# Fileset [2512.08251v1.pdf](https://mdr.nims.go.jp/filesets/1ea816c4-d177-4d43-a2eb-53b9943e473a/download) ## Creator Ryoma Asai, Ryotaro Arita, Takumi Chida, Ryota Masuki, Kazuhiko Kuroki, [Terumasa Tadano](https://orcid.org/0000-0002-8132-2161) ## Rights ## Other metadata [Finite-temperature ab initio structural optimization of the bilayer nickelate superconductor

{La}_{3} {Ni}_{2} O_{7}

](https://mdr.nims.go.jp/datasets/bfda33c4-904e-4ef1-9e15-d4a519a10985) ## Fulltext Finite-Temperature ab initio Structural Optimization of the Bilayer Nickelate SuperconductorLa3Ni2O7Ryoma Asai,1, ∗ Ryotaro Arita,2, 3 Takumi Chida,2 Ryota Masuki,2 Kazuhiko Kuroki,4 and Terumasa Tadano5, †1Department of Applied Physics, The University of Tokyo, Hongo, Bunkyo-ku, Tokyo 113-8656, Japan2Department of Physics, The University of Tokyo, Hongo, Bunkyo-ku, Tokyo 113-0033, Japan3RIKEN Center for Emergent Matter Science, Hirosawa, Wako, Saitama 351-0198, Japan4Department of Physics, The University of Osaka, Machikaneyama, Toyonaka, Osaka 560-0043, Japan5Research Center for Magnetic and Spintronic Materials,National Institute for Materials Science, Sengen, Tsukuba, Ibaraki 305-0047, JapanWe develop a first-principles framework for finite-temperature structural optimization that incorporates vi-brational contributions to the free energy through anharmonic phonon theory. We extend and further improvethe efficiency of the recent approach, enabling its application to systems in which the size of the primitive cellchanges across structural phase transitions. Applying this framework to La3Ni2O7, we establish its pressure-temperature phase diagram and find that the slope of the phase boundary between the high-symmetry and low-symmetry phases is negative, with a magnitude of approximately −60K/GPa. The present results provide atheoretical foundation for discussing how changes in crystal symmetry influence the emergence of supercon-ductivity.Introduction. The discovery of superconductivity with atransition temperature approaching 80K in the bilayer nick-elate La3Ni2O7 under pressure [1–30] has had a tremendousimpact on the field of high-temperature superconductivity.While the pairing mechanism is not yet fully understood, it isalmost certain that the contribution of the Ni 3d3z2−r2 orbitalto the low-energy electronic states plays a crucial role. Be-cause the 3d3z2−r2 orbital extends toward the apical oxygensites, the Ni–O–Ni bond angle along the interlayer directionis expected to have a decisive influence on the emergence ofsuperconductivity. These considerations suggest that the inter-play between orbital character and crystallographic symmetryis one of the key issues to unravel the physics of superconduc-tivity in La3Ni2O7 [31–34].In order to elucidate the relation between crystal structureand superconductivity in La3Ni2O7, a detailed understandingof its pressure–temperature (p–T ) phase diagram is indispens-able. Experimental studies have revealed that at ambient pres-sure the system crystallizes in the Amam = Cmcm (No. 63)structure below room temperature [35–38]. Upon applyingexternal pressure, structural transitions occur toward higher-symmetry phases such as Fmmm (No. 69) or I4/mmm (No.139). Importantly, superconductivity has been observed onlywhen the crystal structure possesses a symmetry higher thanthat of the Amam phase.This observation that higher symmetry favors superconduc-tivity is further supported by recent thin-film experiments [39–45]. In epitaxial films, the in-plane lattice constants are re-duced due to substrate constraints, which effectively increasethe Ni-O-Ni bond angle toward 180◦. Under these conditions,superconductivity has been reported even at ambient pressure.These findings indicate that controlling the bond geometrythrough either pressure or epitaxial strain can decisively in-fluence the stability of the superconducting phase.∗ asai-ryoma238@g.ecc.u-tokyo.ac.jp† Tadano.Terumasa@nims.go.jpAlthough these experimental results strongly indicate thecritical role of structural symmetry, its p–T phase diagram ofLa3Ni2O7 is far from being fully established. In particular, thedetails of the phase diagram, such as which structural phasesare realized at given p and T and where the phase boundarieslie, remain unsettled [46, 47]. Thus, a first-principles deter-mination of the phase diagram is essential to establish phasestability and its relation to superconductivity in La3Ni2O7.To address this problem, one must determine the p–T phasediagram through structural optimization at finite temperatures.Such an optimization requires minimizing the free energy,rather than the total energy, incorporating the contributions ofphonons. Recently, Masuki et al. developed a scheme that en-ables such finite-temperature structural optimization [48, 49].Their method has been successfully applied to the three-stepstructural phase transitions of BaTiO3 as well as to the tran-sitions of polar metals LiBO3 (B=Ta, W, Re, Os), yieldingresults in good agreement with