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1 Reactions of xenon with iron and nickel are predicted in the Earth s inner core Li Zhu 1, Hanyu Liu 1, Chris J. Pickard 2, Guangtian Zou 1, and Yanming Ma 1 * 1 State Key Laboratory of Superhard Materials, Jilin University, Changchun , China 2 Department of Physics and Astronomy, University College London, Gower Street, London, WC1E 6BT, United Kingdom * mym@jlu.edu.cn NATURE CHEMISTRY 1

2 Contents Computational details Supplementary Figures Chemical stabilities of Xe-Fe compounds Chemical stabilities of Xe-Fe/Ni compounds in Fe/Ni rich regimes Electronic band structure of XeFe Formation enthalpy of XeFe Melting curve of XeFe Enthalpy evolution during the metadynamics simulations Projected densities of states Crystal structures of various Xe-Fe compounds Crystal structures of various Xe-Ni compounds Phonon dispersion curves Calculated stress tensors and position correlation functions.. 17 Supplementary Tables Comparison of formation enthalpy of XeFe 3 with XeFe 239 alloy Calculated Bader atomic volumes Calculated structural parameters of various Xe-Fe compounds Calculated structural parameters of various Xe-Ni compounds Formation enthalpies of various XeMFe 3 compounds.. 22 Supplementary References.. 23 NATURE CHEMISTRY 2

3 Computational details Our structural prediction approach is based on a global minimization of free energy surfaces merging ab initio total-energy calculations through CALYPSO (Crystal structure AnaLYsis by Particle Swarm Optimization) methodology as implemented in its same-name CALYPSO code 1,2. The effectiveness of our CALYPSO method has been demonstrated by the successful applications in predicting high-pressure structures of various systems, ranging from elements to binary and ternary compounds 1. Particularly, high-pressure structures of dense hydrogen 3, lithium 4, magnesium 5, bismuth telluride 6, water ice 7, calcium hydride 8, cage-like diamondoid nitrogen 9, lithium-boron compounds 10, and tungsten borides 11 were predicted, among which the high-pressure insulating Aba2-40 (Pearson symbol oc40) structure of lithium and the two low-pressure monoclinic structures of bismuth telluride were confirmed by independent experiments 6,12. The results of the current structural searches were subsequently confirmed using the Ab Initial Random Structure Searching approach to structure prediction 13,14. Structure searching simulations through CALYPSO code were performed at 150, 250, and 350 GPa for XeFe x (x=0.5, 1 6) and at 100, 200 and 350 GPa for XeNi x (x = 0.5, 1 6), respectively, with one to four formula units in the models. Each generation contained 40 structures, and the first generation was produced randomly with symmetry constraint. All structures were locally optimized using the CASTEP code 15. Local optimizations performed during structure search, were done with the conjugate gradients method and were stopped when the enthalpy changes became smaller than ev per cell. The 60% lowest-enthalpy structures of each generation were used to produce the structures in the next generation by local PSO technique, and the remaining 40% structures were randomly generated within symmetry constraint to enhance the structural diversity. The inertia weight is dynamically varied and decreases linearly from 0.9 to 0.4. We keep the self-confidence factor and the swarm confidence factor as constant 2. The magnitudes of the velocities are confined within the range of [-0.2, 0.2]. Typically, the structure searching simulation for each calculation was stopped after we generated 1000 ~ 1200 structures (e.g., about 20 ~ 30 generations). To analyze the results of structure simulations, we selected a number of distinct NATURE CHEMISTRY 3

