg₃ (K6) and KGaHg₃ (K7) yielded—together with KHg₆ [26] and K₂Hg₇ [27], respectively—nice needle-shaped crystals of the ternary Hg-containing derivative K₃Ga_12.65Hg_0.35 of the K₃Ga₁₃-type.

In the corresponding rubidium system Rb–Ga–In, an analogous border phase Rb₃Ga₁₃ is unknown. Here, RbGa₇ [7,13] is the most Ga-rich compound of the binary system Rb–Ga [15,16,17]. This compound, similar to K₃Ga₁₃, exhibits incongruent melting behavior, with a peritectic temperature of 354 ${}^{\circ }$C. The phase diagram substantiates that the growth of the crystals of this compound is only possible from very Ga-rich samples with a Rb content of less than 1%. As the structures of the RbGa₇-type do not take up mercury, the crystallization of well-developed crystals of RbGa₇ was possible from a sample RbGa_3.5Hg_3.5 (Rb7; cf. Table 1). An increased amount of rubidium (Rb:Ga = 1:1, sample RbGaHg₃, Rb8) yielded the Rb-analog compound Rb₃Ga_12.58Hg_0.42 of the K₃Ga₁₃-type (together with the byproduct Rb₅Hg₁₉), even though the pure gallide Rb₃Ga₁₃ is unknown. Likewise, several In-substituted derivatives of this unknown gallide and of RbGa₇ could be obtained, again from triel-rich samples. In accordance with the incongruent melting of both types of compounds, samples with an overall composition $AM$₇ (i.e., 12.5% Rb; Rb1 and Rb2) yielded crystals of ternary KGa₃-type derivatives only [22]. At a reduced Rb content of 6% (A₃M₄₇) and for In poorer samples [Rb₃Ga₃₂In₁₅ (Rb3) still contained KGa₃-type compounds besides elemental In], i.e., from the samples Rb4 (Rb₃Ga₃₇In₁₀) and Rb5a-c (Rb₃Ga₄₂In₅), ternary In-derivatives of RbGa₇ and the K₃Ga₁₃-type structure could be obtained. The In-rich sample RbGa_1.5In_4.5 (Rb4) resulted in the formation of the most In-rich derivative of the RbGa₇-type compounds, RbGa_6.31In_0.69, in addition to RbIn₄ and an excess of elemental In.

The only known binary Ga-rich Cs-gallide is CsGa₇ [18]. Its peritectic isothermal line at 426 °C intersects the liquidus curve at approximately 3% Cs [15,18,19], which again allows the single crystal growth only in the presence of a large excess of gallium. Similar to RbGa₇ described above, well-developed crystals of CsGa₇ could be obtained from a melt of mercury (sample Cs4: CsGaHg₃). Indium substituted derivatives of CsGa₇, with an In content of up to 9%, were again obtained from the samples Cs₃M₄₇ (Cs1 and Cs2). The most In-rich phase (13.6% In) was yielded—together with a ternary variant of the KGa₃-type trielides and elemental In—from the sample CsGa_4.5In_4.5 (Cs3).

2.2. Crystal Structure Refinements

The silver-metallic crystals of the title compounds were fixed under dried paraffin oil in glass capillaries (⊘< 0.1 mm) and centered on a diffractometer equipped with a sealed tube and an image plate detector (Stoe IPDS-2) or a micro-source-equipped CCD detector (Apex Quazar).

The reflections of the plate-like crystals of the binary gallides RbGa₇ and CsGa₇, which were obtained from Hg-melts in the samples Rb7 and Cs4 (cf. ${}^{*}$ in Table 1), could be indexed using the expected trigonal rhombohedral unit cell. The refined unit cell parameters (Table 2) are in very good agreement with the transformed values of Belin for the Rb phase (a = 660.3(2), c = 2856.6(3) pm [7]) and with the powder data obtained by van Vucht for the Cs gallide (a = 662.0(1), c = 2904.5(1) pm [14]). The collected single crystal data show no additional extinction conditions, and the symmetry of the intensities corresponds to the high Laue class, both fitting the expected space group $R\overline{3}m$. The refinement of the (standardized and transformed, program Structure Tidy [30]) atomic coordinates of the structure model published by Belin [7] smoothly converged to low residual values $R1$ below 3% (program Shelxl-2013 [31]; cf. Table 2 and

Table 3. Atomic coordinates and equivalent isotropic displacement parameters/pm${}^2$ for the crystal structures of the RbGa₇-type gallides.

Atom	Wyckoff		Parameter	Compounds
	Position			RbGa7	RbGa6.31In0.69	CsGa7	CsGa6.05In0.95
A	$6c$	$0,0,z$	z	0.30521(3)	0.30501(4)	0.30662(2)	0.30623(2)
			$U_{\mathrm{equiv}.}$	285(3)	247(3)	232(2)	222(2)
M(1)	$6c$	$0,0,z$	z	0.04418(4)	0.04480(3)	0.04448(4)	0.04650(3)
			$U_{\mathrm{equiv}.}$	189(3)	179(3)	232(2)	183(3)
			In content %	0	40(2)	0	79(2)
M(2)	$18h$	$x,-x,z$	x	0.47309(6)	0.47145(6)	0.47370(6)	0.46941(6)
			z	0.24384(2)	0.24282(2)	0.24253(2)	0.24043(2)
			$U_{\mathrm{equiv}.}$	205(2)	190(2)	199(2)	195(2)
			In content %	0	10(1)	0	5(1)
Ga(3)	$18h$	$x,-x,z$	x	0.53448(7)	0.53474(7)	0.53448(7)	0.53470(6)
			z	0.14785(2)	0.14828(2)	0.14812(2)	0.14897(2)
			$U_{\mathrm{equiv}.}$	260(3)	208(2)	249(2)	221(2)

). The data for the isotypic mixed Ga/In trielides with the maximum indium proportions were collected using crystals obtained from the In richer samples Rb6 and Cs3. In agreement with the incorporation of the larger indium atoms, the lattice parameters are slightly (approximately 1%) increased with respect to the pure gallides. During the structure refinement, the Ga(3) position turned out to remain a pure gallium site. The Ga(2) position, which forms the icosahedra together with Ga(3), contains only very small amounts of indium. The four-bonded M(1) position shows the highest In-proportions of up to 80% in the Cs compound. In the final structure refinements, the two statistically occupied positions M(1) and M(2) were treated using constrained positional and thermal parameters. The overall compositions for the two In richest border phases are finally RbGa_6.31In_0.69 (9.9% In) and CsGa_6.05In_0.95 (13.6% In). The crystal data of the four RbGa₇-type compounds are collected in Table 2 and Table 3; selected interatomic distances are listed in Table 4 [32].

Table 2. Crystallographic data and details of the data collection and structure refinement of the RbGa₇-type gallides.

Compound		RbGa7	RbGa6.31In0.69	CsGa7	CsGa6.05In0.95
In/Hg content %		0	9.9(2)	0	13.6(4)
Structure type, Pearson symbol		RbGa7 [7], $tR$48
Crystal system, space group		trigonal, $R\overline{3}m$, No. 166
Lattice parameters, pm	a	661.1(2)	662.96(3)	661.76(8)	668.46(2)
	c	2860.5(8)	2887.1(2)	2904.1(3)	2961.16(14)
Volume of the u.c., ${10}^6$ pm${}^3$		1082.7(7)	1098.94(11)	1101.4(3)	1145.89(9)
Z		6
Density (X-ray), gcm${}^{-3}$		5.28	5.48	5.62	5.77
Diffractometer		APEX	IPDS	APEX	IPDS
Radiation		Mo-$K_{\alpha }$	Ag-$K_{\alpha }$	Mo-$K_{\alpha }$	Ag-$K_{\alpha }$
Absorption coeff. ${\mu }_{\mathrm{Mo}/\mathrm{Ag}-K\alpha }$ /mm${}^{-1}$		32.31	16.60	10.02	14.93
$\theta$ range, deg.		2.1–30.1	1.7–27.9	2.1–30.0	1.6–26.8
No. of reflections collected		8957	12,875	6396	12,050
No. of independent reflections		452	725	454	675
$R_{\mathrm{int}}$		0.0733	0.0491	0.0525	0.0473
Corrections		Lorentz, polarization, absorption:
		Sadabs  [33]	Xshape [34]	Sadabs	Xshape
Structure refinement		Shelxl-2013  [31]
No. of free parameters		20	22	20	22
Goodness-of-fit on $F^2$		1.133	1.285	1.200	1.200
R Values (for refl. with I ≥ 2$\sigma$(I))	R1	0.0291	0.0428	0.0253	0.0349
	$wR$2	0.0865	0.0879	0.0721	0.0715
R Values (all data)	R1	0.0314	0.0481	0.0254	0.0394
	$wR$2	0.0879	0.0897	0.0721	0.0732
Residual elect. density, e${}^{-}\times$10${}^{-6}$pm${}^{-3}$		$+1.5$/$-1.0$	$+2.4$/$-1.8$	$+1.8$/$-1.3$	$+1.7$/$-1.8$

Table 2. Crystallographic data and details of the data collection and structure refinement of the RbGa₇-type gallides.

