Accurate Binding Energies for Lithium Polysulfides and Assessment of

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Accurate Binding Energies for Lithium Polysulfides and Assessment of Density Functionals for Lithium-Sulfur Battery Research Qiu He, Xiaobin Liao, Lixue Xia, Zhaohuai Li, Huan Wang, Yan Zhao, and Donald G. Truhlar J. Phys. Chem. C, Just Accepted Manuscript • DOI: 10.1021/acs.jpcc.9b05235 • Publication Date (Web): 23 Jul 2019 Downloaded from pubs.acs.org on July 23, 2019

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The Journal of Physical Chemistry

Revised July 16, 2019 for JPC C

Accurate Binding Energies for Lithium Polysulfides and Assessment of Density Functionals for Lithium-Sulfur Battery Research Qiu He,a Xiaobin Liao,a Lixue Xia,a Zhaohuai Li,a Huan Wang,*b Yan Zhao,*a and Donald G. Truhlar*c aState

Key Laboratory of Silicate Materials for Architectures, International School of Materials Science

and Engineering, Wuhan University of Technology, Wuhan 430070, People’s Republic of China. bState

Key Laboratory of Advanced Technology for Materials Synthesis and Processing, Center of Smart Materials and Devices, School of Materials Science and Engineering, Wuhan University of Technology No. 122 Luoshi Road, Wuhan 430070, People’s Republic of China.

cDepartment

of Chemistry, Chemical Theory Center, Supercomputing Institute, and Nanoporous

Materials Genome Center, University of Minnesota, 207 Pleasant Street SE, Minneapolis, MN 554550431

ABSTRACT Lithium-sulfur batteries have high theoretical energy density, but a better knowledge of their intimate structural details will be helpful in improving their conductivity and long-term cycling behavior. In order to identify the stationary configurations of lithium polysulfides (Li2Sn, 2 ≤ n ≤ 8) formed in the charging and discharging processes of the lithium-sulfur batteries, ab initio molecular dynamics was employed to sample the configuration space of Li2Sn, followed by optimization of structures by CCSD(T)-F12b/aug-cc-pVDZ. Using the optimized stationary points, we have created the LiSAE38 benchmark database of atomization energies (AEs) of 38 lithium polysulfides isomers by using the higher-level WMS and W3X-L methods. In addition, the performances of 39 density functionals have been assessed against the benchmark AEs and the relative stabilities of Li2Sn isomers. Based on the assessments with the def2-QZVP basis set, the PW6B95, B97-1, B3LYP-D3, TPSS, and DSD-PBEP86 density functionals are the most accurate for the AEs, whereas the mPW2-PLYP-D, DSD-PBEP86, PW6B95, HSE06, and PBEQIDH functionals are most accurate for the relative energies. Local functionals are of special interest because of their lower cost (faster timings in terms of computer resources) for large systems; among the tested local functionals, MN15-L and revM06-L are the most accurate for the calculations of relative stabilities.

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Introduction

Development of lithium-sulfur (Li-S) batteries has become one of the hottest research areas in high-performance energy storage1-5 due to their attractive high specific energy density (~2600 Wh kg−1). However, instead of directly producing the final insoluble Li2S2 or Li2S, the discharging processes undergo multistep reactions, forming Li2Sn (2 < n ≤ 8).6 Most of the Li2Sn species can dissolve in electrolyte, and the larger the n value, the higher the solubility. Unfortunately, the soluble Li2Sn species formed around the S-cathode can diffuse to the Li anode and react with Li to form smaller Li2Sn (3≤ n ≤5), or Li2S2/Li2S. This is called the “shuttle effect”, and it corrodes both the sulfur cathode and lithium metal anode, consumes electrolyte, and causes rapid capacity decay.3, 6-9 The deposition of the insoluble Li2S2/Li2S on the anode aggravates the dendrite growth problem of the lithium metal anode, which leads to safety issues and the irreversible loss of active electrode materials. Understanding the transformation mechanism of Li2Sn is essential for solving the “shuttle effect” problem; however, it is very difficult to obtain the required information experimentally because charging and discharging are dynamical processes in the condensed phase. Therefore computational chemistry has been emerging as a powerful method for the investigation of the issues arising in Li-S battery development.10-29 Density functional theory has been employed to simulate the spectroscopies of lithium polysulfides, including infrared (IR), Raman, and X-ray adsorption spectroscopies (XAS),19-21 and to elucidate the reaction mechanisms. Chen et al.22 optimized the structures of Li2Sn (2 ≤ n ≤ 8) by using the B3LYP density functional, and they calculated the infrared and Raman spectra. Hagen et al.23 calculated the Raman spectra of various Li2Sn by using the B3PW91 functional to compare to in situ Raman data taken during charging and discharging of a Li-S battery. They concluded that several Li2Sn species co-exist in the electrolytes, with equilibria favoring smaller Li2Sn compounds during discharge larger Li2Sn species during charging. Vijayakumar et al.24 conducted XAS measurements of Li2Sn solutions and compared the experimental data to the calculated XAS spectra. They concluded that the formation of soluble Li2Sn was triggered by lithium exchange between solvent molecules and Li2Sn. This kind of conclusion provides very valuable guidance for electrolyte selection for Li-S batteries. An effective way of ameliorating the shuttle effect is to employ an anchoring cathode material with high affinity for Li2Sn to prevent its migration from the S-cathode to the Li2 ACS Paragon Plus Environment

