Formation mechanism of chemically precompressed hydrogen clathrates in metal superhydrides
Shichang Yao, Chongze Wang, Shuyuan Liu, Hyunsoo Jeon, Jun-Hyung Cho
aa r X i v : . [ c ond - m a t . s up r- c on ] F e b Formation mechanism of chemically precompressed hydrogen clathrates in metal superhydrides
Shichang Yao, Chongze Wang, Shuyuan Liu, Hyunsoo Jeon, and Jun-Hyung Cho ∗ Department of Physics, Research Institute for Natural Science,and Institute for High Pressure at Hanyang University, Hanyang University,222 Wangsimni-ro, Seongdong-Ku, Seoul 04763, Republic of Korea (Dated: February 25, 2021)Recently, the experimental discovery of high- T c superconductivity in compressed hydrides H S and LaH at megabar pressures has triggered searches for various superconducting superhydrides. It was experimentallyobserved that thorium hydrides, ThH and ThH , are stabilized at much lower pressures compared to LaH .Based on first-principles density-functional theory calculations, we reveal that the isolated Th frameworks ofThH and ThH have relatively more excess electrons in interstitial regions than the La framework of LaH .Such interstitial excess electrons easily participate in the formation of anionic H cage surrounding metal atom.The resulting Coulomb attraction between cationic Th atoms and anionic H cages is estimated to be strongerthan the corresponding one of LaH , thereby giving rise to larger chemical precompressions in ThH andThH . Such a formation mechanism of H clathrates can also be applied to another experimentally synthesizedsuperhydride CeH , confirming the experimental evidence that the chemical precompression in CeH is largerthan that in LaH . Our findings demonstrate that interstitial excess electrons in the isolated metal frameworks ofhigh-pressure superhydrides play an important role in generating the chemical precompression of H clathrates. I. INTRODUCTION
In recent years, hydrides have attracted much attention the-oretically and experimentally because of their promising pos-sibility for the realization of room-temperature superconduc-tivity (SC) [1, 2]. Motivated by the pioneering idea of NeilAshcroft on high-temperature SC in metallic hydrogen [3]and the incessant theoretical predictions of high supercon-ducting transition temperature T c in a number of hydrides [4–20], experiments have confirmed that sulfur hydride H S andlanthanum hydride LaH exhibit T c around 203 K at ∼ −
260 K at ∼
170 GPa [22, 23], respec-tively. Subsequently, such a conventional Bardeen-Cooper-Schrieffer type SC has also been experimentally observed invarious compressed hydrides at high pressures [24–30]. Forexamples, ThH (ThH ) exhibits T c = 159 −
161 (146) K be-tween 170 and 175 GPa [24], while CeH exhibits a T c of ∼
100 K at 130 GPa [27]. More recently, carbonaceous sul-fur hydride was observed to exhibit a room-temperature SCwith a T c of 288 K at ∼
267 GPa [30]. Therefore, the exper-imental observations of high-temperature SC in either surfur-containing hydrides [21, 30] or superhydrides containing anabnormally large amount of hydrogen [22–28] has launched anew era of high- T c superconductors.Compared to the existence of metallic hydrogen at highpressures over ∼
400 GPa [31, 32], the syntheses of super-hydrides with H-rich clathrate structures have been achievedat relatively much lower pressures, because H atoms canbe “chemically precompressed” by chemical forces betweenmetal atoms and H cages [33]. Using density-functional the-ory (DFT) calculations, the high- T c superconducting phasesof various superhydrides have been predicted to be metastableat higher pressures than a critical pressure P c [34–37]. It isnoticeable that the magnitude of P c reflects the strength ofchemical precompression in superhydrides. Experimentally,the P c value of ThH having a fcc-Th framework [see Fig.1(a)] was measured to be ∼
85 GPa [24], much lower than P c ≈
170 of an isostructural superhydride LaH [22, 23]. Fur- thermore, ThH (CeH ) having a hcp metal framework [seeFig. 1(b)] was observed to exhibit a P c of ∼
