Article pubs.acs.org/JPCB
Computational Electrochemistry. Voltages of Lithium-Ion Battery Cathodes Bo Wang, Sijie Luo, and Donald G. Truhlar* Department of Chemistry, Chemical Theory Center, and Supercomputing Institute, University of Minnesota, 207 Pleasant Street S.E., Minneapolis, Minnesota 55455-0431, United States S Supporting Information *
ABSTRACT: Theoretical studies on the electrode materials in lithium-ion batteries provide information on the structural changes during the charging and discharging processes. In the present study, we tested the M06-L and N12 exchange-correlation functionals on some wellstudied lithium-containing materials. These functionals, which have already shown good performance for a variety of databases, outperform the widely used PBE functional for reproducing the experimental structures and averaged intercalation potentials. It is especially noteworthy that the M06-L functional gives voltages as accurate as those provided by the Heyd−Scuseria−Ernzerhof (HSE06) hybrid functional, but with less computational cost.
1. INTRODUCTION Rechargeable lithium-ion batteries are important for energy storage due to their high output voltage and high-energy storage density. The voltage and energy storage density are determined by the selection of the electrode materials, and various kinds of electrode materials have been investigated.1−6 In order to understand lithium-ion battery properties at the molecular level and to design new materials for lithium-ion batteries, computational modeling provides a useful tool.7,8 Among the available computational methods, Kohn−Sham theory (KST)9 provides the best compromise between accuracy and the computational cost, and therefore it is widely used in lithium-ion battery studies. However, the accuracy of KST depends on the accuracy of the exchange-correlation (xc) functional. Many classes of xc functionals are available, including both local functionals, such as LDA (local density approximation), GGA (generalized gradient approximation), NGA (nonseparable gradient approximation), meta-GGA, and meta-NGA functionals, and nonlocal (“hybrid” and “doubly hybrid”) functionals. Even within a class of xc functional, the performance is not uniform.10 It has been found that the hybrid HSE functional11 is much better than GGA functionals for reproducing the averaged Li intercalation potentials due to the reduction of the electron self-interaction error by Hartree− Fock exchange in hybrid functionals.12 However, local xc functionals, which have no Hartree−Fock exchange, are much more economical for calculations on extended and periodic systems, and, therefore, it is interesting and potentially useful to test some other local functionals that have proved accurate for predicting various chemical properties but that have not yet been tested for cell voltages. In the present study, we test the M06-L13 and N1214 local functionals for their accuracy in © 2015 American Chemical Society
calculating the structures and voltages of lithium-ion battery materials. M06-L is a meta-GGA that includes the kinetic energy density in the functional form, which is another way (complementary to Hartree−Fock exchange) to decrease the self-interaction error. It has shown good performance for a variety of databases.10,15,16 In addition, M06-L shows the correct 2D-3D trends in Au clusters,17 and it provides improved surface formation energies and CO adsorption energies for transition metals.18 The N12 xc functional is an NGA, and it has been found to provide good accuracy for both the energetic and structural properties of both solids and molecules.10 This paper is concerned with validating these local functionals for Licontaining solid materials.
2. METHODS AND COMPUTATIONAL DETAILS The test suite to validate xc functionals is composed of LiCoO2/CoO2, LiTiS2/TiS2, LiMn2O4/Mn2O4, and Li2Ti2O4/ LiTi2O4, where in each case we have listed the discharged state and a delithiated state of a cathode material. Calculation of the voltages versus Li/Li+ also requires calculations on Li metal. The quantum mechanical calculations were carried out using a combination of software packages, in particular a locally modified version of the plane-wave-basis electronic structure program VASP19,20 and the Gaussian-basis-function electronic structure program Gaussian 09.21 Primitive cells were used in Special Issue: Bruce C. Garrett Festschrift Received: April 7, 2015 Revised: May 29, 2015 Published: June 5, 2015 1437
DOI: 10.1021/acs.jpcb.5b03356 J. Phys. Chem. B 2016, 120, 1437−1439
Article
The Journal of Physical Chemistry B
Table 1. Optimized Volumes per Formula Unit (Angstrom3) and Mean Unsigned Percentage Error (MUPE) Using Various Functionals PBE PBEa N12 PBE+Ua M06-L HSE06a expt.a a
GGA GGA NGA GGA+U meta-GGA hybrid-GGA
LiCoO2
CoO2
LiMn2O4
Mn2O4
LiTiS2
TiS2
Li2Ti2O4
LiTi2O4
MUPE
32.70 33.00 31.48 32.86 32.61 31.81 32.17
32.09 33.00 31.15 34.00 31.04 32.00 29.61
67.86 67.84 65.60 73.88 70.92 70.25 69.89
66.77 66.93 65.10 70.74 67.53 65.81 66.36
63.66 63.95 60.09 63.95 63.00 64.92 64.12
60.41 65.00 57.94 65.00 59.37 62.00 57.32
73.33 73.65 70.43 73.65 71.54 74.35 73.44
75.50 75.72 73.08 75.72 74.25 75.50 74.18
2.7 4.2 3.5 5.7 2.2 2.9
Reference 12.