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anode,28-33 and density functional calculations have been employed for fast in silico screening of the organic and inorganic anchoring materials. In a systematic searching of potential anchoring materials for Li2Sn, Seh et al.25 calculated the binding energies of Li2S and LiS with various organic functional groups, and their computations reveal that the oxygen-rich functional groups have strong interactions with LiS and Li2S, with the electron acceptance and donation occurring at lithium and oxygen. Zhang et al.26 reported density functional calculations of the van der Waals binding energies between lithium polysulfides and various two-dimensional layered materials including graphene, transition metal oxides, sulfides, and chlorides, and they found that V2O5 gives the largest anchoring effect for Li2Sn, and that the strong binding interactions also cause the decomposition of Li2Sn. In another density functional study, Zhao et al.27 discovered that phosphorene is a promising anchoring material for Li2Sn. Although Kohn-Sham density functional theory is in principle exact for electronic structure calculations, the accuracy is limited by the need to employ approximate exchange-correlation functionals. Over a hundred approximate functionals have been developed and are employed in the quantum chemistry and solid-state physics communities, but their performances for describing Li2Sn have not been ascertained. In this paper, we report calculations that explore the Born-Oppenheimer potential energy surface (PES) of Li2Sn, and we use high-level wave function theory to create a benchmark database of atomization energies (AEs) of 38 Li2Sn (2 ≤ n ≤ 8) isomers. Atomization energy of a molecule is defined as the energy change when all bonds of the molecule are completely broken, and thus atomization energy is a measure of the stability of the molecule. For the series compounds with the same components, i.e., lithium and sulfur atoms in Li2Sn, calculating the atomization energies provides a way to make systematic studies of stabilities. For two Li2Sn compounds with the same n, the difference in atomization energies is the isomerization energy, which is a measure of relative stability of isomers. A density functional that gives good atomization energies would also be expected to give good heats of formation. To provide a benchmark database for the atomization energy, we calculated the energies by the recently developed WMS34 and W3X-L35 methods, both of which attempt to attain higher accuracy than the widely used (for small systems) CCSD(T) theory. Then 37 popular density functionals are assessed against the new database. The paper is organized as follows: Section 2 describes the methodology. Section 3 presents the results and discussion of the benchmark AEs, the relative stability of Li2Sn isomers, and the 3 ACS Paragon Plus Environment


performances of density functionals. Section 4 concludes the paper and provides some recommendations and warnings for choosing density functionals for the study of Li2Sn. 1. Methodology 2.1 The AIMD/Q exploration of PESs of Li2Sn In order to identify the stationary points on the PES of each Li2Sn, we have employed ab initio molecular dynamics with repeated quenching steps (AIMD/Q) in an NVE ensemble. The forces in AIMD/Q are calculated by PBE-D3 (the PBE functional36, 37 with the D3 molecular mechanics damped dispersion correction38). We also performed AIMD/Q with the M06-L method, and it produces the same sets of stationary points for Li2Sn as PBE-D3. 2.2 Geometry Optimization After the AIMD/Q sampling of the configuration space of Li2Sn, pre-optimizations were carried out using M06-2X/6-311+G(2df,2p) for all the selected configurations using the Gaussian 16 software. A pruned ultrafine grid and tight SCF convergence were used for all preoptimizations. After the density functional pre-optimization step, a final CCSD(T)-F12b/aug-ccpVDZ optimization was carried out for each stationary point on the PESs of Li2Sn, where CCSD(T)-F12b is the explicitly correlated version of the coupled cluster method with single and double excitations and a quasiperturbative treatment of connected triple excitations, as developed by Knizia et al.39, 40 2.3 AE Calculations The AEs of benchmark quality for Li2Sn have been obtained with the W3X-L and WMS methods. W3X-L is wave function based method developed by Chan and Radom,35 and it is a combination of nonrelativistic CCSD(T)/CBS(FC), scalar relativistic effects, and post-CCSD(T) and core-valence correlation contributions with the goal of approximating the CCSDT(Q)/CBS(Full) energy. (Note that CBS(FC) denotes extrapolation to a complete basis set for valence and electrons, and CBS(Full) stands for such extrapolation for both valence and core electrons.) WMS34 is a wave function method developed recently by us in an attempt to approximate results at the complete configuration interaction level in a relatively low-cost way involving no calculations of higher excitation level than CCSD(T), and it has been shown to give better performance for the AEs in the W4-17 database41 with a lower computational cost than W2X,35 which is an efficient method of extrapolation to the CCSD(T)/CBS(Full) level. Most of the 4 ACS Paragon Plus Environment