86 (80) GPa [24–27]. Based on these existing experimental data [22–24], itis most likely that P c changes with respect to metal species:i.e., Group-4 metal hydrides ThH , ThH , and CeH withoccupied f -subshell electrons have lower P c values or largerchemical precompressions compared to a Group-3 metal hy-dride LaH . Our recent DFT calculations for LaH [38] re-vealed that the isolated La framework without H atoms be-haves as an electride at high pressures, where some electronsdetached from La atoms are well localized in interstitial re-gions. These interstitial excess electrons are easily capturedto H atoms, forming a H clathrate structure in LaH . In thepresent study, such an electride feature in the La framework ofLaH is compared with other metal frameworks of the above-mentioned superhydrides ThH , ThH , and CeH . By the es-timation of Coulomb attractions between metal atoms and Hcages, we provide an explanation for the different chemicalprecompressions observed in such superhydrides [22–27], aswill be discussed below.In this paper, using first-principles DFT calculations, weperform a comparative study of chemical precompressions inThH , ThH , CeH , and LaH . We find that the isolatedmetal frameworks of ThH , ThH , and CeH possess moreinterstitial excess electrons than that of LaH at an equalpressure of 300 GPa. Such loosely bound electrons can beeasily captured to form H clathrate structures with attractiveCoulomb interactions between cationic metal atoms and an-ionic H cages. Using the calculated Bader charges [39] andpositions of metal and H atoms in each superhydride, we es-timate a chemical pressure acting on H cage around a metalatom. As a result, ThH , ThH , and CeH are found tohave larger chemical precompressions than LaH , consistentwith the experimentally observed P c values in these superhy-drides [22–27]. It is thus demonstrated that Group-4 metalhydrides occupying f electrons can be more chemically pre-compressed compared to Group-3 metal hydride, thereby con-tributing to lower P c . The present findings illuminate that in-Typeset by REVTEXterstitial excess electrons in the metal frameworks of super-hydrides are of importance to generate the chemical precom-pression of H cages around metal atoms. FIG. 1. Optimized structures of (a) ThH and (b) ThH at 300 GPa.ThH (ThH ) has the fcc (hcp) Th framework with the H (H )cage surrounding each Th atom. There are two (three) different typesof H atoms: i.e., H and H for ThH (H , H , and H for ThH ).The (110) planes are drawn in the fcc and hcp lattices. II. CALCULATIONAL METHODS
Our first-principles DFT calculations were performed usingthe Vienna ab initio simulation package with the projector-augmented wave method [40–42]. Here, we treated Th6 s p f d s , Ce 5 s p f d s , La 5 s p d s and H 1 s as valence electrons, including 6 s p , 5 s p ,and 5 s p semicore electrons for Th, Ce, and La, respec-tively. For the exchange-correlation energy, we employedthe generalized-gradient approximation functional of Perdew-Burke-Ernzerhof [43, 44]. A plane-wave basis was used witha kinetic energy cutoff of 500 eV for ThH and ThH . The k -space integration was done with 24 × ×
24 and 18 × × k points for ThH and ThH , respectively. All atoms wereallowed to relax along the calculated forces until all the resid-ual force components were less than 0.005 eV/ ˚A. We calcu-lated phonon frequencies with the 6 × × × × q pointsfor ThH (ThH ) using the QUANTUM ESPRESSO pack-age [45]. For CeH and LaH , we chose the calculation pa-rameters used in our previous works [38, 46]. III. RESULTS
