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[Takao Tsumuraya](https://orcid.org/0000-0001-9063-9278), [Tsuyoshi Miyazaki](https://orcid.org/0000-0003-3534-4404), Hitoshi Seo

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[Stability of Correlated Insulating States in Molecular Conductors from First-Principles Calculation](https://mdr.nims.go.jp/datasets/9f0c739a-04bf-4124-ab06-301045faa7a8)

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Stability of Correlated Insulating States in Molecular Conductors from First-Principles CalculationStability of Correlated Insulating States in Molecular Conductorsfrom First-Principles CalculationTakao Tsumuraya1,2+ , Tsuyoshi Miyazaki3 , and Hitoshi Seo4,5†1Magnesium Research Center, Kumamoto University, Kumamoto 860-8555, Japan2Department of Materials Science and Engineering, Kumamoto University, Kumamoto 860-8555, Japan3Research Center for Materials Nanoarchitectonics (WPI-MANA), National Institute for Materials Science,Tsukuba, Ibaraki 305-0044, Japan4Condensed Matter Theory Laboratory, RIKEN, Wako, Saitama 351-0198, Japan5RIKEN Center of Emergent Matter Science (CEMS), Wako, Saitama 351-0198, Japan(Received October 20, 2024; accepted October 29, 2024; published online November 20, 2024)Electronic properties of molecular conductors exhibiting antiferromagnetic (AFM) spin order and charge order (CO)owing to electron correlation are studied using first-principles density functional theory calculations. We investigate twosystems, a quasi-two-dimensional Mott insulator β A-(BEDT-TTF)2ICl2 with an AFM ground state, and several membersof quasi-one-dimensional (TMTTF)2X showing CO. The stabilities of the AFM and CO states are compared between theuse of a standard exchange–correlation functional based on the generalized gradient approximation and that of a range-separated hybrid functional; we find that the latter describes these states better. For β A-(BEDT-TTF)2ICl2, the AFM orderis much stabilized with a wider band gap. For (TMTTF)2X, only by using the hybrid functional, the AFM insulating stateis realized and the CO states coexisting with AFM order are stable under structural optimization, whose stability amongdifferent X shows the tendency consistent with experiments.Molecular conductors are studied as typical correlatedelectron systems, taking advantage of their simple electronicstructure near the Fermi level despite their complicatedcrystal structures.1) They present fundamental many-bodyproblems, such as the interplay between strong electron–electron interaction and the underlying lattice structurecontrolled by molecular arrangements, together with rela-tively strong electron–phonon interactions, which can lead tosymmetry breaking. A representative issue is the competitionbetween antiferromagnetic (AFM) ordering and quantum-disordered states such as the spin-gapped or spin-liquidstates.2) Another is the role of the lattice degree of freedomin stabilizing charge ordered (CO) states whose primarydriving force is considered to be the electron–electronrepulsion.3)To elucidate the mechanisms of their rich physicalproperties, effective low-energy models such as theHubbard-type models have extensively been applied.4,5)Their microscopic parameters can now be evaluated in aquantitative level using first-principles band calculationsbased on the density-functional theory (DFT), thanks to themodern computational developments making accessible tolarge number of atoms in the unit cell.6–11) On the other hand,standard first-principles calculations often run into essentialproblems when attempting to directly reproduce insulatingstates of molecular conductors due to symmetry breaking,such as the aforementioned AFM and CO states.This is related to the fact that standard treatments usinglocal density approximation12) or generalized gradientapproximation (GGA)13) have a problem of unphysical self-interaction error14) and generally underestimate the bandgaps. To overcome this difficulty, in studies of inorganicstrongly correlated systems such as transition metal oxides,the “þU” method has been heavily used,15) implementing thestrong correlation effect in the d=f atomic orbitals. However,this method cannot be simply adopted to molecularconductors whose electronic properties are governed