Chem. Mater. 2005, 17, 5957-5969
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Concerning the Structure of Hydrogen Molybdenum Bronze Phase III. A Combined Theoretical-Experimental Study Benoit Braı¨da,†,‡ Stefan Adams,*,§,| and Enric Canadell*,† Institut de Cie´ ncia de Materials de Barcelona, Campus de la UAB, 08193 Bellaterra, Spain, Laboratoire de Chimie The´ orique (UMR-CNRS 7616), UniVersite´ Pierre et Marie Curie, 4 place Jussieu, 75252 Paris Ce´ dex, France, and GZG, Abt. Kristallographie, UniVersita¨t Go¨ttingen, 37077 Go¨ttingen, Germany ReceiVed May 4, 2005. ReVised Manuscript ReceiVed September 20, 2005
The structure of hydrogen molybdenum bronze phase III, HxMoO3 (x ≈ 5/3) has been studied on the basis of a combined experimental-theoretical approach. Our study of the average structure allows us to qualify previous contradictory results concerning the octahedral distortions and show no sign of hydrogen occupation of the intralayer sites. A first-principles theoretical study of this average structure leads to a clear understanding of the nature of the band structure as well as the importance and requirements for metal-metal bonding. The nature of the calculated Fermi surface suggests that a 3b × 2c superstructure may be a better description for phase III, which would be compatible with powder X-ray diffraction (XRD) results. Since the information available from powder XRD data does not allow a direct determination of the superstructure, the problem has been approached by means of first-principles geometry optimizations and molecular dynamics studies. The main structural features of this superstructure (distribution of Mo-Mo bonds, main distortions of the Mo-O network and hydrogen distribution) are described.
Introduction The layered structure of orthorhombic R-MoO3 permits the intercalation of protons onto two different types of sites. Protons can occupy places in the van der Waals (vdW) gaps between double layers of MoO6 octahedra as well as intralayer sites on zigzag chains along the channels (see Figure 1). The resulting nonstoichiometric hydrogen molybdenum bronze (HMB) phases HxMoO3 (0 < x < 2) include phases I (0.23 < x < 0.40), II (0.85 < x < 1.04), and III (1.55 < x < 1.72) that are thermodynamically stable against decomposition under standard conditions.1 Further, known HMB phases comprise the stoichiometric phase IV (x ) 2.0)1 with the maximum hydrogen content as well as various metastable intermediate phases, of which only phase IIa (0.6 < x < 0.8) has so far been characterized structurally.2 Technical interest in the deeply colored mixed electron/proton conductors was raised by a great number of possible applications (e.g., hydrogen-transfer catalysts, electrochromic displays, fuel cells, hydrogen storage, and gas sensors). Phase III is of particular interest as this phase is in equilibrium with H2 partial pressures >10-2 Pa and readily forms from Pt-coated MoO3 by spilloVer in a hydrogen atmosphere.3 * To whom correspondence should be addressed. Phone: +34 93 580 1853. Fax: +34 93 580 57 29. E-mail:
[email protected] (E.C.);
[email protected] (S.A.). † Institut de Cie ´ ncia de Materials de Barcelona. ‡ Universite ´ Pierre et Marie Curie. § Universita ¨ t Go¨ttingen | Present address: Department of Materials Science and Engineering, National University of Singapore, 9 Engineering-Drive-1, Blk. EA #07-40, Singapore 117576.
(1) Birtill, J. J.; Dickens, P. G. Mater. Res. Bull. 1978, 13, 311-316. (2) Adams, St. J. Solid State Chem. 2000, 149, 75-87. (3) Schwitzgebel, G.; Adams, St. Ber. Bunsen-Ges. Phys. Chem. 1988, 92, 1426-1430.
Figure 1. Schematic view of the hydrogen intercalation sites in MoO3.
Our previous detailed characterizations of structures and properties for phase I4,5 and II2,6-7 have shown that the ordered distribution of protons and electronic effects (such as Mo-Mo interactions) give rise to distortions of the host lattice in the form of incommensurate or commensurate superstructures. The structural information on phase III is, however, so far limited. Determinations of the average (4) Adams, St.; Ehses, K. H.; Spilker, J. Acta Cryst. 1993, B49, 958967. (5) Rousseau, R.; Canadell, E.; Alemany, P.; Galva´n, H.; Hoffmann, R. Inorg. Chem. 1997, 36, 4627-4632. (6) Adams, St.; Moretzki, O.; Canadell, E. Solid State Ionics 2004, 168, 281-290. (7) Scherlis, D. A.; Lee, Y. J.; Rovira, C.: Adams, St.; Nieminen, R. M.; Ordejon, P.; Canadell, E. Solid State Ionics 2004, 168, 291-298.