experiments.However, their approach still faced challenges when ap-plied to transitions accompanied by a modification of theunit cell, and convergence was often slow near phase bound-aries. In this work, we overcome these difficulties by extend-ing and accelerating the optimization scheme, and apply itto La3Ni2O7. As a result, we establish its p–T phase dia-gram, where the phase boundary between the high-symmetryI4/mmm phase and the low-symmetry Amam phase isfound to form an almost straight line connecting (p,T ) ∼(0GPa, 630K) and (10GPa, 0K). These results provide asolid theoretical basis for understanding the emergence of su-perconductivity in La3Ni2O7, where lattice symmetry plays adecisive role.Method. We determine the pressure–temperature phase dia-gram of La3Ni2O7 based on the self-consistent phonon (SCP)theory [48–57]. The general formulation of SCP-based struc-tural optimization has been described in Ref. [48]; here wesummarize the essential steps relevant to our study.The SCP theory is an efficient mean-field approach fortreating lattice anharmonicity, i.e., phonon–phonon interac-tions, and has been widely used to calculate finite-temperaturearXiv:2512.08251v1 [cond-mat.supr-con] 9 Dec 2025mailto:asai-ryoma238@g.ecc.u-tokyo.ac.jpmailto:Tadano.Terumasa@nims.go.jphttps://arxiv.org/abs/2512.08251v12phonons. In this framework, the lattice vibration under the an-harmonic interaction Û =∑∞n=0 Ûn, where Ûn denotes then-th order anharmonic term, is approximated by an effectiveharmonic (one-body) phonon Hamiltonian. As a trial Hamil-tonian, we employ Ĥ0 =∑kλ ℏΩkλ(â†kλâkλ + 12), whereΩkλ are variational phonon frequencies. The correspondingvariational free energy is a functional of the trial Hamiltonian,FSCP[Ĥ0] = F0+〈T̂+Û−Ĥ0〉0, and satisfies the variationalinequality F ≤ FSCP[Ĥ0]. Here, T̂ is the kinetic-energy oper-ator, F0 = −β−1 lnZ0, ⟨X⟩0 = Z−10 Tr(Xe−βĤ0)denotesthe thermal average, and Z0 = Tr(e−βĤ0)is the correspond-ing partition function.The optimal effective Hamiltonian is obtained by minimiz-ing this functional, Ĥopt0 = argminĤ0FSCP[Ĥ0], which weperform efficiently in reciprocal space by iteratively updatingĤ0 using interatomic force constants (IFCs) computed for asupercell [55]. Once Ĥopt0 is determined at each temperature,the corresponding value FSCP[Ĥopt0 ] provides an accurate ap-proximation of the exact vibrational free energy F , which isthen used for structural and thermodynamic optimization.To update atomic structures at finite temperatures withinSCP theory, we use the analytical expression for the gradientof FSCP[Ĥopt0 ] with respect to the atomic displacements qλ, asderived by Masuki et al. [48]. After each structure update, theIFCs must be updated to evaluate FSCP[Ĥopt0 ] and its gradientfor the new configuration. These IFC updates are efficientlyperformed using the “IFC renormalization” technique, whichhas been demonstrated to achieve reliable accuracy for variousmaterials, including BaTiO3 [48], ZnO, and GaN [57].The original SCP-based structure optimization algorithmsuccessfully reproduced phase transitions driven by zone-center (q = 0) soft phonons (Fig. 1(a)). However, it couldnot capture transitions induced by soft phonons at nonzerowavevectors (q ̸= 0) that involve changes in unit-cell size(Fig. 1(b)). This limitation arose from the original implemen-tation, which always performed the reciprocal-space updatesof Ĥ0 and FSCP[Ĥ0] using a primitive lattice. In this study,we have overcome this limitation by enabling the code to usea supercell throughout an SCP calculation. By choosing a su-percell so that it becomes commensurate with the relevant softmode, the q ̸= 0 soft mode in the primitive Brillouin zone isfolded to the Γ-point of the supercell, allowing us to reuse theexisting formalism for computing the gradient of FSCP[Ĥopt0 ].Although this extension appears straightforward, the in-crease in the cell size introduces numerical challenges, partic-ularly slower convergence of the structure parameters whenusing the previously adopted Newton’s method. To addressthis, we have improved the optimization algorithm by incor-porating the elements of the GDIIS (Geometry optimizationusing Direct Inversion in the Iterative Subspace) scheme [58–60]. This method couples the rapid RMM-DIIS (ResidualMinimization Method with Direct Inversion in the IterativeSubspace) electronic minimization [61] with a BFGS (Broy-den–Fletcher–Goldfarb–Shanno) step [62], achieving robustconvergence on smooth potential-energy surfaces [63].FIG. 1. Schematic figures representing two types of structuralphase transitions: (a) BaTiO3, whose primitive cell size remainsunchanged before and after the structural phase transition, and (b)CaTiO3, whose primitive cell size changes. The crystal structureswere visualized using VESTA[64].Electronic-structure calculations were performed using theVienna Ab initio Simulation Package (VASP) [65]. We em-ployed the PBE exchange–correlation functional [66] andPAW pseudopotentials [67, 68]. The SCP loop convergencecriterion