bymolecular orbitals, on which “on-site” Coulomb repulsionacts.16) Here we address this issue by choosing an alternativeapproach, rather than implementing the strong Coulombrepulsion effect on the molecular orbitals, but to improve thetreatment of the exchange–correlation term in DFT.In this context, we have shown the efficiency of the use ofa range-separated hybrid functional by Heyd, Scuseria, andErnzerhof (HSE),17,18) for several molecular CO systems.19,20)In the hybrid functional method, the exchange–correlationenergy is evaluated by mixing the exact Fock exchange andexchange-energy functional used in LDA or GGA, whoserange-separated formalism was proposed by HSE for thedescription of solid states. In Refs. 19 and 20, the structuralstability of the CO state was demonstrated, in clear contrastwith the standard approaches in which CO structures areunstable and reduced to the non-CO state by the structuraloptimization process.In this paper, we investigate the correlation-inducedinsulating states further taking account of AFM ordering.AFM insulating states are ubiquitously seen in molecularconductors and are extensively studied within effective modelapproaches;16,21–26) however, due to the difficulties mentionedabove, they have not been properly studied by first-principlescalculations even for typical compounds. In the following,we first investigate the insulating state with AFM order in�0-(BEDT-TTF)2ICl2,27,28) which is a Mott insulator withan AFM transition temperature TN ¼ 22K.29,30) Then wediscuss a series of (TMTTF)2X, showing CO whose transitiontemperature (TCO) varies by changing the anion as X ¼ NbF6(TCO ¼ 165K), AsF6 (100K), and PF6 (70K).31–35) In(TMTTF)2NbF6, at the ground state, CO and AFM spinorder (TN ¼ 10K) coexist, while the other two compoundsundergo a spin-Peierls transition to spin-gapped statesaccompanying lattice distortions. The crystal structures ofthese two systems are shown in Figs. 1(a) and 1(b),respectively, and a schematic phase diagram for the(TMTTF)2X including the salts studied in this paper is drawnin Fig. 1(c). Here we investigate the systems withoutconsidering the lattice distortions due to the spin-PeierlsJournal of the Physical Society of Japan 93, 123704 (2024)https://doi.org/10.7566/JPSJ.93.123704Letters123704-1 ©2024 The Physical Society of Japanmaintain attribution to the author(s) and the title of the article, journal citation, and DOI.©2024 The Author(s)This article is published by the Physical Society of Japan under the terms of the Creative Commons Attribution 4.0 License. Any further distribution of this work mustJ. Phys. Soc. Jpn.Downloaded from journals.jps.jp by （研）物質・材料研究機構 on 11/19/24https://orcid.org/0000-0001-9063-9278https://orcid.org/0000-0003-3534-4404https://doi.org/10.7566/JPSJ.93.123704http://creativecommons.org/licenses/by/4.0/http://crossmark.crossref.org/dialog/?doi=10.7566%2FJPSJ.93.123704&domain=pdf&date_stamp=2024-11-20(SP) transition and investigate the AFM state as its non-distorted counterpart.We perform first-principles calculations based on plane-wave basis sets with the projector-augmented wave meth-od,37,38) within the Vienna ab initio simulation package.39,40)Spin-dependent exchange–correlation functionals are used,and we compare the standard GGA functional by Perdew,Burke, and Ernzerhof (PBE)13) and the hybrid functional byHSE (HSE06).18) The cutoff energies for plane waves wereset to 700 eV for both the PBE and HSE06 functionals.The k-point meshes for �0-(BEDT-TTF)2ICl2 were set to4 � 6 � 2 and 4 � 3 � 2, and for (TMTTF)2X, 4 � 4 � 2 and4 � 2 � 2, for the nonmagnetic (NM) and AFM states,respectively. The density of states (DOS) for the AFM stateswere calculated using the k-point meshes of 4 � 4 � 2.As for the experimental structures, we used that of �0-(BEDT-TTF)2ICl2 measured at 12K,41,42) and those of(TMTTF)2X at 30K.35) The positions of the H and F atomswere optimized by the first-principles calculations. The datafor (TMTTF)2X are within the CO phase, with the so-calledferroelectric CO pattern, losing the inversion symmetry thatthe system holds above TCO,32,43) but above the spin-Peierlstransition temperatures for X ¼ AsF6 and PF6. We alsoperformed structural optimization for the internal coordinates,fixing the lattice parameters and relaxing all the