Quantifying Confidence in Density Functional Theory Predicted Magnetic Ground States
QQuantifying Confidence in Density Functional Theory Predicted Magnetic GroundStates
Gregory Houchins
Department of Physics, Carnegie Mellon University, Pittsburgh, Pennsylvania 15213, USA
Venkatasubramanian Viswanathan ∗ Department of Physics, Carnegie Mellon University, Pittsburgh, Pennsylvania 15213, USA andDepartment of Mechanical Engineering, Carnegie Mellon University, Pittsburgh, Pennsylvania 15213, USA (Dated: June 2, 2017)Density functional theory (DFT) simulations, at the generalized gradient approximation (GGA)level, are being routinely used for material discovery based on high-throughput descriptor-basedsearches. The success of descriptor-based material design relies on eliminating bad candidates andkeeping good candidates for further investigation. While DFT has been widely successfully forthe former, often times good candidates are lost due to the uncertainty associated with the DFT-predicted material properties. Uncertainty associated with DFT predictions has gained prominenceand has led to the development of exchange correlation functionals that have built-in error estimationcapability. In this work, we demonstrate the use of built-in error estimation capabilities within theBEEF-vdW exchange correlation functional for quantifying the uncertainty associated with themagnetic ground state of solids. We demonstrate this approach by calculating the uncertaintyestimate for the energy difference between the different magnetic states of solids and comparethem against a range of GGA exchange correlation functionals as is done in many first principlescalculations of materials. We show that this estimate reasonably bounds the range of values obtainedwith the different GGA functionals. The estimate is determined as a post-processing step and thusprovides a computationally robust and systematic approach to estimating uncertainty associatedwith predictions of magnetic ground states. We define a confidence value (c-value) that incorporatesall calculated magnetic states in order to quantify the concurrence of the prediction at the GGAlevel and argue that predictions of magnetic ground states from GGA level DFT is incompletewithout an accompanying c-value. We demonstrate the utility of this method using a case studyof Li and Na-ion cathode materials and the c-value metric correctly identifies that GGA levelDFT will have low predictability for NaFePO F. Further, there needs to be a systematic test of acollection of plausible magnetic states, especially in identifying anti-ferromagnetic (AFM) groundstates. We believe that our approach of estimating uncertainty can be readily incorporated into allhigh-throughput computational material discovery efforts and this will lead to a dramatic increasein the likelihood of finding good candidate materials.
PACS numbers: 1.15.Mb, 75.10.-b, 7 75.25.-j
I. INTRODUCTION
There is an explosion in the computation of materialproperties based on off-the-shelf density functional the-ory software and this has led to a rapid acceleration inmaterial discovery in a variety of areas such as energy and biology. Increasingly, the results of these calcula-tions are driving material design choices.
There arenumerous success stories of computationally-driven ma-terial discovery largely based on high throughput com-putation of one or more descriptors, in thermoelectrics ,electrocatalysis , battery materials , hydrogen storage ,topological insulators , and magnetic materials. The challenge of using high-throughput materials dis-covery is that typically large searches lead to only a fewcandidates. One example is a high-throughput search ofover 5400 oxide/oxynitride compounds for solar light cap-ture leading to only 15 new candidates. It is now widelyacknowledged that high-throughput discovery based ondensity functional theory calculations, largely at the gen-eralized gradient approximation (GGA) level, is excel- lent at eliminating bad candidates but not as good inspotting and keeping the good candidates. An emergingfrontier is to incorporate uncertainty in order to improvethe predictability and aid in high-throughput discoveryas understanding materials with high uncertainty withina model may lead to the discovory of new phenomena. One of the most useful yet potentially troubling aspectof DFT is the easy accessibility of meta-stables states. In the context of magnetic materials, this means a wholecollection of magnetic states can be attained for thesame material and structure. This makes predicting thetrue magnetic ground state very challenging.