was set to 10−8 eV, and the accurate precision modewas used to suppress numerical errors and ensure reliableforces. For structural optimization of the 24-atom cell, a12×12×3 Monkhorst–Pack k mesh and a 600 eV plane-wavecutoff were used. For the 96-atom supercell, a 7×7×3 k meshwith the same cutoff was employed. Although GGA+U calcu-lations [33, 34] are typically employed in nickelate systems toachieve consistency with the Fermi level, because GGA+Usystematically underestimates lattice constants and predictsunrealistic transition temperatures, the main results presentedhere are based on GGA (U = 0).The second-order IFCs were obtained from phonon calcu-lations using a 2 × 2 × 1 supercell containing 96 atoms. Toobtain the third- and fourth-order IFCs, 80 random configura-tions were generated by ab initio molecular dynamics (AIMD)at 50K with fixed lattice constants, combined with additionalrandom displacements up to 0.04 Bohr. The AIMD simula-tions were carried out with a 2 × 2 × 2 k mesh and a 400 eVcutoff. The cutoff radii of 10 Bohr and 7 Bohr were usedfor the third- and fourth-order IFCs, respectively. The IFCswere fitted using the LASSO method, yielding a fitting errorof 3.55%. The SCP calculations were performed using the2 × 2 × 1 supercell with a 1 × 1 × 1 k-point mesh. This cellsize is large enough to accommodate the atomic distortions as-sociated with the I4/mmm to Amam phase transition. TheSCP structural optimization at each temperature was iterateduntil the maximum atomic displacement became smaller than10−4 Bohr.To benchmark the newly implemented GDIIS-based algo-rithm, we performed structural optimizations at 0GPa usingtwo optimization schemes. We adopted the high-symmetryI4/mmm structure as the initial configuration, slightly per-turbed atoms in the direction of the Amam phase, and startedthe structure optimization at 380K. Since the Amam phase30 25 50 75 100 125 150 175Iteration [steps]10−410−310−210−1100DisplacementNorm[Bohr]Newton MethodGDIIS-based MethodFIG. 2. Iterative convergence behavior of two optimization schemesat 0GPa and 380K, showing the norm of atomic displacements ver-sus iteration number.was more stable at this temperature, the magnitude of the dis-placement developed and eventually converged, as shown inFig. 2. Here, we evaluated the atomic displacement norm rel-ative to the final optimized structure, where the convergencecriterion was defined such that the maximum atomic displace-ment is less than 10−4 Bohr. It is clear that the GDIIS-basedmethod accelerates the convergence by one order of magni-tude, demonstrating its efficiency in structural optimizationsat finite temperatures. We therefore employed the GDIIS-based scheme for all subsequent structural optimizations.Results. Before performing finite-temperature optimiza-tions, we first investigated the structural phase transition atzero temperature (T=0). Although several experimental stud-ies have suggested that the high-symmetry phase belongs tothe Fmmm space group [1, 46, 47], theoretical works havereported that Fmmm does not appear and instead transitionsinto I4/mmm [34, 69]. Therefore, in this study, we focusedon the phase boundary between Amam and I4/mmm andexamined the critical pressure ps.Figures 3(a) and 3(b) display the phonon dispersions atp0 = 0GPa and 13GPa, respectively. Here, p0 representsa pressure of the static DFT calculation. Imaginary phononmodes are present around the M and A points at 0GPa, butthese modes become stable at 13GPa. Figure 3(c) shows thepotential energy surface (PES) along the normal coordinatesQ corresponding to the unstable M-point mode. At ambi-ent pressure, the PES exhibits a double-well structure, whichchanges to single-well under compression. Based on thesePES profiles, we estimated the transition pressure. Followingthe Landau theory [70], the PES is expressed as a function ofthe mode amplitude Q, and the potential energy is expandedup to the fourth order as F (Q) = a2Q2 + b4Q4. For eachpressure, we fitted the PES to this quartic function and plottedthe coefficients a2 in Fig.3 (d). Assuming, as in the Landautheory of phase transitions, that a2(p) = A(p − ps), we per-formed a linear fit of a2(p) and obtained ps from the intercepton the pressure axis. From this analysis, the critical pressureΓ X M ΓZ R A Z0200400600800Frequency[cm−1 ](a)0 GPaΓ X M ΓZ R A Z(b)13 GPa−4 −2 0 2 4Q [u1/2 · Å]012345U(Q;p)[eV](c)Pressure0 GPa2 GPa4 GPa6 GPa8 GPa10 GPa12 GPa0 5 10Pressure [GPa]−0.150−0.125−0.100−0.075−0.050−0.0250.0000.025a 2[eV/(u·Å2 )]ps(d)FIG. 3. Phonon dispersion of the conventional cell of La3Ni2O7under different pressure conditions: (a) 0GPa and (b) 13GPa. Thetwo lowest-energy modes at the M point are highlighted in orange.