atomicpositions.Experimentally, �0-(BEDT-TTF)2ICl2 exhibits insulatingbehavior below room temperature.27) NM band calculationsshow that their conduction band is half-filled,44,45) thereforethe compound is considered as a Mott insulator. Since thenominal charge of BEDT-TTF is +1=2 and the dimerizedstructure (see Fig. 1) brings about the half-filling of theantibonding combination of the HOMO between BEDT-TTFforming dimers, it is a typical dimer-Mott insulator. Atheoretical analysis based on the Hubbard model describingthe HOMO level of BEDT-TTF with the �0-type arrangementshows the stability of the AFM order.23,46)Using the experimental structure, we find an AFM solutionwith a unit cell twice of the primitive cell along the b axiswhich contains four BEDT-TTF molecules. The spin-dependent self-consistent field calculations show an AFMordering between the dimers. Within the PBE functional, themagnetic moments are evaluated as 0.31�B=dimer, whileusing the HSE06 functional, they become 0.58�B=dimer, aslisted in Table I; these are calculated by the sum of magneticmoments on the C and S atoms. The spin density distributionis shown in Fig. 2(a) using the results for the HSE functional,whose spin pattern is common with those within PBE: thedimers with spin-up and spin-down electrons alternate alongthe b-axis. This pattern is consistent with the theoreticalresults using the Hubbard model.23,46) The dashed curves inFig. 2(b) show the band structure of the AFM state withinPBE, which exhibits a band gap of 0.06 eV. Notably, theband gap expands to 0.42 eV when we use the HSE06functional, as shown by the solid curve in Fig. 2(b). We alsoperform structure optimizations using the HSE functionalwith the AFM ordering, which show the band gap and themagnetic moments of 0.39 eV and 0.57�B=dimer, respec-tively, close to the results above.Experiments on TMTTF salts reveal insulating groundstates, despite the calculated NM band structure showing ametallic quarter-filled band at the Fermi level.4,47) Therefore,the correlation effect is considered to be crucial. The weaklydimerized crystal structure brings about an effectively half-filled band and the tendency toward the dimer-Mottinsulating state owing to the electron correlation, i.e., theon-site (intra-molecular) Coulomb repulsion in terms of thequarter-filled Hubbard model with dimerization.48) In addi-tion, the intersite (inter-molecular) Coulomb repulsion termsHClCISbac(a)(b)cbaHCNbSFNbF6 SbF6 AsF6 PF6 Br050100150200250300Temperature [K]AFM SPCODimer-MI(c)MetalAFMFig. 1. (Color online) Crystal structures of (a) � 0-(BEDT-TTF)2ICl2 and(b) (TMTTF)2X. (c) A schematic experimental phase diagram of the(TMTTF)2X compounds: The horizontal axis corresponds to the chemicalpressure with smaller anion to the right. The symbols represent the crossovertemperature to the dimer-Mott insulating (dimer-MI) phase,36) the transitiontemperatures to CO, AFM, and SP phases, respectively.Table I. Magnetic moments per molecular dimer (in �B) calculated byusing the GGA-PBE and HSE06 functionals, for the experimental structures.For the TMTTF salts, the values for the two monomers in the dimer are alsolisted.Functional GGA-PBE HSE06Dimer Monomer Dimer Monomer� 0-ET2ICl2 0.31 0.58(TMTTF)2NbF6 0.06 0.05 0.01 0.48 0.33 0.15(TMTTF)2AsF6 — — — 0.51 0.30 0.21(TMTTF)2PF6 — — — 0.49 0.25 0.24(b)Z Y C E A X 0.2 0.4 0.6 0Energy (eV)abcSHC ClI(a)Fig. 2. (Color online) (a) Spin density distribution of the antiferromag-netic state of � 0-(BEDT-TTF)2ICl2, calculated using the HSE06 functional.(b) Its band structures, where the dashed and solid curves are the results withthe PBE and HSE06 functionals, respectively. The dotted horizontal lines at0 eV show the top of the valence bands.J. Phys. Soc. Jpn. 93, 123704 (2024) Letters T. Tsumuraya et al.123704-2 ©2024 The Physical Society of Japan©2024 The Author(s)J. Phys. Soc. Jpn.Downloaded from journals.jps.jp by （研）物質・材料研究機構 on 11/19/24are considered to be responsible for the CO instability,21)while the role of electron–phonon interaction has beeninvestigated.26,49) We note that the electronic properties ofX ¼ NbF6 are similar to those for X ¼ SbF6,50) with TCO ¼157K and TN ¼ 6K, on which there are more experimentalstudies conducted; the calculations for X ¼ SbF6 were notperformed here due to the lack