4 lowest-enthalpy structures and optimized their structures as a function of pressures using CASTEP code 15 with ultrasoft pseudopotentials 16 and the Perdew Burke Ernzerhof generalized gradient approximation functional 17. To reduce the calculation errors, the cut-off energy for the expansion of wavefunctions into plane waves was set to 600 ev in all calculations, and the Monkhorst Pack grid 18 with a maximum spacing of 2π 0.03 Å 1 was individually adjusted in reciprocal space to the size of each computational cell. The electronic density of states has been calculated with a large Monkhorst-Pack k point set generated from a k mesh for XeFe 3 and hypothetical XeFe 0. To calculate the atomic volume, we used the Bader charge analysis 19. We also employ the hybrid functional of Heyd, Scuseria, and Ermzerhof 20,21 (HSE) to calculate the electronic band-structure for obtaining a more accurate bandgap. The phonon calculations were carried out by using supercell approach 22 as implemented in the phonopy code 23. All structures of Xe-Fe/Ni compounds are demonstrated to be dynamically stable in their respective pressure ranges by phonon calculations. We explored the effects of temperature using the quasiharmonic approximation (QHA) that introduces volume dependence of phonon frequencies as a part of anharmonic effect, for which phonon calculations were performed for all promising structures using the Phonopy code 23. Gibbs free energy (G) is defined at a constant temperature (T) and pressure (p) by the formula: G(T, p) = min [U(V) + F phonon (T, V) + pv], where V is the volume, U is the internal lattice energy and F phonon is the phonon (Helmholtz) free energy. The minimal value for G is found at the equilibrium volume for a given T and p. The metadynamics 24, 25 calculations were carried out for examining the phase stabilities of XeFe 3 and XeNi 3 at 350 GPa and 6000 K. Metadynamics simulation is a powerful technique to improve sampling of a system where ergodicity is hindered by the form of the system's energy landscape. Though the method is able to overcome barriers and therefore can explore a wide range of candidate structures at finite temperatures, it neglects entropic contributions and therefore does not deal with the free energy. Successful applications of the method include several examples of reconstructive NATURE CHEMISTRY 4

5 structural transitions In the present study, the simulation cells were constructed by supercells of Cu 3 Au-structure for XeFe 3 and Pmmn structure for XeNi 3. Brillouin zone was sampled with the use of k mesh. The canonical NVT (N-number of particles, V-volume, T-temperature) ensemble was used for molecular dynamics runs. Each metastep of metadynamics simulations includes 600 time steps with each time step of 1 fs. Extensive metadynamics simulations with more than 200/100 metasteps for XeFe 3 /XeNi 3 at 6000 K and 350 GPa were conducted using Gaussian width δ = 50 (kbar Å 3 ) 1/2 and Gaussian height W = 2500 kbar Å 3, which are typically good parameters for investigation of phase transitions 24, 25. We calculated the melting temperatures of the XeFe 3 compound by using a well-established Z method 28,29, which has received a wide application in melting studies The Z method was developed by Belonoshko and co-authors 28. The method can be employed by using empirical potentials 28 and ab initio pseudopotentials 29. In this method one tries to determine the threshold of thermal stability. If the total energy of a system remains constant, then the temperature drops to the melting temperature. Thus, the connected P-T points on the isochore form a characteristic shape similar to the letter Z. The Z method is a good alternative to the two-phase approach 33 or the coexistence method 34 because it allows one to calculate melting temperatures with a modest computational cost. The method has been demonstrated to be reliable and the achieved result can be in good agreement with that derived from the two-phase method 28. We have applied this technique in combination with first-principles calculations to simulate the melting temperatures of XeFe 3 at different pressures using the simulation cells containing 32 atoms. The Brillouin zone was sampled with a k mesh and the simulations ran at least 10 ps for each calculation with the time step of 1 fs. The formation enthalpy (h f ) of XeM x (M = Fe and Ni) was calculated by using the following formula: H f (XeM x ) = [H(XeM x ) H(Xe) xh(m)] / (1 + x), in which H is the enthalpy of the most stable structure of certain compositions at the given pressure. For pure Fe and Ni, the structures of the ε-phase 35 and face-centered cubic structure 36 were used, respectively. For pure Xe, the hexagonal close-packed structure 37 was considered. NATURE CHEMISTRY 5