(c) Potential energy surface (PES) along the M-point modes at eachpressure. As the pressure increases, the double-well potential be-comes shallower and approaches a single-well shape. (d) Coeffi-cients a2 obtained by fitting the PES at each pressure with the quarticfunction F (Q) = a2Q2 + b4Q4. We performed linear fit of a2(p),and the intercept on the pressure axis (red triangle) was taken as theestimated critical pressure ps.was estimated to be ps = 12.6GPa.Then we performed finite-temperature structural optimiza-tion based on the SCP theory. To reduce the computationalcost associated with large supercells, the relaxation was re-stricted to the internal atomic coordinates while the latticeconstants were fixed at the DFT-optimized values. We car-ried out SCP-based structural optimization at p0 = 0GPaover a range of temperatures. Figure 4(a) shows the tem-perature dependence of the atomic displacements from thehigh-symmetry positions. At high temperatures, the systemremains close to the reference I4/mmm structure, while be-low a structural transition temperature, significant displace-ments appear. The pattern of atomic displacements—namely,in-plane shifts of La and Ni atoms and out-of-plane displace-ments of O atoms—corresponds to the octahedral rotationcharacteristic of the Amam phase. A symmetry analysisusing spglib [71] revealed a clear symmetry reduction fromI4/mmm to Amam at the onset temperature. Furthermore,as shown in Fig. 4(b), the total SCP free energy of the Amamstructure becomes lower than that of the I4/mmm phase at40.000.020.040.06Displacements[Å] La(x)Ni(y)O(z)380 400 420 440 460Temperature [K]−3−2Freeenergy[eV]AmamI4/mmm∆FSCPΓ X M Γ Z R A ZWave vector020406080Phononfrequency(cm−1 )410 K420 K−40−200∆FSCP[meV](a)(b)(c)FIG. 4. (a) Temperature dependence of atomic displacements ob-tained from finite-temperature structural optimization starting fromthe relaxed I4/mmm structure. The onset of finite displacementsindicates the symmetry change to Amam. (b) Temperature depen-dence of the SCP free energy. The free-energy difference ∆FSCP be-tween the Amam (blue) and I4/mmm (red) phases indicates thatthe former becomes thermodynamically stable when ∆FSCP < 0.(c) Phonon band dispersions at T = 410 and 420K, showing thesplitting near the M point associated with the transition.the same temperature, confirming the thermodynamic stabil-ity of the distorted phase. From the onset of atomic displace-ments, the emergence of symmetry breaking, and the cross-ing of free energies, we determine that the finite-temperaturephase transition occurs at 411K under p0 = 0GPa.Figure 4(c) shows the phonon band structures calculatedfrom the SCP frequencies at 0GPa. The spectra at 410K and420K, just below and above the transition temperature, areoverlaid for comparison. Focusing on the M-point modes, onecan clearly see that the degeneracy present at high temperatureis lifted in the low-temperature spectrum, indicating the sym-metry breaking associated with the structural phase transition.For each pressure condition, we repeated the structural op-timization based on the SCP theory, while keeping the lat-tice constants fixed to the DFT-optimized values at the corre-sponding pressure. For each pressure, we extracted the Ts atwhich the Amam-type distortion becomes stable. The result-ing temperature–pressure phase diagram is shown in Fig. 50 2 4 6 8 10 12Pressure [GPa]0100200300400500600Temperature [K]AmamI4/mmmps from PESSCP TsShifted Ts by pvib(Ts)ps from SCPCorrected Ts at 0 GPaFIG. 5. Phase diagram of La3Ni2O7 obtained from self-consistentphonon (SCP) calculations. Blue points denote the transition temper-atures T s calculated under fixed pressures from 0 to 8GPa. The bluedashed line represents a linear fit obtained using bootstrap resam-pling, and the shaded region indicates the corresponding confidenceinterval. The intercept of this fit with the T = 0K axis providesthe transition pressure ps estimated from SCP calculations. The redtriangle indicates the transition pressure ps determined from poten-tial energy surface. Green points show the Ts values corrected bythe vibrational pressure Pvib(Ts), which accounts for the effect offinite-temperature lattice vibrations. The green dashed line corre-sponds to a bootstrap-based linear fit to the pvib-corrected data, andthe shaded band gives its confidence range. The intercept of this linewith the p = 0GPa axis yields the corrected transition temperatureTs at 0GPa..(blue points). These Ts values obtained from the SCP cal-culations at various p0 can be well approximated by a linearfunction, from which the phase boundary between the Amamand I4/mmm phases can be estimated. More specifically,the phase boundary is found to form an almost straight lineconnecting (p0,T ) ∼ (0GPa, 400K) and (9GPa, 0K). Theestimated critical pressure of ∼ 9GPa at 0K is smaller thanps = 12.6GPa obtained from the PES at 0K (red triangle).This difference can be partially attributed to zero-point vibra-tion, which tends to stabilize the high-symmetry I4/mmmphase.Next, we correct the phase boundary by considering the ef-fect of phonon excitations on pressure. The vibrational con-tribution to pressure, which is not included in the static hydro-static pressure p0, was evaluated as follows.pvib(T ) = −∂Fvib∂Ω=1Ω∑qjℏωqj(nqj +12)γqj . (1)Here, Ω