tested in other solid-state calculations.17,18 Since M06-L can semiquantitatively describe the dispersion energy, it greatly improves the geometric results for the cases of CoO2 and TiS2, in which the interactions between layers are mostly of van der Waals type. The N12 functional is an NGA functional, and it performs better than PBE for the intercalation potentials. Although the HSE06 functional gives the best performance among all functionals, it is a hybrid functional and can be quite expensive in simulations (depending on the system and the program, the cost can be orders of magnitude higher than that of the local functionals). M06-L and N12 are local functionals, and therefore, they have the potential to be applied to more complex systems. Since N12 and M06-L have both been validated in past work to perform well for a variety of databases,10 we conclude that they are reasonable functionals to use if only local functionals are practical for the calculations. We note that the PBE geometry employed for TiS2 in this study differs from Chevrier et al.’s results.12 This can be mainly due to the flatness of the potential energy surface, as the intercalation potentials are quite similar despite the different geometries. Since PAW potentials were used in all calculations, and the VASP PAW potentials are optimized only for PBE functionals, we also performed all-electron calculations using Gaussian 09 on several materials and compared the results with those obtained by VASP. A detailed description of the method is given in the Supporting Information. In previous work24 it was shown that calculations using Gaussian 09 can agree well those with plane-wave basis set when both sets of calculations are well converged with respect to the basis. In our cases, Tables S1 and S2 (Supporting Information) indicate that these two programs agree with each other to some extent; however, we do find some differences. For example, Gaussian 09 gives consistently lower intercalation potentials than VASP for Li2Ti2O4 with all functionals. Many aspects of the calculations may contribute to this kind of difference; for example, we have not extrapolated the results to the basis set limit, and PAW potentials are used to replace the core electrons in VASP while all-electron calculations are performed in Gaussian 09. Although such comparisons of results obtained with different kinds of basis sets are less common in the solid-state literature than in the gasphase literature, such differences are not unexpected in the light of gas-phase experience. In the gas phase it is common to see differences between calculations with various practical basis sets and effective core potentials since the results are not at the complete-basis-set limit. In the rest of this paper we present only results obtained with VASP. In the case of LiMn2O4, we tested both the ferromagnetic (FM) and antiferromagnetic (AFM) ordering of the Mn spins. The N12 functional gives the FM ordering as the ground state,
all calculations. The VASP calculations used a 500 eV cutoff energy and 6 × 6 × 6 k points, except for Li for which we used 10 × 10 × 10 k points. VASP’s projector-augmented-wave (PAW) potentials22,23 were used in all calculations. All calculations were spin polarized, and the energy criterion for self-consistency was set to less than 0.0001 eV/unit cell. Both the coordinates of the atoms and the lattice constants were optimized, and the structural relaxation criterion was set to less than 0.001 eV/unit cell. The cell voltages vs Li/Li+ are approximated here as averaged Li intercalation potentials and were calculated as E(LixM) − E(Lix − 1M) − E(Li) (1) F where E is the total electronic energy (including nuclear repulsion), F is the Faraday constant, “M” denotes a cathode host material in which the Li+ can be reversibly inserted and extracted, and x1 and x2 are the numbers of Li atoms contained in the delithiated host and the lithiated host, respectively. Equation 1 neglects zero point energy differences, thermal phonon energies, and entropic effects; this is a reasonable procedure for the present purposes. V=−
3. RESULTS AND DISCUSSION Tables 1 and 2 show the optimized volumes and the averaged Li intercalation potentials calculated using various functionals. Table 2. Averaged Li Intercalation Potentials versus Li/Li+ (eV) and Mean Unsigned Error (MUE) Using Various Functionals LiCoO2 LiMn2O4 PBE PBEa N12 PBE+Ua M06-L HSE06a expt.a a
GGA GGA NGA GGA+U meta-GGA hybrid-GGA
3.34 3.38 3.52 3.85 3.46 4.51 4.1
3.37 3.37 3.59 4.04 3.86 4.25 4.1
LiTiS2 Li2Ti2O4 1.94 1.91 1.86 1.91 2.09 2.06 2.1
1.05 1.05 1.17 1.05 1.23 1.19 1.3
MUE 0.47 0.47 0.37 0.19 0.24 0.18
Reference 12.