6 In order to test the validity of the CASTEP pseudopotentials, we performed full-potential all-electron calculation with the WIEN2K code 38. In the full-potential calculations, the muffin-tin radii were chosen to be 1.9 atomic unit for both Xe and Fe. The plane-wave cutoff was defined as K max R MT = 7, where R MT represents the muffin-tin radius and K max denotes the maximum size of the reciprocal-lattice vectors. Convergence tests resulted in the choices of 2,000 k points in the electronic integration of the Brillouin zone. The exchange-correlation functional was described using PBE 17 of generalized gradient approximation 39. We determined the energy data from all-electron calculations for nine geometries of XeFe 3, Xe and Fe derived from CASTEP structure optimization in the pressure range from 0 to 400 GPa. Note that the all-electron calculations did not involve any structure optimizations. We fit the energy vs. volume data into Birch-Murnaghan equations of state 40 to obtain the formation pressure of XeFe 3. The all-electron method finds a formation pressure of 195 GPa, in excellent agreement with that (195 GPa) derived from CASTEP calculation. Our current test validated the pseudopotentials used in CASTEP at such high pressures (up to 360 GPa). NATURE CHEMISTRY 6

7 Supplementary Figures Supplementary Figure 1 Chemical stabilities of Xe-Fe compounds. Predicted formation enthalpy of various Xe-Fe compounds with respect to the elemental decomposition into Xe and Fe at 0 K and high pressures. The stoichiometries of XeFe x with the half integer of values of x = 1.5, 2.5, 3.5, 4.5, and 5.5 were also considered, but no new stable structures were found. NATURE CHEMISTRY 7

8 Supplementary Figure 2 Chemical stabilities of Xe-Fe and Xe-Ni compounds in Fe and Ni rich regimes. a-b, Predicted formation enthalpy of various Xe-Fe (a) and Xe-Ni (b) compounds with respect to the decomposition into XeFe 3 /XeNi 3 and Fe/Ni at 0 K and high pressures. c-d, Predicted Gibbs free energy of various Xe-Fe (c) and Xe-Ni (d) compounds at 250 GPa relative to the decomposition into XeFe 3 /XeNi 3 and Fe/Ni as a function of temperatures. NATURE CHEMISTRY 8

9 Supplementary Figure 3 The electronic band structure of Cu 3 Au-type XeFe 3 calculated by using the PBE functional at 250 GPa. The dashed line indicates the Fermi energy. This band structure result together with the HSE band structure data in Figure 3 of main text revealed that XeFe 3 is semi-metallic. There is noticeable difference between PBE and HSE data. Both the conduction and valance bands are crossing the Fermi level in the PBE calculation. By contrast, the conduction bands in the HSE results do not show any crossing over the Fermi level. NATURE CHEMISTRY 9

10 Supplementary Figure 4 Formation enthalpy ( H), relative internal energy U, and p V term for Cu 3 Au-type structured XeFe 3 with respect to elemental Xe and Fe as a function of pressure. At zero temperature, Gibbs free energy reduces to enthalpy (H = U + pv). Though the relative internal energies U have a large positive value over a wide pressure range, the inclusion of pv term effectively tunes the formation enthalpy to be negative, leading to the formation of XeFe 3 compound. p V term is negative in the whole pressure range studied since the Cu 3 Au-type structure of XeFe 3 is much denser than that of the mixture of Xe and Fe. The figure clearly shows the important role played by the pressure in initializing the chemical reaction of Fe and Xe. NATURE CHEMISTRY 10

11 Supplementary Figure 5 The melting curve of XeFe 3 obtained with the Z method 28. The isochoric curves for XeFe 3 are shown by blue solid line with open diamonds. After reaching the limit of superheating (indicated by red filled diamonds), the temperature of the system spontaneously drops to the exact temperature of melting (indicated by green filled squares). NATURE CHEMISTRY 11