is the unit-cell volume, ωqj is the SCP frequency attemperature T , and γqj is the corresponding Grüneisen pa-rameter. By shifting the pressure by pvib(Ts), we obtained thecorrected phase boundary (green points). When this correc-tion is applied, the phase boundary shifts toward higher pres-sures, and the resulting Ts(p) curve can still be well approx-5imated by a linear function. The extrapolated y-intercept is∼630 K, which is close to the experimental result 720K[72],indicating that the present approach yields quantitatively rea-sonable results.Conclusion. We have predicted the p–T phase diagramof the superconducting compound La3Ni2O7 by performingfinite-temperature structural optimization based on the SCPtheory. Our calculations reveal that the phase boundary be-tween the low- and high-symmetry phases does not run ver-tically in the pressure axis, but instead exhibits a significanttemperature dependence even at ambient pressure. This indi-cates that a structural phase transition can occur purely as afunction of temperature, without applying external pressure.More broadly, we expect that the ability to perform finite-temperature structural optimization will enable reliable pre-dictions of p–T phase diagrams in a wide variety of materials.The method holds promise as a general theoretical frameworkto guide the discovery of new functional materials exhibitingstructural or electronic phase transitions under external condi-tions.Acknowledgments. This work is supported by Grant-in-Aid for Scientific Research from JSPS, KAKENHI Grant No.25H01246, No. 25H01252, No. 24H00190, JST K-ProgramJPMJKP25Z7, JST-PRESTO JPMJPR23J6, RIKEN TRIP ini-tiative (RIKEN Quantum, Advanced General Intelligence forScience Program, Many-body Electron Systems).[1] H. Sun, M. Huo, X. Hu, J. Li, Z. Liu, Y. Han, L. Tang, Z. Mao,P. Yang, and B. Wang, Signatures of superconductivity near 80K in a nickelate under high pressure, Nature 621, 493 (2023).[2] Y. Zhang, D. Su, Y. Huang, Z. Shan, H. Sun, M. Huo, K. Ye,J. Zhang, Z. Yang, and Y. Xu, High-temperature supercon-ductivity with zero resistance and strange-metal behaviour inLa3Ni2O7−δ , Nat. Phys. 20, 1269 (2024).[3] J. Hou, P.-T. Yang, Z.-Y. Liu, J.-Y. Li, P.-F. Shan, L. Ma,G. Wang, N.-N. Wang, H.-Z. Guo, J.-P. Sun, Y. Uwatoko,M. Wang, G.-M. Zhang, B.-S. Wang, and J.-G. Cheng, Emer-gence of high-temperature superconducting phase in pressur-ized La3Ni2O7 crystals, Chin. Phys. Lett. 40, 117302 (2023).[4] G. Wang, N. N. Wang, X. L. Shen, J. Hou, L. Ma, L. F. Shi,Z. A. Ren, Y. D. Gu, H. M. Ma, and P. T. Yang, Pressure-induced superconductivity in polycrystalline La3Ni2O7−δ ,Phys. Rev. X 14, 011040 (2024).[5] Z. Luo, X. Hu, M. Wang, W. Wú, and D.-X. Yao, Bilayer two-orbital model of La3Ni2O7 under pressure, Phys. Rev. Lett.131, 126001 (2023).[6] Y. Gu, C. Le, Z. Yang, X. Wu, and J. Hu, Effective modeland pairing tendency in the bilayer ni-based superconductorLa3Ni2O7, Phys. Rev. B 111, 174506 (2025).[7] F. Lechermann, J. Gondolf, S. Bötzel, and I. M. Eremin, Elec-tronic correlations and superconducting instability in La3Ni2O7under high pressure, Phys. Rev. B 108, L201121 (2023).[8] Y. Shen, M. Qin, and G.-M. Zhang, Effective bi-layer modelhamiltonian and density-matrix renormalization group study forthe high-tc superconductivity in La3Ni2O7 under high pressure,Chin. Phys. Lett. 40, 127401 (2023).[9] V. Christiansson, F. Petocchi, and P. Werner, Correlated elec-tronic structure of La3Ni2O7 under pressure, Phys. Rev. Lett.131, 206501 (2023).[10] D. A. Shilenko and I. V. Leonov, Correlated electronic structure,orbital-selective behavior, and magnetic correlations in double-layer La3Ni2O7 under pressure, Phys. Rev. B 108, 125105(2023).[11] Z. Liu, M. Huo, J. Li, Q. Li, Y. Liu, Y. Dai, X. Zhou, J. Hao,Y. Lu, and M. Wang, Electronic correlations and partial gapin the bilayer nickelate La3Ni2O7, Nat. Commun. 15, 7570(2024).[12] W. Wú, Z. Luo, D.-X. Yao, and M. Wang, Superexchangeand charge transfer in the nickelate superconductor La3Ni2O7under pressure, Sci. China Phys. Mech. Astron. 67, 117402(2024).[13] Y. Cao and Y.-f. Yang, Flat bands promoted by hund’s rulecoupling in the candidate double-layer high-temperature super-conductor La3Ni2O7 under high pressure, Phys. Rev. B 109,L081105 (2024).[14] X. Chen, P. Jiang, J. Li, Z. Zhong, and Y. Lu, Charge and spininstabilities in superconducting La3Ni2O7, Phys. Rev. B 111,014515 (2025).[15] C. Lu, Z. Pan, F. Yang, and C. Wu, Interlayer-coupling-drivenhigh-temperature superconductivity in La3Ni2O7 under pres-sure, Phys. Rev. Lett. 132, 146002 (2024).[16] H. Oh and Y.-H. Zhang, Type-ii t − j model and sharedsuperexchange coupling from hund’s rule in superconductingLa3Ni2O7, Phys. Rev. B 108, 174511 (2023).[17] Z. Liao, L. Chen, G. Duan, Y. Wang, C. Liu, R. Yu, and Q. Si,Electron correlations and superconductivity in La3Ni2O7 underpressure tuning, Phys. Rev. B 108, 214522 (2023).[18] X.-Z. Qu, D.-W. Qu, J. Chen, C. Wu, F. Yang, W. Li, and G. Su,Bilayer t−J−J⊥ model and magnetically mediated pairingin the pressurized nickelate La3Ni2O7, Phys. Rev. Lett. 132,036502 (2024).