We first optimize the structures of experimentally synthe-sized superhydrides ThH , ThH , CeH , and LaH as afunction of pressure using DFT calculations. These super-hydrides have hydrogen sodalitelike clathrate structures withhigh-symmetry space groups: i.e., Fm m (No. 225) for ThH and LaH , while P6 / mmc (No. 194) for ThH and CeH . Asshown in Fig. 1(a), ThH (LaH ) is constituted by the fccmetal framework, where each Th (La) atom is surrounded bythe H cage consisting of 32 H atoms. Meanwhile, ThH (CeH ) have the hcp metal framework with the H cage sur-rounding a Th (Ce) atom [see Fig. 1(b)]. Note that thereare two (three) species of H atoms composing the H (H )cages in ThH and LaH (ThH and CeH ). The optimizedstructures of these superhydrides show that the lattice con-stants decrease monotonously with increasing pressure (seeFig. S1 in the Supplemental Material). Accordingly, the aver-aged bond lengths d M − H between metal and H atoms decreasewith increasing pressure [see Fig. 2(a)]. We find that d M − H for ThH having the H cage is longer than ThH havingthe H cage at a given pressure. Meanwhile, d M − H for CeH is shorter than that for ThH , possibly because of the smallersize of Ce atom with the atomic number of Z = 58 compared toTh atom with Z = 90. Interestingly, despite the larger atomicnumber of Th than La ( Z = 57), the d M − H values for ThH and ThH are close to that for LaH at a given pressure, im-plying that the former superhydrides have larger chemical pre-compressions than the latter one. We note that the charges ofcationic metal and anionic H atoms are also essential ingre-dients for determining chemical precompression, as discussedbelow. Figure 2(b) displays the averaged H − H bond lengths d H − H for ThH , ThH , CeH , and LaH , which also de-crease monotonously with increasing pressure. It is seen thatthe d H − H values for ThH and LaH are shorter than thosefor ThH and CeH , indicating that the H cages composedof larger number of H atoms give rise to shorter d H − H com-pared to the H cages. (a) (b) FIG. 2. Calculated averaged bond lengths (a) d M − H between metaland H atoms and (b) d H − H between H atoms for ThH , ThH , CeH ,and LaH as a function of pressure. In order to examine the dynamical stability of ThH ,ThH , CeH , and LaH , we calculate their phonon spectraas a function of pressure. As shown in Fig. 3(a), the phononspectrum of ThH , calculated at 130 GPa, exhibits the soft-ening of low-energy phonon modes (marked by arrows) alongthe Γ − L and Γ − K lines. Such H-derived phonon modes fi-nally have imaginary frequencies at 120 GPa [see Fig. 3(b)].This indicates that the fcc-ThH phase becomes unstable aspressure decreases. Therefore, the phonon spectra as a func-tion of pressure allow us to predict the P c values of about 130,110, 100, and 220 GPa for ThH , ThH , CeH [46], andLaH [36], respectively (see Fig. S2 in the Supplemental Ma-terial). We find that ThH and ThH have much lower P c thanLaH , while their P c values are close to that of CeH . Theoverall trend of these predicted P c values in four superhydridesare well consistent with the experimentally measured ones ofabout 85, 86, 80, and 170 GPa for ThH [24], ThH [24],CeH [25–27], and LaH [22, 23], respectively. It was pre-viously pointed out that for LaH , the anharmonic effects onphonons and the quantum ionic zero-point energy are of im-portance for a proper prediction of P c [47]. Therefore, theabove overestimation of predicted P c values is likely due to theignorance of anharmonic and quantum effects [48–50] in thepresent theory. Nevertheless, we can say that ThH , ThH ,and CeH have significantly larger chemical precompressionscompared to LaH . F r equen cy ( c m - ) G X WK G L UW L G X WK G L UW L (a) (b) U FIG. 3. Calculated phonon spectra of ThH at (a) 130 and (b) 120GPa. The arrows in (a) indicate the softened phonon modes. Theimaginary phonon frequencies appear at 120Gpa along the Γ − L line. We next explore the electride-like characteristics of the iso-lated metal frameworks of ThH , ThH , CeH , and LaH .Here, the structure of each metal framework is taken from theoptimized structure of the corresponding superhydride. Thevalence charge densities ρ M (without including of semicoreelectrons) of the metal frameworks of ThH , ThH , CeH ,and LaH , calculated at an equal pressure of 300 GPa, aredisplayed in Figs. 4(a), 4(c), 4(e), and 4(g), respectively. Itis seen that some electrons detached from metal atoms are lo-calized in the interstitial regions around the A and A sites.These interstitial excess electrons of the so-called A and A anions are well confirmed by the corresponding electron lo-calization function (ELF) [51]. Figures. 