Structure of magnesium selenate enneahydrate, MgSeO4·9H2O, from 5 to 250 K using neutron time-of-flight Laue diffraction

Single-crystal neutron diffraction, ab initio calculations and Raman spectroscopy are applied to study the structure and hydrogen bonding of magnesium selenate enneahydrate, a recently discovered new hydration state in M 2+ XO4 hydrates and a unique example of a material with a cation: (oxy)anion:water ratio of 1:1:9, giving novel insights into the polymerization of water in highly hydrated materials.


Introduction
Crystalline hydrates with the general formula M 2+ XO 4 ÁnH 2 O are very well known for n in the range 1-7 with X = S, Se, Cr and Mo, and M 2+ being any of a wide range of first-row transition metals, as well as Mg 2+ and Cd 2+ , although not all integer values of n occur for a given cation or oxyanion composition. Until quite recently, there was a conspicuous gap for n = 8-10, and just one known example with n = 11, being MgSO 4 Á11H 2 O (aka meridianiite). Motivated principally by an interest in 'planetary' cryohydrates (Fortes & Choukroun, 2010), a series of systematic studies have been carried out in an effort to understand the apparently limited occurrence of these materials with n ! 8. This work has revealed new examples of 11-hydrates that are isotypic with the sulfate compound, MgCrO 4 Á11H 2 O Fortes et al., 2013) and MgSeO 4 Á11H 2 O (Fortes, 2015), and shown that hydrates with n = 8 and 9 can be synthesized (Fortes et al., 2012a,b;Fortes, 2014). All of the M 2+ sulfate octa-and enneahydrates prepared thus far are metastable, being crystallized only by rapidly quenching small droplets of aqueous solution in liquid nitrogen, after which they typically undergo solid-state transformations over periods of minutes to hours into other stable hydrates. However, this proved not to be the case for magnesium selenate, where it was determined that MgSeO 4 Á11H 2 O undergoes incongruent melting to form MgSeO 4 Á9H 2 O apparently in equilibrium with aqueous solution, and furthermore that MgSeO 4 Á7H 2 O crystals stored in a freezer formed pseudomorphs after the heptahydrate composed of the enneahydrate (Fortes, 2015).
The solid-liquid phase equilibria in the MgSeO 4 -H 2 O binary system have been a source of some disagreement, there being significant differences between the observations of Meyer & Aulich (1928) and those of Klein (1940). In neither instance was the behaviour below the freezing point of ice characterized, the solubility curves and eutectic being found by extrapolation: the highest stable hydrate found in this system, and the phase believed to be in equilibrium at the eutectic (266 K), was MgSeO 4 Á7H 2 O, the structure of which was recently determined (Fortes & Gutmann, 2014). Nevertheless, attempts to grow single crystals of magnesium selenate enneahydrate by evaporation of an aqueous solution in air at 269 K were successful and this allowed us to carry out a more thorough characterization of the material's properties and to obtain more accurate and complete structural parameters, as reported here.

Sample preparation
An aqueous solution of magnesium selenate was prepared as follows: commercially available aqueous H 2 SeO 4 (Sigma-Aldrich 481513, 40 wt %) was diluted to 25 wt % H 2 SeO 4 (1.72 M) with distilled water (Alfa-Aesar, ACS Reagent Grade, 36645), which was then heated to $ 340 K. To this liquid was added a molar excess of powdered MgO (Sigma Aldrich 342793, > 99% trace metals basis, À325 mesh). Since this synthesis, unlike the reaction with basic magnesium carbonate, is quiescent its progress was followed with a Tecpel hand-held pH meter. Once the pH of the solution stabilized at 8.80, the supernatant liquid was decanted, triply filtered and left to stand. Evaporation in air at room temperature resulted in the precipitation of cm-sized crystals of MgSeO 4 Á6H 2 O. After a further round of recrystallization from distilled water the phase purity of the hexahydrate was verified by X-ray powder diffraction (see xS1 of the supplementary information).
Crystals of MgSeO 4 Á6H 2 O were crushed to a coarse powder and dissolved in distilled water to a concentration of $ 35 wt % MgSeO 4 , the liquid then being left to evaporate in a petri dish in a refrigerated workshop (air temperature = 269 K). Crystals up to 1 cm in length, having a morphology distinct from either the heptahydrate (acicular prisms with a square cross section) or the hexahydrate (pseudo-hexagonal plates, Fig. S1), grew from solution over a period of a few days. One of these was extracted from solution, dried and powdered under liquid nitrogen before being subjected to X-ray powder diffraction analysis using a custom-made Peltier cold stage  mounted on a PANAlytical X'Pert Pro powder diffractometer. The X-ray powder diffraction pattern of this material was identical to that reported previously for MgSeO 4 Á9H 2 O obtained by quenching droplets of the solution in liquid nitrogen (Fortes, 2015).
Subsequent attempts to form single crystals produced only MgSeO 4 Á7H 2 O. However, these crystals transformed to MgSeO 4 Á9H 2 O (as pseudomorphs after the heptahydrate) during 11 days storage in a freezer at 253 K. This material was used to seed the growth of new 9-hydrate single crystals from solution. Two cycles of recrystallization using this seed material yielded good quality single crystals up to 5 mm in length, which were harvested for analysis.
Magnesium selenate enneahydrate crystallizes in the monoclinic prismatic class as rhomboids that are tabular on (0 1 1) and exhibit marked growth sectors (Fig. 1). The facial indices shown in Fig. 2 were determined by X-ray diffraction methods and quantitative analysis of microphotographs, the morphology being modelled using WinXmorph (Kaminsky, 2005  Microphotograph of MgSeO 4 Á9H 2 O crystals in air illustrating their representative platy rhomboidal habit. which corresponds to 9.4 water molecules per formula unit. It is not unusual to find a small excess water content in these circumstances since crystals often occlude small pockets of mother liquor that are inevitably more water-rich. The dried residue was identified by X-ray powder diffraction as thephase of MgSeO 4 (space group Pbnm), whilst elemental analysis using a Jeol JXA8100 microprobe confirmed the compound's stoichiometry, the Mg:Se molar ratio being 1.03 (5). The solid content of the solution in equilibrium with these crystals was determined by drying at 673 K to be 24.3 wt % MgSeO 4 . This is significantly greater than the estimated solubility of MgSeO 4 Á7H 2 O at the same temperature (cf. Klein, 1940), an observation that accords with other evidence we have obtained to suggest that the heptahydrate is actually metastable with respect to the enneahydrate over a narrow temperature range. Our determination of the enneahydrate's solubility leads us to propose a modification of the binary phase diagram (Fig. 3) with a stable eutectic between MgSeO 4 Á9H 2 O and ice around 2 above the previously reported eutectic between MgSeO 4 Á7H 2 O and ice. The presumption that MgSeO 4 Á9H 2 O undergoes incongruent melting to MgSeO 4 Á7H 2 O + liquid remains to be experimentally verified.

