Superconductive Sodalite-Like Clathrate Calcium Hydride at High Pressures

Superconductive sodalite-like clathrate calcium hydride at high pressures

Hui Wanga, John S. Tseb, Kaori Tanakab, Toshiaki Iitakac, and Yanming Maa,1

aState Key Lab of Superhard Materials, Jilin University, Changchun 130012, Peoples Republic of China; bDepartment of Physics and Engineering Physics, University of Saskatchewan, Saskatoon, Saskatchewan, S7N 5E2, Canada; and cComputational Astrophysics Laboratory, Rikagaku Kenkyūjo , 2-1 Hirosawa, Wako, Saitama 351-0198, Japan

Edited by Michael L. Klein, Temple University, Philadelphia, PA, and approved February 17, 2012 (received for review November 5, 2011)

Hydrogen-rich compounds hold promise as high-temperature up to 4 formula units (f.u.). in the model. The enthalpies of can- superconductors under high pressures. Recent theoretical hydride didate structures relative to the products of dissociation into structures on achieving high-pressure superconductivity are com- CaH2 þ solid H2 at the appropriate pressures are summarized posed mainly of H2 fragments. Through a systematic investigation in Fig. 1A. The essential information can be summarized as of Ca hydrides with different hydrogen contents using particle- follows: (i) except for CaH10, stable structures began to emerge swam optimization structural search, we show that in the stoichio- at pressures <50 GPa; (ii) CaH4 was the most stable phase at

metry CaH6 a body-centered cubic structure with hydrogen that pressures between 50 and 150 GPa, whereas at 200 GPa CaH6 forms unusual “sodalite” cages containing enclathrated Ca stabi- had the lowest enthalpy of formation; (iii) the breakup of hydro- lizes above pressure 150 GPa. The stability of this structure is gen molecules depended on the Ca/H ratio, and a higher suscept-

derived from the acceptance by two H2 of electrons donated by ibility to dissociation was observed at higher ratios. Notably, “ ” Ca forming an H4 unit as the building block in the construction calcium hydrides with stoichiometry having an odd number of of the three-dimensional sodalite cage. This unique structure has a hydrogen (e.g., CaH3, CaH5, and CaH7, etc.) were found to be partial occupation of the degenerated orbitals at the zone center. energetically very unfavorable and were excluded in the discus- The resultant dynamic Jahn–Teller effect helps to enhance electron– sions (Supporting Information). PHYSICS phonon coupling and leads to superconductivity of CaH6. A super- The structures of the stable phases for each stoichiometry at conducting critical temperature (Tc) of 220–235 K at 150 GPa 150 GPa are shown in Fig. 1 B)–(D. Three types of hydrogen obtained from the solution of the Eliashberg equations is the high- species, “H4” units, monatomic H þ H2, and molecular H2, were est among all hydrides studied thus far. observed. CaH4 had a tetragonal (I4/mmm, Pearson symbol tI10) structure and included a body-centered arrangement of calcium hydride ∣ sodalite structure Ca and two molecular and four monatomic hydrogen units (Ca2ðH2Þ2ðHÞ4). The structure of CaH6 adopted a remarkable 3¯ urrent studies of the hydrides of simple elements at high pres- cubic form (Im m, Pearson symbol cI14), with body-centered Ca atoms and, on each face, squared H4 units tilted 45° with Csures are motivated by the proposition that a high Tc may be achieved from “precompressed” H2 in dense hydrogen-dominant respect to the plane of the Ca atoms. These H4 units were inter- hydrides (1). Superconductivity has been reported in SiH4 (2), linked to form a sodalite framework with a Ca atom enclathrated although controversy surrounds models of the origin of supercon- at the center of each cage. The next stable polymorph, CaH12, 3¯ ductivity and the structure of the superconducting state (3–5). had a rhombohedral (R , Pearson symbol hR13) structure consisting entirely of molecular H2. Many theoretical structures have the predicted Tc values ranging from 10 to 139 K (6–12). Except for AlH3 (13), in which super- The presence of different types of hydrogen can be rationa- conductivity was predicted but not observed, a common feature lized based on the effective number of electrons contributed of these structures is the ubiquitous presence of molecular hydro- by the Ca atom and accepted by each H2 molecule. Assuming gen fragments. At high pressures, unfavorable electron–electron that the two valence electrons of each Ca atom were completely “ ” “ ” repulsion in the atomic valence shells can be reduced by migra- ionized and accepted by H2 molecules, the formal effectively 1 ∕ tion and localization in the empty interstitial regions (14, 15). added electron (EAE) per H2 for CaH4 was e H2, for CaH6 was ð2∕3Þ ∕ ð1∕3Þ ∕ Introduction of an electronegative element, as was demonstrated e H2, and was e H2 for CaH12. Because H2 already – had a filled σ bond, the added electrons resided in the antibond- in the case of K Ag (16), increases the ability of the electrone- σ gative Ag atoms to accommodate electrons from K into the out- ing orbitals, which weakened the H-H bond (i.e., lengthened ermost 5s and 5p valence shells, thereby forming a bonding the H-H bond length) and eventually resulted in complete disso- network and stabilizing the alloy structures that otherwise would ciation. The presence of H and H2 units in the CaH2n structures not be observed under ambient pressures. Recently, Zurek et al. depended on the number of EAEs. Two formulae were present in (17, 18) demonstrated that charge transfer from lithium or so- each unit cell of CaH4. Because half (two) of each H2 molecule dium to hydrogen molecules under high pressures could lead to was retained, the remaining two H2 molecules had to accommodate four “excess” electrons into their σ orbital, which broke up the formation of new metallic lithium- or sodium-hydride alloys. − Depending on the stoichiometry, the structures consist of “pre- the molecules into monatomic hydrides (H ). The formation of H4 units in CaH6 was not accidental. If molecular hydrogen dissociated” molecular H2 and/or monoatomic hydrogen. atoms were present, each σ orbital of H2 had to accommodate Results and Discussion