4(b), 4(d), 4(f), and4(h) represent the calculated ELFs of the metal frameworksof ThH , ThH , CeH , and LaH , respectively. For theTh framework of ThH , the number of electrons Q A ( Q A )within the muffin-tin sphere of the A ( A ) anion is − − e , larger in magnitude than − − e for theLa framework of LaH [see Figs. 4(a) and 4(g)]. Similarly,as shown in Figs. 4(c) and 4(e), Q A ( Q A ) in ThH is − − e , larger in magnitude than the corresponding valueof − − e in CeH . Therefore, the metal frame-work of ThH (ThH ) exhibits a more electride feature thanthat of LaH (CeH ). It is noted that the magnitudes of Q A and Q A change as a function of pressure (see Figs. S3 andS4 in the Supplemental Material), showing that the electride-like characteristics of metal frameworks are enhanced with in- creasing pressure. Indeed, the localization of interstitial ex-cess electrons also emerges in compressed alkali metals athigh pressures [52–54], in order to reduce Coulomb repul-sions arising from the overlap of atomic valence electrons.Such loosely bound anionic electrons residing in the metalframeworks of compressed superhydrides can be easily cap-tured to H atoms, forming H cages with attractive Coulombinteractions between cationic metal and anionic H atoms. Itis remarkable that the anionic electrons in H cages are mostlysupplied because of the electride nature of metal frameworksat high pressures, rather than due to the different electronega-tivities of metal and H atoms [11]. (a) (b) (c) (d)(f) (g) (h) (e) (e/Å ) (e/Å )(e/Å ) (e/Å ) (cid:39) (cid:23) (cid:39) (cid:23) (cid:39) (cid:23) A (cid:39) (cid:24) (cid:39) (cid:24) (cid:39) (cid:24) (cid:39) (cid:23) (cid:39) (cid:23) (cid:39) (cid:23) (cid:39) (cid:23) (cid:39) (cid:23) (cid:39) (cid:24) (cid:39) (cid:24) (cid:39) (cid:24) (cid:39) (cid:24) (cid:39) (cid:24) FIG. 4. Calculated valence charge density plot of the metal frame-works of (a) ThH , (c) ThH , (e) CeH , and (g) LaH at 300 GPa,together with the ELFs of the metal frameworks of (b) ThH , (d)ThH , (f) CeH , and (h) LaH . The charge densities in (a), (c), (e),and (g) are drawn on the (110) plane with a contour spacing of 0.005 e / ˚A . The ELF in (b), (d), (f), and (h) are drawn with a contour spac-ing of 0.05. A and A indicate the two anions in interstitial regions,and the dashed circles represent the muffin-tin spheres around A and A with the radii of 0.75 (0.60) and 0.75 (0.60) ˚A in ThH andLaH (ThH and CeH ), respectively. Figures 5(a) and 5(b) show the calculated total charge den-sities of ThH and ThH at 300 GPa, respectively. It is seenthat the H atoms in each H cage are bonded to each other withcovalent bonds, where each H − H bond has a saddle point ofcharge density at its midpoint, similar to the C − C covalentbond in diamond [55]. For ThH (TH ), the charge densi-ties ρ H − H at the midpoints of the H − H bonds are 0.911 and0.729 (0.992, 0.769, and 0.601) e / ˚A : see the arrows in Figs.5(a) and 5(b). In order to confirm that the interstitial excesselectrons of the Th framework of ThH (ThH ) are capturedto form the H (H ) cages, we calculate the charge densi-ties of the isolated H (H ) cages without Th atoms. Here,the structure of each isolated H cage is taken from the opti-mized structure of the corresponding superhydride. As shownin Fig. 5(c) [5(d)], we find that ρ H − H decreases as 0.742 and0.621 (0.843, 0.580, and 0.479) e / ˚A , smaller than those inThH (TH ). This indicates that the H − H covalent bonds inThH and TH are strengthened by capturing the interstitialexcess electrons of isolated Th frameworks. (b)(c) (d)(a) (e/Å ) (e/Å )(e/Å ) (e/Å ) FIG. 5. Calculated total charge densities of (a) ThH and (b) ThH at 300 GPa, with a contour spacing of 0.1 e / ˚A . The Bader basins ofTh atoms are also drawn in (a) and (b). The calculated charge den-sities of the isolated H and H cages of ThH and ThH withoutTh atoms are displayed in (c) and (d), respectively. The numbers rep-resent the values of ρ H − H at the midpoints (marked by the arrows) ofthe H − H bonds.