Con-ventionally, only a single energy difference between twostates could be attained when using any one exchange-correlation functional. One naive way to increase thelikelihood of an accurate prediction is to use multiple ex-change correlation functionals. However, the selectionof these functionals is not systematic and unlikely toguarantee realistic uncertainty bounds associated withthe ground state prediction and in the case where GGAlevel DFT is insufficient to capture the correct properties a r X i v : . [ c ond - m a t . m t r l - s c i ] J un the few functionals tested may give inconsistent results.The Bayesian error estimation functional (BEEF) car-ries with it a prediction of uncertainty calibrated to recre-ate the error as mapped to experimental training data bygenerating a collection of GGA level functionals. Thisempirical error estimation has been used to quantify un-certainty in heterogenous catalysis, electrocatalysis and mechanical properties for solid electrolytes. In thiswork, we extend the framework of uncertainty estima-tion and demonstrate a robust approach for quantify-ing uncertainty of the ground states of magnetic mate-rials. Magnetic materials are important in a wide vari-ety of applications such as information storage , colossalmagnetoresistance , spin currents , and medicine. Inour approach, we calculated an ensemble of energy dif-ference between different magnetic states (for e.g., ferro-magnetic and anti-ferromagnetic) of the same material.From that ensemble, we can define a confidence value(c-value) that quantifies the agreement between GGAlevel functionals and ultimately the certainty that onespin state is more energetically favorable than the other.Our approach is computationally efficient, as comparedto doing many different calculations based on differentfunctionals, simulating non-self consistently thousandsof functionals using one self consistent calculation. Wedemonstrate the utility of the developed method and thec-value with a case study of Li-ion and Na-ion cathodematerials. We propose that such a method needs to beused to ensure a sufficient level of agreement for the mag-netic ground state, and thus, the derived properties suchas voltage, electronic conductivity of the magnetic mate-rial.
II. METHODSA. Ground state prediction
The magnetic ground state is defined as the spin con-figuration that minimizes the energy.E GS = min[E( S , S , · · · , S i )]Naively, all other parameters other than spin configu-ration should be fixed and only the spin is changed. Inthe context of DFT, the exact cell parameters and atomslocations that give the lowest energy may vary for differ-ent initial spin states. It is therefore best give a create acollection of initial spin states that are to be comparedand for each spin state, allow the lattice parameters andatomic positions to vary.Another issue in the prediction of magnetic states isthe ambiguity of ”antiferromagnetic” (AFM) for materi-als with certain geometries or with multiple distances ofmagnetic interactions that could lead to different mag-netic couplings. Layered honeycomb materials provide agreat example of a geometry that leads to ambiguity asthey can have a zigzag antiferromagnetic structure (Type I) such as that seen in FePSe or where all three nearest-neighbors are antiferromagnetically coupled (Type II) asseen in MnPSe . Perovskites on the other hand demon-strate the ambiguity of various length scales. Atoms canbe inter-plane AFM and intra-plane FM (A-Type) asin LaMnO at low temperature , intra-plane AFM andinter-plane FM (C-Type)as in CaCrO and SrCrO ,or both intra-plane and inter-plane AFM (G-Type) asin LaTiO . Studies of magnetic states compare typ-ically two (or a few) states: the obvious ferromagnetic(FM) where all spin point in the same direction and oneof the possibly many antiferromagnetic orientations. Forexample, a first-principles study by Baetting et. al. of perovskite multiferics calculated the energy differencefor ferromagnetic and G-type like ferrimagnet (FiM), butdid not study the A-type or C-type couplings. It is worthpointing out that the difference in energy between thesetwo states does not necessarily predict the global groundstate but rather which of the two is more favorable. Inmaterials with a variety of magnetic range interactions,it is conceivable that other AFM states could exist. Thiscan be seen again in the context of perovskites that canhave intra-plane AFM, inter-plane AFM, or both. Itshould therefore be understood that in some cases, sev-eral spin configurations must be tested for an accurateprediction. B. Bayesian Error Estimation