[19] Z. Fan, J.-F. Zhang, B. Zhan, D. Lv, X.-Y. Jiang, B. Normand,and T. Xiang, Superconductivity in nickelate and cuprate su-perconductors with strong bilayer coupling, Phys. Rev. B 110,024514 (2024).[20] K. Jiang, Z. Wang, and F.-C. Zhang, High-temperature super-conductivity in La3Ni2O7, Chin. Phys. Lett. 41, 017402 (2024).[21] M. Nakata, D. Ogura, H. Usui, and K. Kuroki, Finite-energyspin fluctuations as a pairing glue in systems with coexistingelectron and hole bands, Phys. Rev. B 95, 214509 (2017).[22] H. Sakakibara, N. Kitamine, M. Ochi, and K. Kuroki, Possi-ble high Tc superconductivity in La3Ni2O7 under high pres-sure through manifestation of a nearly half-filled bilayer hub-bard model, Phys. Rev. Lett. 132, 106002 (2024).[23] N. Wang, G. Wang, X. Shen, J. Hou, J. Luo, X. Ma, H. Yang,L. Shi, J. Dou, and J. Feng, Bulk high-temperature supercon-ductivity in pressurized tetragonal La2PrNi2O7, Nature 634,579 (2024).[24] M. Shi, D. Peng, Y. Li, S. Yang, Z. Xing, Y. Wang, K. Fan,H. Li, R. Wu, and B. Ge, Spin density wave rather thantetragonal structure is prerequisite for superconductivity inLa3Ni2O7−δ , Nat. Commun. 16, 9141 (2025).[25] J. Yang, H. Sun, X. Hu, Y. Xie, T. Miao, H. Luo, H. Chen,B. Liang, W. Zhu, and G. Qu, Orbital-dependent electron corre-lation in double-layer nickelate La3Ni2O7, Nat. Commun. 15,4373 (2024).[26] S. Ryee, N. Witt, and T. O. Wehling, Quenched pair break-https://doi.org/10.1038/s41586-023-06408-7https://doi.org/10.1038/s41567-024-02515-yhttps://doi.org/10.1088/0256-307X/40/11/117302https://doi.org/10.1103/PhysRevX.14.011040https://doi.org/10.1103/PhysRevLett.131.126001https://doi.org/10.1103/PhysRevLett.131.126001https://doi.org/10.1103/PhysRevB.111.174506https://doi.org/10.1103/PhysRevB.108.L201121https://doi.org/10.1088/0256-307X/40/12/127401https://doi.org/10.1103/PhysRevLett.131.206501https://doi.org/10.1103/PhysRevLett.131.206501https://doi.org/10.1103/PhysRevB.108.125105https://doi.org/10.1103/PhysRevB.108.125105https://doi.org/10.1038/s41467-024-52001-5https://doi.org/10.1038/s41467-024-52001-5https://doi.org/10.1007/s11433-023-2300-4https://doi.org/10.1007/s11433-023-2300-4https://doi.org/10.1103/PhysRevB.109.L081105https://doi.org/10.1103/PhysRevB.109.L081105https://doi.org/10.1103/PhysRevB.111.014515https://doi.org/10.1103/PhysRevB.111.014515https://doi.org/10.1103/PhysRevLett.132.146002https://doi.org/10.1103/PhysRevB.108.174511https://doi.org/10.1103/PhysRevB.108.214522https://doi.org/10.1103/PhysRevLett.132.036502https://doi.org/10.1103/PhysRevLett.132.036502https://doi.org/10.1103/PhysRevB.110.024514https://doi.org/10.1103/PhysRevB.110.024514https://doi.org/10.1088/0256-307X/41/1/017402https://doi.org/10.1103/PhysRevB.95.214509https://doi.org/10.1103/PhysRevLett.132.106002https://doi.org/10.1038/s41586-024-07996-8https://doi.org/10.1038/s41586-024-07996-8https://doi.org/10.1038/s41467-025-63701-xhttps://doi.org/10.1038/s41467-024-48701-7https://doi.org/10.1038/s41467-024-48701-76ing by interlayer correlations as a key to superconductivity inLa3Ni2O7, Phys. Rev. Lett. 133, 096002 (2024).[27] M. Yashima, N. Seto, Y. Oshita, M. Kakoi, H. Sakurai,Y. Takano, and H. Mukuda, Microscopic Evidence for Spin–Spinless Stripe Order with Reduced Ni Moments within abPlane for Bilayer Nickelate La3Ni2O7 Probed by 139La-NQR,J. Phys. Soc. Jpn. 94, 054704 (2025).[28] C. Xia, H. Liu, S. Zhou, and H. Chen, Sensitive dependence ofpairing symmetry on Ni-eg crystal field splitting in the nickelatesuperconductor La3Ni2O7, Nat. Commun. 16, 1054 (2025).[29] Y. Nomura, M. Kitatani, S. Sakai, and R. Arita, Strong-couplinghigh-Tc superconductivity in doped correlated band insulators,Phys. Rev. B 112, L020504 (2025).[30] F. Li, Z. Xing, D. Peng, J. Dou, N. Guo, L. Ma, Y. Zhang,L. Wang, J. Luo, J. Yang, J. Zhang, T. Chang, Y.-S. Chen,W. Cai, J. Cheng, Y. Wang, Y. Liu, T. Luo, N. Hirao, T. Mat-suoka, H. Kadobayashi, Z. Zeng, Q. Zheng, R. Zhou, Q. Zeng,X. Tao, and J. Zhang, Bulk superconductivity up to 96 K inpressurized nickelate single crystals, Nature 10.1038/s41586-025-09954-4 (2025).[31] Q.-G. Yang, D. Wang, and Q.-H. Wang, Possible s±-wavesuperconductivity in La3Ni2O7, Phys. Rev. B 108, L140505(2023).[32] Y.-B. Liu, J.-W. Mei, F. Ye, W.-Q. Chen, and F. Yang, s±-wavepairing and the destructive role of apical-oxygen deficiencies inLa3Ni2O7 under pressure, Phys. Rev. Lett. 131, 236002 (2023).[33] Y. Zhang, L.-F. Lin, A. Moreo, T. A. Maier, and E. Dagotto,Structural phase transition, s±-wave pairing, and magneticstripe order in bilayered superconductor La3Ni2O7 under pres-sure, Nat. Commun. 15, 2470 (2024).[34] B. Geisler, J. J. Hamlin, G. R. Stewart, R. G. Hennig, andP. J. Hirschfeld, Structural transitions, octahedral rotations, andelectronic properties of A3Ni2O7 rare-earth nickelates underhigh pressure, npj Quantum Mater. 9, 38 (2024).[35] Z. Zhang, M. Greenblatt, and J. B. Goodenough, Synthesis,structure, and properties of the layered perovskite La3Ni2O7−δ ,J. Solid State Chem. 108, 402 (1994).