Theoretical Prediction of CH n Crystal Structures under High Pressures

: CH n is the precursor unit for graphene synthesis. We have theoretically predicated a series of CH n structures with n = 1, 2, 4, 6, 8, 10, and 12 at elevated pressures (ambient pressure, 50, 100, 200, 300, 350, and 400 GPa) using evolutionary algorithms. The predicted CH and CH 2 structures are graphane-type and polyethylene over the whole considered pressure range, respectively. The molecular crystalline methane is predicted for the stoichiometry of CH 4 . The combination of methane and H 2 for CH 6 , CH 8 , CH 10 , and CH 12 up to 300 GPa are obtained. At 400 GPa, the mixture of polymer and H 2 for CH 6 , CH 10 , and CH 12 comes into play. From the computed enthalpy, higher pressure and more hydrogen concentration contributed to the decomposition (to carbon and H 2 ) of CH n systems. The total density of states for these CH n structures show that only the CH 12 phase is metallic above 300 GPa. The rotational properties are traced in H 2 and the CH n structures. The CH 4 rotation is more sensitive to the pressure. The H 2 units are nearly freely rotational. Other structures of CH n , including fcc-type and experimentally known structures, are not competitive with the structures predicted by evolutionary algorithms under high pressure region. Our results suggest that the CH n (n > 4) system is a potential candidate for hydrogen storage where H 2 could be released by controlling the pressure.


Introduction
A significant paper by N. W Ashcroft in 2004 [1] entitled "Hydrogen Dominant Metallic Alloys: High Temperature Superconductors?" opened the door for the study of Group 14 hydrides. In the past few tens of years, a number of studies of Group 14 hydrides were performed; both experimental [2][3][4] and theoretical [5][6][7][8] studies have appeared. These studies were mainly focused on EH x with x ≥ 4 systems (E = Si, Ge, and Sn) which have a high atomic fraction of H (≥80%) and were predicted to metalize as hydrogen-dominant metallic alloys at lower pressures than pure hydrogen. These hydrogen-dominant alloys might become superconductors under high pressure with similar properties to those of pure metallic hydrogen. The strategy for the metallization of hydrogen-dominant metallic alloys is that due to the chemical precompression exerted by heavier atoms on hydrogen, the expected pressure needed for the required metallic transition as a preliminary step toward superconductivity might be within current experimental capabilities.
In Group 14, the member carbon is the basis of all known life on Earth, which is different to other members (Si, Ge, Sn, and Pb). We investigated their differences in the Group 14 elements at ambient pressure [9]. Carbon is the only element of Group 14 known to have metastable, kinetically persistent 0D (fullerenes), 1D (nanotubes), 2D (graphene, graphane), and 3D (diamond, graphite) allotropes. This kinetic persistence arises from large barriers to oligo-and polymerization and bond reorganization. Moving down the Group 14 column reveals stark changes in chemical bonding that directly affect the types of structures each element favors. At high pressure, the stable phase of C is the diamond type (up to 500 GPa), which is insulating with a wide gap. Si, Ge, and Sn become metallic at 12, 11, and 1 GPa, respectively, owing to the phase transition from diamond type to β-Sn type (six-coordination) [10].
Based on the difference between carbon and other members, the carbon hydrides (CH n ) exhibit different chemical and physical properties to other Group 14 hydrides. For instance, from the thermodynamic point of view, the heat of formation is negative (exothermic) only for CH 4 . Methane has been detected to exist in several locations of the solar system. According to the first-principle calculation by Martinez-Canales and Bergara [11], there is no evidence for the metallization of CH 4 up to 520 GPa which is quite high, well above the current experiment capabilities.
The high-pressure behavior of carbon hydrides, CH n (n ≥ 4), is just beginning to be studied. In 1996 [12], methane (CH 4 ) was discovered to form four stoichiometric compounds (CH 4 (H 2 ) 2 , (CH 4 ) 2 H 2 , CH 4 (H 2 ) 4 , and CH 4 H 2 ), with hydrogen at pressures up to 30 GPa. These van der Waals compounds exhibit many interesting properties with relevance to many fields such as planetary science, hydrogen storage, energy, environment field, and so on. Experiments have demonstrated that the methane is unstable under high temperature and high pressures. At T = 2000 K and P = 20 GPa, the methane could dissociate to form diamond and hydrogen [7,13].
