Crystalline structures of polymeric hydrocarbon with 3,4-fold helical chains

Molecular hydrocarbons are well-known to polymerize under pressure to form covalently bonded frameworks. Here we predict by ab initio calculations two distinct three-dimensional hydrocarbon crystalline structures composed of 3-fold and 4-fold helical CH chains in rhombohedral () and tetragonal (I41/a) symmetry, respectively. Both structures with 1:1 stoichiometry are found to be energetically more favorable than solid acetylene and cubane, and even more stable than benzene II solid at high pressure. The calculations on vibrational, electronic, and optical properties reveal that the new chiral hydrocarbons are dynamically stable with large bulk moduli around 200 GPa, and exhibit a transparent insulating behavior with indirect band gaps of 5.9 ~ 6.7 eV and anisotropic adsorption spectra. Such forms of hydrocarbon, once synthesized, would have wide applications in mechanical, optoelectronic, and biological materials.


Results
Our newly identified chiral crystalline forms of hydrocarbon are depicted in Fig. 1. The 4-fold chiral structure [ Fig. 1(a)] is bodycentered tetragonal in I4 1 /a symmetry, and the optimized lattice parameters are a 5 6.106 Å and c 5 4.146 Å , with C and H atoms occupying 16f (0.2189, 0.1188, 0.8213) and 16f (0.0770, 0.1018, 0.6617) Wyckoff positions, respectively. This form has eight CH units in the primitive cell and we refer to it as the T 8 -CH. The 3-fold chiral structure in R 3 symmetry, hereafter named as R 6 -CH, has a rhombohedral lattice and six CH units per primitive cell. In hexagonal representation [ Fig. 1(b)], its equilibrium lattice parameters are estimated to be a 5 7.392 Å and c 5 3.671 Å with 18f (0.4202, 0.0351, 0.0448) C and 18f (0.1816, 0.5314, 0.4693) H positions. As shown in Fig. 1, for both T 8 -CH and R 6 -CH structure, the helical CH chains are formed along the c axis, and each chain is connected to neighboring chains of opposite chirality (left-handed indicated by S and right-handed R) by C-C covalent bonds. The intrachain and interchain C-C bond lengths are respectively 1.559 and 1.564 Å in T 8 -CH, 1.561 and 1.543 Å in R 6 -CH, which are all close to 1.530 Å in diamond 19 . Hence, the two chiral hydrocarbons here can be considered as diamond-like CH phases similar to the previously proposed K 4 -CH 19 and Hex-CH 21 ; they all adopt a fully 3D framework with saturated nature of sp 3 carbon. Figure 2(a) shows the calculated total energy versus volume curves of various hydrocarbon phases with CH stoichiometry of 151. We can see that the T 8 -CH structure is as stable as K 4 -CH, with energy about 0.47 and 1.16 eV/CH lower than solid cubane 23,24 and acetylene 25 , respectively. Compared with T 8 -CH, the R 6 -CH structure has lower energy (close to Hex-CH), being even more stable than the benzene II 26 crystal and its hypothetical layered polymer reported by Wen et al. 17 It is noticeable that the equilibrium volumes of K 4 -CH, Hex-CH, T 8 -CH, and R 6 -CH are in the range 9.6-9.9 Å 3 /CH, much smaller than 14.1-20.5 Å 3 /CH of benzene II and solid cubane and acetylene (see Table I). These results suggest great potential for synthesizing the low-energy CH structures through compression of metastable molecular phases of hydrocarbon. Furthermore, we find that among all CH systems considered here the most stable phase is the layered graphane I 18 crystal in an AA stacking (0.04 eV/CH lower in energy than graphane III). The less favorable energetic state of diamond-like CH phases compared to layered graphane is likely attributed to the stronger steric interactions of hydrogens 18 , as evidenced by the nearest H-H distances of 1.56 Å for T 8 -CH and 1.79 Å for R 6 -CH being shorter than 2.51 Å for graphane I.
To understand the pressure effect, the enthalpy difference of each phase to that of benzene II is presented in Fig. 2(b) in a wide pressure range 0-50 GPa. It is found consistent with previous calculations 18 that instead of graphane I, graphane III becomes the most stable phase under pressures above 12 GPa. For diamond-like CH phases, an increasing stabilization with pressure can be seen relative to benzene II and the related polymer. Above 3.5 and 7.0 GPa, the T 8 -CH and K 4 -CH phases (enthalpically almost degenerate in 0-10 GPa) become more stable than benzene II and benzene II polymer, respect-ively. Moreover, the hexagonal phases of R 6 -CH and Hex-CH are more favorable than T 8 -CH or K 4 -CH in enthalpy by about 0.15 eV/ CH in the whole pressure range. Meanwhile, R 6 -CH is found to be preferable to Hex-CH above 25 GPa, and to compete with graphane I above 50 GPa. In view of the above enthalpy results, the experimental syntheses of diamond-like CH phases are thermodynamically possible.
We have fitted the energy versus volume data to the Murnaghan equation of state 27 to obtain the bulk moduli (B 0 ) of different hydrocarbons, as listed in Table I. The predicted B 0 of diamond-like CH phases are significantly higher than those of molecular hydrocarbons, with T 8 -CH and R 6 -CH having the values of 201.7 and 185.2 GPa, respectively. Note that for benzene II we calculate the bulk modulus to be B 0 5 9.8 GPa, which is close to the reported experimental value of ,5.5 GPa 28 .
Phonon calculations give a criterion for the structure stability of a crystal. Therefore, we calculated phonon dispersion curves for the T 8 -CH and R 6 -CH phases at 0 GPa, as presented in Fig. 3(a) and 3(b). The absence of imaginary frequency modes indicates that these two chiral structures are dynamically stable. High frequency C-H stretching phonon modes emerge around 3047 and 2992 cm 21 for T 8 -CH and R 6 -CH, respectively, which can be compared with the observed broad infrared peaks at 2950 and 2920 cm 21 (assigned to the C-H stretching modes involving sp 3 carbon atoms) for amorphous samples recovered from compressed acetylene 6 and benzene 8,9 .  In addition, we also checked the phonon dispersion for both chiral phases under pressure confirming their dynamical stability up to at least 50 GPa.
The electronic band structure calculations within the hybrid functional method 36  .88 eV at 0 GPa, respectively. Hence, both chiral phases are predicted to be optically transparent as previously proposed K 4 -CH (6.07 eV) and Hex-CH (6.54 eV). We have further explored the electronic properties at increasing pressures. The calculated band gaps as a function of pressure for K 4 -CH, Hex-CH, T 8 -CH, and R 6 -CH are shown in Fig. 4(a). According to the results, T 8 -CH remains dielectric at pressures up to at least 50 GPa. The band gap has only a weak dependence on pressure and decreases from 6.67 to 6.42 eV as pressure increases from 0 to 50 GPa. For R 6 -CH, the band gap decreases more rapidly with increasing pressure and reaches 88% of the original band gap at 50 GPa, indicating a stronger pressure dependence similar to those for K 4 -CH and Hex-CH.
We now move on to discuss the optical properties of the T 8 -CH and R 6 -CH phases in terms of the calculated frequency dependent imaginary part of the dielectric matrix 37 . By comparing the optical spectra at 0 GPa obtained with light polarized either along the c axis or in the ab plane [ Fig. 4 Table I | Calculated equilibrium structural parameters (space group, lattice parameters a, b, and c, volume V 0 , bond lengths d C-C ), total energy E tot , bulk modulus B 0 , and electronic band gap E g for various hydrocarbon phases at zero pressure. Energies are given relative to that of graphane I. For benzene II and benzene II polymer, b 5 110.62 and 97.49u, respectively. The C-H bond lengths are around 1.10 Å for all phases  shape and intensity, suggesting anisotropic features for both chiral phases. The application of pressure on the two structures induces an opposite optical response. By increasing pressure the C-C bond length is shortened, the band gap decreases and the whole optical spectrum is almost rigidly shifted toward higher energy. The calculated static dielectric constants for the two chiral phases increase slowly with pressure, with the ab-(c-) components going from 4.045 (4.071) and 3.846 (3.797) at 0 GPa to 4.174 (4.227) and 4.082 (3.934) at 30 GPa for T 8 -CH and R 6 -CH, respectively. To provide more information and characters for possible experimental observation, we have also simulated the x-ray diffraction (XRD) spectra of the various CH phases at 0 GPa. Monochromatic radiation with a wavelength 1.540 56 Å is used, and the results are shown in Fig. 5. Unlike K 4 -CH where the main peak (110) at 2h 5 29.69u is observed, three sharp XRD peaks of (101) at 25.95u, (200) at 29.23u, and (211) at 39.48u with strong intensities are seen for T 8 -CH. Furthermore, we also find significant difference between the two hexagonal phases, with the most prominent peak observed to be (101) at 23.08u for Hex-CH and (110) at 24.06u for R 6 -CH, although both phases share the same peak (20 1) with close positions. Compared with these diamond-like CH phases, the layered ones have specific XRD peaks at relatively small 2h values of 19.70u for graphane I, 18.46u for graphane III, and 20.65u for benzene II polymer, which originate from a large interlayer distance in layered crystal structure. We believe that the above comparison between the chiral hydrocarbons found here and other CH structures may be helpful for identifying them in experiments.