of structural data for the lowtemperature CO phase.Here we seek for AFM self-consistent solutions with a unitcell twice of the primitive cell which contains four TMTTFmolecules, suggested in the literature.10,43) For the exper-imental structure, using PBE, the AFM state was achieved forX ¼ NbF6, whereas for X ¼ AsF6 and PF6, only NM stateswere obtained within the numerical accuracy. The results forX ¼ NbF6 indicate the magnetic moments on the TMTTFmolecules in each dimer to be 0.01�B and 0.05�B (Table I).The band structure is depicted in Fig. 3(a), showing ametallic state, but with negligible charge disproportionationbetween the two molecules as seen from Fig. 3(c).When the hybrid functional is applied, we can now obtainAFM solutions in all the three salts, coexisting with CO,showing different degree of spin polarization in the twoTMTTF molecules within each dimer. Moreover, they allshow insulating states; the band structure for X ¼ NbF6 isdepicted in Fig. 3(a) showing a clear band gap of 0.30 eV.We obtain insulating states for X ¼ AsF6 and PF6 as wellwith band gaps of 0.33 and 0.34 eV, respectively, in contrastto the case of PBE where only metallic states are found for allthe compounds. The AFM pattern coexisting with CO isdrawn in Fig. 3(b), and the estimated magnetic moments arelisted in Table I, together with the PBE results. While thevalues per dimer are similar for the three salts (about 0.5�B),their difference between the two TMTTF molecules is largestin X ¼ NbF6 (0.18�B), next in X ¼ AsF6 (0.09�B), andalmost vanishes in X ¼ PF6 (0.01�B). This is compatible tothe degree of CO and with the transition temperatures TCO,whose structural stability will be investigated next.As mentioned in the introduction, when we use PBEfunctional, the CO structure is lost after the structuraloptimization process. Namely, the structural=electronicdifference between the two TMTTF molecules turnsnegligibly small; therefore in the following we show resultsfor the HSE06 functional. First, for X ¼ NbF6, the CO stateis properly kept. From the structural viewpoint, we can seethis from the difference in the lengths of the central C=Cbond in the two molecules, as listed in Table II. Althoughthe calculated difference, 0.007Å, is smaller than theexperimental value, 0.034Å, the results show a stable COstructure. This difference becomes smaller for X ¼ AsF6(0.002Å) and numerically negligible in X ¼ PF6. Consider-ing the tendency that the structural stability is underestimatedin other CO systems as well,19,20) the variation amongdifferent X is consistent with the experimental CO stability,e.g., reflected by the transition temperatures TCO (see Fig. 1).To see the degree of charge disproportion in the optimizedstructure, we plot the local density of states (LDOS) inFig. 3(c), which is obtained as a summation of the projectedDOS on the orbitals of C and S atoms. The red (blue) linescorrespond to LDOS for the TMTTF molecule with shorter(longer) central C=C bond length; the results for the threesalts are shown together with the results using PBE forX ¼ NbF6 within the experimental structure for comparison.In the occupied states of X ¼ NbF6 with the energy rangefrom −0.6 to 0 eV, there is a clear difference between theLDOS. This indicates the molecules with the shorter bond arecharge-rich. On the contrary, from −1.0 to −0.7 eV and from0.3 to 0.4 eV, the LDOS of the charge-poor moleculesis slightly larger. These results clearly show the chargedisproportionation between the molecules. Such a differencebecomes obscure for X ¼ AsF6 and almost absent in X ¼PF6; this reduction corresponds to the systematic variation inthe bond lengths presented in Table II.Here we compare our results with the experimental results.In �0-(BEDT-TTF)2ICl2, from the temperature dependence of 0  2  4  6LDOS (/eV)(a) (b)(c) AsF6NbF6 PF6−1.2−1.0−0.8−0.6−0.4−0.2 0 0.2 0.4 0.6Y Γ R U Γ XEnergy (eV)NbF6GGA-PBE HSE06 HSE06 HSE06Energy (eV) 0  4  8 12LDOS (/eV)-1.0-0.8-0.6-0.4-0.2 0.0 0.2 0.4 0  4  8 12LDOS (/eV) 0  4  8 12LDOS (/eV)PBE Exp. HSE06  Exp.HSE06  Opt. Fig. 3. (Color online) (a) Band structures for AFM + CO phase in(TMTTF)2NbF6, calculated by GGA-PBE and HSE06 functionals. Theorigin in the longitudinal axis (dotted lines) is taken as the Fermi energy andthe top of the valence