Insights into the high-pressure behaviour of solid bromine from hybrid DFT calculations

Understanding the properties of molecular solids at high pressure is a key element in the development of new solid-state theories. However, the most commonly used generalized-gradient approximation (GGA) of the density functional theory (DFT) often fails to correctly describe the behavior of these systems at high pressure. Here we utilize the hybrid DFT approach to model the properties of elemental bromine at high pressure. The calculations reproduce in very good agreement with experiment the properties of the molecular phase I (Cmca symmetry) and its pressure-induced transition into the non-molecular phase II (Immm). The experimentally yet unobserved transition into phase III (I4/mmm) is predicted to occur at 128 GPa, followed by subsequent formation of an fcc lattice at 188 GPa. Analysis of the structure and electronic properties of the modelled phases indicates that the molecular Cmca phase becomes metallic just at the borderline of its stability, and that both Immm and I4/mmm phases are metallic and quasi-2D. Finally, we show that the incommensurate phases of bromine postulated from experiment are transient species which can be viewed as intermediates in the dissociation process occurring at the boundary of the transition from phase I to phase II.


I. INTRODUCTION
High-pressure research provides fundamental insight into material properties, which is essential for the development and validation of new condensed matter and planetary models. [1,2] One of the ongoing efforts in this field is the study of the process of molecular dissociation under pressures in the range of hundreds of gigapascal. [3,4] In particular, solids consisting of diatomic molecules, such as H2, N2, O2 and the halogens (F2, Cl2, Br2, I2), have been studied at high pressure. Hydrogen, the first element in the periodic table, hosts within its molecule the simplest covalent bond. Theoretical considerations indicated that pressure-induced dissociation of this bond and formation of an alkali-metal type structure would lead to room-temperature superconductivity. [5,6] Despite the many theoretical and experimental studies devoted to this system the pressure of hydrogen dissociation and metallization is still not established. [7][8][9][10][11][12][13] Nitrogen is abundant in nature forming an exceptionally strongly bonded diatomic molecule. However, at high pressure and temperature (>110 GPa and >2,000 K), the triple N≡N intramolecular bond transforms into three N-N single bonds leading to the formation of a polymeric structure, which has been proposed as a high-energy-density material. [14][15][16] In contrast to both H2 and N2, the molecular crystal of oxygen is predicted to transform into a polymeric spiral chain structure only at an extremely high pressure of 1920 GPa. [17][18][19] In general, high pressure leads to a reduction in volume, and a decrease in mean interatomic distances.
This leads to the modification of material properties which as shown by the abovementioned examples can be surprisingly different for seemingly similar molecular crystals. Halogens (F2, Cl2, Br2, I2) are another representative of diatomic molecules whose high-pressure behaviour serves as an important analogue for both hydrogens, as they undergo pressure-induced metallization and dissociation, and oxygen, as their molecules are electron-rich species. Importantly, the respective phase transitions are known to take place in halogens at much lower pressures than for both hydrogen and oxygen. This is particularly true for the heavier halogens, Br2 and I2, which have been the most extensively studied, [20][21][22][23][24][25] while solids of F2 and Cl2 have gained relatively less attention. [26][27][28][29] Iodine is solid at ambient conditions and forms a molecular crystal of Cmca symmetry and 8 atoms (four molecules) in the unit cell. Previous research showed that there is no structural phase transition in this element up to 20.6 GPa when metallization occurs. [25] A later study by Takemura et al. [21] reported molecular dissociation in iodine at 21 GPa when it transforms to a non-molecular orthorhombic structure (phase II) of Immm symmetry. Further experiments revealed that at 43 GPa the Immm structure transforms to a tetragonal phase III of I4/mmm symmetry (Z = 2). Above 55 GPa iodine enters phase IV with a closed-packed fcc structure (Fm3 m). [30] In addition to these phases two incommensurate structures, phase V and VI, have been found in the pressure range 22-26 GPa, that is in the vicinity of the Cmca -Immm phase boundary. [31,32] The emergence of these structures has been linked to the appearance of new Raman band in solid iodine at 23.5 GPa. [33,34] The high-pressure phase transition sequence of bromine is similar to that of iodine, although the phase transitions are shifted to higher pressure. Bromine is liquid at room temperature and atmospheric pressure. It crystallizes at 266 K into a Cmca structure analogous to that of phase I of iodine. [35] Synchrotron X-ray diffraction studies revealed that solid bromine begins to undergo the Cmca to Immm phase at 80±5 GPa. [36,37] Raman scattering experiments were also performed on compressed solid bromine yielding frequency-pressure dependences for the intramolecular vibrational Ag and B3g modes, as well as the intermolecular librational Ag and B3g modes. [35,38] As in the case of iodine incommensurate structures of bromine were postulated to emerge at the Cmca to Immm transition. An incommensurate phase (V) was proposed by T. Kume at. al [33] based on the appearance at 84 GPa of two new Raman modes ( ( ) and the X band) whose frequency decreases with increasing pressure.