The HSE06 and PBE+U results are from Chevrier et al.12 Among the functionals tested, M06-L best reproduces the experimental geometry, with a mean unsigned percentage error (MUPE) of 2.2%, and M06-L and N12 give more accurate intercalation potentials than PBE (though they are still less accurate than the much more expensive HSE06 hybrid functional). The relatively low mean unsigned percentage error of M06-L is especially welcome since it has previously been successfully 1438
DOI: 10.1021/acs.jpcb.5b03356 J. Phys. Chem. B 2016, 120, 1437−1439
Article
The Journal of Physical Chemistry B
(8) Islam, M. S.; Fisher, C. A. Lithium and Sodium Battery Cathode Materials: Computational Insights into Voltage, Diffusion and Nanostructural Properties. Chem. Soc. Rev. 2014, 43, 185−204. (9) Kohn, W.; Sham, L. J. Self-Consistent Equations Including Exchange and Correlation Effects. Phys. Rev. 1965, 140, A1133− A1138. (10) Peverati, R.; Truhlar, D. G. The Quest for a Universal Density Functional: The Accuracy of Density Functionals Across a Broad Spectrum of Databases in Chemistry and Physics. Philos. Trans. R. Soc. A 2014, 372, 20120476. (11) Heyd, J.; Scuseria, G. E.; Ernzerhof, M. Hybrid Functionals Based on a Screened Coulomb Potential. J. Chem. Phys. 2003, 118, 8207−8215; Erratrum 2006, 124, 219906. (12) Chevrier, V. L.; Ong, S. P.; Armiento, R.; Chan, M. K. Y.; Ceder, G. Hybrid Density Functional Calculations of Redox Potentials and Formation Energies of Transition Metal Compounds. Phys. Rev. B 2010, 82, 075122. (13) Zhao, Y.; Truhlar, D. G. A New Local Density Functional for Main Group Thermochemistry, Transition Metal Bonding, Thermochemical Kinetics, and Noncovalent Interactions. J. Chem. Phys. 2006, 125, 194101. (14) Peverati, R.; Truhlar, D. G. Exchange−Correlation Functional with Good Accuracy for Both Structural and Energetic Properties While Depending Only on the Density and its Gradient. J. Chem. Theory Comput. 2012, 8, 2310−2319. (15) Zhao, Y.; Truhlar, D. G. The M06 Suite of Density Functionals for Main Group Thermochemistry, Thermochemical Kinetics, Noncovalent Interactions, Excited States, and Transition Elements: New Functionals and Systematic Testing of Four M06-Class Functionals and 12 Other Functionals. Theor. Chem. Acc. 2008, 120, 215−241. (16) Yang, K.; Zheng, J. J.; Zhao, Y.; Truhlar, D. G. Tests of the RPBE, revPBE, τHCTHhyb, ωB97X-D, and MOHLYP Density Functional Approximations and 29 Others Against Representative Databases for Diverse Bond Energies and Barrier Heights in Catalysis. J. Chem. Phys. 2010, 132, 164117. (17) Ferrighi, L.; Hammer, B.; Madsen, G. K. H. 2D−3D Transition for Cationic and Anionic Gold Clusters: A Kinetic Energy Density Functional Study. J. Am. Chem. Soc. 2009, 131, 10605−10609. (18) Luo, S. J.; Zhao, Y.; Truhlar, D. G. Improved CO Adsorption Energies, Site Preferences, and Surface Formation Energies from a Local Density Functional, M06-L. J. Phys. Chem. Lett. 2012, 3, 2975− 2979. (19) Kresse, G. Furthmuller, Efficiency of Ab-Initio Total Energy Calculations for Metals and Semiconductors Using a Plane-Wave Basis Set. J. Comput. Mater. Sci. 