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Dimanganese(II) hydroxide vanadate, Mn2(OH)[VO4]

a112 Fleming Building, Department of Chemistry, University of Houston, Houston, TX 77204-5003, USA
*Correspondence e-mail: amoeller@uh.edu

(Received 12 May 2014; accepted 3 June 2014; online 11 June 2014)

Dimanganese(II) hydroxide vanadate was obtained from hydro­thermal reactions. The crystal structure of the title compound is isotypic with that of Zn2(OH)[VO4]. Three crystallographically independent Mn2+ ions are present, one (site symmetry .m.) with a distorted trigonal-bipyramidal and two (site symmetries .m. and 1) with distorted octa­hedral coordination spheres. These polyhedra are linked through common edges, forming a corrugated layer-type of structure extending parallel to (100). A three-dimensional framework results via additional Mn—O—V—O—Mn connectivities involving the two different tetra­hedral [VO4] units (each with point-group symmetry .m.). O—H⋯O hydrogen bonds (one bifurcated) between the OH functions (both with point-group symmetry .m.) and the [VO4] units complete this arrangement.

Related literature

Mn2(OH)[VO4] is isotypic with Zn2(OH)[VO4] (Wang et al., 1998[Wang, X., Liu, L. & Jacobson, A. J. (1998). Z. Anorg. Allg. Chem. 624, 1977-1981.]), Zn1.86Cd0.14(OH)[VO4] (Đorđević et al., 2010[Đorđević, T., Stojanović, J. & Karanović, L. (2010). Acta Cryst. E66, i79.]), Cu2(OH)[VO4] (Wu et al., 2003[Wu, C., Lu, C., Zhuang, H. & Huang, J. (2003). Eur. J. Inorg. Chem. pp. 2867-2871.]), but not with the acentric Cu polymorph reported by Zhang et al. (2014[Zhang, S.-Y., He, Z.-Z., Yang, M., Guoa, W.-B. & Tang, Y.-Y. (2014). Dalton Trans. 43, 3521-3527.]).

Experimental

Crystal data
  • Mn2(OH)[VO4]

  • Mr = 241.83

  • Orthorhombic, P n m a

  • a = 14.9112 (10) Å

  • b = 6.1225 (3) Å

  • c = 9.1635 (4) Å

  • V = 836.57 (8) Å3

  • Z = 8

  • Mo Kα radiation

  • μ = 8.04 mm−1

  • T = 293 K

  • 0.14 × 0.02 × 0.02 mm

Data collection
  • Rigaku R-AXIS conversion diffractometer

  • Absorption correction: multi-scan (CrystalClear; Rigaku, 2011[Rigaku, (2011). CrystalClear. Rigaku Corporation, Tokyo, Japan.]) Tmin = 0.777, Tmax = 1.000

  • 11657 measured reflections

  • 1715 independent reflections

  • 1637 reflections with I > 2σ(I)

  • Rint = 0.046

Refinement
  • R[F2 > 2σ(F2)] = 0.045

  • wR(F2) = 0.062

  • S = 1.36

  • 1715 reflections

  • 92 parameters

  • 2 restraints

  • H atoms treated by a mixture of independent and constrained refinement

  • Δρmax = 0.74 e Å−3

  • Δρmin = −0.84 e Å−3

Table 1
Hydrogen-bond geometry (Å, °)

D—H⋯A D—H H⋯A DA D—H⋯A
O3—H3⋯O12 0.96 (2) 1.79 (2) 2.734 (4) 165 (5)
O4—H4⋯O11i 0.97 (2) 2.39 (1) 3.161 (3) 137 (1)
O4—H4⋯O11ii 0.97 (2) 2.39 (1) 3.161 (3) 137 (1)
Symmetry codes: (i) [x-{\script{1\over 2}}, y, -z+{\script{1\over 2}}]; (ii) [x-{\script{1\over 2}}, -y+{\script{1\over 2}}, -z+{\script{1\over 2}}].