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energies in the W4-17 database were obtained by extrapolation to the CCSDTQ5/CBS(Full) or CCSDTQ56/CBS(Full) level of theory, and they should be close to complete configuration interaction, which should be close to experiment. 2.4 Assessment of Density Functionals We have tested 39 density functionals. This includes nine local functionals and 30 hybrid functionals. The tested functionals are listed in Table 1 with the type and reference for each functional. 2.5 Software and Computational Details The AIMD/Q sampling of the stationary points of Li2Sn were performed with the DMol3 module in the Materials Studio 2017R2 package.42 The DND numerical basis set (version 4.4)43 and a global orbital cutoff of 3.3 Å were used for the AIMD/Q simulation, and the SCF tolerance was set to be 1.0×10-5 a.u. As mentioned in Section 2.1, the PBE-D3 and M06-L functionals have been employed for the AIMD/Q sampling. The CCSD(T)-F12b/aug-cc-pVDZ optimization and the W3X-L, W2X, and WMS benchmark calculations were performed with the WMPack44 package and the Molpro program.45 The W3X-L calculation also needs the MRCC program46 for the post-CCSD(T) calculations. The calculations with the two multi-coefficient doubly hybrid density functionals, namely MC3MPW47 and MCCO-MPW,48 were performed with the MLGAUSS module,49 and all the density functional calculations were carried out with the MG3S50 and def2-QZVP51 Gaussian basis sets using a locally modified Gaussian program52 incorporating the MN-GFM module.53 The Materials Studio, Molpro, and Gaussian 16 calculations are all-electron calculations. We also performed plane wave calculations with the VASP program54 and the MN-VFM module;53 these calculations treat the core electrons of S with the projector augmented wave (PAW) method. The valence electrons of the S atom and all electrons of Li atom (i.e., the Li_sv POTCAR) are treated explicitly in all VASP calculations with an energy cutoff of 800 eV. The computational timings for the W3X-L, W2X, and WMS methods and all the tested DFT methods (using def2-QZVP basis set) have been evaluated by the total calculation time for the Li2S4 molecule on a single computing node using six cores of the Intel Xeon E5-2640 v3 CPU and 2300 MW memory. The computational timings relative to an MP2/aug-cc-pVTZ calculation

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of the Li2S4 molecule and are listed in the last row of Table 2 and the last column of a table to be discussed in Section 3.3. 2. Results and Discussion 3.1 AIMD/Q Sampling of Li2Sn We started with computational characterization the stationary points of Li2Sn (2 < n ≤ 8). For each of the small lithium polysulfides, namely Li2S2, Li2S3, and Li2S4, our protocol located only one stationary point for each. By using the AIMD/Q sampling and CCSD(T)-F12b/aug-cc-pVDZ optimizations, we found six isomers for Li2S5 (Figure 1), five isomers for Li2S6 (Figure 2), nine isomers for Li2S7 (Figure 3), and fifteen isomers for Li2S8 (Figure 4). As mentioned in Section 2, all isomers were located through sampling of AIMD/Q trajectories; we carried out two trajectories, each of length 4 ps (Figure S1 is the potential energy profile for an M06/DND AIMD trajectory), and all the structures have been optimized at the CCSD(T)-F12b/aug-ccpVDZ level of theory. Figures with a prefix S are in the supporting information (SI), and, while the Cartesian coordinates of all Li2Sn isomers are also five in the SI. 3.2 Development of the LiSAE38 Atomization Energy Database Due to the two orders of magnitude higher cost of the post-CCSD(T) calculations in the W3X-L method (see the last row of Table 2) than W2X and WMS, we can only afford to do the W3X-L calculations for Li2S2, Li2S3, and Li2S4. Listed in Table 2 are the various components of the W3X-L calculations: Hartree-Fock at the CBS(FC) limit, correlation energy by CCSD/CBS(FC), the corrections due to quasiperturbative connected triple excitations (T/CBS(FC)), core-valence correlation plus the scalar-relativistic-effect (C+R), and the beyondCCSD(T) components. As shown in Table 2, the HF/CBS(DC) components contribute more than 60% of the AEs, and the CCSD/CBS(FC) components contribute more than 30%. Although the quasi-perturbative connected (T) components contribute less than 5% of the AEs, the postCCSD(T) contributions are not negligible, ranging from 0.27 to 0.44 kcal/mol. The latter finding is consistent with a previous study55 showing that CCSD(T)/CBS(FC) is accurate to only about 0.4 kcal/mol for non-multireference (i.e., weakly correlated) systems. Also listed in Table 2 are AEs calculated by the W2X and WMS methods. The AEs of both methods agree very well with those of W3X-L, and WMS gives a mean unsigned deviation (MUD) of only 0.19 kcal/mol, which is lower than the MUD (0.35 kcal/mol) of W2X.


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Furthermore, the MUD is almost a factor-of-two smaller than the average post-CCSD(T) contribution in W3X-L. For the large Li2Sn (5 ≤ n ≤ 8) clusters, we have used AEs of WMS as the reference for the assessment of density functionals. The 38 benchmark AEs of Li2Sn are listed in Table 3, and we named this database LiSAE38. 3.3 Performances of Density Functionals for AEs of the LiSAE38 Database We have tested several density functionals with the Gaussian 16 and VASP software. Two Gaussian basis sets, i.e. MG3S50 and def2-QZVP,51 have been employed in Gaussian 16, and the mean signed errors (MSEs) and MUEs for the tested functionals with Gaussian 16 are listed in Table 4. (The average of the MUEs (labeled as AMUE) obtained with the two basis sets is given in the second last column of Table 4, and the root-mean-squared errors (RMSEs) are listed in the Table S1 of SI.) In each group, the general trends of the performances obtained from the two basis sets are similar, and we sort the results mainly based on the MUEs with the def2-QZVP basis set. Local functionals. As shown in Table 4, among the local functionals, the TPSS functional is superior to other functionals with the def2-QZVP basis set, with an MUE of 1.81 kcal/mol, whereas the GAM functional has an almost as small MUE, 1.89 kcal/mol, with the MG3S basis set. Note that TPSS is a meta functional, while GAM involves fewer ingredients, being a gradient approximation (it is an improved version of the N12 functional). Next in performance are two recent Minnesota functionals, MN15-L and M06-L. The GVWN3 functional (the only LSDA functional in this test) is the least accurate functional of all functionals tested in this work. Functionals with nonlocal exchange. In this group, the PW6B95 and B97-1 functionals are the two best performers, with MUEs of 1.27 and 1.32 kcal/mol, respectively, when using the def2-QZVP basis set. These functionals are both global-hybrid GGAs, and they involve six and 12 adjusted parameters, respectively. The MUEs of the tested Minnesota meta hybrid functionals in this group (M05-2X, M06-HF, M06, M06-2X, M08-HX, M08-SO, and MN15) range from 3.66 to 13.97 kcal/mol. The B3LYP functional shows the worst performance in this group; we will discuss it further at the end of this section. Functionals with nonlocal exchange and nonlocal correlation. The MCCO-MPW doubly hybrid functional performs the best in this group with an MUE of 4.16 kcal/mol, whereas MC3MPW is the worst performer. The basis set effects are quite large for doubly hybrid 7 ACS Paragon Plus Environment