This softening was previously considered to be related to the onset of molecular dissociation as a displacive structural phase transition, and was also observed in the incommensurate phase for iodine.
[16] Synchrotron X-ray data presented by Liu et al. seems to confirm the existence of phase V at 81 GPa, although the authors did not provide any structural details of this phase. [39] According to the previous electrical conductivity measurements, bromine metallization occurs at 60-70 GPa when it is still in the molecular Cmca phase. [40] Ab initio modelling, mostly within the framework of Density Functional Theory (DFT), is an important tool for inspiring and guiding high pressure experiments, as well as aiding data interpretation. Several theoretical studies for iodine and bromine have been reported using the plane-wave pseudopotential approach with the generalized gradient approximation (GGA). [41][42][43][44][45] Based on calculations utilizing the PBE functional with the D2 van der Waals correction, Wu et al. [44] proposed that the splitting of the Ag and B3g intramolecular Raman bands observed in compressed bromine in the 25 -60 GPa pressure range [33] is due to symmetry lowering of the molecular phase from Cmca to C2/m. Their results indicated that bromine should metallize at 42.5 GPa while still in the molecular phase. Recently, four commensurate approximations for the incommensurate phase V of bromine were proposed based on DFT calculations. [45] Although considerable research effort has been put into studying the high-pressure phase transitions of solid halogens, many questions still remain opened. As noted by Dalladay-Simpson et al. [28] large discrepancies exist in the experimental phase boundary pressures of halogens and those modelled with DFT. This might signal that the commonly used GGA approximation might be insufficient to correctly describe the properties of these systems. Indeed, a need for including more advanced functionals in modelling the high-pressure behaviour of hydrogen has been recently put forward. [12] Here we
All geometry parameters (lattice vectors and fractional coordinates) of the studied structures were Although DFT methods are highly successfully at material modelling both at ambient and high pressure, in some cases the most commonly used generalized-gradient approximation (GGA) approximation does not correctly reproduce the actual properties. Hence, methods from higher rungs of the Jacob's ladder of DFT functionals, [62] such as meta-GGA functionals, are needed to correctly modelled such systems to avoid invoking empirical parameters such as the Hubbard U. [63,64] In the case of iodine and bromine it was found that both GGA and meta-GGA methods fail to describe correctly the ground state structure at 1 atm yielding a structure of Cmcm symmetry, composed of monoatomic chains, lower in energy than the experimentally observed molecular Cmca phase. [65] The correct energetic preference (Cmca lower than Cmcm) was obtained only with hybrid functionals.
This surprising result cast doubt on previous theoretical studies on the high-pressure behaviour of bromine, all of which used GGA methods.
In order to evaluate the influence of the functional on the resultant simulated properties of solid bromine we optimized the Cmca molecular structure at 1 atm using the three above-mentioned approximation: GGA (PBE functional), meta-GGA (r 2 SCAN), and hybrid (HSE06) all with the D3 dispersion correction which is essential for the correct reproduction of the experimental geometry. The comparison of geometry parameters obtained with those functionals, given in Fig. 1, clearly shows an improved agreement between theory and experiment when moving from GGA through meta-GGA, to hybrid methods (see Table S1 in the Supplemental Material [66] for more details).    [67] properties of the bromine molecule: the dissociation energy (corrected for zero-point energy motion, D0 in eV), the Br-Br bond distance (r1 in Å), an its stretching frequency (ν in cm -1 ). The energetic difference between the Cmcm and Cmca structures of solid bromine at 1 atm (in meV per atom) obtained with various DFT functionals in this work is also shown. The need for using the hybrid DFT method when modelling elemental bromine is corroborated by the comparison of the dissociation energy Br2 molecule ( Table 1). The GGA and meta-GGA functionals overestimate Br2 stability, whereas the hybrid functional yields better agreement with experiment. In accordance with a previous study [65], we find that only the hybrid functional yields the experimental Cmca structure as lower in energy than the erroneous Cmcm structure which is composed of linear chains of bromine atoms (Table 1). Therefore, we choose this method for further calculations. FIG . 3 (a) Comparison of experimental lattice constants (blue/grey stars for Cmca/Immm, data taken from ref. [37]) with those calculated with the HSE06 functional (red/yellow lines for Cmca/Immm); (b) Comparison of experimental frequencies of Raman bands for the Cmca phase (blue close and open circles for data from ref. [33] and [38], respectively) and those calculated with HSE06 (grey line).