bands, respectively. The green dotted curves depict theHSE06 band structure for the experimental structure. (b) Calculated spindensities with the HSE06 functional. (c) Local DOS for X ¼ NbF6, AsF6, andPF6, in which geometric optimization is performed with the HSE06functional. The LDOS calculated with GGA-PBE for NbF6 salt is basedon the experimental structure.Table II. Anion dependence of the distances of the central C=C bond in(TMTTF)2X, comparing the results after structural optimization using HSE06functional and experimental data at 30K.35)HSE06 Exp.d(C=C) [Å] d(C=C) [Å](TMTTF)2NbF6 1.371 1.3891.364 1.355(TMTTF)2AsF6 1.369 1.3961.367 1.352(TMTTF)2PF6 1.368 1.3921.368 1.371J. Phys. Soc. Jpn. 93, 123704 (2024) Letters T. Tsumuraya et al.123704-3 ©2024 The Physical Society of Japan©2024 The Author(s)J. Phys. Soc. Jpn.Downloaded from journals.jps.jp by （研）物質・材料研究機構 on 11/19/24the resistivity, the activation energy is evaluated as 0.113eV,28) consistent with an optical measurement.51) Our resultsshow a direct band gap of 0.06 eV using PBE functional, and0.4 eV using HSE06; although the comparison with the Mottgap experimentally opened above TN is difficult, we see thatPBE (HSE06) underestimates (overestimates) the gap. As forthe magnetic moments, NMR suggests the ordered momentof almost 1�B per dimer at low temperatures.52) Thecalculated results of 0.31 �B for PBE and 0.58 �B forHSE06 suggest that the latter provides closer estimates.Systematic evaluation of activation energy in TMTTF saltsby conductivity measurements gives 0.043, 0.031, and0.032 eV, for X ¼ SbF6, AsF6, and PF6, respectively.53) Here,instead of X ¼ NbF6 whose data were not available, we referto the value for X ¼ SbF6 that shows similar properties andtransition temperatures as mentioned above. In our calcu-lations insulating states could only be obtained by the HSE06functional, whose band gaps are 0.30, 0.33, and 0.34 eV forX ¼ NbF6, AsF6, and PF6, respectively. Again, although thecomparison with the gap seen in experiments, opened wellabove TN, may not be appropriate, the same tendency thatPBE (HSE06) underestimates (overestimates) the gap is seen.We should also note that in (TMTTF)2X, the quasi-one-dimensional electronic structure is expected to bring aboutlow-dimensional physics when electron correlation plays arole, which we need to be careful about when makingthe comparison between our first-principles evaluation andexperiments. The low dimensionality is known to affectthe magnetic ground state as well, evidenced from the SPinstability in X ¼ AsF6 and PF6 accompanying latticedistortion,26,49,54–57) not considered in our study. Althoughwe could find stable AFM solutions coexisting with CO in allthe three salts within HSE06, the comparison of the magneticstate is available only for X ¼ NbF6. Again, due to theabsence of data for X ¼ NbF6, we refer to an NMR estimatefor X ¼ SbF6:58,59) 0.70�B and 0.24�B for the charge rich andpoor TMTTF molecules, respectively. As shown in Table I,the calculated values are smaller: 0.33�B and 0.15�B.In summary, we investigate the stability of insulating statescaused by symmetry breakings in correlated molecularconductors by using the exchange–correlation functionalbased on the standard GGA (PBE) and a range-separatedhybrid functional (HSE06). The antiferromagnetic insulatingstate of �0-(BEDT-TTF)2ICl2 is reproduced with larger bandgap and magnetic moments with HSE06. In (TMTTF)2X, wefail to obtain the charge ordered state by PBE, but succeededby the use of HSE06; we show a stable charge orderedelectronic state coexisting with antiferromagnetic order, seenexperimentally in X ¼ NbF6. The structural relaxation in theTMTTF salts using HSE06, although the degree of chargeorder is depressed, shows the stability of charge ordered stateseen structurally and electronically with distinct TMTTFmolecules in the unit cell. We note that such structuralstability is important in a sense that now we have a solidground to calculate the dynamical motions of atoms, such asthe molecular vibrations which have been providing richinformation in many experimental studies.Acknowledgements JSPS Grant-in-Aid funded this research for ScientificResearch Nos. 16K17756, 20H04463, 20H05883, 23H01129, 23K03333, and23H04047. 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