Stability of Dense Liquid Carbon Dioxide

Stability of dense liquid carbon dioxide

Brian Boatesa,b, Amanuel M. Teweldeberhana, and Stanimir A. Boneva,b,1

aLawrence Livermore National Laboratory, Livermore, CA 94550; and bDepartment of Physics, Dalhousie University, Halifax, NS, Canada B3H 3J5

Edited by Efthimios Kaxiras, Harvard, Cambridge, MA, and accepted by the Editorial Board July 18, 2012 (received for review December 9, 2011)

We present ab initio calculations of the phase diagram of liquid CO2 debated. The melting curve has only been measured up to 30 GPa and its melting curve over a wide range of pressure and tempera- (17–19) and has not yet been computed. By focusing on liquid ture conditions, including those relevant to the Earth. Several dis- CO2 we avoid the problems of metastability associated with solid tinct liquid phases are predicted up to 200 GPa and 10,000 K based phases. Nevertheless, the evolution of liquid CO2 bonding with on their structural and electronic characteristics. We provide pressure sheds light on the high-T solid phases as well. Impor- evidence for a first-order liquid–liquid phase transition with a cri- tantly, the work presented here answers questions about the sta- tical point near 48 GPa and 3,200 K that intersects the mantle bility of CO2 at thermodynamic conditions relevant to geochem- geotherm; a liquid–liquid–solid triple point is predicted near 45 GPa ical processes. and 1,850 K. Unlike known first-order transitions between thermodynamically stable liquids, the coexistence of molecular and poly- Results and Discussion meric CO2 phases predicted here is not accompanied by metal- Phase Diagram and Liquid Structure. The phase diagram of liquid lization. The absence of an electrical anomaly would be unique CO2 is mapped by a series of first-principles molecular dynamics among known liquid–liquid transitions. Furthermore, the previ- (FPMD) simulations in the NVT ensemble (where N and V are ously suggested phase separation of CO2 into its constituent ele- the number of particles and volume, respectively). The statistical ments at lower mantle conditions is examined by evaluating their data generated from these calculations is subsequently analyzed Gibbs free energies. We find that liquid CO2 does not decompose to obtain thermodynamic, electronic, and structural properties. into carbon and oxygen up to at least 200 GPa and 10,000 K. The predicted new features in carbon dioxide’s high-pressure phase diagram are summarized in Fig. 1 and are discussed in de- high pressure ∣ density functional theory ∣ first principles molecular tail below. dynamics ∣ polymerization The stability regimes for molecular and polymeric liquid CO2 are determined based on an evaluation of the coordination num- t ambient conditions, the sp-valent second-row elements C, ber of carbon atoms with respect to oxygen. Fig. 2 shows the frac- P T N, and O form simple volatile molecules characterized by tions of 2- and 3-coordinated carbon over the entire liquid - A A double and triple bonds. These materials often undergo dramatic range considered. The region of high 2-coordination in Fig. 2 at P T transformations at high pressures into extended single-bonded lower and clearly represents the liquid regime dominated by covalent phases with novel optical, energetic, and mechanical molecular CO2 and corresponds to the molecular boundaries properties (1, 2). The polymerization of solid carbon dioxide has drawn in Fig. 1. Similarly, the region showing high 3-coordination B been studied extensively as a prototype for the evolution of a che- in Fig. 2 represents the region of stability for an emergent poly- mical bond under compression (2–7). CO2 also plays a fundamen- meric