[36] C. Ling, D. Argyriou, G. Wu, and J. Neumeier, Neutron diffrac-tion study of La3Ni2O7: Structural relationships among n =1, 2, and 3 phases Lan+1NinO3n+1, J. Solid State Chem. 152,517 (2000).[37] V. I. Voronin, I. F. Berger, V. A. Cherepanov, L. Y. Gavrilova,A. N. Petrov, A. I. Ancharov, B. P. Tolochko, and S. G.Nikitenko, Neutron diffraction, synchrotron radiation and exafsspectroscopy study of crystal structure peculiarities of the lan-thanum nickelates Lan+1NinOy (n = 1, 2, 3), Nucl. Instrum.Methods Phys. Res., Sect. A 470, 202 (2001).[38] S. Taniguchi, T. Nishikawa, Y. Yasui, Y. Kobayashi, J. Takeda,S.-i. Shamoto, and M. Sato, Transport, magnetic and thermalproperties of La3Ni2O7−δ , J. Phys. Soc. Jpn. 64, 1644 (1995).[39] G. Zhou, W. Lv, H. Wang, Z. Nie, Y. Chen, Y. Li, H. Huang,W.-Q. Chen, Y.-J. Sun, and Q.-K. Xue, Ambient-pressure super-conductivity onset above 40 k in (La, Pr)3Ni2O7 films, Nature640, 641 (2025).[40] E. K. Ko, Y. Yu, Y. Liu, L. Bhatt, J. Li, V. Thampy, C.-T. Kuo,B. Y. Wang, Y. Lee, and K. Lee, Signatures of ambient pres-sure superconductivity in thin film La3Ni2O7, Nature 638, 935(2025).[41] M. Lorenz, M. S. Ramachandra Rao, T. Venkatesan, E. Fortu-nato, P. Barquinha, R. Branquinho, D. Salgueiro, R. Martins,E. Carlos, and A. Liu, The 2016 oxide electronic materials andoxide interfaces roadmap, J. Phys. D: Appl. Phys. 49, 433001(2016).[42] S. Middey, J. Chakhalian, P. Mahadevan, J. W. Freeland, A. J.Millis, and D. D. Sarma, Physics of ultrathin films and het-erostructures of rare-earth nickelates, Annu. Rev. Mater. Res.46, 305 (2016).[43] F. Belviso, V. E. P. Claerbout, A. Comas-Vives, N. S. Dalal,F.-R. Fan, A. Filippetti, V. Fiorentini, L. Foppa, C. Franchini,and B. Geisler, Viewpoint: Atomic-scale design protocols to-ward energy, electronic, catalysis, and sensing applications, In-org. Chem. 58, 14939 (2019).[44] B. Geisler, P. Yordanov, M. E. Gruner, B. Keimer, andR. Pentcheva, Tuning the thermoelectric properties of transitionmetal oxide thin films and superlattices on the quantum scale,Phys. Status Solidi B 259, 2100270 (2021).[45] P. Li, G. Zhou, W. Lv, Y. Li, C. Yue, H. Huang, L. Xu,J. Shen, Y. Miao, W. Song, Z. Nie, Y. Chen, H. Wang, W. Chen,Y. Huang, Z.-H. Chen, T. Qian, J. Lin, J. He, Y.-J. Sun,Z. Chen, and Q.-K. Xue, Angle-resolved photoemission spec-troscopy of superconducting (La, Pr)3Ni2O7/SrLaAlO4 het-erostructures, Natl. Sci. Rev. 12, nwaf205 (2025).[46] L. Wang, Y. Li, S.-Y. Xie, F. Liu, H. Sun, C. Huang, Y. Gao,T. Nakagawa, B. Fu, and B. Dong, Structure responsible for thesuperconducting state in La3Ni2O7 at high-pressure and low-temperature conditions, J. Am. Chem. Soc. 146, 7506 (2024).[47] H. Wang, H. Zhou, and W. Xie, Temperature-dependentstructural evolution of ruddlesden–popper bilayer nickelateLa3Ni2O7, Inorg. Chem. 64, 828 (2025).[48] R. Masuki, T. Nomoto, R. Arita, and T. Tadano, Ab initio struc-tural optimization at finite temperatures based on anharmonicphonon theory: Application to the structural phase transitionsof BaTiO3, Phys. Rev. B 106, 224104 (2022).[49] R. Masuki, T. Nomoto, R. Arita, and T. Tadano, Continuouscrossover between insulating ferroelectrics and polar metals:Ab initio calculation of structural phase transitions of LiBO3(B=Ta, W, Re, Os), Phys. Rev. B 110, 094102 (2024).[50] D. J. Hooton, The use of a model in anharmonic lattice dynam-ics, Philos. Mag. 3, 49 (1958).[51] N. S. Gillis, N. R. Werthamer, and T. R. Koehler, Propertiesof crystalline argon and neon in the self-consistent phonon ap-proximation, Phys. Rev. 165, 951 (1968).[52] T. Tadano and W. A. Saidi, First-principles phonon quasiparti-cle theory applied to a strongly anharmonic halide perovskite,Phys. Rev. Lett. 129, 185901 (2022).[53] P. Souvatzis, O. Eriksson, M. I. Katsnelson, and S. P. Rudin,The self-consistent ab initio lattice dynamical method, Comput.Mater. Sci. 44, 888 (2009).[54] R. Masuki, T. Nomoto, R. Arita, and T. Tadano, Anharmonicgrüneisen theory based on self-consistent phonon theory: Im-pact of phonon-phonon interactions neglected in the quasihar-monic theory, Phys. Rev. B 105, 064112 (2022).[55] T. Tadano and S. Tsuneyuki, Self-consistent phonon calcula-tions of lattice dynamical properties in cubic SrTiO3 with first-principles anharmonic force constants, Phys. Rev. B 92, 054301(2015).[56] L. Monacelli, R. Bianco, M. Cherubini, M. Calandra, I. Errea,and F. Mauri, The stochastic self-consistent harmonic approxi-mation: calculating vibrational properties of materials with fullquantum and anharmonic effects, J. Phys.: Condens. Matter 33,363001 (2021).[57] R. Masuki, T. Nomoto, R. Arita, and T. Tadano, Full optimiza-tion of quasiharmonic free energy with an anharmonic latticemodel: Application to thermal expansion and pyroelectricity ofwurtzite GaN and ZnO, Phys. Rev. B 107, 134119 (2023).