12 Supplementary Figure 6 Enthalpy evolution during the metadynamics simulations at T = 6000 K and P = 350 GPa. a, Evolution of the enthalpy in the metadynamics simulation starting from the Cu 3 Au-type structure for XeFe 3. b, Evolution of the enthalpy in the metadynamics simulation starting from the Pmmn structure for XeNi 3. After typically long simulation runs (200 metasteps for XeFe 3 and 100 metasteps for XeNi 3 ), the two structures remain stable without having any obvious structural changes. These meta-dynamics simulations validated the phase stabilities of XeFe 3 and XeNi 3 at the conditions of the Earth s core. During the metadynamics simulations, the entropic contributions are neglected. NATURE CHEMISTRY 12

13 Supplementary Figure 7 Projected densities of states (DOS) for XeFe 3 at 250 GPa. The Fermi energy is shown as a vertical dashed line. From the DOS, we find a rather weak orbital hybridization between Xe-5p and Fe-3d, seemingly indicating a weak Xe-Fe covalent bonding. However, our charge density calculations did not find any clear charge localization between Xe and Fe. We therefore, do not suggest the covalent bonding in the system. NATURE CHEMISTRY 13

14 a b c d e f g Fe Xe Supplementary Figure 8 Crystal structures of various Xe-Fe compounds. a-g, The predicted energetically best structures of Xe 2 Fe (a), XeFe (b), XeFe 2 (c), XeFe 3 (d), XeFe 4 (e), XeFe 5 (f), and XeFe 6 (g) at 250 GPa. Please refer to Supplementary Table 3 for the detailed structural parameters for various stoichiometries. NATURE CHEMISTRY 14

15 Supplementary Figure 9 Crystal structures of various Xe-Ni compounds. a-g, The predicted energetically best structures of Xe 2 Ni (a), XeNi (b), XeNi 2 (c), XeNi 3 (d), XeNi 4 (e), XeNi 5 (f), and XeNi 6 (g) at 350 GPa. Please refer to Supplementary Table 4 for the detailed structural parameters for various stoichiometries. NATURE CHEMISTRY 15

16 Supplementary Figure 10 Phonon dispersion curves for Cu 3 Au-type structure of XeFe 3 (a) and P-62m structure of XeFe 5 (b) at 250 GPa. The absence of any imaginary frequency in the whole Brillouin zone demonstrates that Cu 3 Au-type structure of XeFe 3 and P-62m structure of XeFe 5 is dynamically stable. NATURE CHEMISTRY 16

17 SUPPLEMENTARY INFORMATION Supplementary Figure 11 The calculated stress tensors as a function of simulation time and position correlation functions. a-b, The calculated stress tensors as a function of simulation time for XeFe3 (a) and XeNi3 (b) at 6000 K. It can be seen that the cell experiences hydrostatic stresses throughout the simulation, suggesting that the cell is elastically stable. c-d, The plots of position correlation functions, p(t), for XeFe3 (c) and XeNi3 (d) at 6000 K. The position correlation function for a chosen atom, i, is given by p(t) =< (ri (t + t0 )! Ri0 ) (ri (t0 )! Ri0 ) >, where ri is the time-varying position of the atom and Ri0 is the position of that atom s lattice site in the perfect starting structure. The angular brackets denote the thermal average, which in practice is evaluated as an average over time origins, t0, and atoms i. For long times t, vibrational displacements become uncorrelated, so that p(t) =< (ri (t + t 0 )! Ri0 ) (ri (t 0 )! Ri0 ) >"< ri! Ri0 > 2, and if all atoms vibrate about the starting lattice sites, < ri! Ri0 >= 0, so that p(t) 0 as t. It is clearly seen that p(t) 0 as t, indicating that the structure of XeFe3/XeNi3 is dynamically stable. NATURE CHEMISTRY

18 Supplementary Tables Supplementary Table 1. Comparison of formation enthalpy of the Cu 3 Au-type XeFe 3 with the earlier proposed XeFe 239 alloy 41. The alloy was constructed by which one Xe atom substitutes one Fe atom in a supercell of hcp Fe lattice containing 240 Fe atoms. Our calculations demonstrated that the XeFe 239 alloy was energetically very unfavorable when compared with the Cu 3 Au-type XeFe 3. H (mev/atom) Pressure (GPa) XeFe 3 XeFe NATURE CHEMISTRY 18