Single-crystal neutron diffraction
Two crystals were dried on filter paper and then loaded together into a thin-walled aluminium foil pouch suspended inside a 6 mm inner diameter vanadium sample tube. The sample holder was transported to the ISIS facility packed in dry ice and screwed onto the end of a cryostat centre-stick. This assembly was inserted into a closed-cycle refrigerator (CCR) on the SXD beamline (Keen et al., 2006) after which the sample was cooled from $ 200 K directly to 5 K.
Time-of-flight Laue data were then collected in a series of four orientations, counting each for $ 2 h (300 mA of beam current). The diffraction spots from both crystals in the beam were indexed with the unit cell obtained at 250 K (Fortes, 2015), after which the intensities were extracted using the three-dimensional profile-fitting method implemented in SXD2001 (Gutmann, 2005). Data were subsequently collected using the same measurement strategy at 100, 175 and 250 K.
Structure refinement was carried out using SHELX2014 (Sheldrick, 2008;Gruene et al., 2014) starting from the atomic coordinates -including estimated H positions -determined by the structure solution from X-ray powder diffraction data. The structure was found to be correct in all aspects and inspection of Fourier difference maps revealed no missing or incorrectly positioned atoms, and no evidence of possible H-atom site disorder, which the previous work had mooted as a possibility. For the 100, 175 and 250 K datasets, no restraints were used and all anisotropic displacement parameters were refined independently. At 5 K, the displacement ellipsoid of the Se Reinterpretation of the binary MgSeO 4 -H 2 O phase diagram in light of the measured solubility of the enneahydrate at 269 K (filled square), observations of the incongruent melting behaviour of the undecahydrate and the apparent metastability of the heptahydrate at low temperatures. Open circles are measurements reported by Klein (1940) and solid lines are assumed to be the equilibrium liquidi and solidi; the dashed lines depict the previously understood low-T behaviour, which we now interpret as being metastable liquidi.  atom became marginally non-positive definite (NPD), which is not wholly unsurprising given the very low temperature and the fact that this is the heaviest atom (by a factor of three) in the structure such that the displacement amplitudes are inevitably very small. This was remedied with the application of a SHELX instruction (ISOR) to force the selenium U ij to behave in a slightly more isotropic fashion.

Raman spectroscopy
Laser-stimulated Raman spectra were measured using a portable B&WTek i-Raman Plus spectrometer equipped with a 532 nm laser (P max = 37 mW at the probe tip) that records spectra over the range 171-4002 cm À1 with an optimal resolution of $ 3 cm À1 . Measurements were carried out on a large single crystal of MgSeO 4 Á9H 2 O in our cold room using the BC100 fibre-optic coupled Raman probe. Background noise was minimized by acquisition of multiple integrations, the time per integration being limited by detector saturation. At 262 K spectra were integrated for a total of 400 s (P = 18 mW). Another set of spectra were acquired on the same crystal after 5 months of storage in a freezer; this measurement at 259 K (1200 s, P = 37 mW) revealed no evidence of changes in hydration state. The temperature of the crystal was then reduced to 78 K by immersion in liquid nitrogen and another integration (1750 s, P = 37 mW) was acquired. Complementary measurements on single crystals of MgSeO 4 Á6H 2 O were obtained at 262 K and on powders of MgSeO 4 (-phase formed at 673 K by dehydration of the enneahydrate) at room temperature.