Following the trend of analyzing Group 14 hydrides, we studied a series of CH n structures with n = 1, 2, 4, 6, 8, 10, and 12 at elevated pressures (up to 400 GPa) by evolutionary algorithm procedures. We examined whether pressure could lead to the formation of new compounds in this series or not. The structural and electronic properties were paid attention to in the current work to check the metallization of CH n systems.

Materials and Methods
The calculations are based on the plane wave/pseudopotential approach using the computer program VASP (Vienna Ab-initio Simulation Package) [14,15], employing the PBE type and the projected-augmented wave (PAW) [16,17] method. The energy cutoff for plane-wave basis was set to 600 eV. The convergence criterion for electronic energy and force is 1.0 × 10 −6 eV/atom and 0.01 eV/Å, respectively. The density of K-mesh was set to 0.04. In geometry optimization, we allow all structural parameters (atomic position, lattice constants, and symmetry) to relax. The evolutionary algorithm USPEX [18][19][20] was employed to find the lowest energy structures; the searching calculations were carried out at 1 atm, 50, 100, 200, 300, 350, and 400 GPa, which is consistent with our DFT calculations.

The Tie Line of CH n Systems
We performed evolutionary prediction methodology for searching ground state CH n structures with x = 1, 2, 4, 6, 8, 10, and 12 at a series of pressures (ambient pressure, 50 GPa, 100 GPa, 200 GPa, 300 GPa, 350 GPa, and 400 GPa). The searched lowest energy CH n structures at series pressure are shown in SI ( Figures S1-S8). The predicted CH and CH 2 (Table S1) structures are graphane-type and polyethylene over the whole considered pressure range, respectively. The predicted structure of CH 3 depends very much on the pressure: at or below 50 GPa, it forms ethane molecules; at 100 GPa, it forms butane, which can be thought of as oligomerized ethane; at 200 GPa polymerization happens, forming polyethylene (200 GPa), polyethylene and methane (300 GPa), or graphene (350 GPa). The predicated CH 4 structures (Table S2) are the molecular crystalline methane. The combination of methane and H 2 for CH 6 (Table S3), CH 8 (Table S4), CH 10 (Table S5), and CH 12 (Table S6) up to 300 GPa are obtained. Interestingly, at 400 GPa, the mixture of polymer and H 2 for CH 6 , CH 10 , and CH 12 come into play. Figure 1 (a tie line curve) shows the static-lattice enthalpies (eV/per C atom) for the most competitive CH n structures at selected pressure. Note that the tie line energy curve has been used in our previous studies [21,22], which may be used to connect two phases, and if the enthalpy of a third falls below it, the first two will react to give the third. The enthalpy of formation of CH n is defined as ∆H (CH n ) = H(CH n ) -H(C) -1/2x·H(H 2 ). The carbon in graphite (0-10 GPa), diamond (10-400 GPa), and solid H 2 structures in the P63/m (0-100 GPa), C2/c (100-300 GPa), Cmca-12 (300-350 GPa), and Cmca (400 GPa) are taken as references. It is important to recognize that in the light-element systems, ion dynamics can significantly change the total energies [23]. On the other hand, DFT static-lattice energies, in general, reproduce experimental phase stabilities well, even for light-element highpressure phases [24], in part owing to the cancellation of the dynamical effects in the energy differences. In the discussions that follow, we use the static-lattice enthalpies. Note that kinetic effects and metastability are not considered at this level.