The known calcium hydrides have a stoichiometry of CaH2 at Author contributions: Y.M. designed research; H.W., J.S.T., K.T., T.I., and Y.M. performed ambient pressure. Here, we explored other calcium hydrides with research; H.W., J.S.T., T.I., and Y.M. analyzed data; and H.W., J.S.T., and Y.M. wrote larger hydrogen contents by compressing a mixture containing the paper. þ þ Ca hydrogen or CaH2 hydrogen. The search for stable CaH2n The authors declare no conflict of interest. (n ¼ 2–6) structures at high pressures was performed using the This article is a PNAS Direct Submission. particle-swam optimization structure prediction algorithm (19) 1To whom correspondence should be addressed. E-mail: [email protected]. in combination with ab initio calculations. For each stoichiometry, This article contains supporting information online at www.pnas.org/lookup/suppl/ calculations were performed at pressures of 50–200 GPa with doi:10.1073/pnas.1118168109/-/DCSupplemental.

www.pnas.org/cgi/doi/10.1073/pnas.1118168109 PNAS ∣ April 24, 2012 ∣ vol. 109 ∣ no. 17 ∣ 6463–6466 Downloaded by guest on October 3, 2021 with those of 1.27 and 1.17 Å in isolated and H4 squares, respectively. This suggested the presence of a weak covalent H…H interaction in CaH6. As will be described below, the H4 unit is the fundamental building block of the sodalite cage. In CaH12, the EAE was ð1∕3Þe∕H2, which could be taken up by H2 without severing the bond. Extending this concept further then predicted that CaH12 and hydride alloys with a high H content (smaller EAE) would be composed predominantly of molecular H2. An important observation in support of the EAE description given above is that the H-H bond in CaH4 lengthened from 0.81 Å at 100 GPa to 0.82 Å at 150 GPa, whereas the H-H bond in CaH12 shortened from 0.81 Å to 0.80 Å. Here, more Ca valence electrons in CaH4 were available for transfer to the H2 σ orbitals than were available in CaH12, thereby more severely weakening the bond in CaH4. This effect was relatively small for CaH12, in which the H-H bond was shortened due to compression. The zero-point motion was not included in the calculation of the formation enthalpy of the various hydrides (Fig. 1A), although it is expected to be very influential due to the presence of large amounts of hydrogen. We estimated the zero-point energies of CaH6 and CaH4 using the quasiharmonic model (20) at 150 GPa. It was found that the inclusion of zero-point motion significantly lowered the formation enthalpy of CaH6 with respect to CaH4 (Fig. S11). As a consequence, CaH6 became more stable at and above 150 GPa. The physical mechanism underlying this effect stemmed from the H4 moieties, which Δ included much longer H-H distances and led to significantly sof- Fig. 1. Enthalpies of formation ( H, with respect to CaH2 and H2) of CaH2n (n ¼ 2–6) and crystal structures. (A) The abscissa x is the fraction of H2 in the tened phonons. This contrasted with other Ca hydrides studied structures. The open, solid, and half-filled symbols indicate that the structures (e.g., CaH4 and CaH12), in which the presence of H2 molecular are composed of H4 units, molecular H2, and the coexistence of H2 and H, units gave rise to higher frequency phonons and, thus, a larger respectively. The metastable structures are indicated by circles. The stable zero-point energy. pressure ranges for CaH4, CaH6, and CaH12 are 50–200 GPa, 150–200 GPa, The three-dimentional sodalite cage in CaH6 is the result of and 100–200 GPa, respectively. The EAE (per H2) is shown in brackets. The interlink of other H4 units via each H atom at the corner of estimated stability fields were determined according to the static enthalpies and may shift upon inclusion of dynamic effects (the zero-point motion of one H4 unit. So, what is the electronic factor promoting the for- the nuclei). (B) Structure of tI10-CaH4.