1996, 6, 15−50. (20) Kresse, G. Furthmuller, Efficient Iterative Schemes for Ab Initio Total-Energy Calculations Using a Plane-Wave Basis Set. J. Phys. Rev. B 1996, 54, 11169−11180. (21) Frisch, M. J.; Trucks, G. W.; Schlegel, H. B.; Scuseria, G. E.; Robb, M. A.; Cheeseman, J. R.; Montgomery, J. A., Jr.; Vreven, T.; Kudin, K. N.; Burant, J. C.; et al. Gaussian 09, version C.01; Gaussian, Inc.: Wallingford CT, 2009. (22) Blochl, P. E. Projector Augmented-Wave Method. Phys. Rev. B 1994, 50, 17953−17978. (23) Kresse, G.; Joubert, D. From Ultrasoft Pseudopotentials to the Projector Augmented-Wave Method. Phys. Rev. B 1999, 59, 1758− 1775. (24) Paier, J.; Hirschl, R.; Marsman, M.; Kresse, G. The Perdew− Burke−Ernzerhof Exchange-Correlation Functional Applied to the G2−1 Test Set Using a Plane-Wave Basis Set. J. Chem. Phys. 2005, 122, 234102. (25) Rodriguez-Carvajal, J.; Rousse, G.; Masquelier, C.; Hervieu, M. Electronic Crystallization in a Lithium Battery Material: Columnar Ordering of Electrons and Holes in the Spinel LiMn2O4. Phys. Rev. Lett. 1998, 81, 4660−4663.
while M06-L yields the AFM state as the ground state, which is consistent with the HSE06 results12 and experiment.25 However, the charge disproportionation of LiMn2O4 to form Mn3+ and Mn4+ found in the experiment25 is not predicted by the M06-L functional. The M06-L functional yields a ground state with Mn ions having identical average valences. By using different initial wave functions for the iterative self-consistentfield calculations, we were able to find a charge disproportionation state, but it is 0.03 eV higher in energy than the ground state. Overall, M06-L and N12 are found to be suitable to study the geometries and delithiation energies in lithium-ion batteries.
4. CONCLUSIONS Theoretical studies on lithium-containing materials improve our understanding of lithium batteries. In the present study, we have tested the M06-L and N12 exchange-correlation functionals for calculating the lattice parameters and voltages of several lithium-containing materials. They have shown better performance than the widely used PBE exchange-correlation functional. The M06-L meta-GGA local functional shows similar performance to that of the HSE hybrid functional, but it has lower computational cost than HSE. The N12 nonseparable-gradient-approximation local functional has a similar computational cost to the PBE generalized-gradient-approximation local functional, but it predicts more accurate voltages for lithium-ion batteries.
■
ASSOCIATED CONTENT
S Supporting Information *
Comparison between VASP and Gaussian calculations and full ref 21. The Supporting Information is available free of charge on the ACS Publications website at DOI: 10.1021/ acs.jpcb.5b03356.
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AUTHOR INFORMATION
Corresponding Author
*E-mail: [email protected]; telephone: 1-612-624-7555. Notes
The authors declare no competing financial interest.
■
ACKNOWLEDGMENTS This work was supported in part by the U.S. Department 21of Energy, Office of Basic Energy Sciences, under Grant No. DESC0008662.