functionals; this is perhaps not too surprising because the nonlocal correlation in these methods is known to be slowly convergent with respect to basis set size. Functionals with nonlocal exchange and molecular mechanics: In this group, B3LYP-D3 is the best performer with the def2-QZVP basis set, whereas the APF-D functional is best with MG3S. Functionals with nonlocal exchange, nonlocal correlation, and molecular mechanics: Among the tested functionals in this group, the DSD-PBEP86 functional is the best performer, with an MUE of 1.81 kcal/mol. General remarks on adding molecular mechanics. The popular B3LYP functional gives unacceptable MUEs above 30 kcal/mol with either the MG3S or def2-QZVP basis sets, but adding the D3 molecular mechanics damped dispersion correction gives a dramatic improvement in that the MUE is reduced to 3.84 kcal/mol for the MG3S basis set and 1.79 kcal/mol for the def2-QZVP basis set. However, for the PW6B95 functional, the molecular mechanics damped dispersion correction worsens the performance by raising the MUE from 3.05 to 8.81 kcal/mol for the MG3S basis set and from 1.27 to 10.63 kcal/mol for the def2-QZVP basis set. Adding molecular mechanics damped dispersion to PBE also makes the performance worse. Therefore, the addition of molecular mechanics terms does not always lead to an improvement. We also see that X-D and the last three rows of Table 4, which all have molecular mechanics damped dispersion, give fairly large MUEs (5.6, 6.9, 10.5, and 17.3 kcal/mol), whereas the APF-D functional, which also has a molecular mechanics term, has excellent performance. Comparisons made with the VASP program. Table 5 presents the performance of six functionals using the VASP program. Among the six tested functionals, GAM shows the best performance, followed by SCAN+rVV10 and BEEF. We note that GAM is very efficient in plane wave codes and shows good SCF convergence in VASP. If we compare the performances of the GAM, N12-SX, and PBE functionals with the def2-QZVP basis set (Table 4) to the ones with plane waves and PAW (Table 5), we can see that they are encouragingly similar, and the plane wave/PAW treatment gives slightly lower MUEs. 3.4 Relative Stability of the Li2Sn Isomers The geometries of the lowest-energy structures of Li2Sn (2 ≤ n ≤ 8) are shown in Figure 5. The Li2S2 molecule is like a bent parallelogram with two lithium and two sulfur atoms located at the opposite vertices, respectively. The rest of the most stable Li2Sn structures (with 3 ≤ n ≤ 8) 8 ACS Paragon Plus Environment

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share a common motif of being basket shaped, with Li2S2 at the bottom while the rest of the sulfur atoms link as a handle bonding to the two sulfur atoms of the bottom tetramer. In the rest of this section, we evaluate the performances of density functionals for the relative energies of stable Li2Sn for those cases (5 ≤ n ≤ 8) where we fund two or more isomers for the same n; we do this by comparing to the relative energies obtained with the WMS method. The MSEs and MUEs of the relative energies for each functional are summarized in Table 6 and Table 7. Based on the results with the def2-QZVP basis set, the performance of the MN15-L functional is the best of the tested local functionals with an MUE of only 0.78 kcal/mol; however, MN15-L and revM06-L have MUEs lower than 0.78 kcal/mol with MG3S. The revM06-L also performs well with def2-QZVP, with an MUE of 0.82 kcal/mol. The rest of the functionals in this group yield MUEs less than 2.0 kcal/mol, except the SVWN functional, which gives an MUE of 2.35 kcal/mol. In the group of functionals with nonlocal exchange, the PW6B95 functional is the best performer for both basis sets with MUEs of 0.32 and 0.41 kcal/mol. Among the four functionals with nonlocal exchange and nonlocal correlation, the PBEQIDH functional is the best performer with MUEs of 0.28 and 0.46 kcal/mol. PW6B95-D3 is the best performer in the four functionals with nonlocal exchange and molecular mechanics dispersion, and it has MUEs of 0.51 and 0.52 kcal/mol, and mPW2-PLYPD and DSD-PBE86 are the best of the three tested functionals with nonlocal exchange, nonlocal correlation, and molecular mechanics dispersion; they have MUEs in the range 0.35-0.69 kcal/mol. For the functionals tested with the VASP program (see Table 7), the N12-SX functional performs the best, with an MUE of 1.24 kcal/mol. The MSEs in Tables 6 and 7 indicates that the local functionals, except MN15-L and revM06-L, systematically underestimates the relative energies. A general observation about the errors in relative energies is that they are much smaller than the errors in total atomization energies, indicating admirable cancellation of errors. However, it is noteworthy that 24 of the 36 functionals in the Gaussian-basis-set table (Table 6) have AMUEs smaller than the lowest MUE in the plane-wave table (Table 7). Figure 6 is a plot of the relative energies for Li2Sn (5 ≤ n ≤ 8) with the def2-QZVP basis set, and only the top one performer in each group of Table 6, the top performer in Table 7, and the 9 ACS Paragon Plus Environment