Computational methods
In order to confirm the veracity of our structure solution, and to aid in interpretation of the Raman spectrum, we carried out a first-principles calculation using density functional theory, DFT (Hohenberg & Kohn, 1964;Kohn & Sham, 1965), as implemented in the Vienna ab initio simulation package, VASP (Kresse & Furthmü ller, 1996). The plane-wave expansion was treated using the projected augmented-wave method, PAW (Blö chl, 1994); with the PAW potentials generated by Kresse & Joubert (1999) and distributed with VASP. The exchange-correlation was accommodated using the PBE generalized gradient corrected functional (Perdew et al., 1997). This form of the generalized gradient approximation (GGA) has been demonstrated to yield results of comparable accuracy to higher-level quantum chemical methods, such as MP2 and coupled-cluster methods, in hydrogen-bonded systems (e.g. Ireta et al., 2004), despite not correctly representing dispersion forces.
Convergence tests were carried out to optimize the k * -point sampling of the Brillouin zone within the Monkhorst-Pack scheme (Monkhorst & Pack 1976) and the kinetic energy cutoff of the plane-wave basis set. It was found that a 3 Â 2 Â 1 k * -point grid combined with a kinetic energy cut-off of 875 eV yielded a total-energy convergence better than 10 À3 eV per atom and pressure converged better than 0.1 GPa. A structural relaxation under zero-pressure athermal conditions was carried out, starting from the experimental crystal structure obtained at 10 K, in which the ions were allowed to move according to the calculated Hellman-Feynman forces and the unit-cell shape was allowed to vary. The relaxation was stopped when the forces on each atoms were less than 5 Â 10 À4 eV Å À1 and each component of the stress tensor was smaller than 0.05 GPa. The phonon spectrum was then computed using the small displacement method as implemented in the PHON code (Alfè, 2009). The construction of the full force-constant matrix requires knowledge of the forcefield induced by displacing each atoms in the primitive cell in the three Cartesian directions. Since there are 132 atoms in the primitive cell, the total number of required displacements for this system would then be 792 (allowing for both positive and negative displacements), although this can be reduced to 198 by exploiting the symmetry elements present in the crystal. We used displacements of 0.02 Å , which are sufficiently small to obtain phonon frequencies that are converged to better than 0.1%. Since, in this instance, we are interested only in the normal modes at the Brillouin zone (BZ) centre, all of the required information could be obtained by computing the force matrix for the primitive unit cell.
The resulting modes at the BZ centre were classified according to the irreducible species of point group C 2h ; this was done using standard group-theory techniques with the help of the program SAM (Kroumova et al., 2003), available at the Bilbao Crystallographic server (Aroyo et al., 2006).

Description of the experimental structure and bonding
The asymmetric unit of MgSeO 4 Á9H 2 O is shown in Fig. 4; selected interatomic distances at various temperatures are listed in Tables 1-4. The more accurately determined atomic coordinates allow us now to make more definitive statements concerning the structure and its relationship to other crystalline hydrates.  Table 1 Bond lengths (Å ) in the polyhedral ions of MgSeO 4 Á9H 2 O as a function of temperature. The structure consists of isolated Mg(H 2 O) 6 2+ octahedra and SeO 4 2À tetrahedra linked by a framework of moderately strong hydrogen bonds (with one exception discussed below, HÁ Á ÁO ranges from 1.75 to 1.87 Å ). The remaining three water molecules are not coordinated to either Mg or Se, occupying 'voids' between the polyhedral ions. The Mg(H 2 O) 6 2+ octahedron is slightly elongated along the Mg-Ow6 vector compared with the other five Mg-O distances (Table 1), which is due to the tetrahedral coordination of Ow6 compared with the trigonal coordination of the remaining Mg-coordinated water molecules. Conversely, the Se-O1 bond is significantly shorter than the other three Se-O distances, which is due to O1 acting as an acceptor of two hydrogen bonds whilst O2 through to O4 accept three hydrogen bonds each (Tables 1 and 3). The same pattern of systematic bondlength variations occurs in both MgSeO 4 Á7H 2 O (Fortes & Gutmann, 2014) and MgSeO 4 Á6H 2 O (Kolitsch, 2002), as well as their sulfate analogues (Baur, 1964;Ferraris et al., 1973;Fortes et al., 2006), depending on the number of hydrogen bonds accepted by a given water molecule or O atom (Table  2). Indeed, such patterns provide useful complementary information of value in deducing the geometry of an otherwise indeterminate hydrogen-bond network from, for example, powder X-ray diffraction data.