From the enthalpy curve in Figure 1, one can see that (I) all of the CH and CH 2 phases are stable relative to carbon and H 2 over the whole given pressure range; (II) the stabilization for CH 4 , CH 6 , and CH 8 are up to 300 GPa, while there is on stabilization for CH 10 and CH 12 above 300 GPa; (III) in the pressure region of 1 atm to 100 GPa, the "flat energy lines" for the CH n (n ≥ 4) are owing to the van der Waals forces in the combination of methane and H 2 ; (IV) higher pressure and more hydrogen concentration contribute to the decomposition of CH n system; (V) when pressure lowers under 200 GPa, the CH 4 , CH 6 , and CH 8 are more stable than other compounds, but under high pressure (p > 200 GPa), the CH 4 becomes a metastable phase, which is consistent with previous studies [13]. We would like to propose that the CH n (n > 4) system is a potential candidate for hydrogen storages since such a system can release H 2 under high pressure. Figure 2b shows the calculated Wigner-Seitz radius (r s ) as a function of pressure (r s -P curve) for the lowest energy structure of carbon, H 2 , and CH n systems. The Wigner-Seitz radius (r s ) is defined by 4πr s 3/3 = 1/ρ, where ρ is the average valence-electron density. In the work, we set up ρ = N/V, where N is the number of valence electrons in the unit cell; V is the volume of the unit cell. Let r s be the standard measure of the average valence electron density for a system. The volume reductions for all these structures are rapid from 0 GPa to 50 GPa due to squeezing out van der Waals space in H 2 and CH n structures, and the phase transition in carbon (graphite to diamond). The computed r s for CH n structures are located between the value of H 2 and carbon. The corresponding slopes reflect the compressible properties. (b) Computed relative compression as a function of pressure. The relative compression is defined by r s (0) r s (p) 3 , where r s (0) and r s (p) is the Wigner-Seitz radius at ambient pressure and certain high pressure, respectively. Figure 3a shows the closest distance between C and C and the closest inter-distance of H and H (Figure 3b) in CH n structures. We can see that there are C-C chemical bonds in carbon, CH, and CH 2 phase, while the carbon atoms in CH n (n ≥ 4) are separated by forming methane molecules. After 350 GPa, the complexes of polymer and H 2 are formed in CH 6 , CH 10 , and CH 12 phase, as shown in the jump of C-C distances. With increasing pressure (0 to 50 GPa), the distance of the closest inter H-H in CH n structures is decreased rapidly due to squeezing the van der Waals space, as shown in Figure 3b.

Electronic Properties of CH n Phase
The Goldhammer-Herzfeld criterion states that a material becomes metallic when the quantity (1 − f αV m ) −1 diverges, where α is the molecular polarizability, Vm is the volume per molecule in the solid, and f is a dimensionless factor determined only by the packing of the molecules in the crystal. The divergence occurs when fα/Vm = 1. For cubic systems, f is 4π/3, and this gives V m = 4πα/3. Using the definition of r s , 4πr 3 s /3 ≡ V m /N ve (where Nve is number of electrons), we then have the following equation for cubic systems, r s = (α/N ve ) 1/3 , where the polarizability of a material, α for C, CH 4 , and H 2 is 7.437, 17.53, and 5.42 bohr, respectively. One readily obtains r s = 1.23, 1.30, and 1.40 bohr for the metallization of CH 4 and H 2 , respectively. Figure 4 shows the computed total density of states for carbon (diamond), H 2 (C2/c), and CH n structures at 300 GPa. One can see that only the CH 12 phase is almost metallic above 300 GPa. The band gap for the CH n phase is decreased with the increasing concentration of H. Note that the free-electron-like shapes are also shown in Figure 4.

The Rotational Solid for CH n
The term-the rotational solid-is mainly used to describe the properties of a molecular crystal, such as solid H 2 [25], solid CH 4 [26], and C60 crystal structure [27]. For molecular crystals, the rotational dynamic might be an important factor for the complicated phase transition at low temperature and pressure.
The rotational barriers in H 2 and CH n (n ≥ 4) structures are examined via scanning potential energy surface (shown in Figures S9-S15). Table 1 and Table S7 show the rotational barriers of H 2 unit and CH 4 unit in corresponding structure at 100 GPa and 300 GPa. As expected, the rotational barriers for H 2 and CH 4 units are elevated with increasing pressure. At ambient pressure, these rotations are free (no energy barriers were found), owing to the dispersion forces. The energy barrier of CH 4 rotation is higher than that of H 2 rotation under same pressure, indicating that the CH 4 rotation is more sensitive to the pressure. From the small rotational barriers for H 2 units, even at 300 GPa, one can know that the these H 2 units are nearly in free rotation. The rotational solid is a good term to describe the CH n phases.