Compound		RbGa7	RbGa6.31In0.69	CsGa7	CsGa6.05In0.95
In/Hg content %		0	9.9(2)	0	13.6(4)
Structure type, Pearson symbol		RbGa7 [7], $tR$48
Crystal system, space group		trigonal, $R\overline{3}m$, No. 166
Lattice parameters, pm	a	661.1(2)	662.96(3)	661.76(8)	668.46(2)
	c	2860.5(8)	2887.1(2)	2904.1(3)	2961.16(14)
Volume of the u.c., ${10}^6$ pm${}^3$		1082.7(7)	1098.94(11)	1101.4(3)	1145.89(9)
Z		6
Density (X-ray), gcm${}^{-3}$		5.28	5.48	5.62	5.77
Diffractometer		APEX	IPDS	APEX	IPDS
Radiation		Mo-$K_{\alpha }$	Ag-$K_{\alpha }$	Mo-$K_{\alpha }$	Ag-$K_{\alpha }$
Absorption coeff. ${\mu }_{\mathrm{Mo}/\mathrm{Ag}-K\alpha }$ /mm${}^{-1}$		32.31	16.60	10.02	14.93
$\theta$ range, deg.		2.1–30.1	1.7–27.9	2.1–30.0	1.6–26.8
No. of reflections collected		8957	12,875	6396	12,050
No. of independent reflections		452	725	454	675
$R_{\mathrm{int}}$		0.0733	0.0491	0.0525	0.0473
Corrections		Lorentz, polarization, absorption:
		Sadabs  [33]	Xshape [34]	Sadabs	Xshape
Structure refinement		Shelxl-2013  [31]
No. of free parameters		20	22	20	22
Goodness-of-fit on $F^2$		1.133	1.285	1.200	1.200
R Values (for refl. with I ≥ 2$\sigma$(I))	R1	0.0291	0.0428	0.0253	0.0349
	$wR$2	0.0865	0.0879	0.0721	0.0715
R Values (all data)	R1	0.0314	0.0481	0.0254	0.0394
	$wR$2	0.0879	0.0897	0.0721	0.0732
Residual elect. density, e${}^{-}\times$10${}^{-6}$pm${}^{-3}$		$+1.5$/$-1.0$	$+2.4$/$-1.8$	$+1.8$/$-1.3$	$+1.7$/$-1.8$

Needle-shaped crystals of the ternary In- and Hg-containing derivatives of K₃Ga₁₃ were obtained from the samples K5 and K7. Their diffraction images exhibit the expected orthorhombic symmetry with extinction conditions compatible with the space group of this structure type ($Cmcm$). The lattice parameters, mainly the length of the b axis, are slightly enlarged with respect to the values of K₃Ga₁₃ itself (a = 644.1(3), b = 1614.3(3), c = 2840.4(7) pm [20]). The structure model of K₃Ga₁₃, which contains eleven Ga and three K sites, was used for the refinement of the structures, and all gallium sites were checked for a possible mixed Ga/In or Ga/Hg occupation. In the case of the In gallides, the cluster site M(11) exhibits the largest substitution. In contrast, the likewise larger Hg atoms are incorporated at the three-bonded position M(1) exclusively. Although the indium and mercury content of the samples K5 and K7 was much higher, the final resulting compositions K₃Ga_12.85In_0.15 (1.2% In) and K₃Ga_12.65Hg_0.35 (2.7% Hg) indicate only a small substitution of gallium by the larger atoms In and Hg. Ternary In and Hg containing derivatives of the unknown gallide Rb₃Ga₁₃ were obtained from the samples Rb4, Rb5a, and Rb8. Here, the proportion of larger M atoms is increased compared to the potassium phases, but the site preferences (In mainly at M(11), Hg at M(1)) are nevertheless similar. The phase width x of the In-derivatives Rb₃Ga_13-xIn_x reaches from x = 0.68 to 1.07; the Hg content in Rb₃Ga_13-xHg_x is x = 0.42. The atomic parameters are given in

Table 5. Atomic coordinates and equivalent isotropic displacement parameters/pm${}^2$ for the crystal structures of the ternary gallides forming the K₃Ga₁₃-type structure.

Atom	Wyckoff	Parameter	Compounds
	Position		K3Ga12.85In0.15	K3Ga12.65Hg0.35	Rb3Ga12.32In0.68	Rb3Ga11.93In1.07	Rb3Ga12.58Hg0.42
A(1)	$0,y,z$	y	0.13795(12)	0.13783(7)	0.13645(12)	0.13671(11)	0.13737(3)
	$8f$	z	0.67963(9)	0.68175(6)	0.67918(8)	0.67911(8)	0.68110(2)
		$U_{\mathrm{equiv}.}$	383(5)	403(3)	423(4)	352(4)	331.4(14)
A(2)	$0,y,z$	y	0.28682(10)	0.28712(7)	0.28734(11)	0.28720(10)	0.28777(3)
	$8f$	z	0.58520(7)	0.58578(4)	0.58676(6)	0.58600(6)	0.58732(2)
		$U_{\mathrm{equiv}.}$	260(3)	281(2)	325(3)	238(3)	253.9(12)
A(3)	$0,y,z$	y	0.48479(12)	0.48475(8)	0.48312(13)	0.48414(12)	0.48259(4)
	$8f$	z	0.13881(6)	0.14082(4)	0.13938(6)	0.13920(6)	0.14143(2)
		$U_{\mathrm{equiv}.}$	263(3)	288(2)	353(4)	279(4)	268.4(12)
M(1)	$0,y,1/4$	y	0.42740(7)	0.42126(2)	0.43336(19)	0.43480(18)	0.42177(2)
	$4c$	$U_{\mathrm{equiv}.}$	197(4)	223.6(12)	330(10)	258(10)	223.9(11)
		In/Hg content %	8(1)	70.2(3)	9(3)	11(3)	84.2(3)
M(2)	$0,y,z$	y	0.29859(5)	0.29712(3)	0.29907(11)	0.29951(10)	0.29733(3)
	$8f$	z	0.07135(3)	0.07256(2)	0.07066(6)	0.07024(6)	0.07232(2)
		$U_{\mathrm{equiv}.}$	156(2)	183.7(11)	286(6)	221(6)	190.8(12)
		In content %	0	0	19(2)	38(2)	0
M(11)	$0,y,1/4$	y	0.27368(6)	0.26919(5)	0.27782(15)	0.27812(13)	0.26952(5)
	$4c$	$U_{\mathrm{equiv}.}$	218(4)	236(2)	317(7)	243(7)	231(2)
		In content %	21.3(14)	0	50(3)	66(3)	0
M(12)	$0,y,z$	y	0.24087(6)	0.24034(3)	0.23987(14)	0.23945(12)	0.23994(4)
	$8f$	z	0.15536(3)	0.15622(2)	0.15540(7)	0.15531(7)	0.15660(2)
		$U_{\mathrm{equiv}.}$	206(2)	230.1(12)	328(7)	248(7)	221.2(13)
		In content %	0	0	12(2)	14(2)	0
Ga(13)	$0,y,z$	y	0.07324(5)	0.07268(3)	0.07614(13)	0.07567(12)	0.07468(4)
	$8f$	z	0.17112(3)	0.17207(2)	0.17079(6)	0.17085(7)	0.17197(2)
		$U_{\mathrm{equiv}.}$	166(2)	196.5(11)	276(4)	198(4)	197.9(12)
Ga(14)	$x,y,1/4$	x	0.20858(13)	0.21106(8)	0.2067(3)	0.2058(3)	0.21094(9)
	$8g$	y	0.03252(4)	0.03251(3)	0.03627(12)	0.03662(11)	0.03434(3)
		$U_{\mathrm{equiv}.}$	162(2)	195.8(11)	275(4)	193(4)	193.9(12)
Ga(15)	$x,y,z$	x	0.30086(9)	0.29812(5)	0.3005(2)	0.3016(2)	0.29682(6)
	$16h$	y	0.17081(3)	0.17127(2)	0.17100(9)	0.17052(8)	0.17117(2)
		z	0.20307(2)	0.20354(2)	0.20351(4)	0.20348(5)	0.20405(2)
		$U_{\mathrm{equiv}.}$	172.4(12)	201.1(10)	281(3)	200(3)	200.3(10)
M(21)	$0,y,z$	y	0.15655(5)	0.15598(3)	0.15621(12)	0.15574(11)	0.15602(3)
	$8f$	z	0.03256(3)	0.03326(2)	0.03155(7)	0.03139(7)	0.03254(2)
		$U_{\mathrm{equiv}.}$	158(2)	193.3(11)	287(6)	222(7)	196.3(12)
		In content %	0	0	8(2)	16(2)	0
Ga(22)	$0,y,z$	y	0.01576(5)	0.01476(3)	0.01861(13)	0.01893(12)	0.01738(4)
	$8f$	z	0.08953(3)	0.09067(2)	0.08906(6)	0.08900(7)	0.09015(2)
		$U_{\mathrm{equiv}.}$	157(2)	187.5(11)	270(4)	196(4)	193.0(12)
Ga(23)	$x,y,z$	x	0.29649(8)	0.29421(5)	0.2977(2)	0.2985(2)	0.29392(6)
	$16h$	y	0.39025(3)	0.38993(2)	0.39220(9)	0.39358(8)	0.39098(2)
		z	0.04646(2)	0.04670(2)	0.04621(4)	0.04628(5)	0.04634(2)
		$U_{\mathrm{equiv}.}$	152.8(11)	182.1(9)	273(3)	196(3)	187.3(9)
Ga(24)	$x,y,z$	x	0.30718(8)	0.30599(5)	0.30700(19)	0.3074(2)	0.30542(6)
	$16h$	y	0.04892(3)	0.04843(2)	0.04845(9)	0.04815(8)	0.04796(2)
		z	0.03563(2)	0.03603(2)	0.03483(4)	0.03484(4)	0.03541(2)
		$U_{\mathrm{equiv}.}$	140.4(11)	169.5(9)	256(3)	175(3)	173.8(9)

[32], and Table 6 contains the crystal data of the new ternary members of the K₃Ga₁₃-type.

Table 4. Selected interatomic distances/pm in the crystal structures of the compounds forming the RbGa₇-type structure.