Data collection: CrystalClear (Rigaku, 2011[Rigaku, (2011). CrystalClear. Rigaku Corporation, Tokyo, Japan.]); cell refinement: CrystalClear; data reduction: CrystalClear; program(s) used to solve structure: SHELXS97 (Sheldrick, 2008[Sheldrick, G. M. (2008). Acta Cryst. A64, 112-122.]); program(s) used to refine structure: SHELXL97 (Sheldrick, 2008[Sheldrick, G. M. (2008). Acta Cryst. A64, 112-122.]); molecular graphics: DIAMOND (Brandenburg, 2012[Brandenburg, K. (2012). DIAMOND. Crystal Impact GbR, Bonn, Germany.]); software used to prepare material for publication: publCIF (Westrip, 2010[Westrip, S. P. (2010). J. Appl. Cryst. 43, 920-925.]).

Supporting information


Comment top

Dimanganese hydroxide vanadate, Mn2(OH)[VO4], is isotypic with Zn2(OH)[VO4] (Wang et al., 1998), Zn1.86Cd0.14(OH)[VO4] (Đorđević et al., 2010) and Cu2(OH)[VO4] (Wu et al., 2003). The crystal structure contains three crystallographically independent Mn2+-ions (Fig. 1). Mn3 (8d) is located on a general position whereas Mn1 and Mn2 occupy a 4c position each, which is of .m. site symmetry. Mn1 is found in a distorted trigonal-bipyramidal coordination and connects to Mn2 and two Mn3 only via the edges of a single trigonal face. The coordination around Mn2 and Mn3 is octahedral. The former shares four common edges with Mn3 on two trans-faces as well as a single edge with Mn1. Mn3 connects to symmetry-related Mn3 positions via trans-edges along [010] and through edge-sharing to two Mn2 and only one Mn1. From this connectivity corrugated layers parallel to (100) with terminal [Mn1O5] units result (Fig. 2). Per unit cell two of these layers are present which are linked through Mn—O—V—O—Mn bridges into a three-dimensional framework.

Two [VO4]3- units are present. Each V-atom and two O per tetrahedron are located on a mirror plane, whereas one O atom per [VO4]-unit is found on a general position, respectively. V1 is coordinated by O11 (8d), O12 (4c), and O13 (4c). Mn2 and Mn3 are linked via O11 whereas O13 connects to two Mn3. O12 connects solely to Mn1. However, the second coordination sphere around V2 is dissimilar with respect to O21 (8d) connecting to Mn1–3 and O23 (4c) linked to Mn2 and two Mn3. Again O22 (4c) connects only to Mn1. The two hydroxide groups are of .m. point group symmetry with one straight and one bifurcated OD···OA contacts, respectively (Table 1).

Based on the evaluation of the Mn—O—Mn connectivities, the difference in the second coordination sphere between the [V1O4] and [V2O4] units, and the dissimilarities found for the two independent hydroxide groups, the more informative structure-related formula should be presented as Mn4(OH)2[VO4]2 with Z = 4 per unit cell.

It is noteworthy, that for Cu2(OH)[VO4] two polymorphs (Pnma and P212121) were reported by Wu et al. (2003) and Zhang et al. (2014), respectively. We have also repeatedly obtained the acentric modification (hydrothermal conditions at 493 K, pH 9–10, NH4VO3 and Cu-acetate or -chloride). Interestingly, Wang et al. (1998) include in a side note that Ni2(OH)[VO4] crystallizes in the P212121 space group as well. Thus, polymorphismn might be related to a size effect or driven by anisotropic magnetic correlations. The latter conjecture is based on the non-magnetic Zn-compound and the "isotropic" S=5/2 spin-system represented by Mn2+. For these compounds only the centrosymmetric modification are known up to now.

Related literature top

Mn2(OH)[VO4] is isotypic with Zn2(OH)[VO4] (Wang et al., 1998), Zn1.86Cd0.14(OH)[VO4] (Đorđević et al., 2010), Cu2(OH)[VO4] (Wu et al., 2003), but not with the acentric Cu polymorph reported by Zhang et al. (2014).

Experimental top

The title compound was obtained as red needle-shaped crystals from reacting H2V3O8 (hydrothermal synthesis, 0.5 mmol V2O5 and 0.5 mmol H2C2O4 at 453 K) with 1.75 mmol Mn(acetate)2.4H2O and 5 mmol LiCl in 20 ml of water. Concentrated NH3-solution was added to adjust the pH to 9.1. The reaction was carried out in a 26 ml Teflon-lined stainless steel autoclave at 493 K for 3 days with subsequent cooling (6 K/h) to room temperature. The final product was washed with distilled water and ethanol alcohol. Source of materials: Mn(acetate)2.4H2O, Sigma; anhydrous H2C2O4, GFS Chemicals; anhydrous LiCl 98+%, Alfa Aesar; V2O5, Alfa Aesar; NH4OH 14.8M, EMD.