10.1021/cm050940o CCC: $30.25 © 2005 American Chemical Society Published on Web 10/27/2005
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structure based on neutron powder diffraction data for D1.68MoO38 and D1.65MoO39 accordingly state that phase III can be described as an intercalate phase of R-MoO3 but contradict each other in crucial details of the hydrogen distribution and host lattice distortion. Similarly, various attempts to clarify the hydrogen distribution by spectroscopic techniques did not lead to conclusive results (see below). Here we present a combination of theoretical and experimental investigations to promote the understanding of the structure-property relationships in the HMB phase III (HxMoO3, where x ≈ 5/3). Experimental Section Powder samples of phase III were prepared by electrochemical intercalation of hydrogen into MoO3 or chemical reduction of an aqueous MoO3 slurry with Zn/HCl. The hydrogen content was determined from the lattice constants based on a calibration curve derived from redox titrations following Choain and Marion.10,11 Room-temperature X-ray powder diffraction (XRD) data were collected in the range 10 < 2Θ < 100° (Cu KR) with a Philips X-pert powder diffractometer. A Mylar foil prevented the sample from contact with air. Temperature-dependent XRD data were recorded for the temperature range 130-400 K in an evacuated cryostat on a Philips multipurpose diffractometer and analyzed by the Rietveld program GSAS.12 Since the localization of protons in complex structures from X-ray powder data is hardly possible, our structure refinements are complemented by bond valence (BV) calculations to identify plausible hydrogen intercalation sites as a starting point for the first-principles calculations. The required set of BV parameters are known from a previous study.6 Energetically preferable potential H intercalation sites should be sites for which the BV sum V(H) from interactions of a H atom to all neighboring oxide ions matches the ideal BV sum Videal ) 1 valence unit, excluding sites too close to Mo cations. Isosurfaces of constant BV mismatch |∆V| ) |V(H) - Videal(H)| thus can be used as a simple tool to visualize regions that a proton might occupy. A more reliable prediction of H intercalation sites can be reached by taking into account also the effect of the proton on the BV sums of the other atoms in the structure, e.g., by determining the root meansquared average BV sum mismatch of all atoms in the structure (generally termed the global instability index GII).13 Again, isosurfaces of constant GII can be used to visualize the preferable regions for a hydrogen intercalation. These isosurfaces are constructed by calculating point by point the GII for hypothetical structures from combinations of the experimentally determined heavy atom positions and a 3D grid of hypothetical hydrogen positions. 6 The first-principles calculations were carried out using a numerical atomic orbitals density functional theory (DFT)14 approach, which has been recently developed and designed for efficient calculations in large systems and implemented in the SIESTA code.15-17 We have used the generalized gradient approximation (8) Dickens, P. G.; Short, A. T.; Crouch-Baker, S. Solid State Ionics 1988, 28-30, 1294-1299. (9) Anne, M.; Fruchart, D.; Derdour, S.; Tinet, D. J. Phys. France 1988, 49, 505-509. Anne, M.; Fruchart, D.; Derdour, S.; Tinet, D. J. Phys. France 1988, 49, 1315. (10) Adams, St., Thesis, Saarbru¨cken, 1991. (11) Choain, C.; Marion, F. Bull. Soc. Chim. Fr. 1963 212. (12) Larson, A. C.; Von Dreele, R. B. General Structure Analysis System (GSAS); Los Alamos National Laboratory Report LAUR 86-748, 2000. (13) Salinas-Sanchez, A.; Garcia-Munoz, J. L.; Rodriguez-Carvajal, J.; SaezPuche, R.; Martinez, J. L. J. Solid State Chem. 1992, 100, 201-211. (14) (a) Hohenberg, P.; Kohn, W. Phys. ReV. 1964, 136, B864-B871. (b) Kohn, W.; Sham, L. J. Phys. ReV. 1965, 140, A1133-A1138.