Figure 4
The asymmetric unit of MgSeO 4 Á9H 2 O at 100 K with ellipsoids drawn at the 50% probability level. This and all subsequent structural representations are drawn using DIAMOND (Putz & Brandenburg, 2006). atoms, is obliged to be ordered. However, the first ring is not so obliged and it was hypothesized by Fortes (2015) that occupational disorder of the H atoms could occur in ring #1 in a manner similar to that which occurs in the water tetramer in Na 2 SO 4 Á10H 2 O (cf. Brand et al., 2009, and references therein).
In sodium sulfate decahydrate, variable ordering of the hydrogen bonds in the square rings has a measureable influence on the material's thermal expansion depending on the cooling rate: high cooling rates freeze in substantial disorder, resulting in a strained unit cell at low temperatures. One feature that led to the inference of possible disorder in ring #1 was the presence of a rather long (i.e. weak) hydrogen bond between Ow9 and Ow6, roughly 4-8% longer than the other OÁ Á ÁO contacts, which was apparent in the original Xray powder refinement (Fortes, 2015). This long hydrogen bond is confirmed by the single-crystal data, the Ow9Á Á ÁOw6 distance through H9B increasing from 2.923 (4) Å at 5 K to 2.984 (14) Å at 250 K; compare these with the value of 3.05 (1) Å obtained from X-ray powder data at 248 K. Additionally, the ADPs of H atoms associated with the three interstitial water molecules (Ow7, Ow8 and Ow9) are, in general, of greater magnitude than of the Mg-coordinated waters and the ADP of H9B is either close to, or is, the largest of all H-atom ADPs at each of the four temperatures. H9BÁ Á ÁOw6, indicated by a dashed yellow rod in Fig. 4, thus appears to be the weakest hydrogen bond in the structure.
As shown in Fig. 6, Fourier maps (F obs ) phased on the refined structure reveal no partially occupied H-atom sites in the tetramer rings at any temperatures, at least down to the level of the background noise (around 5% of the nuclear scattering density due to a H atom); the Ow6 and Ow9 molecules are thus orientationally ordered and remain so throughout cooling and warming.
One hitherto unidentified aspect of the structure of these highly hydrated inorganic salts is the occurrence of extended water polymer networks. In MgSeO 4 Á9H 2 O the water tetramer described above is merely the centre of a larger structure, a centrosymmetric dodecamer, (H 2 O) 12 , comprised of Ow2, Ow5, Ow6, Ow7, Ow8 and Ow9 (Fig. 7a). The aforementioned tetramer is extended by a pair of pentagonal rings with apical chains folded back upon the structure in such a way as to form a pseudo-pentagonal sigmoidal profile (Fig. 7b), the 'missing' edge of the open pentagon being supplied by the edge of the Mg(H 2 O) 6 octahedron.
The position of the neutral dodecamer cluster in relation to its surrounding ionic polyhedra and the remaining three water monomers is shown in Fig. 8.
Similar pentagonal motifs occur in the polymeric water frameworks of all related hydrates with one or more free interstitial water molecule (Fig. 9). In MgSeO 4 Á7H 2 O there is an infinite chain of open-sided pentagons with decorated corners extending along the crystal's c-axis, whereas in MgSeO 4 Á11H 2 O there is a hexadecamer, (H 2 O) 16 , composed of closed pentagonal rings, which has short side-chains extending along the c-axis of the crystal.
The observation of dodecamer and hexadecamer water clusters in crystal structures is not unique (see Song & Ma, 2007;Wang et al., 2010;Li et al., 2012;Ghosh & Bharadwaj, 2004;Jin et al., 2008); indeed there is a dodecamer structure very similar to that described here in copper citrate phenanthroline hexahydrate (Fu et al., 2010). Close examination of their structure, however, reveals that the two pentagons of the dodecamer appear not to be hydrogen bonded to one another. Consequently, their 'novel' self-assembled structure may not be a true 12-mer, whereas ours is a bona fide hydrogenbonded polymeric unit. The two symmetry-inequivalent square rings that occur in the structure of MgSeO 4 Á9H 2 O and the manner in which these bridge to form chains of octahedra (horizontally, along [1 1 0]) and chains of tetrahedra (vertically, along [1 0 1]). The dominance of pentagonal motifs in these water polymer structures is particularly interesting since five-sided rings have been implicated in the unusual properties of supercooled water (Speedy, 1984) and they are a key structural feature of various high-pressure forms of ice, such as ices III, IX, XII and XIV (Londono et al., 1993;Lobban et al., 1998;Salzmann et al., 2006), as well as the clathrate hydrates (e.g. Kirchner et al., 2004). Furthermore, pentagonal motifs are found in hydrogenbonded 'ices' such as ammonia hydrates ; hence the importance of solutes in aqueous solutions (or the substrate; Carrasco et al., 2009) as a structure-forming template cannot be overlooked.

Computational analysis of the structure and bonding
The ab initio zero-pressure athermal relaxed structure agrees very well, for the most part, with the experimental structure described above. The unit-cell dimensions are a = 7.28013, b = 10.50519, c = 17.25609 Å , = 109.2550 and V = 1245.907 Å 3 , values that are each within 1% of the experimental unit cell at 5 K. Fractional atomic coordinates are given in xS2 of the supporting information. Several noteworthy differences between calculations and experiments appear in Tables 1 and 3    and experimental Mg-O distances (R 2 = 0.990). By contrast, the calculated HÁ Á ÁO hydrogen bond lengths are almost all shorter than those observed in the real crystal. Evidently, the over-inflation of the unit cell by the greater Se-O and Mg-O distances is largely cancelled out by the shorter hydrogen bonds. There is, again, a good linear correlation (R 2 = 0.827) between hydrogen-bond angles and, with a single exception, a reasonable correlation (R 2 = 0.703) between HÁ Á ÁO distances from DFT calculations and from the real crystal (Fig. 10). The outlier in this instance is one of the bonds comprising the water tetramer, H9BÁ Á ÁOw6: DFT calculations give a length of 1.8275 Å for this contact, very similar to other hydrogenbonded distances in the structure; however, this differs markedly from the length found in our single-crystal neutron diffraction refinements ($ 1.98 Å ) and inferred from the Xray powder diffraction refinements ($ 2.05 Å ).
It becomes a matter of interest, therefore, to ascertain the energetic properties of the hydrogen bonds in the structure. The properties of the hydrogen bonds may be described quantitatively from the topology of the electron density according to Bader's Quantum Theory of Atoms-in-Molecules, QTAIM (Bader, 1990). Of interest to us are the saddle points where the gradient in the electron density, r(r), vanishes, the so-called bond critical points (BCPs). Useful metrics of the bond strength and character are the electron density at the BCP, (r BCP ), and the Laplacian of the electron density at the BCP, r 2 (r BCP ), which represents the 3 Â 3 Hessian matrix of second partial derivatives of the electron density with respect to the coordinates. The eigenvalues of this matrix, 1 , 2 and 3 (which sum to r 2 ) are the principal axes of 'curvature' of the electron density perpendicular to the bond ( 1 , 2 ) and along the bond ( 3 ). A negative Laplacian at the bond critical point generally corresponds to a concentration of electron density, which is characteristic of a covalent bond, whereas ionic bonds and hydrogen bonds have a positive Laplacian, indicative of a depletion in electron density. We have used the program AIM-UC (Vega & Almeida, 2014) to compute the locations and properties of the electron density at the HÁ Á ÁO BCPs in magnesium selenate enneahydrate ( Table 5).
The dissociation energy of the hydrogen bond may be estimated with varying degrees of accuracy from the electron density (e.g. Vener et al., 2012). The total energy density is the sum of the local kinetic and potential electronic energies, G(r) and V(r), respectively, at the BCP (Bader & Beddall, 1972) where the potential energy is related to the Laplacian of the electron density via the local form of the virial theorem (Bader, 1990) and the kinetic energy is obtained by partitioning of the electron density (e.g. Abramov, 1997)