Refinement top

Hydrogen atom positions were found from difference Fourier maps and were refined by restricting the O—H distance (DFIX 1.0 0.02 O3 H3 O4 H4) and using the ride-on option for the isotropic displacements with Uiso(H) = 1.5 × Ueq(O).

Structure description top

Dimanganese hydroxide vanadate, Mn2(OH)[VO4], is isotypic with Zn2(OH)[VO4] (Wang et al., 1998), Zn1.86Cd0.14(OH)[VO4] (Đorđević et al., 2010) and Cu2(OH)[VO4] (Wu et al., 2003). The crystal structure contains three crystallographically independent Mn2+-ions (Fig. 1). Mn3 (8d) is located on a general position whereas Mn1 and Mn2 occupy a 4c position each, which is of .m. site symmetry. Mn1 is found in a distorted trigonal-bipyramidal coordination and connects to Mn2 and two Mn3 only via the edges of a single trigonal face. The coordination around Mn2 and Mn3 is octahedral. The former shares four common edges with Mn3 on two trans-faces as well as a single edge with Mn1. Mn3 connects to symmetry-related Mn3 positions via trans-edges along [010] and through edge-sharing to two Mn2 and only one Mn1. From this connectivity corrugated layers parallel to (100) with terminal [Mn1O5] units result (Fig. 2). Per unit cell two of these layers are present which are linked through Mn—O—V—O—Mn bridges into a three-dimensional framework.

Two [VO4]3- units are present. Each V-atom and two O per tetrahedron are located on a mirror plane, whereas one O atom per [VO4]-unit is found on a general position, respectively. V1 is coordinated by O11 (8d), O12 (4c), and O13 (4c). Mn2 and Mn3 are linked via O11 whereas O13 connects to two Mn3. O12 connects solely to Mn1. However, the second coordination sphere around V2 is dissimilar with respect to O21 (8d) connecting to Mn1–3 and O23 (4c) linked to Mn2 and two Mn3. Again O22 (4c) connects only to Mn1. The two hydroxide groups are of .m. point group symmetry with one straight and one bifurcated OD···OA contacts, respectively (Table 1).

Based on the evaluation of the Mn—O—Mn connectivities, the difference in the second coordination sphere between the [V1O4] and [V2O4] units, and the dissimilarities found for the two independent hydroxide groups, the more informative structure-related formula should be presented as Mn4(OH)2[VO4]2 with Z = 4 per unit cell.

It is noteworthy, that for Cu2(OH)[VO4] two polymorphs (Pnma and P212121) were reported by Wu et al. (2003) and Zhang et al. (2014), respectively. We have also repeatedly obtained the acentric modification (hydrothermal conditions at 493 K, pH 9–10, NH4VO3 and Cu-acetate or -chloride). Interestingly, Wang et al. (1998) include in a side note that Ni2(OH)[VO4] crystallizes in the P212121 space group as well. Thus, polymorphismn might be related to a size effect or driven by anisotropic magnetic correlations. The latter conjecture is based on the non-magnetic Zn-compound and the "isotropic" S=5/2 spin-system represented by Mn2+. For these compounds only the centrosymmetric modification are known up to now.

Mn2(OH)[VO4] is isotypic with Zn2(OH)[VO4] (Wang et al., 1998), Zn1.86Cd0.14(OH)[VO4] (Đorđević et al., 2010), Cu2(OH)[VO4] (Wu et al., 2003), but not with the acentric Cu polymorph reported by Zhang et al. (2014).

Computing details top

Data collection: CrystalClear (Rigaku, 2011); cell refinement: CrystalClear (Rigaku, 2011); data reduction: CrystalClear (Rigaku, 2011); program(s) used to solve structure: SHELXS97 (Sheldrick, 2008); program(s) used to refine structure: SHELXL97 (Sheldrick, 2008); molecular graphics: DIAMOND (Brandenburg, 2012); software used to prepare material for publication: publCIF (Westrip, 2010).