Atoms		Distance in				Label	Freq.	CN
		RbGa7	RbGa6.31In0.69	CsGa7	CsGa6.05In0.95
A	- M(2)	372.5(1)	378.0(1)	377.3(1)	388.5(1)		3×
	- Ga(3)	374.4(1)	378.0(1)	383.8(1)	391.9(1)		3×
	- M(2)	375.5(1)	378.4(1)	380.8(1)	388.7(1)		6×
	- M(1)	384.4(1)	385.7(1)	385.5(1)	390.2(1)		3×
	- A	414.2(1)	416.2(1)	412.4(1)	418.0(1)		3×
	- M(1)	434.1(1)	437.1(1)	434.4(1)	443.2(1)		3×	15(+3) + 3
M(1)	- M(1)	252.7(2)	258.7(2)	258.3(2)	275.4(2)	a
	- M(2)	256.8(1)	260.1(1)	258.9(1)	266.5(1)	b	3×
	- A	384.4(1)	385.7(1)	385.5(1)	390.2(2)		3×
	- A	434.1(1)	437.1(1)	434.4(1)	443.2(1)			4 + 3(+1)
M(2)	- M(1)	256.8(1)	260.1(1)	258.9(1)	266.5(1)	b
	- Ga(3)	263.9(1)	264.1(1)	263.9(1)	264.6(1)	c	2×
	- M(2)	277.2(1)	274.7(1)	278.7(1)	272.9(1)	d	2×
	- Ga(3)	283.5(1)	282.5(1)	282.9(1)	281.2(1)	e
	- A	372.5(1)	378.0(1)	377.3(1)	388.5(1)
	- A	375.5(1)	378.4(1)	380.8(1)	390.2(2)		2×	6 + 3
Ga(3)	- Ga(3)	254.3(1)	254.5(1)	254.5(1)	255.6(1)	f	2×
	- Ga(3)	262.2(2)	262.4(1)	262.4(1)	264.6(1)	g	2×
	- M(2)	263.9(1)	264.1(1)	263.9(1)	264.6(1)	c	2×
	- M(2)	283.5(1)	282.5(1)	282.8(1)	281.2(1)	e
	- A	374.4(1)	378.0(1)	383.8(1)	391.9(1)			7 + 1

Table 4. Selected interatomic distances/pm in the crystal structures of the compounds forming the RbGa₇-type structure.

Atoms		Distance in				Label	Freq.	CN
		RbGa7	RbGa6.31In0.69	CsGa7	CsGa6.05In0.95
A	- M(2)	372.5(1)	378.0(1)	377.3(1)	388.5(1)		3×
	- Ga(3)	374.4(1)	378.0(1)	383.8(1)	391.9(1)		3×
	- M(2)	375.5(1)	378.4(1)	380.8(1)	388.7(1)		6×
	- M(1)	384.4(1)	385.7(1)	385.5(1)	390.2(1)		3×
	- A	414.2(1)	416.2(1)	412.4(1)	418.0(1)		3×
	- M(1)	434.1(1)	437.1(1)	434.4(1)	443.2(1)		3×	15(+3) + 3
M(1)	- M(1)	252.7(2)	258.7(2)	258.3(2)	275.4(2)	a
	- M(2)	256.8(1)	260.1(1)	258.9(1)	266.5(1)	b	3×
	- A	384.4(1)	385.7(1)	385.5(1)	390.2(2)		3×
	- A	434.1(1)	437.1(1)	434.4(1)	443.2(1)			4 + 3(+1)
M(2)	- M(1)	256.8(1)	260.1(1)	258.9(1)	266.5(1)	b
	- Ga(3)	263.9(1)	264.1(1)	263.9(1)	264.6(1)	c	2×
	- M(2)	277.2(1)	274.7(1)	278.7(1)	272.9(1)	d	2×
	- Ga(3)	283.5(1)	282.5(1)	282.9(1)	281.2(1)	e
	- A	372.5(1)	378.0(1)	377.3(1)	388.5(1)
	- A	375.5(1)	378.4(1)	380.8(1)	390.2(2)		2×	6 + 3
Ga(3)	- Ga(3)	254.3(1)	254.5(1)	254.5(1)	255.6(1)	f	2×
	- Ga(3)	262.2(2)	262.4(1)	262.4(1)	264.6(1)	g	2×
	- M(2)	263.9(1)	264.1(1)	263.9(1)	264.6(1)	c	2×
	- M(2)	283.5(1)	282.5(1)	282.8(1)	281.2(1)	e
	- A	374.4(1)	378.0(1)	383.8(1)	391.9(1)			7 + 1

2.3. Band Structure Calculations

DFT calculations of the electronic band structure were performed for the three binary gallides RbGa₇, CsGa₇, and K₃Ga₁₃ and the two model compounds “CsGa₆In” (M(1) fully occupied by In; crystal data taken from CsGa_6.05In_0.95, Table 2 and Table 3) and “Rb₆Ga₂₅Hg” (M(1) fully occupied by Hg; data from Rb₃Ga_12.58Hg_0.42, Table 5 and Table 6). $\alpha$-rhombohedral boron (for the data, cf. [35]) was calculated likewise for comparison. In the FP-LAPW (Full-Potential Linear Augmented Plane Wave) method implemented in the program Wien2k [36], the exchange-correlation contribution is described by the Generalized Gradient Approximation (GGA) of Perdew, Burke, and Ernzerhof [37]. Muffin-tin radii were chosen as 121.7 pm (2.3 a.u.) for all heavy atoms and 79.4 pm (1.5 a.u.) for boron. The cutoff energies used are $E_{\max }^{\mathrm{pot}}$ = 190 eV (potential) and $E_{\max }^{\mathrm{wf}}$ = 170 eV (interstitial PW). The calculations were carried out on a Monkhorst–Pack grid of 21 × 21 × 21 (for the small primitive unit cells of RbGa₇-type/$\alpha$-rhomb. B) and 16 × 16 × 3 for the larger orthorhombic K₃Ga₁₃-type compounds (cf. Table 9 for further details). Electron densities and Fermi surfaces were calculated and visualized using the programs XCrysDen [38] and DrawXTL [39]. Bader AIM analyses of the electron density maps were performed to evaluate the charge distribution between the atoms and the heights and positions of the bond, ring, and cage critical points [40] using the program Critic2 [41,42]. Selected results of these analyses (Bader volumes and charges) are summarized in Table 9. The total (tDOS) and some selected partial densities of state (pDOS) are depicted in Figure 3 (CsGa₇ and CsGa₆In) and Figure 4 (K₃Ga₁₃). The band structure and key energy-selected valence electron densities of CsGa₇ and $\alpha$-rhombohedral boron can be found in Figure 5.

3. Results and Discussion

3.1. Synthesis, Crystal Growth, and Phase Widths

The most Ga-rich alkali gallides RbGa₇ [7,13] and CsGa₇ [18] both exhibit a peritectic melting behavior. As their peritectic lines at 354/426 ${}^{\circ }$C (for A = Rb/Cs) meet the liquidus curves at very low A proportions (<3%, [15,18,19]), single crystal growth of these compounds requires a large excess of gallium. Due to the lack of a suitable position in the structure of the RbGa₇-type, these compounds—in contrast to the A₃M₁₃ phases (see below)—do not take up any mercury, and the crystallization of well-developed plate-shaped crystals of both hepta-gallides was possible from Hg-rich melts applying slow cooling rates of 2 ${}^{\circ }$Ch${}^{-1}$ (samples RbGa_3.5Hg_3.5 (Rb7), and CsGaHg₃ (Cs4); cf. Table 1). Similarly, indium-substituted derivatives AGa_7-xIn_x of RbGa₇ and CsGa₇ were obtained from triel-rich samples of overall composition A₃M₄₇, inevitably with elemental Ga or In as byproducts. The most In-rich phases of this structure type, which contain x = 0.69/0.95 In and represent the border compositions of the phase width x, were still yielded from the A and In richer samples RbGa_1.5In_4.5 and CsGa_4.5In_4.5, in these cases with elemental indium and ternary BaAl₄- and KGa₃-type trielides as by-products.

Similar to RbGa₇ and CsGa₇, the most Ga-rich gallide of the K–Ga system, K₃Ga₁₃ [20] also melts incongruently at 510 ${}^{\circ }$C. Here, the liquidus curve is touched at approximately 6% K [18,29]. In accordance, needle-shaped single crystals of the most In-rich ternary K₃Ga₁₃-type phase, which however contains only 1.2% of In, were obtained from the triel-rich samples with the appropriate A content, like, e.g., K₃Ga₄₂In₅, applying slow cooling rates. The latter K-poor sample also proves the lack of mixed $AM$₇ phases for the alkali element potassium. Although the analog binary gallide Rb₃Ga₁₃ is unknown, In-containing ternary phases of the K₃Ga₁₃-type structure, within the small composition range $x_1$ = 0.69 to $x_2$ = 1.07 (for Rb₃Ga_13-xIn_x), were yielded from triel-rich samples Rb₃M₄₇ of variable In proportions ($x_1$: sample Rb₃Ga₄₂In₅, Rb5a; $x_2$: sample Rb₃Ga₃₇In₁₀, Rb4).

Additionally, attempts to crystallize the incongruently melting potassium and rubidium gallides (which were initially obtained as by-products of our work on mixed Hg/triel intermetallics [43]) from liquid mercury yielded (together with the Hg-rich potassium mercurides like KHg₆, K₂Hg₇ and Rb₅Hg₁₉) the ternary Hg-containing K₃Ga₁₃-type derivatives K₃Ga_12.65Hg_0.35 (from samples of as diverse compositions as KGaHg₃ and KGa₃Hg₃) and Rb₃Ga_12.58Hg_0.42 (from the sample RbGaHg₃, Rb8). As described above, an increased Ga:Rb ratio (sample RbGa₃Hg₃, Rb7) favors the formation of the Ga-rich gallide RbGa₇.