liquid and corresponds to the polymeric phase boundaries T P tal role in the physics and chemistry of the Earth interior and its shown in Fig. 1. Another regime arises at high and low where climate (8–14). However, the thermodynamic, chemical, and phy- we see a surge in the fraction (above 33%, see Fig. S1) of 1-coordinated carbon (short-lived CO units). The region classified as sical properties of CO2 at the high temperature (above 2,000 K) P T and pressure conditions relevant to planetary interiors remain a dissociated metallic liquid refers to - conditions where all largely unknown. covalent bonding is highly unstable (20); it is the presence of ─ A critical factor for the Earth’s climate is the concentration of C C chemistry that promotes metallization in this regime (see Figs. S2 and S3). The range of pressures over which the polymer- CO2 in the atmosphere, which is controlled by a complicated dy- namical cycle involving terrestrial reservoirs and fluxes (8). The ization takes place progressively decreases as the temperature is lowered, indicating a possible first-order liquid–liquid phase tran- vast majority of CO2 is stored in the mantle primarily in the form of Ca and Mg carbonates (8–13). Experimental (11) and theore- sition (LLPT) at lower temperatures. To substantiate such a claim, we have carried out a detailed evaluation of thermody- tical (13) works suggest that CO2 is produced at high pressure (P) and temperature (T) during decarbonating reactions with silica namic and structural properties across the transition. in subducted basalts and is subsequently released into the ocean – PðV Þ and atmosphere during volcanic activity (8). Moreover, reactions First-Order Liquid Liquid Phase Transition. The equation of state (EOS) for several isotherms is shown in Fig. 3A. A clear between silica and free CO2 may also take place under such T ≤ 3;000 conditions, leading to the formation of silicon carbonates (15). plateau exists in the EOS for K, indicating the coexistence of two distinct liquid phases with different densities at the Whether free CO2 is stable or decomposes into oxygen and dia- ΔV ∼ 5% mond in the mantle is currently unclear (11, 12, 16, 17). There- transition pressure; the volume change at 3,000 K is . Calculations on a 200 K grid bracketed the critical point at Tc ¼ fore, understanding the stability of CO2 is a major challenge in 3;200 P ¼ 48 establishing the more general issue of terrestrial cycles of C and K and GPa. Previous calculations of the molecular fluid (21) found short-lived CO2 dimers, which are likely precur- CO2. Furthermore, the presence of CO2 fluid is believed to be sors to the gradual dissociation found above Tc. Fig. 3A, Inset responsible for partial melting and rheological weakening of the mantle (9, 10) and even earthquake aftershocks (14). However, the range within the mantle where CO2 remains fluid is hitherto Author contributions: B.B. and S.A.B. designed research; B.B. and A.M.T. performed unknown. research; B.B. and A.M.T. analyzed data; and B.B. and S.A.B. wrote the paper. The difficulties in determining the CO2 phase diagram for The authors declare no conflict of interest. P < 10 GPa and T>600 K arise mainly due to the rich poly- This article is a PNAS Direct Submission. E.K. is a guest editor invited by the Editorial Board. morphism and metastability of its solid phases. The structure of 1To whom correspondence should be addressed. E-mail: [email protected]. high-temperature phases and the existence of intermediate bond- This article contains supporting information online at www.pnas.org/lookup/suppl/ ing phases between molecular and polymeric CO2 are still highly doi:10.1073/pnas.1120243109/-/DCSupplemental.