Figures top
[Figure 1] Fig. 1. A projection of the crystal structure of Mn2(OH)[VO4] along [011]. Displacement ellipsoids are drawn at the 75% probability level.
[Figure 2] Fig. 2. The Mn–Mn connectivity per layer with each connecting line representing a link exclusively via edge-sharing (left). Polyhedral representation for Mn1 (red), Mn2 (yellow) and Mn3 (orange) (right). Displacement ellipsoids are drawn at the 75% probability level.
Dimanganese(II) hydroxide vanadate top
Crystal data top
Mn2(OH)[VO4]F(000) = 912
Mr = 241.83Dx = 3.840 Mg m3
Orthorhombic, PnmaMo Kα radiation, λ = 0.71075 Å
Hall symbol: -P 2ac 2nCell parameters from 18226 reflections
a = 14.9112 (10) Åθ = 3.3–33.1°
b = 6.1225 (3) ŵ = 8.04 mm1
c = 9.1635 (4) ÅT = 293 K
V = 836.57 (8) Å3Needle, red
Z = 80.14 × 0.02 × 0.02 mm
Data collection top
Rigaku R-AXIS conversion
diffractometer
1715 independent reflections
Radiation source: fine-focus sealed tube1637 reflections with I > 2σ(I)
Graphite monochromatorRint = 0.046
Detector resolution: 10.0000 pixels mm-1θmax = 33.1°, θmin = 3.5°
profile data from ω–scansh = 2222
Absorption correction: multi-scan
(CrystalClear; Rigaku, 2011)
k = 99
Tmin = 0.777, Tmax = 1.000l = 1314
11657 measured reflections
Refinement top
Refinement on F2Primary atom site location: structure-invariant direct methods
Least-squares matrix: fullSecondary atom site location: difference Fourier map
R[F2 > 2σ(F2)] = 0.045Hydrogen site location: difference Fourier map
wR(F2) = 0.062H atoms treated by a mixture of independent and constrained refinement
S = 1.36 w = 1/[σ2(Fo2) + (0.0121P)2 + 1.9324P]
where P = (Fo2 + 2Fc2)/3
1715 reflections(Δ/σ)max < 0.001
92 parametersΔρmax = 0.74 e Å3
2 restraintsΔρmin = 0.84 e Å3
Crystal data top
Mn2(OH)[VO4]V = 836.57 (8) Å3
Mr = 241.83Z = 8
Orthorhombic, PnmaMo Kα radiation
a = 14.9112 (10) ŵ = 8.04 mm1
b = 6.1225 (3) ÅT = 293 K
c = 9.1635 (4) Å0.14 × 0.02 × 0.02 mm
Data collection top
Rigaku R-AXIS conversion
diffractometer
1715 independent reflections
Absorption correction: multi-scan
(CrystalClear; Rigaku, 2011)
1637 reflections with I > 2σ(I)
Tmin = 0.777, Tmax = 1.000Rint = 0.046
11657 measured reflections
Refinement top
R[F2 > 2σ(F2)] = 0.0452 restraints
wR(F2) = 0.062H atoms treated by a mixture of independent and constrained refinement
S = 1.36Δρmax = 0.74 e Å3
1715 reflectionsΔρmin = 0.84 e Å3
92 parameters
Special details top

Geometry. All e.s.d.'s (except the e.s.d. in the dihedral angle between two l.s. planes) are estimated using the full covariance matrix. The cell e.s.d.'s are taken into account individually in the estimation of e.s.d.'s in distances, angles and torsion angles; correlations between e.s.d.'s in cell parameters are only used when they are defined by crystal symmetry. An approximate (isotropic) treatment of cell e.s.d.'s is used for estimating e.s.d.'s involving l.s. planes.

Refinement. Refinement of F2 against ALL reflections. The weighted R-factor wR and goodness of fit S are based on F2, conventional R-factors R are based on F, with F set to zero for negative F2. The threshold expression of F2 > σ(F2) is used only for calculating R-factors(gt) etc. and is not relevant to the choice of reflections for refinement. R-factors based on F2 are statistically about twice as large as those based on F, and R-factors based on ALL data will be even larger.