[58] Ö. Farkas and H. B. Schlegel, Methods for optimizing largemolecules. part iii. an improved algorithm for geometry op-timization using direct inversion in the iterative subspacehttps://doi.org/10.1103/PhysRevLett.133.096002https://doi.org/10.7566/JPSJ.94.054704https://doi.org/10.1038/s41467-025-56206-0https://doi.org/10.1103/dygc-94fqhttps://doi.org/10.1038/s41586-025-09954-4https://doi.org/10.1038/s41586-025-09954-4https://doi.org/10.1103/PhysRevB.108.L140505https://doi.org/10.1103/PhysRevB.108.L140505https://doi.org/10.1103/PhysRevLett.131.236002https://doi.org/10.1038/s41467-024-46622-zhttps://doi.org/10.1038/s41535-024-00648-0https://doi.org/10.1006/jssc.1994.1059https://doi.org/10.1006/jssc.2000.8721https://doi.org/10.1006/jssc.2000.8721https://doi.org/10.1016/S0168-9002(01)01036-1https://doi.org/10.1016/S0168-9002(01)01036-1https://doi.org/10.1143/JPSJ.64.1644https://doi.org/10.1038/s41586-025-08755-zhttps://doi.org/10.1038/s41586-025-08755-zhttps://doi.org/10.1038/s41586-024-08525-3https://doi.org/10.1038/s41586-024-08525-3https://doi.org/10.1088/0022-3727/49/43/433001https://doi.org/10.1088/0022-3727/49/43/433001https://doi.org/10.1146/annurev-matsci-070115-032057https://doi.org/10.1146/annurev-matsci-070115-032057https://doi.org/10.1021/acs.inorgchem.9b01785https://doi.org/10.1021/acs.inorgchem.9b01785https://doi.org/10.1002/pssb.202100270https://doi.org/10.1093/nsr/nwaf205https://doi.org/10.1021/jacs.3c13094https://doi.org/10.1021/acs.inorgchem.4c03042https://doi.org/10.1103/PhysRevB.106.224104https://doi.org/10.1103/PhysRevB.110.094102https://doi.org/10.1080/14786435808243224https://doi.org/10.1103/PhysRev.165.951https://doi.org/10.1103/PhysRevLett.129.185901https://doi.org/10.1016/j.commatsci.2008.06.016https://doi.org/10.1016/j.commatsci.2008.06.016https://doi.org/10.1103/PhysRevB.105.064112https://doi.org/10.1103/PhysRevB.92.054301https://doi.org/10.1103/PhysRevB.92.054301https://doi.org/10.1088/1361-648X/ac066bhttps://doi.org/10.1088/1361-648X/ac066bhttps://doi.org/10.1103/PhysRevB.107.1341197(GDIIS), Phys. Chem. Chem. Phys. 4, 11 (2002).[59] Ö. Farkas and H. B. Schlegel, Methods for geometry optimiza-tion of large molecules. i. an O(N2) algorithm for solving sys-tems of linear equations for the transformation of coordinatesand forces, J. Chem. Phys. 109, 7100 (1998).[60] Ö. Farkas and H. B. Schlegel, Methods for optimizing largemolecules. ii. quadratic search, J. Chem. Phys. 111, 10806(1999).[61] P. Pulay, Improved scf convergence acceleration, J. Comput.Chem. 3, 556 (1982).[62] C. G. Broyden, The convergence of a class of double-rank min-imization algorithms 1. general considerations, IMA J. Appl.Math. 6, 76 (1970).[63] T. Chida and R. Masuki, Algorithm improvement for finite-temperature structural optimization using anharmonic phonontheory, Unpublished report (2024).[64] K. Momma and F. Izumi, VESTA3 for three-dimensional visu-alization of crystal, volumetric and morphology data, J. Appl.Crystallogr. 44, 1272 (2011).[65] G. Kresse and J. Furthmüller, Efficient iterative schemes forab initio total-energy calculations using a plane-wave basis set,Phys. Rev. B 54, 11169 (1996).[66] J. P. Perdew, K. Burke, and M. Ernzerhof, Generalized gradientapproximation made simple, Phys. Rev. Lett. 77, 3865 (1996).[67] P. E. Blöchl, Projector augmented-wave method, Phys. Rev. B50, 17953 (1994).[68] G. Kresse and D. Joubert, From ultrasoft pseudopotentials tothe projector augmented-wave method, Phys. Rev. B 59, 1758(1999).[69] Z. Ouyang, M. Gao, and Z.-Y. Lu, Absence of electron-phononcoupling superconductivity in the bilayer phase of La3Ni2O7under pressure, npj Quantum Mater. 9, 80 (2024).[70] L. D. Landau, On the theory of phase transitions, Zh. Eksp.Teor. Fiz. 7, 19 (1937).[71] A. Togo, K. Shinohara, and I. Tanaka, Spglib: a software li-brary for crystal symmetry search, Sci. Technol. Adv. Mater.:Methods 4, 2384822 (2024).[72] J. Song, D. Ning, B. Boukamp, J.-M. Bassat, andH. J. M. Bouwmeester, Structure, electrical conductivityand oxygen transport properties of ruddlesden–popper phasesLnn+1NinO3n+1 (Ln = La, Pr and Nd; n = 1, 2 and 3), J. Mater.Chem. A 8, 22206 (2020).https://doi.org/10.1039/B108658Hhttps://doi.org/10.1063/1.477393https://doi.org/10.1063/1.480484https://doi.org/10.1063/1.480484https://doi.org/10.1002/jcc.540030413https://doi.org/10.1002/jcc.540030413https://doi.org/10.1093/imamat/6.1.76https://doi.org/10.1093/imamat/6.1.76https://www.merit.t.u-tokyo.ac.jp/merit/en/training/pdf/report/jihatsu_chida_masuki_eng.pdfhttps://doi.org/10.1107/S0021889811038970https://doi.org/10.1107/S0021889811038970https://doi.org/10.1103/PhysRevB.54.11169https://doi.org/10.1103/PhysRevLett.77.3865https://doi.org/10.1103/PhysRevB.50.17953https://doi.org/10.1103/PhysRevB.50.17953https://doi.org/10.1103/PhysRevB.59.1758https://doi.org/10.1103/PhysRevB.59.1758https://doi.org/10.1038/s41535-024-00689-5https://doi.org/10.1080/27660400.2024.2384822https://doi.org/10.1080/27660400.2024.2384822https://doi.org/10.1039/D0TA06731Hhttps://doi.org/10.1039/D0TA06731H Finite-Temperature ab initio Structural Optimization of the Bilayer Nickelate Superconductor La3Ni2O7 Abstract References