3.2. Crystal Structure Descriptions

In accordance with the high Ga content and the large electronegativity of gallium, the RbGa₇-type structure [7] shows distinct structural similarities with the $\alpha$-rhombohedral allotrope of boron: analogous to the structure of $\alpha$-rhombohedral boron, the two binary hepta-gallides AGa₇ (A = Rb, Cs) and their slightly In-substituted derivatives contain hexagonal close-packed layers of empty closo [Ga(2,3)₁₂] icosahedra (green polyhedra in Figure 1). Within these layers, the icosahedra (point group $\overline{3}m$) are condensed via [Ga(3)₃] triangles (magenta planes in Figure 1c). In contrast to the structure of $\alpha$-boron, where the icosahedra of adjacent layers are directly connected via exo-bonds, the [Ga₁₂] icosahedra in AGa₇ are further connected via dumbbells of four-bonded M(1) atoms to form a three-dimensional network. According to the likewise rhombohedral symmetry, the icosahedra stacking sequence along [001] is |:ABC:|. Compared to an ideal cubic close packing, the intra-layer distances between the icosahedra (cf. the lengths of the a axes of 660-670 pm) are much smaller than the inter-layer distances (1026 pm for RbGa₇). This is evidently a consequence of the [M(1)₂] dumbbells, which occupy the octahedral voids of the icosahedra packing. The shortest interatomic Ga–Ga distances a of 252.7/258.3 pm (for RbGa₇/CsGa₇) are found within these dumbbells, which are formed by the four-bonded Ga(1) atoms (Table 4).

Due to the preferred indium substitution of this site (cf. the “coloring” discussion in Section 3.4), this bond length increases significantly with the increasing In proportion of M(1). Nevertheless, with respect to the high In substitution of 80% in CsGa_6.05In_0.95, the M(1)–M(1) distance of 275.4 pm still represents a short and strong bond. The bonds b, which connect the dumbbells with the icosahedra, are also considerably short and lengthen similarly with the In content of the M(1) position. The endohedral bonds within the [Ga₁₂] icosahedra (c-f) vary between 254.3/255.6 and 283.5/282.9 pm for the two binary gallides. Due to the only very minimal substitution of the cluster-forming position M(2) (and the always pure gallium site Ga(3)), these distances are mainly unchanged in the two ternary mixed derivatives. Remarkably, the endohedral bond f is much shorter than the three other distances within the cluster, so that the icosahedra are slightly “stretched“ along the crystallographic three-fold axis. The Ga(3)–Ga(3) distances within the connecting [Ga(3)₃] triangles (g: 262.4-264.6 pm for all four compounds) are also in the range of the intra-cluster values (cf. The discussion of the electronic structure and the comparison with the related $\alpha$-rhombohedral boron structure in Section 3.3).

The Rb/Cs cations (Wyckoff position $6c$: $3m$ symmetry) take the elongated tetrahedral voids of the icosahedra packing, which is apparent from the four icosahedra arranged around the yellow cation coordination polyhedron (CCP) in Figure 1b. According to the subdivided doubled formula:

$$\underset{\mathrm{tetrah}.\mathrm{v}.}{\underbrace{{\mathrm{Rb}}_2}}\underset{\mathrm{octah}.\mathrm{v}.}{\underbrace{{\left[\mathrm{Ga}\right(1)}_2]}}\underset{\mathbf{ABC}}{\underbrace{\left[{\mathrm{Ga}(2,3)}_{12}\right]}}$$

the dense icosahedra packing provides one octahedral ([Ga₂] dumbbell) and two tetrahedral voids (Rb) per sphere/icosahedron. The coordination sphere of the alkali cations thus consists of 15(+3) M anion atoms (Table 4), of which 4 × 3 are the [Ga₃] faces of the four adjacent icosahedra and 3(+3) are the M(1) atoms of the three surrounding dumbbells (Figure 1b). The Rb–M distances vary from 372 to 385 pm (+ 3×M at ≈ 440 pm). As expected, the corresponding values in the Cs compounds are at 377 to 418 pm (again + 3× ≈ 440 pm) somewhat larger (Table 4). The shortest A–A distances are 414 to 418 pm (3×), and the Rb/Cs cations form As-analog layers among each other, which are of course stacked along c similar to As in rhombohedral $\alpha$-arsenic.

Table 6. Crystallographic data and details of the data collection and structure refinement of the ternary K₃Ga₁₃-type compounds.

Compound		K3Ga12.85In0.15	K3Ga12.65Hg0.35	Rb3Ga12.32In0.68	Rb3Ga11.93In1.07	Rb3Ga12.58Hg0.42
In/Hg content, %		1.2(1)	2.7(2)	5.2(2)	8.2(3)	3.2(1)
Structure type, Pearson symbol		K3Ga13 [20], $oS$128
Crystal system, space group		orthorhombic, $Cmcm$, no. 63
Lattice parameters, pm	a	645.55(3)	642.98(3)	650.67(3)	653.2(1)	644.6(1)
	b	1626.1(1)	1641.0(1)	1654.3(1)	1665.9(1)	1662.7(1)
	c	2853.8(1)	2841.1(2)	2894.3(1)	2900.1(3)	2871.7(2)
Volume of the u.c., ${10}^6$ pm${}^3$		2995.7(3)	2997.8(3)	3115.5(3)	3156.1(5)	3077.8(4)
Z		8
Density (X-ray), gcm${}^{-3}$		4.57	4.74	5.09	5.10	5.26
Diffractometer		IPDS	APEX	IPDS	IPDS	APEX
Radiation		Ag-K${}_{\alpha }$	Mo-K${}_{\alpha }$	Ag-K${}_{\alpha }$	Ag-K${}_{\alpha }$	Mo-K${}_{\alpha }$
Absorption coeff. ${\mu }_{\mathrm{Mo}/\mathrm{Ag}-K\alpha }$, mm${}^{-1}$		12.42	26.70	16.49	16.23	35.19
$\theta$ range, deg.		1.1–24.6	1.4–30.1	1.1–25.7	1.9–26.8	1.4–30.2
No. of reflections collected		26,396	25,920	22,235	22,457	26,694
No. of independent reflections		2798	2413	3271	3710	2511
$R_{\mathrm{int}}$		0.0792	0.0411	0.1544	0.1906	0.0348
Corrections		Lorentz, Polarization, Absorption:
		Xshape [34]	Sadabs [33]	Xshape	Xshape	Sadabs
Structure refinement		Shelxl-2013  [31]
No. of free parameter		93	92	96	96	92
Goodness-of-fit on $F^2$		1.086	1.194	1.175	0.975	1.236
R Values (for refl. with I ≥ 2$\sigma$(I))	R1	0.0398	0.0238	0.0873	0.0770	0.0223
	$wR$2	0.0603	0.0586	0.1575	0.1132	0.0561
R Values (all data)	R1	0.0626	0.0250	0.1137	0.1617	0.0250
	$wR$2	0.0649	0.0591	0.1690	0.1355	0.0570
Residual elect. density, e${}^{-}\times$10${}^{-6}$pm${}^{-3}$		$+0.9$/$-1.4$	$+2.6$/$-1.4$	$+3.2$/$-1.7$	$+1.9$/$-2.3$	$+1.9$/$-1.3$

Table 6. Crystallographic data and details of the data collection and structure refinement of the ternary K₃Ga₁₃-type compounds.

Compound		K3Ga12.85In0.15	K3Ga12.65Hg0.35	Rb3Ga12.32In0.68	Rb3Ga11.93In1.07	Rb3Ga12.58Hg0.42
In/Hg content, %		1.2(1)	2.7(2)	5.2(2)	8.2(3)	3.2(1)
Structure type, Pearson symbol		K3Ga13 [20], $oS$128
Crystal system, space group		orthorhombic, $Cmcm$, no. 63
Lattice parameters, pm	a	645.55(3)	642.98(3)	650.67(3)	653.2(1)	644.6(1)
	b	1626.1(1)	1641.0(1)	1654.3(1)	1665.9(1)	1662.7(1)
	c	2853.8(1)	2841.1(2)	2894.3(1)	2900.1(3)	2871.7(2)
Volume of the u.c., ${10}^6$ pm${}^3$		2995.7(3)	2997.8(3)	3115.5(3)	3156.1(5)	3077.8(4)
Z		8
Density (X-ray), gcm${}^{-3}$		4.57	4.74	5.09	5.10	5.26
Diffractometer		IPDS	APEX	IPDS	IPDS	APEX
Radiation		Ag-K${}_{\alpha }$	Mo-K${}_{\alpha }$	Ag-K${}_{\alpha }$	Ag-K${}_{\alpha }$	Mo-K${}_{\alpha }$
Absorption coeff. ${\mu }_{\mathrm{Mo}/\mathrm{Ag}-K\alpha }$, mm${}^{-1}$		12.42	26.70	16.49	16.23	35.19
$\theta$ range, deg.		1.1–24.6	1.4–30.1	1.1–25.7	1.9–26.8	1.4–30.2
No. of reflections collected		26,396	25,920	22,235	22,457	26,694
No. of independent reflections		2798	2413	3271	3710	2511
$R_{\mathrm{int}}$		0.0792	0.0411	0.1544	0.1906	0.0348
Corrections		Lorentz, Polarization, Absorption:
		Xshape [34]	Sadabs [33]	Xshape	Xshape	Sadabs
Structure refinement		Shelxl-2013  [31]
No. of free parameter		93	92	96	96	92
Goodness-of-fit on $F^2$		1.086	1.194	1.175	0.975	1.236
R Values (for refl. with I ≥ 2$\sigma$(I))	R1	0.0398	0.0238	0.0873	0.0770	0.0223
	$wR$2	0.0603	0.0586	0.1575	0.1132	0.0561
R Values (all data)	R1	0.0626	0.0250	0.1137	0.1617	0.0250
	$wR$2	0.0649	0.0591	0.1690	0.1355	0.0570
Residual elect. density, e${}^{-}\times$10${}^{-6}$pm${}^{-3}$		$+0.9$/$-1.4$	$+2.6$/$-1.4$	$+3.2$/$-1.7$	$+1.9$/$-2.3$	$+1.9$/$-1.3$

The orthorhombic structure of the K₃Ga₁₃-type consists of a three-dimensional network of two crystallographically different empty closo clusters, [M₁₂] icosahedra (green polyhedra in Figure 2), and [M₁₁] clusters (blue polyhedra), which are connected among each other and with two additional M atoms M(1) and M(2) via $2e2c$ bonds.