Recently, Baysian Ensemble Error Functional with vander Waals correlations (BEEF-vdW) has provided away to systematically estimate the uncertainty of a DFTcalculation. This empirically fit functional is a gener-ated from a generalized gradiant approximation (GGA)exchange energy and Perdew-Burke-Ernzerhof (PBE),Perdew-Wang68 local density approximation (LDA), andvdWDF2 non-local (nl) correlation contributions. Theexchange enhancement factor, is fit using an expansionin terms of Legendre Polynomials B m , given by F GGAx ( s ) = (cid:88) m a m B m [ t ( s )] , (1) t ( s ) = 2 s s − , − ≤ t ≤ . (2)Therefore, the exchange-correlation energy is given by E xc = (cid:88) m (cid:90) (cid:15) UEGx ( n ) B m [ t ( s )] d r + α c E LDA − c + (1 − α c ) E P BE − c + E nl − c (3)The parameters of the functional a m and α c are opti-mized with respect to a collection of experimental data.To generate an ensemble of functionals a distribution ofthese parameters is generated. Therefore, once a self con-sistent DFT calculation has been performed using thebest fit parameters, the converged electron density n ( r )can be used along with the spread of α c and a m to gen-erate an ensemble of energies non-self-consistently usingEquation (3). The spread of these values is tuned tocreate a spread in energies that recreates the error of thebest fit DFT calculation with respect to the experimentaltraining data. The data sets include molecular formationenergies and reaction energies, molecular reaction barri-ers, noncovalent interactions, solid state properties suchas cohesive energies and lattice constants, and chemisorp-tion on solid surfaces. In this way, the error estimationhas been trained to predict how uncertain the predictionis with respect to known uncertainties in experiment. III. BENCHMARKING
It is worth highlighing that the BEEF-vdW functionalwas trained on data sets that did not explicitly includeany magnetic properties in the training sets. We canbenchmark the accuracy of DFT calculations using theBEEF-vdW functional in capturing the magnetic prop-erties of materials by comparing the calculated magne-tization to experiment measurements and other GGAlevel functionals. The calculation of atomic magnetic mo-ments is carried out in the trivial way by integrating thedifference of spin up and spin down electron densities con-verged from a spin polarized calculation. In this way,the predicted magnetic moment is a z-projection of thespin-only magnetic moment, neglecting orbital magneti-zation. The approximation of spin only magnetizationcan be made since the the orbital moment in the caseof transition metals is quenched from the delocalizationand band formation of electrons in the bulk. The cal-culation of the z-projection of spin only can be justifiedthrough the ability to derive a Stoner model through thisformalism, as well as this method is in agreementwith covalent description of magnetism. We thereforeset the initial guess for magnetic moments for each atombased on a spin only estimation that depends only on thenumber of unpaired electrons, n , M = gµ B S z = µ B n, where the gyromagnetic ratio, g is two and the z-component of spin for each unpaired electron is .We find the BEEF-vdW functional predicts the mag-netic moments with similar accuracy to other GGA func-tionals. Table I shows the various GGA functional pre-dictions of magnetic moments for various materials usingthe experimental lattice parameters. The overestimationin magnetic moment for bulk Cr is well known in DFTdue to the fact that the experimental ground state isan incommensurate spin density wave , while the er-ror in CuCr O compared to experiments is due to thenoncolinear structure of the ground state. The DFTcalculation converged to the collinear version of the ex-perimentally seen magnetic configuration, properly cap-turing the total magnitude of the spins rather than thez-projection.
TABLE I. The magnetic moment per magnetic ion in µ B aspredicted by BEEF-vdW functional in the first row comparedto experimental measurements in the second. In the case ofCuCr O the total magnetic moment is given.Fe Cr Ni FePO -qFePO -o LaMnO CuCr O BEEF 2.33 1.62 0.61 4.294.03 3.89 5.00PBE 2.13 1.23 0.60 4.314.00 3.85 5.00RPBE 2.21 1.77 0.61 4.334.02 3.89 5.00PBEsol 2.01 0.74 0.58 4.303.97 3.77 5.00Expt. 2.22 a b a c d e aa Ref. 41 b Ref. 39 c Ref. 42 d Ref. 43 e Ref. 44
IV. PREDICTION OF MAGNETIC ORDERING
In the case of new materials without a known spinstructure, DFT can be used to predict the most ener-getically favorable configuration. We demonstrate thissearch for the correct magnetic state with previouslycharacterized materials to demonstrate our method.Starting from both ferromagnetic and antiferromagneticspin states, we optimize the lattice constants by start-ing from an experimentally derived unit cell and min-imizing the energy with respect to volume by scalingall lattice constants uniformly. At lease five differentvolumes were tested with the volumes varying with astrain parameter with respect to the experimental cellof x = V /V = 0 . , . , . , . , .
10 in most cases and x = 0 . , . , . , . , .