Figure 2. Comparison of spin-lattice relaxation times T1 and spin-spinrelaxation times T2 as determined from 1H NMR by Slade et al.27 (16 MHz, filled symbols) and Ritter et al.25 (15 MHz, open symbols, T1 only) vs temperature T. The inset displays the same T1 data as log(T1) vs 1000/T.
to DFT and, in particular, the functional of Perdew, Burke, and Ernzerhof.18 Only the valence electrons are considered in the calculation, with the core being replaced by norm-conserving scalar relativistic pseudopotentials19 factorized in the Kleinman-Bylander form.20 A specially optimized basis set based on recent soft-potential techniques,21,22 of the single-ζ kind including polarization orbitals, has been used for all atoms. The energy cutoff of the real space integration mesh was 100 Ry. The Brillouin zone (BZ) was sampled using grids of (1 × 2 × 3) and (18 × 18 × 5) k points23 for determination of the density and the Fermi surface (FS), respectively.
Results and Discussion Properties of Phase III. 1H NMR data by various authors24-26 accordingly indicate that the protons are mobile for T > 150 K but could not provide conclusive information on the hydrogen distribution. The comparison of the temperature dependences of the spin-spin relaxation times reported by Slade et al.27 and by Ritter et al.25 shown in Figure 2 demonstrates that the data are fully consistent, whereas the authors had presented them as evidence for their conflicting hydrogen distribution models. Incoherent inelastic neutron scattering data28,29 are consistent with H atoms that occupy sites in the vdW gaps exclusively or at least mainly as OH2 groups and are bonded to the terminal O atoms of (15) Soler, J. M.; Artacho, E.; Gale, J. D.; Garcı´a, A.; Junquera, J.; Ordejo´n, P.; Sa´nchez-Portal, D. J. Phys.: Condens. Matter. 2002, 14, 27452779. (16) http://www.uam.es/siesta/. (17) For reviews on applications of the SIESTA approach, see: (a) Ordejo´n, P. Phys. Status Solidi B 2000, 217, 335-356. (b) Sa´nchez-Portal, D.; Ordejo´n, P.; Canadell, E. Struct. Bonding 2004, 113, 103-170. (18) Perdew, J. P.; Burke, K.; Ernzerhof, M. Phys. ReV. Lett. 1996, 77, 3865-3868. (19) Trouiller, N.; Martins, J. L. Phys. ReV. B 1991, 43, 1993-2006. (20) Kleinman, L.; Bylander, D. M. Phys. ReV. Lett. 1982, 48, 1425-1428. (21) Anglada, E.; Soler, J. M.; Junquera, J.; Artacho, E. Phys. ReV. B 2002, 66, 205101. (22) Artacho, E.; Sa´nchez-Portal, D.; Ordejo´n, P.; Garcı´a, A.; Soler, J. M. Phys. Status Solidi B 1999, 215, 809-817. (23) Monkhorst, H. J.; Park, J. D. Phys. ReV. B 1976, 13, 5188-5192. (24) Slade, R. C. T.; Halstead, T. K.; Dickens, P. G. J. Solid State Chem. 1980, 34, 183-192. (25) Ritter, C.; Mu¨ller-Warmuth, W.; Scho¨llhorn R. J. Chem. Phys. 1985, 83, 6130-6138. (26) Sotani, N.; Eda, K.; Kunimoto, M. J. Chem. Soc., Faraday Trans. 1990, 86, 1583-1586. (27) Slade, R. C. T.; Halstead, T. K.; Dickens, P. G.; Jarman, R. H. Solid State Commun. 1983, 45, 459-463.
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Figure 3. Total conductivity of a powder sample of phase III (redrawn after ref 35).