The [M₁₂] icosahedra are located around the Wyckoff position $4a$ of the orthorhombic space group $Cmcm$ and thus exhibit point group symmetry $\frac{2}{m}$... Three of the four M(2N) (N=1 − 4) atoms forming these icosahedra are pure Ga sites; only M(21) contains very small amounts of In in the ternary rubidium trielides. As expected, the intra-cluster bond lengths r to z (cf.

Table 7. Selected interatomic distances/pm of the cluster-forming Ga/In/Hg atoms in the crystal structures of compounds forming the K₃Ga₁₃-type structure (inter-cluster bonds; cf. Table 8).

Atoms		Distance in					Label	Freq.	CN
		K3Ga12.85In0.15	K3Ga12.65Hg0.35	Rb3Ga12.32In0.68	Rb3Ga11.93In1.07	Rb3Ga12.58Hg0.42
M(11)	- M(1)	250.0(2)	249.6(1)	257.3(4)	261.0(4)	253.14(9)	a
	- M(12)	275.3(1)	270.6(1)	280.9(2)	282.1(2)	272.69(7)	i	2×
	- Ga(15)	289.2(1)	282.8(1)	295.9(2)	298.6(2)	284.18(6)	j	4×
	- A(1)	406.4(2)	405.3(1)	409.8(2)	411.2(2)	408.66(6)		4×	7 + 4
M(12)	- M(2)	257.5(1)	255.3(1)	264.1(3)	266.2(3)	260.18(8)	e
	- Ga(15)	263.1(1)	260.1(1)	265.7(2)	267.4(2)	261.23(6)	k	2×
	- M(11)	275.3(1)	270.6(1)	280.9(2)	282.1(2)	272.69(7)	i
	- Ga(13)	276.3(1)	278.8(1)	274.5(3)	276.6(3)	278.29(9)	l
	- A(2)	382.5(1)	381.4(1)	383.8(1)	386.1(2)	381.56(5)		2×
	- A(1)	384.5(1)	385.5(1)	390.4(2)	392.4(2)	387.87(5)		2×
	- A(3)	399.4(2)	403.5(1)	405.1(3)	410.3(3)	405.79(9)			5 + 5
Ga(13)	- Ga(22)	250.9(1)	250.1(1)	255.0(3)	255.5(3)	253.55(8)	f
	- Ga(15)	266.8(1)	266.3(1)	268.0(2)	269.7(2)	266.15(6)	m	2×
	- Ga(14)	270.5(1)	267.9(1)	273.9(2)	273.8(2)	270.56(6)	n	2×
	- M(12)	276.3(1)	278.8(1)	274.5(3)	276.6(3)	278.29(9)	l
	- A(1)	344.3(2)	346.6(1)	352.5(3)	354.6(3)	353.53(9)
	- A(3)	365.2(1)	363.4(1)	371.2(1)	372.0(1)	367.45(4)		2×	6 + 3
Ga(14)	- M(1)	254.2(1)	260.5(1)	255.7(3)	256.3(3)	264.10(6)	b
	- Ga(15)	268.4(1)	269.1(1)	267.4(2)	268.1(2)	268.76(6)	o	2×
	- Ga(14)	269.3(2)	271.4(1)	269.0(4)	268.9(4)	271.95(12)	p
	- Ga(13)	270.5(1)	267.9(1)	273.9(2)	273.8(2)	270.56(6)	n	2×
	- A(1)	367.8(2)	366.3(1)	376.5(3)	379.1(2)	373.03(8)		2×
	- A(3)	377.0(2)	367.0(1)	383.0(2)	384.5(2)	373.27(6)		2×	6 + 4
Ga(15)	- Ga(15)	257.1(1)	259.6(1)	259.6(3)	259.3(3)	261.95(8)	g
	- M(12)	263.1(1)	260.1(1)	265.7(2)	267.4(2)	261.23(6)	k
	- Ga(13)	266.8(1)	266.3(1)	268.0(2)	269.7(2)	266.15(6)	m
	- Ga(15)	267.9(1)		269.1(3)	269.8(3)	263.93(8)	q
	- Ga(14)	268.4(1)	269.1(1)	267.4(2)	268.1(2)	268.76(6)	o
	- M(11)	289.2(1)	282.8(1)	295.9(2)	298.6(2)	284.18(6)	j
	- A(1)	343.1(2)	344.7(1)	351.1(2)	353.4(2)	350.48(7)
	- A(2)	366.6(2)	365.3(1)	368.5(2)	371.3(2)	366.31(7)
	- A(3)	376.4(2)	377.2(1)	384.6(2)	384.7(2)	384.45(7)			6 + 3
M(21)	- M(2)	256.1(1)	257.1(1)	262.1(3)	264.7(3)	261.25(8)	d
	- Ga(24)	264.6(1)	264.4(1)	267.9(2)	269.4(2)	266.66(6)	r	2×
	- Ga(23)	271.9(1)	273.5(1)	272.7(2)	273.5(2)	273.98(7)	s	2×
	- Ga(22)	280.8(1)	283.4(1)	282.0(3)	282.6(3)	283.73(8)	t
	- A(2)	367.8(1)	366.5(1)	374.3(1)	375.2(1)	370.62(4)		2×
	- A(2)	397.3(2)	400.9(1)	405.4(3)	404.8(3)	407.99(9)			6 + 3
Ga(22)	- Ga(13)	250.9(1)	250.1(1)	255.0(3)	255.5(3)	253.55(8)	f
	- Ga(24)	256.7(1)	256.6(1)	258.8(2)	259.5(2)	257.00(5)	u	2×
	- Ga(23)	272.1(1)	274.0(1)	276.5(2)	276.2(2)	278.64(7)	v	2×
	- M(21)	280.8(1)	283.4(1)	282.0(3)	282.6(3)	283.73(8)	t
	- A(3)	355.7(1)	355.1(1)	361.2(1)	362.3(1)	359.06(4)		2×
	- A(1)	358.6(2)	360.1(2)	365.8(3)	368.1(3)	366.62(9)			6 + 3
Ga(23)	- M(2)	252.8(1)	253.7(1)	257.4(2)	259.6(2)	256.32(6)	c
	- Ga(23)	262.8(1)	264.6(1)	263.3(3)	263.3(3)	265.68(8)	w
	- Ga(24)	263.0(1)	269.7(1)	263.2(2)	263.7(2)	263.67(6)	x
	- Ga(24)	268.3(1)	269.7(1)	269.3(2)	268.7(2)	270.58(6)	y
	- M(21)	271.8(1)	273.5(1)	272.7(2)	273.5(2)	273.98(7)	r
	- Ga(22)	272.1(1)	274.0(1)	276.5(2)	276.2(2)	278.64(7)	v
	- A(2)	335.2(2)	338.0(1)	345.4(2)	348.3(2)	346.14(7)
	- A(3)	360.2(2)	362.6(1)	364.5(2)	365.2(2)	365.6(1)
	- A(1)	404.7(2)	408.4(2)	409.5(3)	410.2(2)	411.84(8)			6 + 3
Ga(24)	- Ga(24)	249.0(1)	249.5(1)	251.2(3)	251.6(3)	250.86(8)	h
	- Ga(22)	256.7(1)	256.6(1)	258.8(2)	259.5(2)	257.00(5)	u
	- Ga(24)	258.2(1)		257.6(3)	258.0(3)	258.44(8)	z
	- Ga(23)	263.0(1)	263.9(1)	263.2(2)	263.7(2)	263.68(6)	y
	- M(21)	264.6(1)	264.4(1)	267.9(2)	269.4(2)	266.66(6)	r
	- Ga(23)	268.3(1)	269.7(1)	269.3(2)	268.7(2)	270.57(6)	y
	- A(2)	326.9(2)	329.2(1)	334.9(2)	336.3(2)	335.50(6)
	- A(3)	336.3(2)	339.3(1)	345.0(2)	344.7(2)	346.77(7)			6 + 2

for individual values) between all cluster-forming atoms have values of 257 to 279 pm larger than the respective exo-bonds c, d, f, and h. Similar to the structures of the RbGa₇-type, the icosahedra form hexagonal close packed layers (A) in the a-b plane. Along the a axis, they are directly connected via two adjacent exo-bonds h (i.e., via [Ga(24)₄] rings) to form chains running along [100]. In contrast, no direct bonds occur between the icosahedra along [010]; the connection in this direction is achieved by the four-bonded M(2) atoms only. These differences in the icosahedra connectivity lead to a large b:a ratio of approximately 2.52, which deviates strongly from the ideal orthohexagonal ratio of $\sqrt{3}$ = 1.73. The icosahedra layers are stacked identically (AA’) along c, whereby the layers at $z=0$ (A) and $z=\frac{1}{2}$ (A’) are related by, e.g., the mirror plane at $z=\frac{1}{4}$.

This mirror plane between the icosahedra layers is occupied by layers of [M₁₁] clusters, which are built up by the atoms of the five M(1N) (N = 1–5) sites. Again, this cluster is preferentially formed by gallium. The [M₁₁] clusters of point group $m2m$ symmetry (center at $0,y,\frac{1}{4}$, Wyckoff position $4c$) exhibit a closo shape, as all boundary surfaces are (somewhat distorted) triangles. The shape of that half of the cluster formed by the pure Ga-sites is close to an icosahedron, whereas the opposite half around the M(11) position, which is preferentially occupied by the minority substituent In, is strongly distorted. The endohedral M–M bonds (i to q) are again always longer than the exo-bonds (a, b, e-g; Table 7). The largest endohedral bonds j of 282–299 pm (and the large angles ∠(Ga–M(11)–Ga) of up to 160${}^{\circ }$) illustrate the distortion of the closo cluster in the vicinity of the M(11) atoms. The indented shape of the cluster around this position and the therewith increased bond lengths around M(11) are evidently relevant for the Ga/In distribution in the In-substituted derivatives of the K₃Ga₁₃-type structure (cf. the further discussion in Section 3.4). Similar to the icosahedra, the [M₁₁] clusters are also directly exo-bonded along the a axis via four-membered rings (exo-bonds q, Ga(15)–Ga(15)). Along b, additional M atoms (in this case, the three-bonded M(1) atoms) are connecting the clusters. The two [M₁₁] cluster layers per unit cell are related by inversion centers, and their shift relative to the icosahedra layers A is similar to a B and C stacking of a close sphere packing. Thus, the cluster centers are overall stacked along c according to |:ABA’C:|.