14808–14812 ∣ PNAS ∣ September 11, 2012 ∣ vol. 109 ∣ no. 37 www.pnas.org/cgi/doi/10.1073/pnas.1120243109 Downloaded by guest on September 25, 2021 Depth [km] Similar to Fig. 2, the fractions of 2-, 3-, and 4-coordinated car- 0 500 1,000 1,500 2,000 2,500 3,000 bon are shown for temperatures below and above Tc in Fig. 3 B 10,000 C CO 33% dissociated metallic liquid and , respectively. At 3,000 K, the 2-coordinated (molecular) regime fraction drops discontinuously at the transition pressure, accom- 50% 33% 8,000 panied by a sharp rise in 3-coordinated (polymeric) carbon. Immediately following the LLPT, carbon atoms are predomi- 67% nantly 3-coordinated, while a smaller fraction is either 4-coordi-

6,000 50% nated or forming unstable CO2 molecules. The liquid character shifts continuously with pressure from 3- to 4-coordination before molecular liquid freezing into tetrahedral-like amorphous solids, consistent with polymeric liquid 4,000 the local structure of the proposed underlying crystalline phases T Widom line (3, 6, 22, 23). The mixture of 3- and 4-coordinated carbon in the Temperature [K] Temperature new liquid phase bears analogy to the low-T amorphous solid

2,000 geotherm (5, 7); a similar analogy also exists between polymeric liquid and polymeric solids low-T amorphous nitrogen (24). Fig. 3C shows the gradual pro- gression of the atomic coordination as CO2 dissociates well above molecular solids 0 the critical point. The presence of an LLPT gives a clear picture 0 20 40 60 80 100 120 140 of the evolution of the CO2 bonding with pressure. For pressures Pressure [GPa] below the sharp transition near 45 GPa, there are no changes in the nature of the CO2 molecules (no intermediate bonding re- Fig. 1. Proposed phase diagram of CO2. Dotted lines represent different gime was found). Based on this, it is reasonable to expect that contours of 1- (CO-like), 2- (molecular), and 3-coordinated (polymeric) carbon the highly contested (25–27) intermediate bonding suggested (as described in the text). Percentages given adjacent to curves reflect coor- T dination contour values. Diamonds indicate the location of a first-order li- (6) for high- solid phases (largely elongated and nonlinear mo- quid–liquid phase boundary. Squares denote melting points determined lecules) does not exist as a thermodynamically stable phase. In- through single-phase simulations. Uncertainties in melting temperatures deed, the transition pressure for the LLPT coincides with that of are derived from the spacing between neighboring isotherms. Experimental the solid phase polymerization. melting measurements at lower pressures are also shown (black line) (17–19). The subject of LLPTs is fundamental in the theory of liquids. The Earth’s geotherm is given by the region shaded in green, taken from First-order transitions in thermodynamically stable (not super- ref. (9) and references therein. cooled) liquids are known for only a few systems (28–30) where they coincide with or are driven by metallization. Even in cases −1 shows the isothermal compressibility κT ¼ −V ð∂V ∕∂PÞT , where they are sharp, but not necessarily first-order, liquid tran- computed from polynomial fits of PðV Þ. Below Tc, the compres- sitions are accompanied by electrical anomalies (31), which can sibility diverges at the phase boundary—a direct consequence of drive a rapid change in volume. In this regard, the LLPT in CO2 is the plateau in PðV Þ. Slightly above Tc, κT is continuous with a unique as it takes place between two insulating phases. In the maximum at the transition pressure—the expected behavior of absence of C─C bonding, the computed density functional theory thermodynamic response functions just above a critical point. (DFT)-generalized gradient approximation (GGA) band gap The determination of κT maxima at other temperatures above remains relatively unchanged across the transition at approxi- Tc allows one to construct a Widom line (locus of response func- mately 1.5 eV. The volume reduction due to the formation of tion maxima)—a natural extension of a first-order phase bound- 3-coordinated sp2 bonded networks is apparently sufficient to ary above its critical point (shown in Fig. 1). establish coexistence. However, it is not as large as in the case A 10,000 1.0 9,000 0.9 C 0.8 8,000 0.7 7,000 0.6 6,000 0.5 5,000 0.4

Temperature [K] Temperature 4,000 0.3 0.2 3,000 PHYSICS 0.1 2,000 0.0 10,000 B 0.6 9,000

8,000 0.5

7,000 0.4 6,000 0.3 5,000

Temperature [K] Temperature 4,000 0.2

3,000 0.1 2,000 0.0 20 40 60 80 100 120 140 Pressure [GPa]

Fig. 2. Fraction of (A) 2-coordinated (molecular) and (B) 3-coordinated (polymeric) carbon mapped over the entire P-T range considered. (C) Visualization of the polymeric liquid at 3,000 K and 60 GPa.