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REFERENCES
(1) Wakihara, M.; Yamamoto, O. Lithium Ion Batteries: Fundamentals and Performance; Wiley: New York, 1998. (2) Huggins, R. A. Advanced Batteries: Materials Science Aspects; Springer: New York, 2009. (3) Lithium-Ion Batteries: Advanced Materials and Technologies; Yuan, X., Liu, H., Zhang, J., Eds.; CRC Press: Boca Raton, FL, 2011. (4) Goodenough, J. B.; Park, K. The Li-Ion Rechargeable Battery: A Perspective. J. Am. Chem. Soc. 2013, 135, 1167−1176. (5) Larcher, D.; Tarascon, J.-M. Towards Greener and More Sustainable Batteries for Electrical Energy Storage. Nat. Chem. 2015, 7, 19−29. (6) Croguennec, L.; Palacin, M. R. Recent Achievements on Inorganic Electrode Materials for Lithium-Ion Batteries. J. Am. Chem. Soc. 2015, 137, 3140−3156. (7) Wang, L.; Maxisch, T.; Ceder, G. A First-Principles Approach to Studying the Thermal Stability of Oxide Cathode Materials. Chem. Mater. 2007, 19, 543−552. 1439
DOI: 10.1021/acs.jpcb.5b03356 J. Phys. Chem. B 2016, 120, 1437−1439
Computational Electrochemistry. Voltages of Lithium-Ion Battery Cathodes Bo Wang, Sijie Luo, and Donald G. Truhlar* Department of Chemistry, Chemical Theory Center, and Supercomputing Institute, University of Minnesota, 207 Pleasant Street S.E., Minneapolis, Minnesota 55455-0431, United States S Supporting Information *
ABSTRACT: Theoretical studies on the electrode materials in lithium-ion batteries provide information on the structural changes during the charging and discharging processes. In the present study, we tested the M06-L and N12 exchange-correlation functionals on some wellstudied lithium-containing materials. These functionals, which have already shown good performance for a variety of databases, outperform the widely used PBE functional for reproducing the experimental structures and averaged intercalation potentials. It is especially noteworthy that the M06-L functional gives voltages as accurate as those provided by the Heyd−Scuseria−Ernzerhof (HSE06) hybrid functional, but with less computational cost.
1. INTRODUCTION Rechargeable lithium-ion batteries are important for energy storage due to their high output voltage and high-energy storage density. The voltage and energy storage density are determined by the selection of the electrode materials, and various kinds of electrode materials have been investigated.1−6 In order to understand lithium-ion battery properties at the molecular level and to design new materials for lithium-ion batteries, computational modeling provides a useful tool.7,8 Among the available computational methods, Kohn−Sham theory (KST)9 provides the best compromise between accuracy and the computational cost, and therefore it is widely used in lithium-ion battery studies. However, the accuracy of KST depends on the accuracy of the exchange-correlation (xc) functional. Many classes of xc functionals are available, including both local functionals, such as LDA (local density approximation), GGA (generalized gradient approximation), NGA (nonseparable gradient approximation), meta-GGA, and meta-NGA functionals, and nonlocal (“hybrid” and “doubly hybrid”) functionals. Even within a class of xc functional, the performance is not uniform.10 It has been found that the hybrid HSE functional11 is much better than GGA functionals for reproducing the averaged Li intercalation potentials due to the reduction of the electron self-interaction error by Hartree− Fock exchange in hybrid functionals.12 However, local xc functionals, which have no Hartree−Fock exchange, are much more economical for calculations on extended and periodic systems, and, therefore, it is interesting and potentially useful to test some other local functionals that have proved accurate for predicting various chemical properties but that have not yet been tested for cell voltages. In the present study, we test the M06-L13 and N1214 local functionals for their accuracy in © 2015 American Chemical Society
calculating the structures and voltages of lithium-ion battery materials. M06-L is a meta-GGA that includes the kinetic energy density in the functional form, which is another way (complementary to Hartree−Fock exchange) to decrease the self-interaction error. It has shown good performance for a variety of databases.10,15,16 In addition, M06-L shows the correct 2D-3D trends in Au clusters,17 and it provides improved surface formation energies and CO adsorption energies for transition metals.18 The N12 xc functional is an NGA, and it has been found to provide good accuracy for both the energetic and structural properties of both solids and molecules.10 This paper is concerned with validating these local functionals for Licontaining solid materials.