Figure 8
The broader context of the neutral dodecamer's relationship to the ionic polyhedra and to particular planes that were identified previously as common forms in the macroscopic morphology of the crystals.

Figure 9
Polymeric water structures found in this and related hydrate structures revealing the propensity of these chains or clusters to adopt pentagonal motifs.
Espinosa et al. (1998) proposed that the hydrogen-bond energy, E HB , could be obtained simply from the potential energy density and this expression continues to be used widely, whereas Mata et al. (2011) subsequently suggested that a more accurate value could be found from the kinetic energy density A subsequent analysis by Vener et al. (2012) found that equation (4) systematically overestimates E HB compared with the spectroscopically determined hydrogen-bond energies. However, the value given by equation (5) appears to yield reasonably accurate values of E HB . In Table 6 we detail E HB as calculated using equations (4) and (5). Furthermore, we give a 'corrected' value of E HB based on equation (4) and the tabulated results in Vener et al. (2012) such that E HB (corrected) = 0.465E HB + 16.58. The mean values of E HB in the right-hand column thus represent our most accurate determination of the hydrogen-bond dissociation energy in this compound. As shown in Table 4, the hydrogen bonds in these hydrates may be classified according to the donor and acceptor species, which of itself reveals interesting trends in the ways these structures are organized with increasing hydration state. Notably, the interstitial water molecules that occur once the  Table 5 Properties of the electron density at the BCPs in the hydrogen bonds as determined from the DFT calculations.
Electron density, (r), is reported in e Å À3 , whereas the Laplacian, r 2 (r), and the eigenvalues of the Hessian matrix, 1 , 2 and 3 , are given in e Å À5 .

Figure 10
Correlation between the experimentally determined and calculated O-HÁ Á ÁO bond angles (left) and HÁ Á ÁO bond lengths (right). The filled circle on the right represents the H9BÁ Á ÁOw6 bond, which appears to be significantly weaker and longer than other water-water hydrogen bonds in the structure.
cation is 'saturated' serve more often than not to bridge between the cationic and anionic polyhedra instead of forming hydrogen bonds with other interstitial water molecules. Categorization of the hydrogen-bond energies in Table 6 according to the types listed in Table 4 shows that the weakest hydrogen bonds (mean = 32.4 AE 3.3 kJ mol À1 ) are those between Mg-coordinated waters and sulfate O atoms. The hydrogen bonds donated by free interstitial waters to the sulfate O atoms are similarly weak, 32.9 AE 2.5 kJ mol À1 . On the other hand, hydrogen bonds formed between water molecules are significantly stronger; those between Mg-coordinated water and interstitial water have a mean energy of 39.0 AE 2.3 kJ mol À1 and the single hydrogen bond between two interstitial waters has an energy of 36.7 kJ mol À1 . The exception, again, is the H9BÁ Á ÁOw6 bond; whilst this is not quite the weakest hydrogen bond in the structure it is much the weakest (by $ 25%) of the water-water hydrogen bonds.

The Raman spectrum
MgSeO 4 Á9H 2 O crystallizes in the centrosymmetric space group P2 1 /c having a primitive cell with C 2h point-group symmetry and four formula units per unit cell; all ions and molecules are located on sites of C 1 symmetry. Based on a consideration of the normal vibrational modes of the ionic polyhedra and the neutral water molecules, we have carried out a factor group analysis by the correlation method to determine the symmetry species of all Raman-active modes. Allowing for the modes corresponding to translation of the entire crystal, we find that there are 198 normal modes summarized as À opt (Raman) = 99 A g + 99B 1g .
The majority of these vibrations are associated with the water molecules, grouped into three distinct portions of the spectrum (Fig. 11). The highest frequency modes, observed between $ 3100-3600 cm À1 , correspond with the O-H symmetric and asymmetric stretch, 1 and 3 , respectively. At higher temperatures, the asymmetric potential well in which the H atoms are oscillating leads to thermal broadening of the peaks, although it is nonetheless possible to identify five discrete Lorentzian contributions to the high-frequency feature. When the temperature is reduced to 78 K, however, the peaks sharpen substantially, making it possible to clearly identify at least seven individual bands (Fig. 12).
In the mid-frequency range, around 1600-1700 cm À1 , a rather weak feature, comprising three identifiable peaks, is produced by the symmetric bending mode, 2 , of water. In the low-frequency range, below 1000 cm À1 , there are numerous librational and rotational modes of water but these are so weak and diffuse that they are manifested (at 259 K) as rather broad 'bumps' between the much sharper and more intense selenate peaks. As with the stretching modes, cooling results in sharpening of these features such that discernible peaks appear between $ 540 and 770 cm À1 (see Fig. 11, upper spectrum, and