Boates et al. PNAS ∣ September 11, 2012 ∣ vol. 109 ∣ no. 37 ∣ 14809 Downloaded by guest on September 25, 2021 90 1.0 A B 3,000 K 2-coordinated 0.02 0.8 3-coordinated

] 47.5 GPa

-1 4-coordinated 80 3,000 K 5,000 K 0.6 [GPa 47.5 GPa T

κ 0.01 3,500 K 0.4 70

30 40 50 60 70 80 Coordination fraction 0.2 Pressure [GPa]

60 0.0 ∆ V ~ 5% 1.0 C 5,000 K Pressure [GPa] 50 0.8

0.6 40 0.4 3,000 K; 64 molecules 3,000 K; 32 molecules 3,500 K; 32 molecules Coordination fraction 0.2 30 5,000 K; 32 molecules

0.0 78910 30 40 50 60 70 3 Volume [Å /atom] Pressure [GPa]

Fig. 3. P-V equation of state along several isotherms near the liquid–liquid phase transition. (Inset) Corresponding isothermal compressibilities of carbon dioxide with pressure. (B) and (C) Fraction of 2-, 3-, and 4-coordinated carbon atoms as functions of pressure along the 3,000 and 5,000 K isotherms, respectively.

of nitrogen (29) (ΔV ∼ 14%) where the LLPT is accompanied by polymeric solid. Unless melting can be induced by interaction metallization. with minerals, CO2 would not play a role in the rheological weak- The high-pressure melting curve is calculated here using a ening and partial melting of mantle rocks at these depths. single-phase simulation method where liquids are gradually compressed along isotherms. This technique can lead to supercooling Free Energies and Phase Separation. Previous studies (16, 17) have and the melting curve presented in Fig. 1 should be interpreted as reported that recovery of CO2 from high P (30–80 GPa) and T a lower bound to the true melting temperatures. Comparison with (1,500–3,000 K) at room temperature yields diamond and ϵ-oxy- experimental measurements at low P gives an estimate for the gen. From this, it was inferred that molecules in the high-T state error, which is relatively small and close to the uncertainty in are unstable and that CO2 may even undergo a phase separation our calculations. Importantly, a lower bound for the melting transition at those conditions. To address this issue, we have com- curve is sufficient to determine that free CO2 in the lower mantle puted the finite-temperature Gibbs free energies of the relevant (above 60 GPa) would exist in the form of a covalently bonded CO2, C, and O phases over a wide range of P-T conditions.

-0.4 -0.4 AB3,000 K 4,000 K 6,000 K -0.5 10,000 K -0.6 [eV/atom]

mix -0.8 -0.6 H ∆

-1.0 -0.7 [eV/atom]

mix C 0.5 G ∆ -0.8

0.0 /atom] B

-0.9 [k -0.5 mix S ∆

-1.0 -1.0

40 60 80 100 200 40 60 80 100 200 Pressure [GPa] Pressure [GPa]

Fig. 4. Gibbs free energies of mixing of liquid CO2 along several isotherms; negative values indicate the mixed phase is favored. (B) Corresponding enthalpies ΔH, and (C) entropies ΔS of mixing of liquid CO2. Arrows and diamonds indicate the shift in the result when thermodynamic integration is used to evaluate liquid entropies.