the MoO3 double layers. A fraction of up to 10% of OH groups (as suggested from a NMR profile analysis for x ) 1.71 at 130 K24) can, however, not be excluded. Quasielastic neutron scattering data30,31 are best fitted assuming that the H in phase III belong to two distinct but slowly exchanging populations. Again the majority of the H is thought to exist as OH2 groups in the vdW gap, whose motions are restricted to 4-fold reorientations among the four sites surrounding each terminal oxygen. A temperature-dependent minority (10% at 255 K to 45% at 349 K) that undergoes self-diffusion is identified with OH groups. The exchange between the populations is assumed to be fast enough so that the populations could not be distinguished on the time scale of the 1H NMR studies.30 In contrast to first reports on conductivity measurements suggesting that phase III should be semiconducting,32 conductivity data for powder samples33,34 clarify that phase III is metallic throughout the investigated temperature range from 110K - 270 K (see Figure 3). At higher temperatures, conductivity data become unreliable due to the low stability of the phase in contact with metallic electrodes. Figure 3 moreover reveals a local maximum of the conductivity at T ≈ 220 K. Accordingly measurements of sound velocity and ultrasonic sound attenuation in phase III (x ≈ 1.6) ceramics indicate a phase transition around T ≈ 210 K.35 Measurements of the magnetic susceptibility by Sotani et al.36 in the temperature range 0-100 K indicate an additional cusp at T ≈ 65 K that is interpreted by the authors as a sign of a spinglass transition. While the above-mentioned low-temperature phase transitions are not related to significant changes in the lattice constants, X-ray powder data show a clear second-order phase transition at T ≈ 340 K (see Figure 4) that mainly affects the direction perpendicular to the layers.4 The temperature dependence of the spontaneous deformation (28) Dickens, P. G.; Birtill, J. J.; Wright, C. J. J. Solid State Chem. 1979, 28, 185-193. (29) Powell A. V.; Pointon, M. J.; Dickens, P. G. J. Solid State Chem. 1994, 113, 109-115. (30) Slade, R. C. T.; Hirst, P. R.; West, B. C.; Ward, R. C.; Magerl, A. Chem. Phys. Lett. 1989, 155, 305-312. (31) Slade, R. C. T.; Hirst, P. R.; Pressman, H. A. J. Mater. Chem. 1991, 1, 429-435. (32) Barbara, T. M.; Gammie, G.; Lyding, J. W.; Jonas, J. J. Solid State Chem. 1988, 75, 183-187. (33) Heusing, S., Diploma thesis, Saarbru¨cken, 1990. (34) Endres, F.; Schwitzgebel, G. Electrochim. Acta 1998, 43, 431-433. (35) Bamberg J., Thesis, Saarbru¨cken, 1987. (36) Sotani, N.; Fukumoto, T.; Eda, K.; Kunimoto, M.; Nakagawa, M. Chem. Lett. 1999, 593-599.
Figure 4. Temperature dependence of the lattice constants in phase III. The variation of the spontaneous deformation ∆a/a below Tc ) 339 K with the reduced temperature (Tc - T)/T (see inset) yields a critical exponent of the phase transition of 2β ) 0.72(3). (Redrawn including previously published data by Adams et al.4,37)
associated with this phase transition yields a critical exponent of the order parameter of β ) 0.36(2). In the temperature range 330-340 K, this phase transition is also observed as a broad λ-type anomaly in differential scanning calorimetry measurements.10 Despite the wealth of experimental information on the properties of HMB phase III, the lack of clear structural information has led to controversial discussions on their interpretation. Thus a more fundamental insight into the structure of phase III is required for a consistent discussion of structure-property relationships. Average Structure of Phase III. Structure determinations are restricted to powder methods, since homogeneous single crystals of a sufficient quality for structure determinations cannot be produced for phase III. As mentioned above, two structure determinations based on neutron powder diffraction data for D1.65MoO3 at 4.2 K by Anne et al.9 and for D1.68MoO3 at 293K by Dickens et al.8 exist in the literature. According to Anne et al. (see Figure 5a) the distortion of the MoO6 octahedra resembles the one that had been found previously for phase I and only half of the deuterium is localized on a well-defined crystallographic site with 41% occupancy. An overlapping nuclear density of the two equivalent deuterium sites along a connecting line between the apical O atoms of the two adjacent octahedral layers suggests a high local mobility. In the refinement by Dickens et al. (see Figure 5b), the MoO6 octahedron is significantly less distorted and all the deuterium atoms are located on two sites with occupancies of 0.44 and 0.41, but significantly higher atomic displacement parameters are found for the D site that had not been observed by Anne et al. H-O-H bond angles in Dickens
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Figure 5. Comparison of the published structure refinements for phase III based on neutron powder diffraction data (a) by Anne et al.,9 (b) by Dickens et al.,8 and (c) of X-ray refinements from this work. (Mo, magenta; O, red; H, orange, yellow). Only one average position could be refined for H from X-ray powder data. In b, the size of the H atoms spheres corresponds to the isotropic ADP parameters for the two distinct H atoms given in ref 8.