reference WMS results are presented. (All the raw relative energies, the Cartesian coordinates of all Li2Sn (2 ≤ n ≤ 8) isomers, and the plot of relative energies with the MG3S basis set are given in SI.) For the six Li2S5 isomers, the energy difference between the third and fourth isomers in this series is small (less than 2 kcal/mol, see Table S2 and S6). However, the geometries of these two isomers are quite different (Figure 1). For the third isomer of Li2S5, two lithium and three sulfur atoms form a twisted pentagon at the bottom and the remaining two sulfur atoms bond to one of the adjacent lithium and sulfur, respectively. The fourth isomer of Li2S5 has a structure similar to that of Li2S4, but it has an extra sulfur atom bonded to the “handle” (see Figure 1). For the five Li2S6 isomers, the relative energies of the fourth and fifth isomers are very close, ~1 kcal/mol. Figure 2 shows that the fourth isomer of Li2S6 has a six-member ring containing two lithium atoms and four sulfur atoms at the bottom and the remaining two sulfur atoms bridging across and bonding to the two sulfur atoms of the bottom ring. The fifth isomer has a three-member ring and a four-member ring, where two lithium atoms belong to different rings, respectively. These two rings are linked by a Li-S bond. For the Li2S7 isomers, all the local functionals, except MN15-L, revM06-L, N12, and GAM, give the incorrect global minimum (see Figure 3 and Tables S5 and S9), they predict that the global minimum is isomer Li2S7_5. Different from the basket-shaped global minimum obtained from the WMS method, the fifth isomer (see Figure 3) of Li2S7 has a caged shape and 2.5 kcal/mol higher than the global minimum (see the WMS row of Table S5). In the Li2S8 series, we calculated the relative energies of 15 isomers, in which the energy differences between third to eleventh isomers are less than 5.5 kcal/mol, Figure 4). The geometries for these isomers can be classified into two groups: (1) 4-member-ring basket (isomers 3, 4, 8, 9) and (2) 5-member-ring basket (isomers 5, 6, 7, 10, 11). As shown in Figure S2 and Tables S6 and S10, most of the local functionals underestimate the relative energies of high-energy isomers, but MN-15 and revM06-L do not suffer this deficiency. 3. Conclusion and Perspective Understanding the electronic structure of lithium polysulfides formed during charging and discharging of lithium-sulfur (Li-S) batteries is essential to designing improvements in the performance of the Li-S batteries. In this study, we built the LiSAE38 benchmark database of


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AEs for 38 Li2Sn (2 ≤ n ≤ 8) isomers using the W3X-L and WMS methods, and we used it to test and validate density functionals for the study of Li-S batteries. Thirty-nine density functionals have been tested against the LiSAE38 database. The general order of performance of the density functionals is similar with the two basis sets, but identifying the performers with the smallest mean errors does depend on the basis set. Since robustness with respect to basis set is a desirable quality, we base our final evaluations in this section on the AMUEs. Based on the AMUEs, we found that the PW6B95 functional is the best performer for AEs in LiSAE38, with an AMUE of 2.2 kcal/mol. For the medium-sized Li2Sn (5 ≤ n ≤ 8) isomers, the mPW2-PLYP-D, PW6B95, HSE06, and PBEQIDH functionals perform best for the relative energies, with MUEs of 0.4 kcal/mol. Adding a damped dispersion correction to the B3LYP functional significantly improves its accuracy, but one should be cautious for as to whether one should add a dispersion correction to the PW6B95 functional. With the VASP program, the GAM functional is recommended for both relative energies and AE calculations of Li2Sn. We caution that the widely used B3LYP and PBE functionals are not recommended for lithium-sulfur battery studies. As shown in Table 4, PBE already overbinds Li2Sn, and adding damped dispersion corrections by molecular mechanics, as done in PBE-D3, worsens the performance, so we do not recommend the use of PBE-D3 for the investigation of the Li2Sn system. Other important issues in understanding the role and behavior of the lithium polysulfides are solvent effects in the electrolyte, ionic states, and the binding energies between the lithium polysulfides and the electrode surfaces. Currently, most of the theoretical work, like the work here, has focused on calculations in vacuum and the neutral states. We expect that the functionals that perform well in the vacuum will also perform well for neutral molecules in solution. Hence the present study of the neutral states in vacuum should provide useful guidance to extend the studies to look at this additional issue. Additional studies of ionic states and electrode interactions would also be useful as the present assessment is not directly related to those issues. Supporting Information The Supporting Information is available free of charge on the ACS Publication website at DOI:xxxx.