(C) Structure of cI14-CaH6.(D) Structure mation of these H4 units? To answer this question, the electron of hR13-CaH12. Monatomic H, molecular H2, and Ca atoms are shown as cyan, localization functions (ELF) of a hypothetical bare bcc Ca lattice green, and royal blue, respectively. The green cylinders and gray dashed lines with the H atoms removed and CaH6 hydride (Fig. 3A and 3B) are drawn to represent molecular H2 and the sodalite cage, respectively. were examined. In bare body-centered cubic (bcc) Ca, regions with ELF values of 0.58 were found to localize at the H atom ð2∕3Þe. This led to a physically unfavorable structure with signif- sites in the H4 units on the faces of the cube. The ELF of CaH6 icant weakening of the intramolecular H-H bonds. There is, how- hydride suggested that no bonds were present between the Ca ever, an alternative lower energy structure, the one predicted and H. A weak “pairing” covalent interaction with an ELF of here. From molecular orbital theory, a square H4 unit should pos- 0.61, however, was found between the H atoms that formed a sess a half-filled degenerated “nonbonding” orbital (Fig. 2A). square H4 lattice. Their formation resulted from the accommo- This orbital, in principle, can accommodate up to two electrons dation by H2 of excess electrons from the Ca. An electron topo- without detrimentally weakening the H-H bond (Fig. 2B). It logical analysis also showed the presence of a bond-critical should be noted that at 150 GPa, the H…H distance of H4 in point (21) along the path connecting neighboring H atoms. the cI14 structure of CaH6 was 1.24 Å, which was substantially The integrated charge within the H atomic basin was 1.17 e, which shorter than both the monoatomic H…H distance of 1.95 Å corresponded to a charge transfer of 1.02 e from each Ca. A par- and the H…H2 distance of 1.61 Å in CaH4 but in good agreement tially ionized Ca was also clearly supported by the band structure and the density of states as reported in Fig. 3C and Fig. S15.At 150 GPa, Ca underwent an s–d hybridization with an electron transferred from the 4 s to the 3 d orbital. In CaH6, the Ca site 3¯ symmetry was m m (Oh) and the Ca 3 d manifold was clearly split into the eg and t2g bands, with the lower energy eg band partially occupied. A comparison of the band structures of CaH6, “H6” (Ca0H6), and bare Ca (CaH0) provided additional supporting evidence. Even without the presence of Ca, the valence band width of the hypothetical H6 (Fig. 3D) was 15.2 eV, comparable to 16.4 eV for CaH6 (Fig. 3C). In comparison, the valence band width of

2− the bare bcc Ca was only 4.3 eV (Fig. 3E). The band structure Fig. 2. Hückel energy-level diagrams of H4 and H4 units. (A) Hückel energy- 2− of CaH6 near the Fermi level was modified from H6 due to level diagram of H4.(B) Hückel energy-level diagram of H4 . The partial occupation of electrons on the degenerate orbitals of H4 units can lead to a JT the the hybridization between Ca 3 d and H 1 s orbital; however, distortion, but no JT distortion is expected for a closed shell electronic struc- the trend in the electronic band dispersions from −2 to −16.4 eV ture as shown in B. was remarkably similar to that of H6 from 3 to −15.2 eV.