2. METHODS AND COMPUTATIONAL DETAILS The test suite to validate xc functionals is composed of LiCoO2/CoO2, LiTiS2/TiS2, LiMn2O4/Mn2O4, and Li2Ti2O4/ LiTi2O4, where in each case we have listed the discharged state and a delithiated state of a cathode material. Calculation of the voltages versus Li/Li+ also requires calculations on Li metal. The quantum mechanical calculations were carried out using a combination of software packages, in particular a locally modified version of the plane-wave-basis electronic structure program VASP19,20 and the Gaussian-basis-function electronic structure program Gaussian 09.21 Primitive cells were used in Special Issue: Bruce C. Garrett Festschrift Received: April 7, 2015 Revised: May 29, 2015 Published: June 5, 2015 1437
DOI: 10.1021/acs.jpcb.5b03356 J. Phys. Chem. B 2016, 120, 1437−1439
Article
The Journal of Physical Chemistry B
Table 1. Optimized Volumes per Formula Unit (Angstrom3) and Mean Unsigned Percentage Error (MUPE) Using Various Functionals PBE PBEa N12 PBE+Ua M06-L HSE06a expt.a a
GGA GGA NGA GGA+U meta-GGA hybrid-GGA
LiCoO2
CoO2
LiMn2O4
Mn2O4
LiTiS2
TiS2
Li2Ti2O4
LiTi2O4
MUPE
32.70 33.00 31.48 32.86 32.61 31.81 32.17
32.09 33.00 31.15 34.00 31.04 32.00 29.61
67.86 67.84 65.60 73.88 70.92 70.25 69.89
66.77 66.93 65.10 70.74 67.53 65.81 66.36
63.66 63.95 60.09 63.95 63.00 64.92 64.12
60.41 65.00 57.94 65.00 59.37 62.00 57.32
73.33 73.65 70.43 73.65 71.54 74.35 73.44
75.50 75.72 73.08 75.72 74.25 75.50 74.18
2.7 4.2 3.5 5.7 2.2 2.9
Reference 12.
tested in other solid-state calculations.17,18 Since M06-L can semiquantitatively describe the dispersion energy, it greatly improves the geometric results for the cases of CoO2 and TiS2, in which the interactions between layers are mostly of van der Waals type. The N12 functional is an NGA functional, and it performs better than PBE for the intercalation potentials. Although the HSE06 functional gives the best performance among all functionals, it is a hybrid functional and can be quite expensive in simulations (depending on the system and the program, the cost can be orders of magnitude higher than that of the local functionals). M06-L and N12 are local functionals, and therefore, they have the potential to be applied to more complex systems. Since N12 and M06-L have both been validated in past work to perform well for a variety of databases,10 we conclude that they are reasonable functionals to use if only local functionals are practical for the calculations. We note that the PBE geometry employed for TiS2 in this study differs from Chevrier et al.’s results.12 This can be mainly due to the flatness of the potential energy surface, as the intercalation potentials are quite similar despite the different geometries. Since PAW potentials were used in all calculations, and the VASP PAW potentials are optimized only for PBE functionals, we also performed all-electron calculations using Gaussian 09 on several materials and compared the results with those obtained by VASP. A detailed description of the method is given in the Supporting Information. In previous work24 it was shown that calculations using Gaussian 09 can agree well those with plane-wave basis set when both sets of calculations are well converged with respect to the basis. In our cases, Tables S1 and S2 (Supporting Information) indicate that these two programs agree with each other to some extent; however, we do find some differences. For example, Gaussian 09 gives consistently lower intercalation potentials than VASP for Li2Ti2O4 with all functionals. Many aspects of the calculations may contribute to this kind of difference; for example, we have not extrapolated the results to the basis set limit, and PAW potentials are used to replace the core electrons in VASP while all-electron calculations are performed in Gaussian 09. Although such comparisons of results obtained with different kinds of basis sets are less common in the solid-state literature than in the gasphase literature, such differences are not unexpected in the light of gas-phase experience. In the gas phase it is common to see differences between calculations with various practical basis sets and effective core potentials since the results are not at the complete-basis-set limit. In the rest of this paper we present only results obtained with VASP. In the case of LiMn2O4, we tested both the ferromagnetic (FM) and antiferromagnetic (AFM) ordering of the Mn spins. The N12 functional gives the FM ordering as the ground state,