Other Possible Structures for CH n Systems
In 2009, Strobel et. al. [4] reported novel molecular compound formation from silanehydrogen mixtures (SiH 4 -2H 2 ) and proposed that the molecular compound adopts an fcc lattice. The study on SiH 4 -2H 2 offers possible structural information for the methanehydrogen mixtures, even though the C is different to Si, as mentioned.
The fcc primitive cell with CH 4 unit (one CH 4 in the cell) is shown in Figure 5. There are two tetrahedral holes (at site 2 and 3) and one octahedral hole (at site 1) along the diagonal-red dot line in the primitive cell. When only the one site is H 2 -occupied, the stoichiometry of the phase is CH 4 H 2 . The CH 4 -2H 2 can be obtained via occupying one more. The sites will be all occupied in CH 4 -3H 2 . In the work, we pick up the CH 4 -H 2 containing one occupied octahedral hole, CH 4 -2H 2 , with occupied octahedral hole and tetrahedral hole. Note that there are similar energies for other possible (H 2 -occupied) fcc isomers. Though these structures begin with fcc type, they will not adopt perfect fcc-type at high pressure any more (see Figure S16). This might be attributed to the properties of rational solid. The calculated enthalpy (with respect to carbon and H 2 ) of these fcc-type CH n structures as a function of pressure is shown in Figure 6. For comparison, the corresponding CH n structures by USPEX are also inserted as dot lines. Due to van der Waals interaction, it is not surprising that the relative enthalpies of these structures are close together in the low-pressure zone, while they are separated with elevating pressure. As one can see, the structures searched for by evolutionary algorithm are more stable than their corresponding fcc type structures, indicating that the fcc-type CH n structures are not competitive (Table S8). These fcc-type CH n structures at 300 GPa are still semiconductors or insulators. Experimentally, the observed CH 4 -H 2 in the range of 7.9-30 GPa is consistent with the wurtzite structure. The CH 4 -2H 2 (which is stable between 5.8 and 7 GPa) is that of a MgZn 2 (Laves phase; H 2 occupy the Zn sites and the CH 4 is in the Mg sites) structure.
Based on the observation in experiment, the CH 4 -H 2 wurtzite phase and CH 4 -2H 2 MgZn 2 phase are also examined, as shown in Figure 7. The enthalpy curve shows that the CH 4 -H 2 wurtzite phase is more stable than the corresponding fcc type below 180 GPa, while the CH 4 -2H 2 MgZn 2 phase is more favorable than fcc-type CH 4 -2H 2 phase over the whole range of pressure. Note the blue line (CH 4 -H 2 wurtzite) undergoes a big jump, indicating that there might be a phase transition (see Figures S17 and S18). In a word, the considered fcc-type CH n structures and the experimental known structures are not competitive with the corresponding CH n structures obtained by evolutionary algorithms above 50 GPa. However, the relative enthalpies of these considered CH n structures are close together in the low-pressure zone. These phenomena are possibly owing to the rotational solid or liquid-like.

Conclusions
We have systematically investigated the CH n crystalline structures with n = 1, 2, 4, 6, 8, 10, and 12 under various pressures (ambient pressure, 50, 100, 200, 300, 350, and 400 GPa) by evolutionary algorithms procedures (USPEX). CH and CH 2 are the graphane-type and polyethylene, respectively, over the whole considered pressure range. The predicted stoichiometry of CH 4 is molecular crystalline methane. The combination of methane and H 2 for CH 6 , CH 8 , CH 10 , and CH 12 up to 300 GPa are obtained. The mixture of polymer and H 2 for CH 6 , CH 10 , and CH 12 come into play at 400 GPa. From the computed static-lattice enthalpy, the CH n (n ≥ 4) phase decomposes to carbon and H 2 under high pressure (above 300 GPa). In addition, the concentration of hydrogen also contributes to the decomposition (of carbon and H 2 ). Based on the calculations, we propose that the CH n (n > 4) system is a potential candidate for hydrogen storages since such a system can release H 2 under high pressure. The calculated total density of states for the CH n structures shows that only the CH 12 phase is almost metallic above 300 GPa. The rotational properties are traced in H 2 and the CH n structures. The energy barrier of CH 4 rotation is higher than that of H 2 rotation under the same pressure, indicating that the CH 4 rotation is more sensitive to the pressure. From the small rotational barriers for H 2 units, even at 300 GPa, one can know that the H 2 units are nearly freely rotational. Other structures of CH n , including fcc-type and experimentally known structures, are not competitive with the structures predicted by evolutionary algorithms under high pressure.