6464 ∣ www.pnas.org/cgi/doi/10.1073/pnas.1118168109 Wang et al. Downloaded by guest on October 3, 2021 Fig. 3. ELF and band structure. (A) ELF of CaH0 (cI14 structure with H removed). (B) ELF of CaH6.(C) Band structure of CaH6.(D) Band structure of Ca0H6 (cI14

structure with Ca removed). (E) Band structure of CaH0. The horizontal dotted lines indicate the Fermi level. PHYSICS

Sodalite cage was constructed from linking of H4 units and distortions that lower the symmetry of the molecules, which is this topologically resulted in the formation of H6 faces. In fact, favorable for electron–phonon processes (24, 25). α2 ðωÞ a primitive cell of CaH6 can be seen as composed of a H6 hexagon Tc was calculated based on the spectral function F by and a Ca. Band structure analysis suggested that four conduction numerically solving the Eliashberg equations (26), which consist bands of “Ca0H6” sodalite cage (Fig. 3D) were half-filled and of coupled nonlinear equations describing the frequency-depen- the addition of maximal two electrons to H6 gave rise to partial dent order parameter and renormalization factor. The Coulomb Γ occupancy of the degenerate bands at as indicated in CaH6 repulsion is taken into account in terms of the Coulomb pseudo- (Fig. 3C). Partial occupancy of a degenerate orbital results in potential, μ, scaled to a cutoff frequency (typically six times the – an orbitally degenerate state and is subject to Jahn Teller (JT) maximum phonon frequency) (27). At 150 GPa, the predicted T distortion (22) (Fig. 2). The JT effect involves coupling between c values were 235 K and 220 K using typical values for μ of 0.1 and the electron and nuclear degrees of freedom, leading to distor- 0.13, respectively. EPC calculations were also performed for tions in the structure, and it lifts the orbital degeneracy. If the distortion is dynamic, JT vibrations can contribute to supercon- 200 GPa and 250 GPa, in which the calculated Tc was found to ductivity. The same mechanism has been invoked to explain the decrease with pressure (201 K at 200 GPa and 187 K at 250 GPa superconductivity in B-doped diamond (23). To investigate this possibility, electron–phonon coupling (EPC) calculations on the sodalite structure of CaH6 at 150 GPa were performed. The phonon dispersion curves, phonon linewidth γðωÞ, EPC parameter λ, and Eliashberg spectral function α2FðωÞ were calculated. A gap at 430 cm−1 (Fig. 4) separated the phonon spectrum into two regions: The lower frequency branches were associated with the motions of both Ca and H, whereas the higher frequency branches were mainly associated with H atoms. The combined contribution (19% and 81%, respectively) gave an EPC parameter λ of 2.69. The calculated phonon linewidths (Fig. 4) showed that the EPC was derived primarily from the T2g and Γ Eg modes at the zone center . Incidentally, these two bands, respectively, were the in-plane breathing and rocking vibrations of the H atoms belonging to the H4 unit. The corresponding atomic vibrations led to distortions in the square planar structure. More- over, both phonon branches showed significant phonon softening along all symmetric directions. A large EPC also benefited from the high density of states at the Fermi level caused by a Van Hove Fig. 4. Phonon band structure and Eliashberg spectral function. Phonon dis- singularity at Γ (Fig. 3C). The very large EPC was unprecedented persion curves of cI14 at 150 GPa (Left). Circles indicate the phonon linewidth with a radius proportional to the strength. Phonons with a larger linewidth at for the main group hydrides. Previous calculations on a variety Γ t e 960 −1 – belong to the 2g and g modes, as indicated by the circles at cm and of systems predicted an EPC in the range of 0.5 1.6 (11). The 1;960 cm−1, respectively. Eliashberg electron–phonon coupling spectral func- mechanism suggested here is not inconsistent with the mechan- tion α2FðωÞ at 150 GPa (Right). Dashed line is the integration of the electron– isms found in JT-induced superconductivity in alkali intercalated phonon coupling strength as a function of phonon frequency. The horizon C60, in which the intramolecular vibrations are responsible for lines are drawn as a guide.