In an alternative description of the structure, the short direct exo-bonds f (along c) and h and q (along a) are connecting the two cluster types to undulated $4^4$ sheets (D) running in the a–c plane around $y=0$ and $y=\frac{1}{2}$ (cf. the dashed magenta lines in Figure 2). Along b, the stacking sequence |:DD’:| follows the unit cell C-centering. In this direction, the cluster layers are connected by the additional atoms M(1) and M(2) only.

The tetrahedrally bonded M(2) atoms ($8f$, two atoms per 2 f.u. ) are connecting three icosahedra via the strong bonds c (2×) and d and one [M₁₁] cluster (bond e). The corresponding bond lengths are among the shortest found within the polyanionic network (253–266 pm;

Table 8. Selected interatomic distances/pm of the K and Rb atoms (top), as well as M(1) and M(2) (bottom) in the crystal structures of compounds forming the K₃Ga₁₃-type structure (intra-cluster bonds; cf. Table 7).

Atoms		Distance in					Label	Freq.	CN
		K3Ga12.85In0.15	K3Ga12.65Hg0.35	Rb3Ga12.32In0.68	Rb3Ga11.93In1.07	Rb3Ga12.58Hg0.42
A(1)	- Ga(15)	343.1(2)	344.7(1)	351.1(2)	353.5(2)	350.5(1)		2×
	- Ga(13)	344.3(2)	346.6(1)	352.5(3)	354.6(3)	353.5(1)
	- Ga(22)	358.6(2)	360.1(2)	365.8(3)	368.1(3)	366.6(1)
	- Ga(14)	367.8(2)	366.3(1)	376.5(3)	379.1(2)	373.0(1)		2×
	- M(12)	384.5(1)	385.5(1)	390.4(2)	392.4(2)	387.9(1)		2×
	- M(1)	394.7(1)	387.8(1)	401.5(2)	403.9(2)	390.8(1)		2×
	- A(2)	362.2(3)	366.6(2)	365.9(3)	368.5(3)	367.5(1)		2×
	- A(3)	397.0(2)	396.7(1)	397.8(2)	400.8(2)	395.8(1)		2×
	- A(1)	401.6(5)	387.8(3)	409.9(4)	411.2(4)	395.8(1)
	- Ga(23)	404.7(2)	408.4(2)	409.5(3)	410.2(2)	411.9(1)		2×
	- M(11)	406.4(2)	405.3(1)	409.8(2)	411.2(4)	408.7(1)		2×	14 + 4
A(2)	- Ga(24)	326.9(2)	329.2(1)	334.9(2)	336.3(2)	335.5(1)		2×
	- Ga(23)	335.2(2)	338.0(1)	345.4(2)	348.3(2)	346.1(1)		2×
	- M(2)	353.6(1)	352.0(1)	358.4(1)	360.1(1)	354.6(1)		2×
	- A(1)	362.2(3)	366.6(2)	365.9(3)	368.5(3)	367.5(1)
	- Ga(15)	366.6(2)	365.3(1)	368.5(2)	371.3(2)	366.3(1)		2×
	- M(21)	367.8(1)	366.5(1)	374.3(1)	375.2(1)	370.6(1)		2×
	- M(12)	382.5(1)	381.4(1)	383.8(1)	386.1(2)	381.6(1)		2×
	- M(21)	397.2(2)	400.9(1)	374.3(1)	375.2(1)	408.0(1)
	- A(3)	401.7(3)	405.7(2)	409.1(3)	411.0(3)	412.3(1)			13 + 2
A(3)	- M(1)	330.7(2)	327.2(1)	330.6(2)	331.7(2)	327.8(1)
	- Ga(24)	336.3(2)	339.3(1)	345.0(2)	344.7(2)	346.8(1)		2×
	- Ga(22)	355.7(1)	355.1(1)	361.2(1)	362.3(1)	359.0(1)		2×
	- M(2)	358.8(2)	363.9(1)	363.7(3)	366.9(3)	366.4(1)
	- Ga(23)	360.2(2)	362.6(1)	364.5(2)	365.2(2)	365.6(1)		2×
	- Ga(13)	365.2(1)	363.4(1)	371.2(1)	372.0(1)	367.4(1)		2×
	- Ga(15)	376.4(2)	377.2(1)	384.6(2)	384.7(2)	384.5(1)		2×
	- Ga(14)	377.0(2)	370.0(1)	383.0(2)	384.5(2)	373.3(1)		2×
	- A(1)	397.0(2)	396.7(1)	397.8(2)	400.8(2)	395.8(1)		2×
	- M(12)	399.4(2)	403.5(1)	405.1(3)	410.3(3)	405.8(1)
	- A(2)	401.7(3)	405.7(2)	409.1(3)	411.0(3)	412.2(1)			15 + 3
M(1)	- M(11)	250.0(2)	249.6(1)	257.3(4)	261.0(4)	253.1(1)	a
	- Ga(14)	254.2(1)	260.5(1)	255.7(3)	256.3(3)	264.1(1)	b	2×
	- A(3)	330.7(2)	327.2(1)	330.6(2)	331.7(2)	327.8(1)		2×
	- A(1)	394.7(1)	387.8(1)	401.5(2)	403.9(2)	390.8(1)		4×	3 + 6
M(2)	- Ga(23)	252.8(1)	253.7(1)	257.4(2)	259.6(2)	256.3(1)	c	2×
	- M(21)	256.1(1)	257.1(1)	262.1(3)	264.7(3)	261.3(1)	d
	- M(12)	257.5(1)	255.3(1)	264.1(3)	266.2(3)	260.2(1)	e
	- A(2)	353.6(1)	352.0(1)	358.4(1)	360.1(1)	354.6(1)		2×
	- A(3)	358.8(2)	363.9(1)	363.7(3)	366.9(3)	366.4(1)			4 + 3

). In the rubidium compounds, for which the pure gallide is not known, this position is nevertheless partly (19–38%) occupied by In, which evidently expands the structure to fit the larger Rb⁺ cations and thus stabilize also a Rb phase of this structure type. With respect to the |:ABA’C:| cluster packing, the M(2) atoms take one half of the tetrahedral voids.

The atoms M(1) at the Wyckoff position $4c$ (i.e., one M(1) per 2 f.u.), which is the only position taken by mercury in the ternary Hg-substituted derivatives, exhibit a pseudo-trigonal planar geometry. The very short bonds a (250–261 pm) and b (254–264 pm, Table 8) connect three [M₁₁] clusters. M(1) is thus located within the B/C[M₁₁] cluster layers taking the common trigonal face of two tetrahedral voids in the h.c.p. stacked slabs of the cluster arrangement.

Three crystallographically different K/Rb cations are incorporated in the polyanionic network, each of them located at an $8f$ position. Their CCPs are depicted as yellow polyhedra at the left-hand side of Figure 2. The A(1) cations are coordinated by 14 triel atoms, of which 12 M atoms belong to four different clusters. They thus occupy one-half of the tetrahedral voids of the cluster packing. The K/Rb atoms A(2) and A(3) with similar coordination numbers of 13 and 15 jointly occupy one large octahedral void within the cluster arrangement, which is accordingly, as described above, elongated along the b axis. Thus, the occupation of the N (two per doubled f.u. A₆M₂₆) octahedra and the $2N$ (4) tetrahedra voids between N (2) clusters can be summarized to:

$$\underset{\frac{1}{2}\mathrm{tetrah}.\mathrm{v}.}{\underbrace{\mathrm{K}(1{)}_2}}\underset{\mathrm{octah}.\mathrm{voids}}{\underbrace{\left[\mathrm{K}\right(2)+{\mathrm{K}\left(3\right)]}_2}}\left[\mathrm{Ga}\right(1\left)\right]\underset{\frac{1}{2}\mathrm{tetrah}.\mathrm{v}.}{\underbrace{{\left[{\mathrm{Ga}\left(2\right)}^{-}\right]}_2}}\underset{{\mathbf{ABA}}^{\prime }\mathbf{C}}{\underbrace{\left[{\mathrm{Ga}}_{11}\right]\left[{\mathrm{Ga}}_{12}\right]}}$$

The shortest K–M distances within the CCPs amount to 340 to 345 pm; the respective values for the Rb phases are only slightly larger (350 to 354 pm; Table 8). These somewhat shorter distances compared to those in the RbGa₇-type compounds, the decreased coordination numbers (13, 14 and 15) compared to 15 + 3 for the RbGa₇-type, and the volume differences of the CCPs (RbGa₇: 178.6×10${}^6$ pm${}^3$; K₃Ga₁₃: 163.9 [K(1)], 155.5 [K(2)] and 145.0×10${}^6$ pm${}^3$ [K(3)]) are in accordance with the observation that the latter structure type occurs only with the larger cations Rb⁺ and Cs⁺. The K₃Ga₁₃-type is formed with the smaller cations K⁺ and Rb⁺, for the latter cation only with a small In/Hg content expanding the polyanion. Similar to the RbGa₇-type structure, this expansion primarily concerns the distances between the exo-bonded cluster layers D (b direction), as the larger In and Hg atoms mainly take the inter-cluster sites (cf. the discussion of the Ga/In/Hg site preferences in Section 3.4).

3.3. Electronic Structures of the Pure Gallides

Both structure types of the Ga-rich gallides presented within this work are formally electron-precise compounds, if the Zintl concept is extended by Wade’s electron counting rules.