Supplementary Materials: The following are available online at https://www.mdpi.com/article/ 10.3390/cryst11121499/s1, Figure S1: CH 2 structures under various pressure by USPEX, Table S1: Calculated volume (Å), density (g/cm 3 ), r s of CH 2 structures by USPEX. The Z and electron number in corresponding unit cell are shown here, Figure S2: CH 4 structures under various pressure by USPEX, Table S2: Calculated volume (Å), density (g/cm 3 ), r s of CH 4 structures by USPEX. The Z and electron number in corresponding unit cell are shown here, Figure S3: CH 4 -H 2 structures under various pressure by USPEX, Table S3: Calculated volume (Å), density (g/cm 3 ), r s of CH 4 -H 2 structures by USPEX. The Z and electron number in corresponding unit cell are shown here, Figure S4: CH 4 -2H 2 structures under various pressure by USPEX, Table S4: Calculated volume (Å), density (g/cm 3 ), r s of CH 4 -2H 2 structures by USPEX. The Z and electron number in corresponding unit cell are shown here, Figure S5: CH 4 -3H 2 structures under various pressure by USPEX, Table S5: Calculated volume (Å), density (g/cm 3 ), r s of CH 4 -H 2 structures by USPEX. The Z and electron number in corresponding unit cell are shown here, Figure S6: CH 4 -4H 2 structures under various pressure by USPEX, Table S6: Calculated volume (Å), density (g/cm 3 ), r s of CH 4 -4H 2 structures by USPEX. The Z and electron number in corresponding unit cell are shown here, Figure S7: CH x structures at 1 atm by USPEX, Figure S8: CH 2 structures by USPEX, Figure S9: Scanned potential energy surfaces of rotation in two solid H 2 structures (at 100 GPa and 300 GPa, respectively), Figure S10: Scanned potential energy surfaces of rotation in two solid CH 4 structures (at 100GPa and 300GPa, respectively), Figure S11: Scanned potential energy surfaces of rotation in two solid CH 4 -H 2 fcc (at 100GPa and 300GPa, respectively) structures, Figure S12: Scanned potential energy surfaces of rotation in two solid CH 4 -H 2 structures (at 100 GPa and 300 GPa, respectively) by USPEX, Figure S13: Scanned potential energy surfaces of rotation in two solid CH 4 -2H 2 structures (at 100 GPa and 300 GPa, respectively) by USPEX, Figure S14: Scanned potential energy surfaces of rotation in two solid CH 4 -3H 2 structures (at 100 GPa and 300 GPa, respectively) by USPEX, Figure S15: Scanned potential energy surfaces of rotation in two solid CH 4 -4H 2 structures (at 100 GPa and 300 GPa, respectively) by USPEX, Table S7: The biggest barriers of rotation in H 2 and CH x structures, Figure S16: Simulated XRD pattern of a perfect fcc CH 4 structure and optimized "fcc" CH 4 structure at 300 GPa. The Synchrotron with a wave length by 1.000 Å is used, Figure S17: Simulated XRD pattern of CH 4 -H 2 wurtzite structure at 300 GPa and 350 GPa. Here, the radiation source is from Synchrotron with a wave length by 1.000 Å, Figure S18: Simulated XRD pattern of CH 4 -H 2 wurtzite structure (the starting structure and optimized at 5 GPa) and CH 4 -H 2 by USPEX at 5 GPa, Table S8: Computed the relative enthalpy of three CH 4