05 in others where conver-gence was an issue. Each cell volume and its corre-sponding energy, converged to < . to find the strain parameter correspond-ing the minimum energy. Once the cell paramters areidentified, the internal coordinates of the atoms are al-lowed to relax to a maximum force of less than 0.01ev/˚A. All calculations are performed using the BEEF-vdW functional in GPAW, a real space grid implemen-tation of the projector augmented-wave method , witha 8 × × h = 0 .
18 ˚A. After the cell and atomic coordinates areoptimized, the converged electron density for this config-uration is used to generate an ensemble of 2000 energyvalues non-self consistently. Doing this for two magneticstates and subtracting the ensemble of energies element-wise provides an ensemble of energy difference for errorestimation. The results of our calculations, where wehave compared FM and the lowest lying AFM state, areshown in Table II. For comparison, the same optimizationprocedure was repeated for three other functionals at theGGA level: Perdew-Burke–Ernzerhof (PBE) , RevisedPerdew-Burke-Ernzerhof (RPBE) and PBEsol .The materials used were chosen to represent a rangeof crystal structures, complexity, and elements, as wellas different mechanisms of magnetism. Lattice dis-tortions play a large role in the case of LaMnO , and CuCr2O4 . Direct exchange accounts for the fer-romagnetic nature of Fe and Ni, while indirect ex-change acounts for the AFM tendencies of the oxidesFePO , LaMnO , and CuCr O .The optimization procedure involves at least ten self-consistent calculations just for cell optimizations. To testthe possibility of predicting the magnetic ground statewithout optimizing the lattice constants to save computa-tion time, we performed only an internal geometry relax-ation on the experimental cell parameters. The results ofthis are in Table III. The two methods of optimizing andusing the experimental lattice constant give very similarresults as seen in Table IVIn most cases, the lowest lying states were simple ferro-magnetic and anti-ferromagnetic configurations. In thecase of CuCr O , however, the antiferromagnetic cou-pling between copper and chromium prevented any DFTcalculation from converging to completely ferromagneticstate. The two antiferromagnetically aligned states areseen in Figure 2. V. PREDICTION CONFIDENCE
In order to measure the confidence of the predictionof the magnetic ground state, we define a c-value as thepercentage of the ensembles that support the hypothesisof the best fit functional. For example, in the case of apredicted ferromagnetic state this would be c = 1 N ens N ens (cid:88) i (cid:89) j Θ( E AF M j ,i − E F M,i ) (4)where N ens is the number of functionals used, the sum isover functionals, the product is over all magnetic statesother than the predicted state, and Θ( x ) is the Heav-iside step function. The method for calculating the c-value requires the entire dataset of energy differences forall of the possible magnetic states and therefore cannotbe easily recalculated or utilized for future independentstudies. We therefore can approximate the c-value bymodeling the spread of energy differences between thepredicted ground state and other possible magnetic stateas a normal distribution with ensemble mean µ and cal-culated standard deviation σ . We can then integrate the -1 -0.5 0 0.5 1 E=E FM -E AFM (eV) A r b i t r a r y U n i t s FIG. 1. A histogram of the n=2000 ensemble of energydifferences with 50 bins. The area under the histogram tothe right of 0 represents the c-value and the area of the grayshaded region represents the appoximate c-value. The nor-mal distribution guess created from the calculated mean andstandard deviation is also shown in good agreement with rawdata. normalized distribution as an approximation of the Heav-iside function. That is for a material that we have againpredicted to be ferromagnetic, the approximate c-valueis c ≈ (cid:89) j (cid:90) ∞ d x (cid:113) πσ j e − ( x − µj )22 σ j (5)Again, the product is over all magnetic states other thanthe predicted state.The two methods of calculating prediction c-value giveresults consistent to within .01 and therefore can be usedrelatively interchangeably. The approximate c-value isexpected to get worse as the number of tested magneticstates increases. Although the method of counting theexact number of functionals in agreement is the mostaccurate, it requires access to the raw data. The approx-imation to a normal distribution, however, requires onlythe mean and standard deviation for each energy differ-ence between all magnetic states tested and has negligibledeviation from the exact calculated confidence. A depic-tion of this c-value as well as the approximate c-value inthe case of only two magnetic states tested can be seenin Figure 1.The c-value of a particular magnetic ground stateof a material may be used to understand when GGA-level DFT is giving a reliable prediction versus when ahigher-order theory is needed. In the case of Fe, Ni,FePO , and LaMnO the c-values are larger than 0.9indicating a nearly unanimous prediction of ferromag-netism in these materials. The high confidence is likely TABLE II. Calculated magnetic energy difference for optimized structures E FM − E AFM , unless otherwise noted, of variouscrystal structures is given so that the prediction of BEEF-vdW can be compared to that of other GGA-level functionals.The BEEF-vdW energy difference is accompanied by the ensemble standard deviation. The mean of the ensemble of energydifference generated by BEEF-vdW, c-value incorperating all magnetic states, as well as the approximated c-value also shownfor each material. All energies are in eVMaterial BEEF-vdW PBE RPBE PBEsol exp EnsembleMean c-value Approx.c-valueFeBCCIm¯3m − . ± .