• For the RbGa₇-type, the ionic separation:
  2 RbGa₇ → 2 Rb⁺ + 2^4b[Ga(1)]⁻ + [Ga₁₂]⁰
  reveals that the four-bonded Ga(1)⁻ atoms of the dumbbells already equalize the cation charge. The [Ga₁₂] icosahedra carry no formal charge, as they are connected via triangular closed 2e3c bonds, similar to the bonding situation within the icosahedra layers of elemental $\alpha$-rhombohedral boron.
• For the K₃Ga₁₃-type structure:
  2 K₃Ga₁₃ → 6 K⁺ + 2^4b[Ga(1)]⁰ + 2^4b[Ga(2)]⁻ + [Ga₁₂]²⁻ + [Ga₁₁]²⁻
  shows that the charge of six cations (per 2 f.u.) is compensated by the two types of closo clusters (both $q=-2$) and again two tetrahedrally bonded Ga(2)⁻ atoms. Ga(1)⁰, with a trigonal-planar coordination, carries no charge and is thus electron-deficient.

Although the chemical bonding in both compounds thus seems to be straightforward and one expects closed-shell compounds with (pseudo) band gaps, both structures show different discrepancies and unexpected features, which are worth a closer inspection:

• The calculated DOS of CsGa₇ and CsGa₆In (Figure 3; cf. Section 2.3 and

  Table 9. Details of the calculation of the electronic structure of the binary alkali trielides CsGa₇ and K₃Ga₁₃ ($r_{\mathrm{MT}}$: muffin tin radius; $K_{\max }$: maximal wavevector for the PWin the interstitium; BCP: bond critical point; IBZ: irreducible part of the Brillouin zone; $V_{\mathrm{BB}}$: volume of the Bader basins; d: bond lengths).

  Compound			CsGa7	‘CsGa6In’	$\mathit{\alpha }$-rhomb.B	K3Ga13	’Rb3Ga12.5Hg0.5’
  Crystal data			Table 2 and Table 3		[35]	[20]	Table 6 and Table 5
  $r_{\mathrm{MT}}$ (all atoms)			121.7 pm (2.3 a.u.)		79.4 pm (1.5 a.u.)	121.7 pm (2.3 a.u.)
  $r_{\mathrm{MT}}·$K${}_{\max }$			8.0
  k-points/BZ			9261			768
  k-points/IBZ			891			144
  Monkhorst-Pack-Grid			21×21×21			16×16×3
  DOS plot			Figure 3			Figure 4
  Band structure plot			Figure 5	–	Figure 5	–	–
  Atoms:		label
  Charge distribution		A(1)	+0.662 (35.3)	+0.666 (36.3)	-	+0.743 (23.1)	+0.703 (28.0)
  after Bader		A(2)	-	-	-	+0.731 (21.4)	+0.678 (26.7)
  ($V_{\mathrm{BB}}$ $\left[{10}^6{\mathrm{pm}}^3\right]$)		A(3)	-	-	-	+0.730 (21.6)	+0.706 (26.3)
  		Ga/In/Hg(1)	−0.095 (24.7)	−0.060 (29.4)	-	−0.215 (29.3)	−0.593 (33.8)
  		Ga(2)	−0.109 (22.4)	−0.132 (22.6)	+0.082 (7.01)	−0.047 (24.2)	−0.063 (24.6)
  		Ga(3)	−0.080 (18.7)	−0.071 (19.0)	−0.082 (7.65)	-	-
  		Ga(11)	-	-		−0.109 (23.7)	+0.020 (21.8)
  		Ga(12)	-	-	-	−0.201 (26.8)	−0.179 (26.2)
  		Ga(13)	-	-	-	−0.141 (22.8)	−0.142 (22.6)
  		Ga(14)	-	-	-	−0.244 (23.4)	−0.087 (21.8)
  		Ga(15)	-	-	-	−0.195 (23.4)	−0.186 (22.9)
  		Ga(21)	-	-	-	−0.047 (23.6)	−0.037 (24.3)
  		Ga(22)	-	-	-	−0.179 (22.7)	−0.172 (22.7)
  		Ga(23)	-	-	-	−0.165 (22.7)	−0.131 (22.8)
  		Ga(24)	-	-	-	−0.197 (20.9)	−0.189 (20.9)
  Bonds:		label
  Electron density at		a	0.343 (258.3)	0.337 (275.4)	-	0.386 (250.0)	0.420 (253.1)
  BCP$\left[e^{-}{10}^{-6}{\mathrm{pm}}^{-3}\right]$		b	0.334 (258.9)	0.338 (266.5)	1.078 (166.8)	0.357 (254.2)	0.355 (264.1)
  (d [pm])		c	0.292 (263.9)	0.290 (264.6)	0.817 (180.7)	0.373 (252.8)	0.352 (256.3)
  		d	0.241 (278.7)	0.260 (272.9)	0.790 (175.4)	0.359 (256.1)	0.329 (261.3)
  		e	0.236 (282.9)	0.242 (281.2)	0.765 (179.9)	0.339 (257.5)	0.323 (260.2)
  		f	0.332 (254.5)	0.327 (255.6)	0.791 (178.4)	0.382 (250.9)	0.364 (253.6)
  		g/m	0.296 (262.4)	0.287 (264.6)	0.529 (202.2)	0.284 (266.8)	0.282 (266.2)

  for the details of the band structure calculations) unexpectedly exhibit no (pseudo) band gaps, but show a broad minimum of the tDOS clearly below the Fermi level.
• Conversely, K₃Ga₁₃ is indeed the expected semiconductor with a (DFT immanent as well) small band gap (cf. tDOS in Figure 4). Thus, In this case, the possibility of a Ga↦Hg exchange is against the expectation.

Both for CsGa₇ and K₃Ga₁₃, the broad valence band region of approximately 12 eV in width is formed by Ga-s and -p states. The pDOS of selected Ga atoms show very similar dispersion, shape, and s/p mixing for the tetrahedrally bonded Ga⁻ atoms Ga(1)/Ga(2) (in CsGa₇/K₃Ga₁₃). The cluster-forming atoms (e.g., Ga(2) and Ga(3)/Ga(24) as an example in K₃Ga₁₃) exhibit a larger band dispersion and a stronger mixing of s and p states, with a pronounced pDOS of those p states, which contribute to exo-bonds, directly below $E_F$. The pDOS of In(1) in “CsGa₆In” (thin lines in Figure 3) does not deviate significantly from Ga(1) in the pure gallide, which shows the very similar bonding features of the heavier triel (see the discussion in Section 3.4).

The calculated valence electron densities ($\rho$) and the results of their Bader analyses fit those of the trielides AM₃ and A₂M₃ discussed in [22], as well as those of the complex cluster structure of K₃Ga_9.6In_1.4 [25]: all endo- and exohedral Ga–Ga bonds carry bond critical points (cf. the individual values in Table 9); ring and cage critical points are again located at the triangular faces and the cluster centers, respectively. For both exo- and endohedral bonds, the bond lengths (d) and the electron densities (${\rho }_{\mathrm{BCP}}$) are correlated, and the absolute values fit the values plotted in Figure 7 of [25]. The associated Laplacians (${\nabla }^2{\rho }_{\mathrm{BCP}}$) are negative for all 2e2c exo-bonds. The positive sign for the endohedral Ga–Ga bonds indicates the delocalized (polyaromatic) character of the cluster shell (cf. the discussion in [25]). The Bader charges (q) and volumes ($V_{\mathrm{BB}}$) of the atomic basins are also in line with the expectations (cf. Table 9): All alkali cations carry positive charges of +0.66 to +0.74. All Ga atoms are negatively charged (−0.05 to −0.24), whereby the small differences are mainly due to the varying number (and distance) of the coordinating A cations and are particularly independent of the formal charges obtained from the above-mentioned ionic partitioning.

• Although this partitioning suggests the closed-shell character for both Ga-rich gallides, CsGa₇ and its In-derivative show no (pseudo) band gaps. Instead, the broad minimum of the tDOS between $-0.55$ and $-0.05$ eV is due to a steep Ga(3)-$p_{x^{\prime }}$-type band crossing the Fermi level (cf. the Ga(3)-$p_{x^{\prime }}$ fatband representation of the band structure in Figure 5a; note the orientation of the local coordinate system ${}^{\prime }$ at Ga(3) in Figure 5e). The shape of the E-selected (E = −0.23 to −0.36 eV) valence electron density (dark blue surfaces in Figure 5d) fits the schematic illustration of Ga(3)-$p_{x^{\prime }}$ orbitals exhibiting the $\sigma$-bonding character within the six-membered “equator” of the icosahedra layer (Figure 5e). The electron density in the equivalent plot of $\alpha$-rhombohedral boron $E_{\min }$ = −0.66 eV) shows a very similar shape. In turn, flat Ga(3)-s states with small $\sigma$-bonding features within the [Ga(3)]₃ rings (cf. the cyan surfaces in Figure 5c) are shifted down the Fermi level at the M and K point of the hexagonal Brillouin zone, which results in the unexpected large DOS at $E_F$. The population of these states, as well as the depopulation of the Ga(3)-$p_{x^{\prime }}$ band, which exhibits the $\pi$-antibonding character within the 2e3c [Ga₃] triangle, evidently contributes to the differences of the bonding in CsGa₇ and $\alpha$-rhombohedral boron: For the latter, the edges of the B₃ ring (g; 202 pm) are much larger than all other exo- and endohedral B–B bonds (cf. the discussion in [44] and the references therein), whereas the triangle of the 2e3c bond (edge g) in CsGa₇ is even somewhat shorter than the intra-cluster bonds/icosahedra edges (cf. small inset table in Figure 5). The two Fermi surfaces of the bands intersecting $E_F$ are a homogeneous tube along $\Gamma$-Z corresponding to the steep Ga(3)-$p_{x^{\prime }}$ band and a complicated surface at the K/M border of the Brillouin zone, which belongs to the Ga(3)-s states at the Fermi level. The two surfaces are well separated, and superconducting properties are not to be expected (and not yet examined).
• The much more complex band structure of K₃Ga₁₃ shows the expected “zero” band gap (cf. tDOS in Figure 4). The pronounced tDOS directly above $E_F$ exhibits the Ga(1)-$p_z$ character only and thus represents the empty orbital of the trigonal-planar coordinated electron-deficient Ga(1). In the Hg-substituted derivative, where Hg takes exactly this site, these states are accordingly missing (cf. the tDOS of the model system Rb₆Ga₂₅Hg in Figure 4, green line).