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A PDF file contains a potential energy time series for a M06-L/DND AIMD trajectory (Figure S1), a figure of the relative energies of Li2Sn (5 ≤ n ≤ 8) obtained from 9 functionals with MG3S basis set (Figure S2); the RMSE of the tested functionals on the Li2Sn database (Table S1 and S2); the relative energies of Li2Sn (5 ≤ n ≤ 8) (Table S3−S14); the Cartesian coordinates of 38 Li2Sn (2 ≤ n ≤ 8) isomers (Table S15−S19).

 AUTHOR

INFORMATION

Corresponding Authors

*E-mail: [email protected], [email protected], [email protected] ORCID Qiu He: 0000-0002-5782-4962 Lixue Xia: 0000-0001-6006-4055 Xiaobin Liao: 0000-0002-2455-832X Huan Wang: 0000-0002-4931-3898 Yan Zhao: 0000-0002-1234-4455 Donald G. Truhlar: 0000-0002-7742-7294  ACKNOWLEDGMENTS

This work was supported in part by the Thousand Innovative Talents Plan of the Chinese Government and by the Nanoporous Materials Genome Center by the U.S. Department of Energy, Office of Basic Energy Sciences, Division of Chemical Sciences, Geosciences, and Biosciences under award DE-FG02-17ER16362.  REFERENCES

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Figure 1. Isomers of Li2S5. The WMS relative energy (kcal/mol) is shown below each isomer.



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Figure 5. Lowest-3nergy configurations of Li2Sn (2 ≤ n ≤ 8) from the W3X-L//CCSD(T)-

F12/aug-cc-pVDZ and WMS//CCSD(T)-F12/aug-cc-pVDZ calculations. Purple and yellow balls denotes lithium and sulfur atoms, respectively.


Page 24 of 36

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Figure 6. Relative Energies of lithium polysulfide isomers obtained from nine functionals (colored open polygons) with def2-QZVP basis set comparing with that of the WMS method (red dot). (a) Li2S5, (b) Li2S6, (c) Li2S7, and (d) Li2S8.



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Table 1. Tested density functionals Method

Xa

Year

Type b

ref(s).

local functionals GVWN3c

0

1980

LSDA

56,57

PBE

0

1996

GGA

36, 37

N12

0

2012

NGA

58

GAM

0

2015

NGA

59

TPSS

0

2003

meta-GGA

60

M06-L

0

2006

meta-GGA

61

MN15-L

0

2016

meta-NGA

62

HLE17

0

2017

meta-GGA

63

revM06-L

0

2017

meta-GGA

64

functionals with nonlocal exchange B3LYP

20

1994

global-hybrid GGA

65-67

B97-1

21

1998

global-hybrid GGA

68

PBE0

25

1999

global-hybrid GGA

69

HSE06

25–0

2005

screened-hybrid GGA

70, 71

N12-SX

25–0

2012

screened-hybrid NGA

72

τ-HCTHh

15

2002

global-hybrid meta-GGA

73

TPSSh

10

2003


60, 74

BMK

42

2004


75

PW6B95

28

2005


76

M05-2X

56

2005


77

M06

27

2008


78

M06-2X

54

2008


78

M06-HF

100

2006


79

M08-HX

52.23

2008


80


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M08-SO

52.23

2008


80

MN15

44

2016

global-hybrid meta-NGA

81

revM06

40.41

2018


82

functionals with nonlocal correlation BEEF

0

2012

GGA + nonlocal correlation

83

SCAN+rVV10

0

2016

meta-GGA + nonlocal correlation

84

functionals with nonlocal exchange and nonlocal correlation MC3MPW

38

doubly hybrid GGA

47

MCCO-MPW

32

doubly hybrid GGA

48

PBE0-DH

50

2011

doubly hybrid GGA

85

PBEQIDH

69.336

2014

doubly hybrid GGA

86

functionals with nonlocal exchange and molecular mechanics B3LYP-D3c

20

1994

global-hybrid GGA + MM

38, 65-67

B97X-D

22.036–100

2008

range-separated-hybrid GGA + MM

87

APF-D

22.95

2012

global-hybrid GGA + MM

88

PW6B95-D3c

28

2005

global-hybrid meta-GGA + MM

38, 76

functionals with nonlocal exchange, nonlocal correlation, and molecular mechanics DSD-PBEP86

68

2011

doubly hybrid GGA + MM

89

mPW2-PLYP-Dd

55

2006


90, 91

B2-PLYP-D3e

53

2011


92, 93

a X is the percentage of Hartree–Fock exchange. When a range is given, the percentage changes continuously

from the first value at small interelectronic separation to the second value at large interelectronic separation. b Abbreviations: LSDA: local spin-density approximation; GGA: generalized gradient approximation; NGA:

nonseparable gradient approximation; MM: molecular mechanics c The keyword for this functional in the Gaussian 16 package is SVWN. It is a combination of the Gáspár

exchange functional and correlation functional no. III of Vosko et al. c This functional uses D3(0)-type damped dispersion. d This functional uses D2-type damped dispersion. e This functional uses D3(BJ)-type damped dispersion.