Figure 11
Raman spectrum of MgSeO 4 Á9H 2 O at 259 and 78 K, measured on the (0 1 1) face of a single crystal. Pertinent groups of vibrational modes contributing to the observed features are labelled. The raw data are given in Table S3 of the supporting information. Table 6 Energetic properties of the hydrogen bonds.
The local kinetic energy density, G(r), the local potential energy density, V(r), and the total energy density, H(r), at the BCP are all given in atomic units. † The hydrogen-bond energies, E HB , are in units of kJ mol À1 . symmetric deformation modes, 2 (E); the triply degenerate asymmetric stretch, 3 (F 2 ); and the triply degenerate asymmetric deformation modes, 4 (F 2 ). The site symmetry of the SeO 4 2À ions in the crystal (C 1 ) break the degeneracy of these internal modes, leading to nine modes of symmetry A. Under the factor-group symmetry of the unit-cell (C 2h ), each of these nine components are further split into modes of symmetry A g , B g , A u and B u , of which only the first two are Raman active. Consequently, we would expect to find 9A g and 9B g vibrational modes in the Raman spectrum due to the SeO 4 2À ion. The extent to which nearly degenerate bands are dispersed depends on the distortion of the selenate ion from ideal T d symmetry.
The highly symmetric 1 (A g ) mode of SeO 4 2À is the sharpest and most intense feature, occurring at 841.3 cm À1 at 259 K and shifting to 845.9 cm À1 at 78 K (compare with 844.1 cm À1 in MgSeO 4 Á6H 2 O). The asymmetric stretching modes are at a slightly higher frequency, the six expected components contributing to the scattering between $ 850 and 900 cm À1 , which can be deconvolved into three broad bands (Table 7). The spread of band centres for these 3 modes, Á 3 , are $ 35 cm À1 in the enneahydrate and 43 cm À1 in the hexahydrate; these differences are attributable to the difference in Se-O bond lengths noted in Table 2, which are due to the number of hydrogen bonds donated to each apical oxygen.
These shifts match the general trend observed in both Mg-and Fe-sulfate hydrates of a shift in 1 to lower frequency and a smaller dispersion of 3 with increasing hydration number (Wang et al., 2006;Chio et al., 2007). Similar trends have been found in ab initio calculations of hydrated tetrahedral oxyanions (Pye & Walker, 2011). Predictably, given the overestimate of the ab initio Se-O bond length, the calculated vibrational frequencies differ substantially from the observations; 1 (A g ) = 783.34 cm À1 and 1 (B g ) = 783.01 cm À1 . Similarly, the calculated asymmetric stretching frequencies fall between 789.14 and 843.15 cm À1 , red-shifted by 50-70 wavenumbers.
The four components of 2 are coalesced into a single broad band centred at 352.6 cm À1 , whilst 4 appears as a band at 403.1 cm À1 . Once again, cooling blue-shifts the band centres by around 10 cm À1 . The observed vibrational frequencies of the selenate ion in MgSeO 4 Á9H 2 O are very similar to those of the fully hydrated free ion in aqueous solution, for which 1 = 837 cm À1 , 2 = 348 cm À1 , 3 = 875 cm À1 and 4 = 414 cm À1 (Walrafen, 1963), but differ systematically from those we have obtained for MgSeO 4 Á6H 2 O and -MgSeO 4 , and from other anhydrous and hydrated selenate crystals reported previously (Scheuermann & Schutte, 1973;Berger, 1976;Ti et al., 1976;Park & Frech, 1989;Wildner et al., 2004). In other words, the selenate ion in the 9-hydrate is closer than   the hexahydrate to ideal T d symmetry, which supports the crystallographic analysis presented earlier.
The remaining low-frequency features, specifically a peak at 425 cm À1 and a pair of overlapping peaks at 210-235 cm À1 , are assigned to stretching and deformation of the Mg(H 2 O) 6 octahedra. The Raman spectroscopic data are given in xS3 of the supporting information.

Crystal morphology
Examination of Fig. 8  As shown in Fig. 13, a fair approximation of the crystal's morphology can be obtained by a more straightforward analysis based on interplanar spacing density, namely the Bravais-Friedel-Donnay-Harker, BFDH, model (Bravais, 1866;Friedel, 1907;Donnay & Harker, 1937): we have used the implementation of this model coded in WinXMorph (Kaminsky, 2007). Note that the predicted crystal has a blockier habit; however, our crystals were grown on the bottom of petri dishes and it seems likely that they would be less platy if they were to be grown suspended in solution. Nevertheless, the predicted forms agree well with those observed in Figs. 1 and 2.