Fractional atomic coordinates and isotropic or equivalent isotropic displacement parameters (Å2) top
xyzUiso*/Ueq
Mn10.07126 (4)0.25000.08355 (7)0.01159 (13)
Mn20.28875 (4)0.25000.15984 (6)0.01046 (12)
Mn30.36181 (3)0.00226 (7)0.12336 (5)0.01003 (9)
V10.42573 (4)0.75000.18749 (7)0.00722 (12)
V20.16571 (4)0.75000.02724 (7)0.00691 (12)
O110.37896 (13)0.5171 (3)0.1122 (2)0.0134 (4)
O120.0956 (2)0.25000.1322 (3)0.0170 (6)
O130.46095 (19)0.25000.1544 (3)0.0137 (6)
O210.17131 (13)0.0131 (3)0.1339 (2)0.0111 (4)
O220.06661 (19)0.25000.0537 (4)0.0189 (6)
O230.25215 (17)0.25000.3867 (3)0.0082 (5)
O30.27510 (18)0.25000.0716 (3)0.0100 (5)
H30.2151 (17)0.25000.110 (5)0.015*
O40.05393 (18)0.25000.3099 (3)0.0106 (5)
H40.0080 (16)0.25000.341 (5)0.016*
Atomic displacement parameters (Å2) top
U11U22U33U12U13U23
Mn10.0092 (2)0.0174 (3)0.0081 (3)0.0000.0003 (2)0.000
Mn20.0112 (3)0.0121 (3)0.0080 (2)0.0000.0020 (2)0.000
Mn30.01168 (18)0.00767 (17)0.01075 (18)0.00054 (14)0.00104 (15)0.00112 (15)
V10.0066 (2)0.0081 (3)0.0070 (3)0.0000.0004 (2)0.000
V20.0068 (3)0.0071 (3)0.0068 (3)0.0000.0012 (2)0.000
O110.0142 (9)0.0143 (9)0.0116 (9)0.0049 (8)0.0008 (8)0.0010 (8)
O120.0171 (14)0.0240 (16)0.0100 (13)0.0000.0001 (12)0.000
O130.0081 (12)0.0102 (12)0.0228 (15)0.0000.0031 (11)0.000
O210.0132 (8)0.0105 (9)0.0095 (9)0.0002 (7)0.0027 (7)0.0017 (7)
O220.0096 (12)0.0270 (17)0.0201 (15)0.0000.0036 (12)0.000
O230.0087 (11)0.0092 (12)0.0067 (11)0.0000.0020 (10)0.000
O30.0095 (11)0.0093 (12)0.0113 (12)0.0000.0005 (11)0.000
O40.0087 (12)0.0112 (13)0.0118 (13)0.0000.0004 (11)0.000
Geometric parameters (Å, º) top
Mn1—O122.010 (3)Mn3—O132.137 (2)
Mn1—O222.074 (3)Mn3—O11i2.177 (2)
Mn1—O42.091 (3)Mn3—O21ii2.2793 (18)
Mn1—O21i2.243 (2)Mn3—O23ii2.2981 (19)
Mn1—O212.243 (2)V1—O12iii1.683 (3)
Mn2—O32.131 (3)V1—O13iv1.717 (3)
Mn2—O232.149 (3)V1—O111.731 (2)
Mn2—O112.162 (2)V1—O11v1.731 (2)
Mn2—O11i2.162 (2)V2—O22vi1.654 (3)
Mn2—O212.391 (2)V2—O21vii1.7513 (19)
Mn2—O21i2.391 (2)V2—O21i1.7513 (19)
Mn3—O32.0487 (18)V2—O23viii1.777 (3)
Mn3—O4ii2.0827 (19)
O12—Mn1—O2292.83 (13)O21ii—Mn3—O23ii84.16 (8)
O12—Mn1—O4176.71 (12)O12iii—V1—O13iv111.06 (15)
O22—Mn1—O490.46 (12)O12iii—V1—O11108.39 (9)
O12—Mn1—O21i94.73 (8)O13iv—V1—O11109.04 (9)
O22—Mn1—O21i133.34 (5)O12iii—V1—O11v108.39 (9)
O4—Mn1—O21i82.99 (7)O13iv—V1—O11v109.04 (9)
O12—Mn1—O2194.73 (8)O11—V1—O11v110.93 (14)
O22—Mn1—O21133.34 (5)O22vi—V2—O21vii107.05 (9)