14810 ∣ www.pnas.org/cgi/doi/10.1073/pnas.1120243109 Boates et al. Downloaded by guest on September 25, 2021 Enthalpies are obtained directly from FPMD using time averages demixing that we have presented is rather conclusive. Considera- of the energy and pressure. In order to cover the wide P-T range tions beyond idealized thermodynamic conditions and/or pure of interest, liquid entropies are first calculated from the FPMD CO2 may be required in order to interpret some of the previous vibrational density of states using an approximate but efficient experimental measurements. approach, the accuracy of this method has been examined in previous work (32). The results for the Gibbs free energy of mixing Materials ΔG A ΔG 0 FPMD simulations were performed using finite-temperature DFT (33, 34) mix, are shown in Fig. 4 ; mix < indicates that the CO2 system is thermodynamically stable. For the lower temperatures within the Perdew-Burke-Ernzerhof (35) generalized gradient approximation ΔG using the Vienna ab initio Simulation Package (36, 37). We used Born–Oppen- and pressures, the absolute values of mix are well beyond the uncertainty of the method. For the highest temperature (10,000 K heimer dynamics, a Nosé-Hoover thermostat, and 32- and 64-molecule super- and 200 GPa), where the entropic contributions are most signifi- cells. Simulations were equilibrated for 1-2 ps and run for an additional 10 ps cant, we have carried out rigorous thermodynamic integration. using a 0.75 fs ionic time-step (convergence verified with 0.30 fs). For the car- The entropy of mixing increases slightly by approximately 0.25 k bon (oxygen) atoms, we employed a four-electron (six-electron) projector B augmented wave (PAW) pseudopotential (PP) with a 1.50 (1.52) Bohr core when computed with thermodynamic integration (Fig. 4C), lead- ΔG radius and 500 eV plane-wave cutoff. Convergence was verified with C and ing to an even more negative mix. P T O PAW PPs with a 1.10 Bohr core radii and 875 eV cutoff, as well as an all- Our results show that for all - conditions considered, electron oxygen PP. including along the geotherm, carbon dioxide does not phase Atomic coordination is defined here as the number of oxygen atoms ΔG ≪ 0 separate ( mix ), consistent with experiments (11) on the surrounding a given carbon atom within a given cut-off distance. The first reduction of carbonates at high pressures (28–62 GPa) and tem- minimum of the C─O pair correlation function is chosen as this distance; jus- peratures (1,490–2,000 K) where free CO2 was produced with no tification is provided in ref. (20). ΔG ¼ traces of diamond in the samples. Other measurements on the The Gibbs free energies of mixing are calculated as mix G –ð1∕3ÞG –ð2∕3ÞG G X ¼ decomposition of carbonates (12) and vibrational frequencies CO2 C O, where X is the free energy of C, O, or CO2. of quenched CO2 (16, 17) may also be explained by the presence The internal energy, pressure, and temperature are FPMD ensemble (time) of the LLPT; indeed, their pressure and temperature conditions averages. The vibrational contributions to G of liquid C, O, and CO2 were compare well with our LLPT. We propose that the observed computed as described in ref. 32, where the method was successfully used to predict mixing-demixing behavior in Na-Li and Ca-Li liquid alloys. destabilization of molecular CO2 at these conditions could be the result of liquid polymerization rather than phase separation Thermodynamic integration (TI) is a method for computing differences in the Helmholtz free energies of two systems related with a potential energy and that the quenched reactive polymeric fluid may very well crys- UðλÞ¼U þ λðU − U Þ U tallize into its constituent elements. Furthermore, an analysis of function of the form 0 1 0 . Here 0 is the potential energy of a reference system with known free energy and U1 is the potential energy the dense fluid’s vibrational frequencies is provided as supple- of the system under investigation. We have carried out TI in two steps: (i) mental material. from an ideal gas (U0 ¼ 0) to a classical system with pair interactions fit to The presence of 1-coordinated carbon at low P and high T led U ¼ U U ¼ U DFT forces ( 1 cl); and (ii) from the classical ( 0 cl) to the actual us to consider the possible phase separation of CO2 into CO and U ¼ U DFT system ( 1 DFT). O under these conditions. However, the calculated free energy of Moreover, the free energies in this work have been checked (using FPMD mixing lies within our uncertainty in TS at these temperatures. configurations) with the successful Heyd–Scuseria–Ernzerhof (HSE) hybrid Therefore, more accurate calculations must be performed before functional (38) as some phases considered are metallic. The maximum HSE ΔG 0 5 ∕ any conclusions can be drawn. correction to mix is approximately . eV atom when a metallic phase The results presented here establish the phase diagram of of liquid oxygen is present. These corrections do not affect any conclusions liquid carbon dioxide at high pressure and temperature. Given drawn here; in fact, they further support them. In all other cases, the correc- 10 ∕ the important role that CO2 plays in geochemical processes, a tions are negligible (< meV atom). useful extension of our work would be to investigate how the mineral environment in the Earth’s interior will perturb the prop- ACKNOWLEDGMENTS. We wish to thank E. Schwegler and S. Hamel for erties of pure CO2 predicted here. Variations in the C─O stoi- discussions. This work was supported by the Lawrence Livermore National chiometric ratio may also yield interesting results, with additional Laboratory (LLNL), the Natural Sciences and Engineering Research Council of Canada, and the Killam Trusts. Computational resources were provided relevance to diamond-anvil cell experiments. The computed free by the Canadian Foundation for Innovation, ACEnet, WestGrid, and LLNL. energies of mixing of CO2 lie outside the uncertainties of the The work at LLNL was performed under the auspices of the US Department computational method. Therefore, the evidence for lack of of Energy under contract No. DE-AC52-07NA27344.