Behaviour on warming
Determination of the unit-cell parameters at four temperatures between 5 and 250 K (Table 8) allows some initial comments to be made with respect to the magnitude and anisotropy of the thermal expansion. All three crystallographic axes expand normally on warming, albeit to substantially different degrees; the b-axis expands very little and the c-axis expands rather more (0.019 versus 0.062 Å , respectively, in absolute terms). The monoclinic angle is apparently saturated below $ 100 K and then shrinks marginally (by 0.15 ) on warming to 250 K.
Since we judge there to be comparatively little ambiguity in the thermal expansion at 150 K, which is not necessarily the case at low temperatures (where saturates) or at high temperatures (where the behaviour of the b-axis is unclear), we may be reasonably confident in the accuracy (if not the precision) of the derived magnitudes and orientation of the thermal expansion tensor's principal axes. At 150 K, a simple point-by-point derivative yields 1 ' 34 AE 10 Â 10 À6 K À1 , 2 ' 12 AE 11 Â 10 À6 K À1 , 3 ' 16 AE 10 Â 10 À6 K À1 , and the volume thermal expansion, V ' 58 AE 18 Â 10 À6 K À1 . Whilst 2 is obliged by convention to be parallel with the b-axis, 1 and 3 are free to adopt any orientation they wish in the ac plane, whilst remaining orthogonal to one another; we find that 3 is tilted by just over 20 from the c-axis. Although the precision of these expansion coefficients is poor, it is clear that the structure expands highly anisotropically and that the magnitude of V is similar to that of other water-rich salt hydrates.
Short X-ray powder diffraction scans were collected from MgSeO 4 Á9H 2 O at nine temperatures on warming from 251.4 to 298.5 K (10-35 2, 15 min each) in order to evaluate the thermal stability of the material through its expected dissociation point and to identify any products of its (partial) melting. This was done by crushing single crystals of MgSeO 4 Á9H 2 O with an LN 2 -cooled steel pestle and mortar and loading the powder into a Peltier-cooled X-ray cold stage  at $ 260 K. The operation of this stage allows fairly accurate control of the sample temperature in the range from 250 to 300 K (depending on the ambient temperature inside the diffractometer enclosure) by the simple expedient of adjusting the power supplied to the Peltier element. Unexpectedly, MgSeO 4 Á9H 2 O persisted up to 298.5 K, albeit with significant variations in some peak shapes -including some slight splitting -and intensities that are attributable to changes in the surface texture of the specimen.  Table 8 Unit-cell parameters of MgSeO 4 Á9H 2 O as a function of temperature and in the athermal limit from DFT calculations.

Figure 13
Morphology of the MgSeO 4 Á9H 2 O crystal calculated using the BFDH method in WinXMorph and depicted in the same orientation as the crystal photographed in Fig. 1.
Overall, the sample spent 30 min in air at temperatures ! 290 K with no obvious signs of transforming to another hydrate. Further work will be necessary to determine the material's behaviour in aqueous solution above 270 K and in air above 300 K.  Klein (1940) and its structure determined by Krivovichev (2007). What is more surprising, given the extensive degree of isotypism amongst M 2+ XO 4 ÁnH 2 O compounds, is that magnesium selenate enneahydrate has a different unit-cell metric (and presumably a different structure) to magnesium sulfate enneahydrate, although both are monoclinic crystals. Nonetheless, there are precedents; the M 2+ (NO 3 ) 2 Á6H 2 O series of compounds exhibit a remarkable diversity, not only of symmetry but also supramolecular connectivity, even between substances where the cation radii are quite similar (Ferrari et al., 1967;Braibanti et al., 1969;Bigoli et al., 1971). Amongst inorganic substances, true enneahydrates -by which we mean materials in which the cation to water ratio is one to nine -are comparatively rare; known examples fall into several distinct groups with various types of cation coordination polyhedra and different degrees of water polymerization.

Ninefold coordinated trivalent cations (r ionic > 1 Å )
Compounds with the general formula M 3+ (X À ) 3 Á9H 2 O, where M is a lanthanide element and X is a halogen element (James et al., 1927;Sokolova et al., 1986;Timofte et al., 2005a,b) or halogen oxyanion, typically BrO 3 À (Poulet et al., 1975;Gallucci et al., 1982;Gerkin & Reppart, 1987;Abbasi & Eriksson, 2006). The cation is ninefold coordinated by water to form a tricapped trigonal prism: the water molecules are isolated monomers, forming hydrogen bonds solely with the anions. Since the cation radius varies only slightly from approximately 1.05 to 1.20 Å across the lanthanide series, all known examples of this class of hydrates are isotypic.

Ninefold coordinated tetravalent cations
Compounds with the general formula M 4+ (XO 4 2À ) 2 Á9H 2 O, which have examples where M may be Th or U, and X may be S or Se. These materials have, for the most part, not been well characterized crystallographically; the older literature reports goniometric measurements (Topsøe, 1874;Fock, 1900;Krause, 1901), but the non-H structure of Th(SO 4 ) 2 Á9H 2 O was only determined quite recently (Albrecht et al., 2011). In this instance the cation is ninefold coordinated by seven water molecules and two sulfate O atoms, leaving two interstitial lattice water molecules.