O4—Mn1—O2182.99 (7)O22vi—V2—O21i107.05 (9)
O21i—Mn1—O2191.77 (10)O21vii—V2—O21i111.86 (13)
O3—Mn2—O23159.81 (11)O22vi—V2—O23viii106.89 (15)
O3—Mn2—O1181.85 (7)O21vii—V2—O23viii111.82 (8)
O23—Mn2—O11110.69 (7)O21i—V2—O23viii111.82 (8)
O3—Mn2—O11i81.85 (7)V1—O11—Mn2142.49 (11)
O23—Mn2—O11i110.69 (7)V1—O11—Mn3i119.19 (10)
O11—Mn2—O11i98.30 (11)Mn2—O11—Mn3i94.93 (8)
O3—Mn2—O2180.28 (7)V1viii—O12—Mn1158.72 (19)
O23—Mn2—O2184.84 (7)V1iv—O13—Mn3134.78 (5)
O11—Mn2—O21160.92 (7)V1iv—O13—Mn3i134.78 (5)
O11i—Mn2—O2185.76 (7)Mn3—O13—Mn3i90.43 (11)
O3—Mn2—O21i80.28 (7)V2ix—O21—Mn1116.62 (10)
O23—Mn2—O21i84.84 (7)V2ix—O21—Mn3x123.94 (10)
O11—Mn2—O21i85.76 (7)Mn1—O21—Mn3x92.06 (7)
O11i—Mn2—O21i160.92 (7)V2ix—O21—Mn2130.42 (10)
O21—Mn2—O21i84.69 (10)Mn1—O21—Mn291.37 (7)
O3—Mn3—O4ii176.11 (11)Mn3x—O21—Mn292.42 (7)
O3—Mn3—O1386.67 (8)V2vi—O22—Mn1160.9 (2)
O4ii—Mn3—O1394.01 (8)V2iii—O23—Mn2121.73 (14)
O3—Mn3—O11i83.40 (9)V2iii—O23—Mn3x122.58 (9)
O4ii—Mn3—O11i100.35 (9)Mn2—O23—Mn3x98.57 (9)
O13—Mn3—O11i95.13 (10)V2iii—O23—Mn3xi122.58 (9)
O3—Mn3—O21ii93.88 (9)Mn2—O23—Mn3xi98.57 (9)
O4ii—Mn3—O21ii82.29 (9)Mn3x—O23—Mn3xi84.45 (9)
O13—Mn3—O21ii89.97 (10)Mn3i—O3—Mn395.53 (11)
O11i—Mn3—O21ii174.06 (7)Mn3i—O3—Mn299.80 (10)
O3—Mn3—O23ii91.24 (7)Mn3—O3—Mn299.80 (10)
O4ii—Mn3—O23ii87.68 (7)Mn3xi—O4—Mn3x95.73 (11)
O13—Mn3—O23ii173.63 (10)Mn3xi—O4—Mn1102.51 (10)
O11i—Mn3—O23ii90.60 (9)Mn3x—O4—Mn1102.51 (10)
Symmetry codes: (i) x, y+1/2, z; (ii) x+1/2, y, z1/2; (iii) x+1/2, y+1, z+1/2; (iv) x+1, y+1, z; (v) x, y+3/2, z; (vi) x, y+1, z; (vii) x, y+1, z; (viii) x+1/2, y+1, z1/2; (ix) x, y1, z; (x) x+1/2, y, z+1/2; (xi) x+1/2, y+1/2, z+1/2.
Hydrogen-bond geometry (Å, º) top
D—H···AD—HH···AD···AD—H···A
O3—H3···O120.96 (2)1.79 (2)2.734 (4)165 (5)
O4—H4···O11xii0.97 (2)2.39 (1)3.161 (3)137 (1)
O4—H4···O11xiii0.97 (2)2.39 (1)3.161 (3)137 (1)
Symmetry codes: (xii) x1/2, y, z+1/2; (xiii) x1/2, y+1/2, z+1/2.
Hydrogen-bond geometry (Å, º) top
D—H···AD—HH···AD···AD—H···A
O3—H3···O120.963 (19)1.79 (2)2.734 (4)165 (5)
O4—H4···O11i0.968 (19)2.387 (14)3.161 (3)136.6 (4)
O4—H4···O11ii0.968 (19)2.387 (14)3.161 (3)136.6 (4)
Symmetry codes: (i) x1/2, y, z+1/2; (ii) x1/2, y+1/2, z+1/2.
 

Acknowledgements

Financial support by the Texas Center for Superconductivity at the University of Houston is gratefully acknowledged.

References

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