20 -0.34 -0.48 -0.41 FM a -0.50 0.997 0.994CrBCCIm¯3m 0 . ± .
18 0.03 0.06 0.01 AFM c − . ± .
03 -0.06 -0.06 -0.06 FM c -0.07 0.998 0.997FePO α -quartzP3
21 0 . ± .
01 0.02 -0.01 0.00 AFM d olivinePnma 0 . ± .
011 0.03 0.02 0.05 AFM e perovskitePbnm − . ± .
02 -0.25 -0.11 -0.05 FM f -0.07 0.994 0.998CuCr O spinelI4 / amd − . ± .
06 -0.07 -0.11 -0.05 AFM1 g -0.10 0.965 0.969 a Ref.
Ref. This is a difference between nonmagentic (NM) and AFM c Ref.
Ref.
Ref.
Ref. The AFM state is A-type g Ref. This is a difference between two FiM states. See Figure 2 demonstrating the success of DFT in describing simpledirect exchange ferromagnetism and indirect exchangeantiferromagnetism as discussed earlier. However, the c-value is not always close to 1 as seen in the case of Cr.It is likely this number can be understood due to thefact that magnetic ground state of Cr is actually an in-commensurate spin density wave as pointed out earlierand a proper understanding of Cr or any material with aspin density wave would require an extension of densityfunctional theory as suggested by Capelle et. al .The utility of this c-value may not only lie in con-firming when GGA has given you a prediction with highconfidence, but it may also identify materials and mate-rial classes that have long been studied at the GGA level,but cannot be reliably understood due to the disagree-ment illuminated by the confidence value. By moving to more accurate, more computationally intensive methods,a truer understanding of these materials may lead to thediscovery of new emergent phenomena. VI. CASE STUDY
We demonstrate the application of our method to thecase of two possible cathode materials. Density func-tional theory provides a simple way of calculating the the-oretical voltage of an intercalation cathode with respectto a metal anode using the Nernst equation, V = − ∆GF ,where ∆G is the Gibbs free energy per mole of the lithi-ation reaction and F is Faraday’s constant. The voltageis written in this way so that a calculation of ∆G perstoichiometric formula unit in eV directly relates to volt- TABLE III. Calculated magnetic energy differences E FM − E AFM in meV of various crystal structures using non-optimized experimental lattice parameters.Material BEEF-vdW PBE RPBE PBEsol expFeBCCIm¯3m − . ± . . ± . − . ± . α -quartzP3
21 27 . ± . olivinePnma 32 . ± . perovskitePbnm − . ± . O spinelI4 / amd − . ± . aa This is a difference between two antiferromagnetic states. SeeFigure 2
TABLE IV. Comparison of the prediction confidence givenby optimized and experiment lattice constants. The thirdcolumn shows the reduction in energy attained from the DFTlattice optimization procedure rounded to the nearest 10 meV.The energy is given in eV.Optimized Experimental Energy differenceFe 0.997 1.000 -0.00Cr 0.630 0.650 -0.01Ni 0.998 0.998 -0.00FePO -q 0.952 1.000 -0.01FePO -o 0.999 1.000 -0.01LaMnO O age in V. It has been shown previously that volume andentropic effects on the free energy change of lithium inter-calation are on the scale of 10 − eV and 10 − eV respec-tively compared to the change in internal energy scaleon the order of eV. Therefore, it is common to use thechange in internal energy at zero Kelvin to estimate thevoltage.Most Li-ion and Na-ion cathode materials typically ex-
AFM1AFM2 -0.3 -0.2 -0.1 0 0.1
E=E
AFM1 -E AFM2 (eV) A r b i t r a r y U n i t s FIG. 2. The two converged magnetic states and the distri-bution of energy differences. For this structure all cases withinitially ferromagnetic spin structure converge to one of thesetwo antiferromagnetic cases. hibit magnetism in some form due to the presence ofmagnetic transition metal ions. Previous works to iden-tify the magnetic ground state of new Li-ion and Na-ioncathode materials have shown inconsistent result for afew GGA-level functionals used.