To provide an explanation for the variation of P c in ThH ,ThH , CeH , and LaH , we estimate chemical precompres-sion by calculating the attractive Coulomb forces between ametal atom and its surrounding H atoms [see the lower panelin Figs. 1(a) and 1(b)]. This simple estimation is basedon the complete screening of the electric field arising frommetal atoms within H cages. Using the Bader [39] analysis,we calculate the cationic charge Q M inside the Bader basin[see Figs. 5(a) and 5(b)] of metal atom in each superhydride.For ThH , ThH , CeH , and LaH , we obtain Q M valuesof 1.486, 1.464, 1.199, and 1.036 e , respectively (see Fig. 6).Assuming that Q M is the point charge at the position of cor-responding metal atom and the anionic charge ( − Q M ) of Hatoms is uniformly distributed on the spherical shell with a ra-dius of d M − H , we evaluate the magnitudes of Coulomb forcesacting on the H atoms composing the H or H cage, anddivide it by the surface area of the spherical shell. Figure6 shows such estimated chemical pressures of ThH , ThH ,CeH , and LaH at 300 GPa, with ratios relative to the valueof LaH . We find that the chemical pressures of ThH andThH (CeH ) are about two (one and half) times higher than that of LaH , indicating that the former Group-4 metal hy-drides have larger chemical precompressions to attain lower P c values than the latter Group-3 metal hydride. Consideringthat the d M − H values of ThH and ThH are close to that ofLaH [see Fig. 2(a)], the higher chemical pressures in Th su-perhydrides are likely attributed to more cationic and anioniccharges compared to LaH . As shown in Fig. 6, the chem-ical pressures of four superhydrides are well consistent withtheir variations of Q M . It is noted that the estimated chemi-cal pressure of CeH is lower than that of isostructural ThH at 300 GPa (see Fig. 6), while the predicted value of P c =100 GPa for the former is lower than that (110 GPa) for thelatter. This inconsistency of chemical precompression and P c between CeH and ThH may be due to the delocalized na-ture of Ce 4 f electrons [46], which could lower P c via a morehybridization with the H 1 s state. Nevertheless, despite theircrude simulations, the estimated relative chemical pressuresof ThH , ThH , CeH , and LaH are in reasonable agree-ment with the variation of experimentally measured P c val-ues [22–27]. LaH ThH ThH CeH FIG. 6. Calculated chemical pressures of ThH , ThH , CeH , andLaH at 300 GPa. The cationic charges Q M of metal atoms obtainedfrom Bader charge analysis are also given. IV. SUMMARY
Using first-principles DFT calculations, we have conducteda comparative study of chemical precompressions in experi-mentally synthesized superhydrides including ThH , ThH ,CeH , and LaH . We found that these superhydrides formH clathrates by capturing excess electrons in interstitial re-gions of their isolated fcc- and hcp-metal frameworks. Bytaking into account the attractive Coulomb interactions be-tween cationic metal atom and its surrounding H atoms, weestimated chemical precompressions in ThH , ThH , CeH ,and LaH . It was found that ThH , ThH , and CeH havelarger chemical precompressions than LaH , consistent withthe variation of experimentally measured P c values [22–27].Our findings not only demonstrated that interstitial excesselectrons in the metal frameworks of superhydrides play animportnt role in generating the chemical precompression of Hcages around metal atoms, but also have important implica-tions for the exploration of new superhydrides which can besynthesized at moderate pressures below ∼
100 GPa.
V. SUPPLEMENTARY MATERIAL
See Supplemental Material for the lattice constants ofThH , ThH , CeH , and LaH , the phonon spectrum ofThH , and the valence charge densities of the Th frameworksof ThH and ThH . VI. AUTHOR’S CONTRIBUTIONS
S. Y., C. W., and S. L. contributed equally to this work.
VII. ACKNOWLEDGEMENT
This work was supported by the National ResearchFoundation of Korea (NRF) grant funded by the Ko-rean Government (Grants No. 2019R1A2C1002975, No.2016K1A4A3914691, and No. 2015M3D1A1070609). Thecalculations were performed by the KISTI SupercomputingCenter through the Strategic Support Program (Program No.KSC-2020-CRE-0163) for the supercomputing application re-search.