Discussion
In summary, we have predicted by ab initio calculations two new 3D hydrocarbon framework structures composed of 3-fold and 4-fold helical CH chains in R 3 and I4 1 /a symmetry, respectively. These saturated crystalline phases are dynamically stable and have large bulk moduli of ,200 GPa. We confirm their increasing stabilization with pressure relative to the molecular solids such as acetylene, cubane, and benzene with 151 stoichiometry in the pressure range 0-50 GPa. Calculations on the electronic and optical properties  show that both chiral hydrocarbons exhibit a wide-gap insulating behavior and anisotropic adsorption spectra. The simulated XRD patterns show the distinct structural feature with respect to the layered CH phases. The present results will stimulate future experiments on the high-pressure polymerization of molecular hydrocarbons to synthesize the amazing chiral CH phases, which may have wide applications in mechanical, optoelectronic, and biological materials.

Methods
The calculations were performed using the density functional theory within the local density approximation (LDA) 29,30 as implemented in the Vienna ab-initio simulation package (VASP) 31 . We adopted the projector augmented wave (PAW) method 32 to describe the electron-ion interaction. The plane wave cutoff energy was set to 800 eV. Brillouin zone integration was carried out at Monkhorst-Pack 33 k-point meshes with a grid spacing of 2p 3 0.02 Å 21 . The geometries were optimized by a conjugate gradient algorithm until the Hellmann-Feynman forces on the ions are less than 10 23 eV/Å . Phonon calculations were based on the supercell approach 34 using the PHONOPY code 35 . The HSE06 hybrid functional method 36 was employed to calculate the electronic and optical properties. The frequency dependent dielectric matrix was obtained by neglecting local field effects 37 .