all calculations. The VASP calculations used a 500 eV cutoff energy and 6 × 6 × 6 k points, except for Li for which we used 10 × 10 × 10 k points. VASP’s projector-augmented-wave (PAW) potentials22,23 were used in all calculations. All calculations were spin polarized, and the energy criterion for self-consistency was set to less than 0.0001 eV/unit cell. Both the coordinates of the atoms and the lattice constants were optimized, and the structural relaxation criterion was set to less than 0.001 eV/unit cell. The cell voltages vs Li/Li+ are approximated here as averaged Li intercalation potentials and were calculated as E(LixM) − E(Lix − 1M) − E(Li) (1) F where E is the total electronic energy (including nuclear repulsion), F is the Faraday constant, “M” denotes a cathode host material in which the Li+ can be reversibly inserted and extracted, and x1 and x2 are the numbers of Li atoms contained in the delithiated host and the lithiated host, respectively. Equation 1 neglects zero point energy differences, thermal phonon energies, and entropic effects; this is a reasonable procedure for the present purposes. V=−
3. RESULTS AND DISCUSSION Tables 1 and 2 show the optimized volumes and the averaged Li intercalation potentials calculated using various functionals. Table 2. Averaged Li Intercalation Potentials versus Li/Li+ (eV) and Mean Unsigned Error (MUE) Using Various Functionals LiCoO2 LiMn2O4 PBE PBEa N12 PBE+Ua M06-L HSE06a expt.a a
GGA GGA NGA GGA+U meta-GGA hybrid-GGA
3.34 3.38 3.52 3.85 3.46 4.51 4.1
3.37 3.37 3.59 4.04 3.86 4.25 4.1
LiTiS2 Li2Ti2O4 1.94 1.91 1.86 1.91 2.09 2.06 2.1
1.05 1.05 1.17 1.05 1.23 1.19 1.3
MUE 0.47 0.47 0.37 0.19 0.24 0.18
Reference 12.
The HSE06 and PBE+U results are from Chevrier et al.12 Among the functionals tested, M06-L best reproduces the experimental geometry, with a mean unsigned percentage error (MUPE) of 2.2%, and M06-L and N12 give more accurate intercalation potentials than PBE (though they are still less accurate than the much more expensive HSE06 hybrid functional). The relatively low mean unsigned percentage error of M06-L is especially welcome since it has previously been successfully 1438
DOI: 10.1021/acs.jpcb.5b03356 J. Phys. Chem. B 2016, 120, 1437−1439
Article
The Journal of Physical Chemistry B
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while M06-L yields the AFM state as the ground state, which is consistent with the HSE06 results12 and experiment.25 However, the charge disproportionation of LiMn2O4 to form Mn3+ and Mn4+ found in the experiment25 is not predicted by the M06-L functional. The M06-L functional yields a ground state with Mn ions having identical average valences. By using different initial wave functions for the iterative self-consistentfield calculations, we were able to find a charge disproportionation state, but it is 0.03 eV higher in energy than the ground state. Overall, M06-L and N12 are found to be suitable to study the geometries and delithiation energies in lithium-ion batteries.
4. CONCLUSIONS Theoretical studies on lithium-containing materials improve our understanding of lithium batteries. In the present study, we have tested the M06-L and N12 exchange-correlation functionals for calculating the lattice parameters and voltages of several lithium-containing materials. They have shown better performance than the widely used PBE exchange-correlation functional. The M06-L meta-GGA local functional shows similar performance to that of the HSE hybrid functional, but it has lower computational cost than HSE. The N12 nonseparable-gradient-approximation local functional has a similar computational cost to the PBE generalized-gradient-approximation local functional, but it predicts more accurate voltages for lithium-ion batteries.
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ASSOCIATED CONTENT
S Supporting Information *
Comparison between VASP and Gaussian calculations and full ref 21. The Supporting Information is available free of charge on the ACS Publications website at DOI: 10.1021/ acs.jpcb.5b03356.
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AUTHOR INFORMATION
Corresponding Author
*E-mail: [email protected]; telephone: 1-612-624-7555. Notes
The authors declare no competing financial interest.
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ACKNOWLEDGMENTS This work was supported in part by the U.S. Department 21of Energy, Office of Basic Energy Sciences, under Grant No. DESC0008662.
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REFERENCES
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DOI: 10.1021/acs.jpcb.5b03356 J. Phys. Chem. B 2016, 120, 1437−1439