VIII. DATA AVAILABILITY
The data that support the findings of this study are availablefrom the corresponding author upon reasonable request. ∗ Corresponding author: [email protected] [1] T. Bi, N. Zarifi, T. Terpstra, and E. Zurek, Reference Module inChemistry, Molecular Science and Chemical Engineering (El-sevier, New York, 2019).[2] J. A. Flores-Livas, L. Boeri, A. Sanna, G. Profeta, R. Arita, andM. I. Eremets, Phys. Rep. , 1 (2020) and references therein.[3] N. W. Ashcroft, Phys. Rev. Lett. , 1748 (1968).[4] E. Zurek, R. Hoffmann, N. W. Ashcroft, A. R. Oganov, and A.O. Lyakhov, Proc. Natl. Acad. Sci. USA , 42 (2009).[5] Y. Xie, Q. Li, A. R. Oganov, and H. Wang, Acta Cryst. C ,104 (2014).[6] J. Hooper and E. Zurek, J. Phys. Chem. C , 13322 (2012).[7] H. Wang, J. S. Tse, K. Tanaka, T. Litaka, and Y. Ma, Proc. Natl.Acad. Sci. USA , 6463 (2012).[8] D. Duan, Y. Liu, F. Tian, D. Li, X. Huang, Z. Zhao, u. Yu, B.Liu, W. Tian, and T. Cui, Sci. Rep. , 6968 (2014).[9] Y. Li, J. Hao, H. Liu, Y. Li, and Y. Ma, J. Chem. Phys. ,174712 (2014).[10] X. Feng, J. Zhang, G. Gao, H. Liu, and H. Wang, RSC Adv. ,59292 (2015).[11] F. Peng, Y. Sun, C. J. Pickard, R. J. Needs, Q. Wu, and Y. Ma,Phys. Rev. Lett. , 107001 (2017).[12] H. Liu, I. I. Naumov, R. Hoffmann, N. W. Ashcroft, and R. J.Hemley, Proc. Natl. Acad. Sci. USA , 6990 (2017).[13] Y. Sun, J. Lv, Y. Xie, H. Liu, and Y. Ma, Phys. Rev. Lett. ,097001 (2019).[14] H. Xie, Y. Yao, X. Feng, D. Duan, H. Song, Z. Zhang, S. Jiang,S. A. T. Redfern, V. Z. Kresin, C. J. Pickard, and T. Cui, Phys.Rev. Lett. , 217001 (2020).[15] L. Liu, C. Wang, S. Yi, K. W. Kim, J. Kim, and J.-H. Cho, Phys.Rev. B , 140501(R) (2019).[16] C. Wang, S. Yi and J.-H. Cho, Phys. Rev. B , 104506 (2020).[17] A. P. Durajski, R. Szcze¸´sniak, Y. Li, C. Wang, and J.-H. Cho,Phys. Rev. B , 214501 (2020).[18] C. Heil, S. diCataldo, G. B. Bachelet, and L. Boeri, Phys. Rev.B , 220502(R) (2019).[19] Y. Quan, S. S. Ghosh, and W. E. Pickett, Phys. Rev. B ,184505 (2019).[20] D. A. Papaconstantopoulos, M. J. Mehl, and P.-H. Chang, Phys.Rev. B , 060506(R) (2020). [21] A. P. Drozdov, M. I. Eremets, I. A. Troyan, V. Ksenofontov, andS. I. Shylin, Nature (London) , 73 (2015).[22] M. Somayazulu, M. Ahart, A. K. Mishra, Z. M. Geballe, M.Baldini, Y. Meng, V. V. Struzhkin, and R. J. Hemley, Phys. Rev.Lett. , 027001 (2019).[23] A. P. Drozdov, P. P. Kong, V. S. Minkov, S. P. Besedin, M.A. Kuzovnikov, S. Mozaffari, L. Balicas, F. F. Balakirev, D.E. Graf, V. B. Prakapenka, E. Greenberg, D. A. Knyazev, M.Tkacz, and M. I. Eremets, Nature (London) , 528 (2019).[24] D. V. Semenok, A. G. Kvashnin, A. G. Ivanova, V. Svitlyk, V.Y. Fominski, A. V. Sadakov, O. A. Sobolevskiy, V. M. Pudalov,I. A. Troyan, and A. R. Oganov, Mater. Today , 36 (2020).[25] X. Li, X. Huang, D. Duan, C. J. Pickard, D. Zhou, H. Xie, Q.Zhuang, Y. Huang, Q. Zhou, B. Liu, and T. Cui, Nat. Commun. , 3461 (2019).