1. Eremets MI, Gavriliuk AG, Trojan IA, Dzivenko DA, Boehler R (2004) Single-bonded 13. Oganov AR, Ono S, Ma Y, Glass CW, Garcia A (2008) Novel high-pressure structures of cubic form of nitrogen. Nat Mater 3:558–563. MgCO3, CaCO3 and CO2 and their role in Earth’s lower mantle. Earth Planet Sci Lett

2. Iota V, Yoo C-S, Cynn H (1999) Quartzlike carbon dioxide: An optically nonlinear 273:38–47. PHYSICS extended solid at high pressures and temperatures. Science 283:1510–1513. 14. Miller SA, et al. (2004) Aftershocks driven by a high-pressure CO2 source at depth. 3. Serra S, Cavazzoni C, Chiarotti GL, Scandolo S, Tosatti E (1999) Pressure-induced solid Nature 427:724–727.

carbonates from molecular CO2 by computer Simulation. Science 284:788–790. 15. Santoro M, et al. (2011) Silicon carbonate phase formed from carbon dioxide and silica – 4. Santoro M, Lin JF, Mao H-K, Hemley RJ (2004) In situ high P-T Raman spectroscopy and under pressure. Proc Natl Acad Sci USA 108:7689 7692. laser heating of carbon dioxide. J Chem Phys 121:2780–2787. 16. Tschauner O, Mao H-K, Hemley RJ (2001) New transformations of CO2 at high pressures and temperatures. Phys Rev Lett 87:075701. 5. Santoro M, et al. (2006) Amorphous silica-like carbon dioxide. Nature 441:857–860. 17. Litasov KD, Goncharov AF, Hemley RJ (2011) Crossover from melting to dissociation of 6. Iota V, et al. (2007) Six-fold coordinated carbon dioxide VI. Nat Mater 6:34–38. CO2 under pressure: Implications for the lower mantle. Earth Planet Sci Lett 7. Montoya JA, Rousseau R, Santoro M, Gorelli F, Scandolo S (2008) Mixed threefold and 309:318–323. fourfold carbon coordination in compressed CO2. Phys Rev Lett 100:163002. 18. Giordano VM, Datchi F, Dewaele A (2006) Melting curve and fluid equation of state of 8. Sleep NH, Zahnle K (2001) Carbon dioxide cycling and implications for climate on carbon dioxide at high pressure and high temperature. J Chem Phys 125:054504. ancient Earth. J Geophys Res 106:1373–1399. 19. Giordano VM, Datchi F (2007) Molecular carbon dioxide at high pressure and high 9. Isshiki M, et al. (2004) Stability of magnesite and its high-pressure form in the lower- temperature. Europhys Lett 77:46002. – most mantle. Nature 427:60 63. 20. Boates B, Hamel S, Schwegler E, Bonev SA (2011) Structural and optical properties of 10. Dasgupta R, Hirschmann MM (2006) Melting in the Earth’s deep upper mantle caused liquid CO2 for pressures up to 1 TPa. J Chem Phys 134:064504. – by carbon dioxide. Nature 440:659 662. 21. Tassone F, Chiarotti GL, Rousseau R, Scandolo S, Tosatti E (2005) Dimerization of CO2 at 11. Takafuji N, Fujino K, Nagai T, Seto Y, Hamane D (2006) Decarbonation reaction of high pressure and temperature. Chemphyschem 6:1752–1756. magnesite in subducting slabs at the lower mantle. Phys Chem Miner 33:651–654. 22. Santoro M, et al. (2012) Partially collapsed cristobalite structure in the nonmolecular 12. Seto Y, Hamane D, Nagai T, Fujino K (2008) Fate of carbonates within oceanic plates phase V in CO2. Proc Natl Acad Sci USA 109:5176–5179. subducted to the lower mantle, and a possible mechanism of diamond formation. Phys 23. Datchi F, Mallick B, Salamat A, Ninet S (2012) Structure of polymeric carbon dioxide Chem Miner 35:223–229. CO2-V. Phys Rev Lett 108:125701.