Page 28 of 36

Table 2. Components of AEs (kcal/mol) for Li2Sn (n = 2 − 4) of the W3X-L method and AEs of W3X-L, W2X, and WMS W3X-L

W2X

WMS

HF /CBS

CCSD /CBS

(T) /CBS

C+R a

post-(T) b

AE

AE

AE

Li2S2

153.82

76.98

7.82

1.97

0.27

240.87

240.59

241.23

Li2S3

199.34

106.17

12.29

2.61

0.34

320.75

320.40

320.75

Li2S4

241.20

135.07

16.82

3.23

0.44

396.76

396.32

396.59

106.07

12.31

2.60

0.35 0.35

0.19

71

20

average c MUD d timing e

7791

7862

a C+R denotes the sum of the core-valence correlation and scalar relativistic corrections. b post-(T) denotes the sum of the post-CCSD(T) components of the W3X-L method. c This row gives the average of the beyond-HF/CBS contributions. d In this table, MUD denotes mean unsigned deviation from W3X-L. e These are relative timings. See section 2.5 for their definition.


Page 29 of 36 1 2 3 4 5 6 7 8 9 10 11 12 13 14 15 16 17 18 19 20 21 22 23 24 25 26 27 28 29 30 31 32 33 34 35 36 37 38 39 40 41 42 43 44 45 46 47 48 49 50 51 52 53 54 55 56 57 58 59 60


Table 3. Benchmark AEs of the LiSAE38 Database Li2Sn

AE (kcal/mol)

Method

Li2Sn

AE (kcal/mol)

Method

Li2S2

240.87

W3X-L

Li2S7_6

589.44

WMS

Li2S3

320.75

W3X-L

Li2S7_7

588.79

WMS

Li2S4

396.76

W3X-L

Li2S7_8

587.46

WMS

Li2S5_1

464.43

WMS

Li2S7_9

575.52

WMS

Li2S5_2

460.39

WMS

Li2S8_1

663.48

WMS

Li2S5_3

457.86

WMS

Li2S8_2

661.92

WMS

Li2S5_4

456.82

WMS

Li2S8_3

660.02

WMS

Li2S5_5

445.18

WMS

Li2S8_4

659.62

WMS

Li2S5_6

431.22

WMS

Li2S8_5

659.52

WMS

Li2S6_1

532.60

WMS

Li2S8_6

657.43

WMS

Li2S6_2

532.02

WMS

Li2S8_7

657.44

WMS

Li2S6_3

528.34

WMS

Li2S8_8

657.07

WMS

Li2S6_4

515.36

WMS

Li2S8_9

656.35

WMS

Li2S6_5

515.94

WMS

Li2S8_10

655.69

WMS

Li2S7_1

597.27

WMS

Li2S8_11

654.60

WMS

Li2S7_2

597.05

WMS

Li2S8_12

650.07

WMS

Li2S7_3

596.45

WMS

Li2S8_13

646.75

WMS

Li2S7_4

594.89

WMS

Li2S8_14

642.18

WMS

Li2S7_5

594.66

WMS

Li2S8_15

634.33

WMS



Page 30 of 36

Table 4. Performance of Density Functionals for the LiSAE38 database (kcal/mol) Using Gaussian Basis Sets a MG3S MSE

def2-QZVP MUE

MSE

MUE

average AMUE

timingb

local functionals TPSS

−2.79

2.92

−0.97

1.81

2.4

1

GAM

−0.83

1.89

5.66

6.28

4.1

1

MN15-L

11.00

11.00

11.41

11.41

11.2

1

revM06-L

11.22

11.22

15.13

15.13

13.2

1

M06-L

16.97

16.97

17.47

17.47

17.2

1

PBE

22.00

22.00

23.92

23.95

23.0

1

N12

33.16

33.41

31.16

31.53

32.5

1

GVWN3

154.04

154.04

154.48

154.48

154.3

1

functionals with nonlocal exchange PW6B95

−3.05

3.05

−1.27

1.27

2.2

1

B97-1

0.20

0.79

0.96

1.32

1.1

1

BMK

2.43

3.09

−1.36

1.94

2.5

1

τ-HCTHh

−2.45

2.45

−2.03

2.03

2.2

1

M08-SO

4.34

4.34

3.62

3.66

4.0

1

M06

4.28

4.29

3.87

3.92

4.1

1

M06-2X

1.72

2.04

3.92

3.92

3.0

1

M06-HF

6.79

6.80

4.96

5.19

6.0

1

PBE0

−7.59

7.59

−5.31

5.31

6.5

1

HSE06

−9.00

9.00

−6.61

6.61

7.8

3

M08-HX

5.41

5.41

6.74

6.74

6.1

1

MN15

16.02

16.02

7.81

7.81

11.9

1

TPSSh

−12.55

12.55

−10.65

10.65

11.6

1


Page 31 of 36 1 2 3 4 5 6 7 8 9 10 11 12 13 14 15 16 17 18 19 20 21 22 23 24 25 26 27 28 29 30 31 32 33 34 35 36 37 38 39 40 41 42 43 44 45 46 47 48 49 50 51 52 53 54 55 56 57 58 59 60