Wang et al. PNAS ∣ April 24, 2012 ∣ vol. 109 ∣ no. 17 ∣ 6465 Downloaded by guest on October 3, 2021 μ¼0 13 ∕ for . ), with a pressure coefficient (dTc dP)of played by pressure in effectively overcoming the kinetic barrier to −0 33 ∕ . K GPa. Tc of the order of 200 K is among the highest formation in the synthesis of hydrides. for all reported hydrides. Methods The predicted high Tc for CaH6 is very encouraging, but it must be viewed with caution. Formally, there is no upper limit Our structure prediction approach is based on a global minimization of free – energy surfaces merging ab initio total-energy calculations via particle swarm to the value of Tc within the Midgal Eliashberg theory of superconductivity. Two practical factors must be considered. The cal- optimization technique as implemented in CALYPSO (crystal structure analy- culation of the EPC is based on the harmonic approximation and sis by particle swarm optimization) code (19). Our CALYPSO method unbiased by any known structural information has been benchmarked on various without consideration of electron correlation effects. Because known systems 19 with various chemical bondings and had several successful strong electron–phonon coupling in CaH6 arises from the proxi- prediction of high-pressure structures of Li, Mg, and Bi2Te3 (30–32), among mity of the electronic and structural instabilities, anharmonicity which the insulating orthorhombic (Aba2, Pearson symbol oC40) structure of of the atomic motions can lead to renormalization of the vibra- Li and the two low-pressure monoclinic structures of Bi2Te3 have been con- tional modes, as demonstrated in a study of AlH3. In that study, firmed by independent experiments (32, 33). The underlying ab initio struc- lower renormalized frequencies were found to reduce the EPC tural relaxations were carried out using density functional theory within the and suppress superconductivity (28). On the other hand, the elec- Perdew–Burke–Ernzerhof exchange-correlation (34) as implemented in the tron–phonon matrix elements may be enhanced by anharmonic Vienna Ab Initio Simulation Package (VASP) code (35). The all-electron pro- vibrations, as in the case of disordered materials (29). In a recent jector-augmented wave method (36) was adopted with 1s and 3p64s2 trea- hybrid functional study of C60 anions, the inclusion of Hartree– ted as valence electrons for H and Ca, respectively. Electronic properties, Fock exchange contributions was shown to have little effect on lattice dynamics and electron-phonon coupling were studied by density func- the structural properties and phonon frequencies but resulted tional (linear-response) theory as implemented in the QUANTUM ESPRESSO in a strong increase in the electron–phonon coupling (24). package (37). More computational details can be found in the Supporting Information. The formation of a hydrogen sodalite cage with enclathrated calcium in CaH6, reported here for hydrogen-rich compounds, ACKNOWLEDGMENTS. H. W. and Y. M. acknowledge Prof. Aitor Bergara for the provides an unexpected example of a good superconductor cre- valuable discussions and are thankful to the financial support by Natural ated by the compression of a mixture of elemental calciumþ Science Foundation of China (NSFC) under 11104104, the China 973 Program hydrogen or CaH2 þ hydrogen. This superconductor can also be (2011CB808200), NSFC under 11025418 and 91022029, Changjiang Scholar viewed as consisting of unique square H4 units and electron- and Innovative Research Team in University (IRT1132), and the research fund donating calcium atoms subject to JT effects. Dense supercon- of Key Laboratory of Surface Physics and Chemistry (SPC201103). T. I. was supported by Ministry of Education, Culture, Sports, Science and Technology ductive states, such as those reported here, may be favored in of Japan (20103001–20103005). The calculations were performed in the com- other mixtures of elemental metals þ hydrogen or any hydrideþ puting facilities at Rikagaku Kenkyūjo Integrated Cluster of Clusters system hydrogen upon compression. This work highlights the major role (Japan) and the High Performance Computing Center of Jilin University.