Sixfold coordinated trivalent cations (r ionic < 0.65 Å )
Compounds of the general formula M 3+ (X À ) 3 Á9H 2 O, where examples are known with M = Fe, Al, Cr and Ga, and X = Br À , NO 3 À or ClO 4 À (Hair & Beattie, 1977;Hermansson, 1983;Gubrynowicz & Strö mich, 1987;Lazar, Ribá r, Divjaković & Mé szá ros, 1991;Lazar, Ribá r & Prelesnik, 1991;Hendsbee et al., 2009;Schmidt et al., 2014;Hennings et al., 2014a,b). The cation in these materials is sixfold coordinated by water, forming isolated octahedra, leaving three interstitial water molecules. The presence of several interstitial water molecules provides opportunities for the formation of polymerized structures, the size and complexity of which are dictated largely by the geometry of the anion. The perchlorates form simple trimers, (H 2 O) 3 , whereas the nitrates form an octamer (H 2 O) 8 comprised of a decorated square ring, as well as having trimers and monomers; the bromides form the largest polymeric units in this class, consisting of branched (H 2 O) 9 chains.
4.4. Poly-coordinated divalent cations (r ionic 0.7-1.3 Å ) These have the general formula M 2+ (X À ) 2 Á9H 2 O. Crystal structures have been determined for examples where X may be Br À , I À and ClO 4 À (Hennings et al., 2013(Hennings et al., , 2014cSchmidt et al., 2014). There is, however, a group of divalent metal nitrate enneahydrates with as-yet unknown structures, which includes Mg, Fe, Co, Cu, Ni, Zn and Cd (e.g. Funk, 1899); the fact that these substances typically crystallize from solution at or below 260 K is a fair explanation for the lack of any impetus to study their structures. Similarly, perchlorate enneahydrates of both Ni and Co have been reported (Goldblum & Terlikowski, 1912), although they may not be isostructural with the Srbearing analogue (Hennings et al., 2014c) due to the large differences in cation radius.
The very large Sr 2+ cation (r ionic = 1.31 Å ) in Sr(ClO 4 ) 2 Á9H 2 O is ninefold coordinated by seven water molecules and two perchlorate O atoms [in a similar fashion to Th(SO 4 ) 2 Á9H 2 O], leaving two interstitial lattice water molecules. This arrangement results in by far the largest and most intricate polymeric water structure in any of these four groups of 9-hydrates, consisting of a folded (H 2 O) 8 ring in the ab plane linked by the ninth H 2 O to form infinite chains, (H 2 O) 1 , extending along the c-axis. The Ca 2+ ion (r ionic = 1.12 Å ) in CaBr 2 Á9H 2 O is in eightfold coordination to water, leaving a single interstitial H 2 O. The water molecules in this compound form a branched octamer chain and isolated monomers. Finally, the Mg 2+ ion (r ionic = 0.72 Å ) in MgBr 2 Á9H 2 O and MgI 2 Á9H 2 O exhibits sixfold coordination to water leaving three interstitial lattice water molecules. In this compound, which is arguably the most closely related to MgSeO 4 Á9H 2 O, one finds a branched (H 2 O) 9 chain similar, but not identical, to that in Al(Br) 3 Á9H 2 O.
Although substances with a divalent cation-to-water ratio of 1:9 are not so exotic (even if there are apparently a number with unknown structures), MgSeO 4 Á9H 2 O is quite unusual in being the only example we know of with a cation:(oxy)anion:water ratio of 1:1:9. Since it is apparent from the study of other enneahydrates in the four groups detailed above that the anions exercise a degree of control over the polymerization of the water network (at least in the solid state) we should not necessarily expect to find similar polymeric structures in MgSeO 4 Á9H 2 O to those already described.

Summary and conclusion
We have completed the structure of MgSeO 4 Á9H 2 O using neutron single-crystal diffraction methods, confirming the hydrogen-atom positions that had previously been estimated from the heavy-atom structure obtained by X-ray powder diffraction. The degree of agreement with the earlier powder study is a useful and satisfying validation of the methods employed in that work. Moreover, the ab initio calculations reproduce the majority of the salient features to high accuracy. This complete structural analysis allows us to understand a range of other observed features, from the crystal morphology through to the anisotropy of the thermal expansion and the frequency of certain vibrational features.
After a thorough comparison of all compounds likely to share structural similarities, we find that MgSeO 4 Á9H 2 O is, for now, unique in having a cation-to-anion-to-water ratio of 1:1:9 and in being built around a 12-membered water polymer consisting largely of pentagonal rings.
The stability of the 9-and 11-hydrates with respect to one another and to the previously known heptahydrate remain to be confirmed by detailed study of the binary phase diagram at low temperature. Nonetheless, we have made some observations that suggest that the 9-hydrate is the stable phase in aqueous solution at $ 269 K (and is perhaps the stable phase co-existing with ice at the eutectic), whereas the heptahydrate seems likely to be metastable (i.e. its solubility is greater than the 9-hydrate) at or around 269 K. Upon warming, MgSeO 4 Á11H 2 O transforms to the 9-hydrate; crystals of MgSeO 4 Á7H 2 O stored in a freezer at 253 K for 2 weeks were discovered to have transformed entirely to the 9-hydrate; crystals of MgSeO 4 Á9H 2 O stored in the same freezer for 5 months were found to be unchanged. Our work provides the means to identify MgSeO 4 Á9H 2 O crystals on the basis of their morphology, crystal structure and vibrational spectrum, which will aid in a re-evaluation of the binary phase diagram.