19 Supplementary Table 2. Calculated Bader atomic volumes of Xe and Fe atoms in the Cu 3 Au-type XeFe 3 at 250 and 350 GPa. The atomic radii of the Fe and Xe atoms in XeFe 3 differ more than 15% (16.78 % at 250 GPa, and 16.18% at 350 GPa), violating the Hume-Rothery rules governing the formation of solid solution. Pressure (GPa) Atom Volume (Å 3 ) 250 Xe (1a) Fe (3c) Xe (1a) Fe (3c) 6.57 NATURE CHEMISTRY 19

20 Supplementary Table 3. Calculated structural parameters of various Xe-Fe compounds at 250 GPa. Space group Lattice Parameters (Å, ) Atomic coordinates (fractional) Atom X Y Z Xe 2 Fe Pbcm a = Xe1 (4d) b = Xe2 (4d) c = Fe (4d) XeFe P-1 a = Xe1 (2i) b = Xe2 (2i) c = Fe1 (2i) α = Fe2 (2i) β = γ = XeFe 2 Pnma a = Xe (4c) b = Fe1 (4c) c = Fe2 (4c) XeFe 3 Pm-3m a = Xe (1a) Fe (3c) XeFe 4 P2 1 /m a = Xe (2e) b = Fe1 (2e) c = Fe2 (2e) β = Fe3 (2e) Fe4 (2e) XeFe 5 P-62m a = Xe (1a) c = Fe1 (2c) Fe2 (3g) XeFe 6 P2 1 /m a = Xe (2x) b = Fe1 (2x) c = Fe2 (2x) β = Fe3 (2x) Fe4 (2x) Fe5 (2x) Fe6 (2x) NATURE CHEMISTRY 20

21 Supplementary Table 4. Calculated structural parameters of various Xe-Ni compounds at 250 GPa Space group Lattice Parameters (Å, ) Atomic coordinates (fractional) Atom X Y Z Xe 2 Ni P4/nmm a = Xe1 (2c) c = Xe2 (2c) Ni2 (2c) XeNi P4/nmm a = Xe (2c) c = Ni (2c) XeNi 2 Pmmn a = Xe1 (2a) b = Xe2 (2b) c = Ni1 (2a) Ni2 (2a) Ni3 (2b) Ni4 (2b) XeNi 3 Pmmn a = Xe (2a) b = Ni1 (2b) c = Ni2 (4f) XeNi 4 P2 1 /m a = Xe (2x) b = Ni1 (2x) c = Ni2 (2x) β = Ni3 (2x) Ni4 (2x) XeNi 5 P-62m a = Xe (1a) c = Ni1 (2c) Ni2 (3g) XeNi 6 P2 1 /m a = Xe (2x) b = Ni1 (2x) c = Ni2 (2x) β = Ni3 (2x) Ni4 (2x) Ni5 (2x) Ni6 (2x) NATURE CHEMISTRY 21

22 Supplementary Table 5. The formation enthalpies of various XeMFe 3 (M = Ni, O, S, and Xe) compounds at 250 and 350 GPa. The formation enthalpy ( H) was calculated by using the formula: H(XeMFe 3 ) = H(XeMFe 3 ) H(XeFe 3 ) H(M). The crystal structures of XeMFe 3 were constructed via the Cu 3 Au-type XeFe 3 structure, in which the M atom was placed at the center of the cubic cell. The results demonstrated that the XeMFe 3 were energetically very unfavorable when compared with the Cu 3 Au-type XeFe 3 and the elemental M. If Fe replaced Xe in the Cu 3 Au-type XeFe 3 structure, the structure transforms to exactly a highly packed fcc structure. All these calculations rationalized the phase stability of Cu 3 Au-type structure of XeFe 3 under conditions of the Earth s inner core. H (ev per formula unit) Pressure (GPa) XeNiFe 3 XeOFe 3 XeSFe 3 Xe 2 Fe NATURE CHEMISTRY 22

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