1. Ashcroft NW (2004) Hydrogen dominant metallic alloys: High temperature super- 20. Ma Y, Tse JS (2007) Ab initio determination of crystal lattice constants and thermal conductors? Phys Rev Lett 92:187002. expansion for germanium isotopes. Solid State Commun 143:161–165. 2. Eremets MI, Trojan IA, Medvedev SA, Tse JS, Yao Y (2008) Superconductivity in hydro- 21. Richard FWB (1990) Atoms in molecules: a quantum theory (Oxford University Press, gen dominant materials: Silane. Science 319:1506–1509. Oxford). 3. Degtyareva O, Proctor JE, Guillaume CL, Gregoryanz E, Hanfland M (2009) Formation 22. Jahn HA, Teller E (1937) Stability of polyatomic molecules in degenerate electronic of transition metal hydrides at high pressures. Solid State Commun 149:1583–1586. states. I. Orbital degeneracy. Proc Roy Soc A 161:220–235. 4. Hanfland M, Proctor JE, Guillaume CL, Degtyareva O, Gregoryanz E (2011) High-pres- 23. Ma Y, et al. (2005) First-principles study of electron-phonon coupling in hole- and sure synthesis, amorphization, and decomposition of silane. Phys Rev Lett 106:095503. electron-doped diamonds in the virtual crystal approximation. Phys Rev B 72:014306. 5. Strobel TA, et al. (2011) High-pressure study of silane to 150 GPa. Phys Rev B 83:144102. 24. Laflamme Janssen J, Côté M, Louie SG, Cohen ML (2010) Electron-phonon coupling in 6. Feng J, et al. (2006) Structures and potential superconductivity in SiH4 at high C60 using hybrid functionals. Phys Rev B 81:073106. “ ” pressure: En route to metallic hydrogen . Phys Rev Lett 96:017006. 25. Schlüter M, Lannoo M, Needels M, Baraff GA, Tománek D (1992) Superconductivity in 7. Tse JS, Yao Y, Tanaka K (2007) Novel superconductivity in metallic SnH4 under high alkali intercalated C60. J Phys Chem Solids 53:1473–1485. pressure. Phys Rev Lett 98:117004. 26. Eliashberg GM (1960) Interactions between electrons and lattice vibrations in a 8. Gao G, et al. (2008) Superconducting high pressure phase of germane. Phys Rev Lett superconductor. Sov Phys JETP 11:696. 101:107002. 27. Yao Y, Tse JS, Tanaka K, Marsiglio F, Ma Y (2009) Superconductivity in lithium under 9. Martinez-Canales M, et al. (2009) Novel structures and superconductivity of silane high pressure investigated with density functional and Eliashberg theory. Phys Rev B under pressure. Phys Rev Lett 102:087005. 79:054524. 10. Kim DY, Scheicher RH, Mao H-K, Kang TW, Ahuja R (2010) General trend for 28. Rousseau B, Bergara A (2010) Giant anharmonicity suppresses superconductivity in pressurized superconducting hydrogen-dense materials. Proc Natl Acad Sci USA AlH3 under pressure. Phys Rev B 82:104504. 107:2793–2796. 29. Garland JW, Bennemann KH, Mueller FM (1968) Effect of lattice disorder on the super- 11. Li Y, et al. (2010) Superconductivity at ∼100 K in dense SiH4ðH2Þ2 predicted by first conducting transition temperature. Phys Rev Lett 21:1315. principles. Proc Natl Acad Sci USA 107:15708–15711. 30. Lv J, Wang Y, Zhu L, Ma Y (2011) Predicted novel high-pressure phases of lithium. Phys 12. Jin X, et al. (2010) Superconducting high-pressure phases of disilane. Proc Natl Acad Sci Rev Lett 106:015503. USA 107:9969–9973. 13. Goncharenko I, et al. (2008) Pressure-induced hydrogen-dominant metallic state in 31. Li P, Gao G, Wang Y, Ma Y (2010) Crystal structures and exotic behavior of magnesium – aluminum hydride. Phys Rev Lett 100:045504. under pressure. J Phys Chem C 114:21745 21749. 14. Neaton JB, Ashcroft NW (1999) Pairing in dense lithium. Nature 400:141–144. 32. Zhu L, et al. (2011) Substitutional alloy of Bi and Te at high pressure. Phys Rev Lett 15. Tamblyn I, Raty J-Y, Bonev SA (2008) Tetrahedral clustering in molten lithium under 106:145501. pressure. Phys Rev Lett 101:075703. 33. Guillaume CL, et al. (2011) Cold melting and solid structures of dense lithium. Nat Phys 16. Tse JS (2005) Crystallography of selected high pressure elemental solids. Zeitschrift für 7:211–214. Kristallographie 220:521–530. 34. Perdew JP, Burke K, Ernzerhof M (1996) Generalized gradient approximation made 17. Zurek E, Hoffmann R, Ashcroft NW, Oganov AR, Lyakhov AO (2009) A little bit of simple. Phys Rev Lett 77:3865. lithium does a lot for hydrogen. Proc Natl Acad Sci USA 106:17640–17643. 35. Kresse G, Furthmüller J (1996) Efficient iterative schemes for ab initio total-energy n>1 18. Baettig P, Zurek E (2011) Pressure-stabilized sodium polyhydrides: NaHn ( ). Phys calculations using a plane-wave basis set. Phys Rev B 54:11169. Rev Lett 106:237002. 36. Blöchl PE (1994) Projector augmented-wave method. Phys Rev B 50:17953. 19. Wang Y, Lv J, Zhu L, Ma Y (2010) Crystal structure prediction via particle-swarm opti- 37. Giannozzi P, et al. (2009) QUANTUM ESPRESSO: A modular and open-source software mization. Phys Rev B 82:094116. project for quantum simulations of materials. J Phys: Condens Matter 21:395502.

6466 ∣ www.pnas.org/cgi/doi/10.1073/pnas.1118168109 Wang et al. Downloaded by guest on October 3, 2021