[26] N. P. Salke, M. M. Davari Esfahani, Y. Zhang, I. A. Kruglov, J.Zhou, Y. Wang, E. Greenberg, V. B. Prakapenka, J. Liu, A. R.Oganov, and J.-F. Lin, Nat. Commun. , 4453 (2019).[27] W. Chen, D. V. Semenok, X. Huang, H. Shu, X. Li, D. Duan, T.Cui, and A. R. Oganov, arXiv:2101.01315 (2021).[28] P. P. Kong, V. S. Minkov, M. A. Kuzovnikov, S. P. Besedin,A. P. Drozdov, S. Mozaffari, L. Balicas, F. F. Balakirev, V. B.Prakapenka, E. Greenberg, D. A. Knyazev, and M. I. Eremets,arXiv:1909.10482 (2019).[29] D. Zhou, D. V. Semenok, D. Duan, H. Xie, W. Chen, X. Huang,X. Li, B. Liu, A. R. Oganov, and T. Cui, Sci. Adv. , eaax6849(2020).[30] E. Snider, N. Dasenbrock-Gammon, R. McBride, M. Debessai,H. Vindana, K. Vencatasamy, K. V. Lawler, A. Salamat, and R.P. Dias, Nature (London) , 373 (2020).[31] R. P. Dias and I. F. Silvera, Science , 715 (2017).[32] P. Loubeyre, F. Occelli, and P. Dumas, Nature (London) ,631 (2020).[33] N. W. Ashcroft, Phys. Rev. Lett , 187002 2004.[34] Z. M. Geballe, H. Liu, A. K.Mishra, M. Ahart, M. Somayazulu,Y. Meng, M. Baldini, and R. J. Hemley, Angew. Chem., Int. Ed. , 688 (2018).[35] H. Liu, I. I. Naumov, Z. M. Geballe, M. Somayazulu, J. S. Tse,and R. J. Hemley, Phys. Rev. B , 100102(R) (2018). [36] C. Wang, S. Yi, and J.-H. Cho, Phys. Rev. B , 060502(R)(2019).[37] A. M. Shipley, M. J. Hutcheon, M. S. Johnson, R. J. Needs, andC. J. Pickard, Phys. Rev. B , 224511 (2020).[38] S. Yi, C. Wang, H. Jeon, and J.-H. Cho, Phys. Rev. Materials ,024801 (2021).[39] R. F. W. Bader, Acc. Chem. Res. , 9 (1985).[40] G. Kresse and J. Hafner, Phys. Rev. B , 13115 (1993).[41] G. Kresse and J. Furthm¨uller, Comput. Mater. Sci. , 15 (1996).[42] P. E. Bl¨ochl, Phys. Rev. B , 17953 (1994).[43] J. P. Perdew, K. Burke, M. Ernzerhof, Phys. Rev. Lett. , 3865(1996); , 1396(E) (1997).[44] L. K´yvala and D. Legut, Phys. Rev. B , 075117 (2020).[45] P. Giannozzi, S. Baroni, N. Bonini, M. Calandra, R. Car, C.Cavazzoni, D. Ceresoli, G. L. Chiarotti, M. Cococcioni, I.Dabo, et al. J. Phys.: Condens. Matter , 395502 (2009).[46] H. Jeon, C. Wang, S. Yi, and J.-H. Cho, Sci. Rep. , 16878(2020). [47] I. Errea, F. Belli, L. Monacelli, A. Sanna, T. Koretsune, T.Tadano, R. Bianco, M. Calandra, R. Arita, F. Mauri, and J. A.Flores-Livas, Nature (London) , 66 (2020).[48] I. Errea, M. Calandra, C. J. Pickard, J. Nelson, R. J. Needs, Y.Li, H. Liu, Y. Zhang, Y. Ma, and F. Mauri, Phys. Rev. Lett. ,157004 (2015).[49] I. Errea, M. Calandra, C. J. Pickard, J. R. Nelson, R. J. Needs,Y. Li, H. Liu, Y. Zhang, Y. Ma, and F. Mauri, Nature (London) , 81 (2016).[50] A. P. Durajski, Sci. Rep. , 38570 (2016).[51] B. Silvi and A. Savin, Nature (London) , 683 (1994).[52] M.-S. Miao and R. Hoffmann, J. Am. Chem. Soc. , 3631(2015).[53] J. Wang, Q. Zhu, Z. Wang, and H. Hosono, Phys. Rev. B ,064104 (2019) and referenes therein.[54] Z. Zhao, S. Zhang, T. Yu, H. Xu, A. Bergara, and G. Yang,Phys. Rev. Lett.122