Boates et al. PNAS ∣ September 11, 2012 ∣ vol. 109 ∣ no. 37 ∣ 14811 Downloaded by guest on September 25, 2021 24. Boates B, Bonev SA (2011) Electronic and structural properties of dense liquid and 31. Brazhkin VV, Popova SV, Voloshin RN (1997) High-pressure transformations in simple amorphous nitrogen. Phys Rev B 83:174114. melts. High Pressure Res 15:267–305. 25. Bonev SA, Gygi F, Ogitsu T, Galli G (2003) High-pressure molecular phases of solid 32. Teweldeberhan AM, Bonev SA (2011) Structural and thermodynamic properties of carbon dioxide. Phys Rev Lett 91:065501. liquid Na-Li and Ca-Li alloys at high pressure. Phys Rev B 83:134120. – 26. Gorelli FA, Giordano VM, Salvi PR, Bini R (2004) Linear carbon dioxide in the high- 33. Hohenberg P, Kohn W (1964) Inhomogeneous electron gas. Phys Rev 136:B864 B871. 34. Kohn W, Sham LJ (1965) Self-consistent equations including exchange and correlation pressure high-temperature crystalline phase IV. Phys Rev Lett 93:205503. effects. Phys Rev 140:A1133–A1138. 27. Datchi F, Giordano VM, Munsch P, Saitta AM (2009) Structure of carbon dioxide phase 35. Perdew JP, Burke K, Ernzerhof M (1996) Generalized gradient approximation made IV: Breakdown of the intermediate bonding state scenario. Phys Rev Lett 103:185701. simple. Phys Rev Lett 77:3865–3868. 28. Katayama Y, et al. (2000) A first-order liquid–liquid phase transition in phosphorus. 36. Kresse G, Hafner J (1993) Ab initio molecular dynamics for liquid metals. Phys Rev B – Nature 403:170 173. 47:558–561. – 29. Boates B, Bonev SA (2009) First-order liquid liquid phase transition in compressed 37. Kresse G, Furthmüller J (1996) Efficiency of ab-initio total energy calculations for nitrogen. Phys Rev Lett 102:015701. metals and semiconductors using a plane-wave basis set. Comput Mater Sci 6:15–50. 30. Scandolo S (2003) Liquid–liquid phase transition in compressed hydrogen from first- 38. Heyd J, Scuseria GE, Ernzerhof M (2003) Hybrid functionals based on a screened principles simulations. Proc Natl Acad Sci USA 100:3051–3053. Coulomb potential. J Chem Phys 121:2780–2787.

14812 ∣ www.pnas.org/cgi/doi/10.1073/pnas.1120243109 Boates et al. Downloaded by guest on September 25, 2021