75 GPa. [42,43] We predict the yet unobserved transformations of Immm to the I4/mmm structure at 128 GPa, and the transition of this tetragonal structure to an fcc arrangement at 188 GPa. These transitions pressures are about 25 % higher than obtained from GGA modelling. [45] Because our calculations correctly reproduce the high-pressure stability of the molecular Cmca phase, we are able to simulate the properties of this phase in the whole pressure range covered by experiment.
As can be seen in Fig. 3a the theoretical lattice constants of the molecular Cmca phase are in excellent agreement with those obtained by x-ray diffraction, the same accordance is also found for the one pressure point for which experimental data was reported for the Immm structure. Moreover, the

C. Chemical bonding and electronic properties of compressed bromine
The excellent agreement between our results and experiment lends credibility to further analysis of the properties of the high-pressure phases of bromine, in particular the chemical bonding picture which is not directly accessible through experiment. The Cmca phase, shown again in Fig. 4a, is predicted to retain its molecular structure up to the phase transition to Immm. Although considerable pressureinduced shortening of intermolecular contacts (marked d2 and d3, see Fig. 4a) is observed up to this pressure, they still remain more than 10 % longer than the intramolecular Br-Br bond (Fig. 4b). These contacts link the molecules into planes separated by an even larger distance (dinter). We find good accordance of the calculated d1 -d3 distances with ambient pressure neutron diffraction data, [68] and high-pressure EXAFS experiments. [69] However we do not observe any considerable pressureinduced lengthening of the Br-Br bond (d1 distance) as postulated from the latter. Comparison of EXAFS data with our calculations and the experimental ambient-pressure values (Fig. 4b) suggests that the lengthening observed in the EXAFS experiments might be an artefact resulting from the underestimation of the Br-Br bond length at pressure below 20 GPa.
The molecular character of solid bromine is lost upon the transition from Cmca to Immm, as evidenced by the near equalization of the intra-and intermolecular contacts (Fig. 4b). While in the former phase each Br atom forms one short bond (d1 = 2.25 Å at 90 GPa) and three longer contacts (two at a distance form a distorted square lattice (Fig. 4a). The Immm phase has a layered character, as the distance between planes formed by the d2D contacts (dinter = 2.86 Å at 0 GPa) is considerably larger than these contacts. Upon compression of Immm the difference between d2D' and d2D" decreases from 7 % at 90 GPa to 4 % at 128 GPa. At this pressure a phase transition to the I4/mmm structure is predicted, in which equalization of the two distances occurs, and hence a perfect square lattice (with d2D = 2.44 Å) is formed. The I4/mmm has still layered character as the difference between d2D and dinter is sizeable competitive in terms of enthalpy with Cmca and Immm: the Fmm2-28 phase (28 atoms in the unit cell) previously proposed by Li et al. [45], and a monoclinic Cm-10 structure (10 atoms in the unit cell) which is a distorted variant of the Fmm2-20 structure also described by these authors.
Previous GGA calculations indicated a large pressure range of stability (52 -79 GPa) for the commensurate approximations of the incommensurate phases. [45] In contrast, our results, shown in

IV. CONCLUSIONS
The results obtained in this study indicate that in contrast to GGA the hybrid DFT method is capable of reproducing, with high accuracy, the behavior of compressed solid bromine. This enables the prediction of the yet unobserved transition from phase II (Immm) to III (I4/mmm) at 128 GPa, and subsequent formation of the fcc arrangement at 188 GPa, giving an incentive for future experiments.
We find that the molecular Cmca phase becomes metallic at pressures higher than previously reported, [40] and that the incommensurate structures are indeed transient species which can be viewed intermediates in the dissociation process marked by the Cmca to Immm transformation. Partial dissociation of Br2 molecules in those phases enhances the metallic properties. We show that both