3.4. Aspects of In/Hg Distribution (“Coloring”)

On the Ga↦In/Hg exchange, the sizes of the anion atoms M increase by approximately 25/16 pm (metallic radius $r_M$ after Gschneidner: 141.1/166.3/157.3 pm for Ga/In/Hg [45]). In the case of In, the electronegativity ($\chi$) decreases marginally (from 1.81 to 1.78 after Pauling [46]), whereas Hg exhibits an increased value of 2.00. These variable atomic properties become apparent in the results of the calculations of the weakly-substituted model variants CsGa₆In and Rb₆Ga₂₅Hg, where the changed M sites exhibit larger Bader basins $V_{\mathrm{BB}}$ for both In and Hg, whereas the negative charges $-q_{\mathrm{BB}}$ are increased for Hg only (Table 9).

Due to the comparatively small amounts of substituting elements In or Hg, the Bader charges and volumes calculated for the binary gallides CsGa₇ and K₃Ga₁₃ may suggest the “coloring” of the polyanion, i.e., the positions preferentially taken by the larger/more electronegative M atoms. Accordingly, In the case of K₃Ga₁₃, the Ga position with the largest atomic basin and one of the most pronounced negative charges (due to its sixfold coordination by A⁺ cations) is selectively occupied by Hg [M(1)]. In contrast, the preferred position for a Ga↦In substitution is not straightforward for both title compounds, even though the site occupations are also quite selective. Here, geometric arguments, i.e., (i) a possible expansion of the 3D polyanion and (ii.) the avoidance of an incorporation of the larger “doping” elements in the stiff (parts of the) closo clusters are evidently the main parameters for the “coloring” of the 3D triel polyanion:

• For example, the sizes and charges of the three Ga sites in CsGa₇ are very similar (Table 9), but indium strongly prefers the tetrahedral “dumbbell” position M(1) and therewith expands the structure mainly along the most flexible direction, i.e., perpendicular to these layers of tightly connected [Ga₁₂] icosahedra. The positions within these layers, especially the central 3.4.6.4. nets containing the six-membered “equators” of the icosahedra, do not accommodate any In atoms.
• Similar arguments apply to the preferred mixed sites in the K₃Ga₁₃-type structure: the substitution of Ga by In occurs at the “extra” positions M(1) and M(2), which are connecting the cluster layers D along the b axis, and mainly at the indented position M(11) of the [M₁₁] closo cluster, which exhibits comparable large M–Ga bonds (i and j; Table 7). The simultaneous occupation of these sites allows for a slight expansion of the structure perpendicular to the layers D, i.e., along [010] (cf. the structure description in Section 3.2).

The fact, that—for both structure types—the maximum amount of In increases with the size of the A cations also corroborates the relevance of such geometric parameters for the Ga/In “coloring” in 3D polyanions. The preference of heteropolar bonds, which is the most dominating factor for the “coloring” in polyanions consisting of similar amounts of two atom types with different electronegativities [9,21,47], plays no part in the slightly substituted title compounds.

3.5. Comparison with Other In-/Hg-/Au-Containing Gallides

The coloring pattern observed for CsGa₇ also meets the observation for the complex structure of the mixed Ga/In trielides K₃Ga_11-xIn_x, where indium also occupies all four-bonded inter-cluster sites; in this case, its presence is even crucial for the formation of this type of complex polyanion [25].

The RbGa₃-type, which is known for K, Rb, and Cs gallides, also exhibits a 3D network of all-exo-bonded closo clusters, in this case, [Ga(2,3)₈] dodecahedra, which are connected by direct inter-cluster exo-bonds and further 4b Ga(1)⁻ [5,6,14,48]. The maximum possible In content x in A(Ga_1-xIn_x)₃ again decreases with the decreasing size of the counter cation (x=1/0.3/0.08 for Cs/Rb/K). Indium prefers to occupy the M(2) position in the “equator” of the dodecahedra, which connects the clusters to layers via long exo-bonds. The four-bonded site M(1) connecting the cluster layers is occupied in second place (e.g., In content in RbGa_2.08In_0.92: M(1/2/3) = 30/56/5%). Band structure calculations of RbGa₃ again yielded closely similar Bader volumes for the three different atom types, whereas the four-bonded Ga(1) exhibits a decreased negative charge [22]. As already discussed by Henning and Corbett [23], the “coloring” in RbGa₃ is determined by the preservation of the short strong Ga–Ga bonds ($d_{\mathrm{Ga}\left(3\right)--\mathrm{Ga}\left(3\right)}$ = 244.2 pm) in the dumbbells at the top and bottom of the dodecahedra. The related $\pi$-bonding contributions are most effective for those positions, where the cluster curvature is high. This position takes up indium atoms only at an overall large In content. Similar to the K₃Ga₁₃ title phase, a transition element derivative with a reduced number of v.e. is reported for RbGa₃ as well [23], even though the compound is again electron precise, and the calculated band structure shows a small band gap accordingly. Here, smaller (r = 144 pm) and much more electronegative ($\chi$ = 2.54), but electron poorer gold atoms statistically take all three atom positions. In accordance with the importance of size criteria, the site preference of the smaller Au atoms (Au content: M(1) >M(3) ≫M(2)) is interchanged with respect to that of the larger In atoms (In content: M(2) >M(1) ≫M(3)). For RbGa₃, the reduced v.e. number causes a pronounced increase of the strong Ga(3)–Ga(3) bonds of the dumbbells within the dodecahedron. The similar, but small Hg content of the K₃Ga₁₃-type causes a shift of the Hg-containing three-bonded site M(1) associated with an increase of the bond b from 253 to 260.5 pm and the angle Ga(14)–M(1)–Ga(14) from 95.3 to 91${}^{\circ }$.

For gallides with only partially connected closo clusters, indium strongly prefers to take the position of the “non-connecting” atom in the cluster. These sites show the largest Bader volumes and carry the most negative charge, as they are coordinated by the maximum number of cations [22]. The inert pair effect (stronger s/p separation of In compared to Ga) may be an additional parameter for this observation as is the vast number of alkali-rich thallides with isolated cluster anions ([49] and the references therein). This type of indium preference is observed in the ternary mixed Ga/In trielides of the Cs₂In₃-type, which exhibit single layers of closo [M₆] octahedra connected via four exo-bonds [50,51,52]. Indium occupies the two “tips” of the octahedra, which are not involved in the exo-bonds [22]. A similar “coloring” applies in the two structures of the new mixed Ga/In trielides (Rb/Cs)₇M₁₅, which are stabilized by In contents of 33–46% (Rb) and 26–56% (Cs). Here, the polyanions are double layers of exo-bonded pentagonal bipyramidal closo clusters [M₇]³⁻ connected via four-bonded M−. Again, In atoms terminate the layers, as they prefer to occupy the “unconnected” corners of the clusters (cf. Cs₂In₃-type), and the “dumbbells” within the [M₇] polyhedra are formed by Ga atoms exclusively (cf. RbGa₃-type).

For mixed Ga/In trielides containing isolated clusters, like the Cs₈Ga₁₁-type [10,12], atom sizes are also the most relevant criteria for the Ga↦In substitution, and the site with the largest Bader volume (and the longest M–Ga bonds) is preferentially occupied by the larger In atoms. On the other hand, mercury atoms with their increased electronegativity again prefer to take the position with the most negative Bader or Mulliken charges [53].

4. Summary

In the course of systematic “coloring” studies of mixed trielides, small amounts of Ga in the binary most Ga richest electron-precise cluster compounds RbGa₇/CsGa₇ and K₃Ga₁₃/“Rb₃Ga₁₃” were substituted by the larger element In and the similarly larger, but also more electronegative and electron poorer Hg. Due to the incongruent melting of all phases, well-developed single crystals were grown from Ga/In/Hg-rich melts, and their structures were refined by means of single crystal X-ray data. In RbGa₇/CsGa₇, up to 9.9/13.6% of Ga could be substituted by In, which occupies up to 80% of the four-bonded M₂²⁻ dumbbells connecting the $\alpha$-B like [Ga₁₂] icosahedra layers. In contrast, a Ga↦Hg substitution is not possible, which opened up the possibility to crystallize the yet not fully characterized binary hepta-gallides from elemental Hg. Though electron precise, the calculated DOS of CsGa₇ and CsGa₆In unexpectedly exhibit no (pseudo) band gaps. An analysis of the band structures, energy-selected electron densities, and the Fermi surfaces (with $\alpha$-rhombohedral boron for comparison) shows that Ga(3)-$p_{x^{\prime }}$-like bands with $\sigma$-bonding characteristic within the layer of the six-membered “equator” of the icosahedra and the [Ga₃] rings are crossing $E_F$ and are thus only partly occupied. Conversely, Ga(3)-s states with $\sigma$-bonding features within the [Ga(3)₃] rings are shifted towards the Fermi level and result in the unexpected large DOS at $E_F$. In K₃Ga₁₃/“Rb₃Ga₁₃”, the overall 1.2/5.2–8.2% of indium mainly fits in the indented position of the closo [Ga₁₁]²⁻ clusters, which allow for larger endohedral M–M bonds. In the Hg-substituted derivatives, the unique, only trigonal-planar bonded Ga(1)⁰ site is nearly completely substituted by the chemically much different mercury, resulting in an overall Hg content/M of 2.7/3.2% (for A = K, Rb). The Ga/In/Hg/(Au) distribution in both types of title compounds is comparable with those in the structures of other 3D anionic networks, in the mixed Ga/In trielides containing cluster layers and in the isolated clusters of the Cs₈Ga₁₁-type.