et al. refinement suggests that the H2O group should contain crystallographically distinct H atoms. Our X-ray Rietveld refinement results (Table 1) for the host lattice are largely in accordance with results of Dickens et al. for the host lattice (cf. parts b and c of Figure 5), while the localization of mobile H from X-ray data are surely less reliable than the findings from the neutron refinements. BV calculations (see Figure 6) confirm that the hydrogen should be mainly bonded to the terminal O(3) atoms (i.e., the apical O atoms pointing toward the vdW gap) and only a minor fraction might be bonded to O(2) atoms (i.e., the O atoms that connect corner-sharing MoO6 octahedra along the b direction). In harmony to the findings of Dickens et al. BV calculations suggest that H(2) might be less well localized. The GII isosurfaces in Figure 7 emphasize that all energetically preferable H intercalation sites (i.e., sites that lead to nearly ideal BV sums for both the intercalated proton and the oxide anions) are located within the vdW gaps either along the O(3)‚‚‚O(3) connecting lines or, to a lesser extent, also along connecting lines between O(3) and O(2) sites of the neighboring layer. There are, however, no signs for an occupation of the intralayer sites along O(2)‚‚‚O(2) zigzag lines that the hydrogen atoms occupy in phase I (x ≈ 1/3). Band Structure, Metal-Metal Bonding, and FS. To get an approximate assessment of the factors influencing the electronic structure of phase III, we carried out a firstprinciples study for the new average structure of this phase
employing a rigid band approach. First results based on preliminary structure data have been published elsewhere,7 so that we will only summarize here those parts required for further discussion. In our description of the electronic structure we will employ a local Cartesian axis system centered on the Mo atoms with its y-axis parallel to b, and its z-axis parallel to c, so that the x-axis is perpendicular to the layers. The three t2g orbitals are then the yz, xy, and xz orbitals (see Scheme 1). The double layers of corner-sharing MoO6 octahedra in phase III are composed of two individual MoO4 layers. For each MoO4 layer, the Mo yz orbital spreads into a twodimensional (2D) band because it makes π-type interactions with p orbitals of O along the two main directions of the lattice. In contrast, the Mo xy and xz orbitals lead to 1D bands along orthogonal directions (b and c, respectively) because they make π-type interactions with the p orbitals of O along one direction but δ-type interactions along the orthogonal direction. When condensing the layers to the double octahedral layers, the energy splitting between the in-phase and out-of-phase combination of the t2g orbitals is important only for the xz orbitals, because they point toward each other and, thus, can lead to Mo-Mo bonding and antibonding combinations. A key aspect for our discussion is that an axial distortion of the octahedra leading to two very different Mo-O apical bonds raises the xy and xz orbitals leaving the yz orbital as the lowest level (cf. Scheme 1). The two axial
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Table 1. Comparison of Refinement Results for Phase IIIa RG: C2/m
Rp ) 0.086
″
Rp ) 0.15
RWP ) 0.114 RWP ) 0.21
″
RI ) 0.15
a ) 13.927(2) Å a ) 13.86(1) Å a ) 13.986(6) Å c ) 4.0588(3) Å c ) 4.059(1) Å c ) 4.065(1) Å
b ) 3.7735(1) Å b ) 3.773(1) Å b ) 3.780(1) Å β ) 93.939(5)° β ) 93.9(1)° β ) 93.99(2)°
atom type Mo(1) O(1) O(2) O(3) H(1) H(2)
x 4i
0.07046(8) 0.068(1) 0.106(2) 4i 0.9271(5) 0.925(1) 0.917(1) 4i 0.5776(5) 0.577(1) 0.614(1) 4i 0.2265(5) 0.225(1) 0.249(1) 8j 0.210(6) 0.238(2) 0.242 8j 0.236(2) -
y 0
z
0.2187(6) 0.214(2) 0.265(5) 0 0.273(4) 0.277(3) 0.233 0 0.217(4) 0.225(3) 0.275(7) 0 0.229(3) 0.247(3) 0.299(3) 0.38(4) 0.82(4) 0.346(3) 0.901(4) 0.32 0.914(5) 0.349(5) 0.611(5) -
N 1
Uiso 0.0030(4) 0.020(3)
1
0.007(2) 0.005(3) 0.05(1) 1 0.007b 0.009(3) 0.12(2) 1 0.007b 0.016(4) 0.030(8) 0.5 0.007b 0.44(2) 0.027(6) 0.41(4) 1.5(2) 0.41(2) 0.053(8) -
a Within each entry the first line corresponds to the results of our own XRD Rietveld refinement for H1.68MoO3 at room temperature (bold), the second to literature data by Dickens et al.8 for D1.68 MoO3 at 293 K (italics), and the third to D1.64MoO3 at 4.2K after Anne et al.9 b Constrained to equal the value for the preceding atom.