We explore a fluori-nated iron phosphate cathode for both Li-ion and Na-ionand demonstrate the utility of our developed method toexplore the magentic states of the material. The relevantreactions during operation areLiFePO F + Li + + e − (cid:10) Li FePO FNaFePO F + Na + + e − (cid:10) Na FePO Fwhere the forward reaction represents discharge andthe backward reaction represents charge. Ramzan et.al have previously tested the FM and one of the possi-ble AFM states of both the Na and Li-ion cathode ma-terials above using the Perdew-Burke-Erzernhof (PBE)functional as well as the PBE-functional with a Hub-bard correction of U=4.95 eV and J=0.95 eV (PBE+U).It is worth highlighting that there are multiple differ-ent lengths between two nearest Fe atoms which couldlead to a collection of magnetic couplings either director mediated by oxygen, phosphorus, and/or fluorine. Weexplicitly consider 3 interaction lengths that can be spinaligned or anti-aligned. This leads to 2 = 8 possiblestates. We therefore test the 7 possible AFM states aswell as the FM state to properly predict the magneticground state.Ramzan et. al found a disagreement between PBEand PBE+U for the magnetic arrangement of both TABLE V. We show the predicted magnetic ground state for optimized structures using BEEF-vdW along with PBE andPBE+U from Ramsan et. al . We also present the c-value which incorporates all possible magnetic configurations.Material BEEF-vdW PBE a PBE+U a c-value modifiedc-valueLiFePO F AFM b AFM AFM 0.925 0.993NaFePO F AFM b AFM AFM 0.567 0.991 a Ref.
AFM interaction at all 3 length scales Li FePO F and Na FePO F but agreement that theground state is AFM for LiFePO F and NaFePO F. Wewere able to recreate these predictions of AFM groundstates using the PBE as well as BEEF-vdW, but show avery low c-value of 0.567 for Na F ePO F. The full resultsof magnetic prediction and c-values can be seen in Ta-ble V. The example of NaFePO F clearly demonstratesthe utility of the c-value as a robust and computationallyefficient metric to identify consensus at the GGA level.To further illustrate the importance of comparing mul-tiple AFM states, Table V includes a c-value encompass-ing all 8 calculated magnetic states as well as a modi-fied c-value that only includes FM and the lowest energyAFM states, a comparison more like what is convention-ally seen. The modified c-value is much higher than themore precise c-value, showing that there is relative cer-tainty that the state is AFM but much less can be saidabout which specific AFM state is the ground state.In the context of calculating the theoretical voltagefor the intercalation cathode materials, the energy dif-ferences are typically small and thus, using the wrongmagnetic state may not greatly affect the prediction.However, in other properties relevant to understandingthe materials such as electronic conductivity, density ofstates, magnet moment for each Fe atom, and net mag-netization, the correct magnetic state is vital. Only sys-tematic searches, both in the possible magnetic statesand spanning a collection of GGA functional, can givea reasonable expectation of getting these properties cor-rect.
VII. CONCLUSION
We presented a computationally efficient way to quan-tify uncertainty in the prediction of magnetic groundstates of materials. We demonstrated the success of thismethod for various crystal structures, magnetic order-ing and material classes. The method was then appliedin the case of Li-ion and Na-ion cathode materials thatcontained magnetic transition metal ions. Our resultsshow that in order to fully predict and understand themagnetic ground state of a material with GGA-level den-sity functional theory, there must be a systematic sam-pling of both GGA functionals and of possible magneticstates. Our method provides a way to do both and there-fore quantify relative certainty that one particular mag-netic arrangement is more energetically favorable thanthe other. More importantly, it provides a simple frame-work with which to quote the certainty that there is aconsensus at the GGA level for the predicted magneticconfigurations.
ACKNOWLEDGMENTS
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