Figure 6. BV maps of volume regions that are accessible to protons with a hydrogen BV sum mismatch of (a) |∆V(H)| ) 0.3 valence units or (b) |∆V(H)| ) 0.38 valence units, requesting that the V(H) contains significant contributions from bonds to at least two oxide anions.
Mo-O bonds in the phase III average structure are not very different, i.e., 2.17 and 2.02 Å (note that in phase I these values are as different as 1.66 and 2.40 Å),4 so that the three Mo t2g orbitals should be sufficiently similar in energy and the 2D bands originating from the yz orbitals and the 1D bands originating from the xy and xz orbitals will overlap at
the bottom of the d-block bands of the ideal double layer, although the xz bands are expected to by lower if metalmetal bonding really occurs. The shape of the bands in Figure 8 can now be easily understood. Since there are two double octahedral layers per unit cell and the interlayer interactions are weak, most bands appear in pairs. After noting some weakly avoided crossings it is clear that the partially filled bands of Figure 8 are two pairs of 2D bands (two bands per double octahedral layer) and one pair of 1D bands along b (one band per double octahedral layer). At the bottom of the t2g block bands, there is another pair of 1D completely filled bands along c. As predicted by our simple model, the partially filled 2D bands are based on the Mo yz orbitals whereas the partially filled 1D bands along b are based on the xy orbitals of Mo. The pair of filled bands originate from the other set of 1D bands (based on the Mo xz orbitals) and describe two Mo-Mo bonds per unit cell (one per double octahedral layer). In the average structure, the values of the two direct Mo-Mo distances per unit cell are 2.55 and 3.11 Å, the first one being a typical Mo-Mo single-bond distance.38 Thus, the two lowlying bands are a clear fingerprint of Mo-Mo bonding along the zigzag Mo chains in the c direction. Two of the 3.33 electrons per double layer filling the t2g bands are thus used to form a Mo-Mo bond and 1.33 are left to partially fill the 2D and 1D bands causing the metallic properties. The two pairs of 2D bands (one of each pair is located in either double octahedral layer) are split by approximately 1 eV because of the distortions in the double octahedral lattice induced by the Mo-Mo bond formation. This is also the case for the two sets of 1D xy bands. The reason why only two of the xz bands (one per layer) are filled is simply due to the large splitting accompanying the Mo-Mo bond formation. As shown in Figure 8, two pairs of 2D bands and one pair of 1D bands are partially filled thus leading to the existence of a FS. This FS (see Figure 9) results from the hybridization of open contributions associated with the 1D bands and closed contributions associated with the lowest 2D bands. There is also a smaller closed contribution due to the second pair of 2D bands. The warping along the interlayer direction is extremely small. The warped open lines of the FS of Figure 9 are quite well nested by a vector which is approximately (1/3)b* + (1/2)c* and consequently, a modulation corresponding to a 3b × 2c superstructure could occur,39 providing some stabilization for the electrons not implicated in the Mo-Mo bond formation, and affecting the transport properties. Problems with the Average Structure and Requirements for a Better Description. Although the discussion of the average structure already helps rationalizing the properties of phase III, it can obviously not explain the crucial correlation of the properties (especially the transport properties) with order-disorder transitions and the electronic structure. The transitions observed around 220 K in conductivity, sound velocity, ultrasonic attenuation, etc.40 are (37) Adams, St., Habilitation Thesis, Go¨ttingen, 2000. (38) Cotton, F. A.; Wilkinson, G. AdVanced Inorganic Chemistry, 5th ed.; Wiley-Interscience: New York, 1988; p 808. (39) Canadell, E.; Whangbo, M.-H. Chem. ReV. 1991, 91, 965-1034.
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Figure 7. Maps of regions in the host lattice of phase III, where the intercalation of a hypothetical proton would lead to a global instability index GII of (a) 0.065, (b) 0.07, (c) 0.14, (d) 0.17, or (e, f) 0.21. Graphs a-e show projections of the vdW gap region along the x axis. The projection of the 0 < x