M05-2X

11.62

11.62

13.97

13.97

12.8

1

revM06

15.31

15.31

16.34

16.34

15.8

1

N12-SX

18.12

18.12

17.03

17.07

17.6

3

B3LYP

−34.63

34.63

−32.37

32.37

33.5

1

functionals with nonlocal exchange and nonlocal correlation MCCO-MPW c

−4.16

4.16

PBEQIDH

−12.87

12.87

−4.62

PBE0-DH

−14.00

14.00

−9.33

MC3MPW c

−29.22

29.22

4.2

2

4.62

8.7

2

9.33

11.7

2

29.2

1

functionals with nonlocal exchange and molecular mechanics B3LYP-D3

−3.84

3.84

−1.58

1.79

2.8

1

APF-D

0.52

1.29

2.73

3.08

2.2

1

−10.53

10.53

−10.38

10.38

10.5

3

8.78

8.81

10.56

10.63

9.7

1

X-D PW6B95-D3

functionals with nonlocal exchange, nonlocal correlation, and molecular mechanics DSD-PBEP86

−9.44

9.44

1.36

1.81

5.6

2

B2-PLYP-D3

−10.68

10.68

−3.14

3.15

6.9

2

mPW2P-LYP-D

−20.95

20.95

−13.74

13.74

17.3

2

a In this table, MSE and MUE are the mean signed error and mean unsigned error relative to the reference values in

Table 3. Rows are sorted according to MUEs with the def2-QZVP basis set. The top five performers for each basis set and for the AMUE are in bold font. b These are relative timings. See section 2.5 for their definition. The timings in this table are rounded to one

significant figure because we get slightly different results (~20% variations) when we run them with different software and different runtime options, so more than one significant figure is not warranted. c These composite methods are defined to use specific basis sets in each of their steps, with MG3S being the largest

basis set.



Table 5. Performance of Density Functionals for the LiSAE38 database (kcal/mol) Using plane wave/PAW. a MSE

MUE

GAM

3.45

5.08

SCAN+rVV10

5.57

5.80

BEEF

−9.47

9.47

N12-SX

13.41

14.75

PBE

22.30

22.35

HLE17

−31.15

31.42

a The reference AEs are in Table 3.


Page 32 of 36

Page 33 of 36 1 2 3 4 5 6 7 8 9 10 11 12 13 14 15 16 17 18 19 20 21 22 23 24 25 26 27 28 29 30 31 32 33 34 35 36 37 38 39 40 41 42 43 44 45 46 47 48 49 50 51 52 53 54 55 56 57 58 59 60


Table 6. Performance of Density Functionals for the Relative Energies (kcal/mol) of Li2Sn Isomers Using Gaussian Basis Sets. a MG3S MSE

def2-QZVP

MUE

average

MSE

MUE

AMUE

local functionals MN15-L

0.32

0.62

0.52

0.78

0.7

revM06-L

0.35

0.64

0.57

0.82

0.7

M06-L

−1.26

1.43

−1.07

1.26

1.3

N12

−1.45

1.59

−1.36

1.49

1.5

GAM

−1.33

1.66

−1.17

1.55

1.6

TPSS

−1.74

1.75

−1.60

1.61

1.7

PBE

−2.02

2.10

−1.91

1.99

2.0

GVWN3

−2.31

2.45

−2.20

2.35

2.4

functionals with nonlocal exchange PW6B95

−0.03

0.32

0.10

0.41

0.4

HSE06

−0.14

0.39

−0.03

0.44

0.4

PBE0

0.03

0.41

0.15

0.51

0.5

B97-1

−0.20

0.61

−0.10

0.60

0.6

τ-HCTHh

−0.58

0.74

−0.48

0.65

0.7

N12-SX

0.22

0.56

0.35

0.68

0.6

M05-2X

0.61

0.70

0.65

0.73

0.7

M06

−0.32

0.76

−0.18

0.73

0.7

B3LYP

−0.50

0.81

−0.37

0.78

0.8

TPSSh

−0.92

0.94

−0.79

0.80

0.9

revM06

0.61

0.73

0.77

0.88

0.8

M06-2X

1.06

1.12

1.16

1.23

1.2

MN15

1.22

1.32

1.35

1.43

1.4



Page 34 of 36

M08-SO

1.17

1.36

1.26

1.45

1.4

M08-HX

1.26

1.43

1.31

1.48

1.5

BMK

1.24

1.59

1.28

1.67

1.6

M06-HF

2.70

2.90

2.27

2.52

2.7

functionals with nonlocal exchange and nonlocal correlation PBEQIDH

0.21

0.28

MC3MPW

0.39

0.56

PBE0-DH

0.54

0.59

MCCO-MPW

−0.58

0.67

0.43

0.46

0.4 0.6

0.70

0.74

0.7 0.7

functionals with nonlocal exchange and molecular mechanics PW6B95-D3

0.03

0.51

0.17

0.52

0.5

B3LYP-D3

−0.46

0.87

−0.33

0.74

0.8

APFD

−0.45

0.83

−0.34

0.76

0.8

X-D

2.04

2.14

2.12

2.22

2.2

functionals with nonlocal exchange, nonlocal correlation, and molecular mechanics mPW2-PLYP-D

−0.23

0.53

−0.03

0.35

0.4

DSD-PBEP86

−0.45

0.69

−0.19

0.42

0.6

B2-PLYP-D3

−0.95

1.07

−0.73

0.88

1.0

a The reference relative energies are in calculated from the data in Table 3. Errors are sorted according to MUEs

with the def2-QZVP Basis Set. The top five performers for each basis set are in bold font.


Page 35 of 36 1 2 3 4 5 6 7 8 9 10 11 12 13 14 15 16 17 18 19 20 21 22 23 24 25 26 27 28 29 30 31 32 33 34 35 36 37 38 39 40 41 42 43 44 45 46 47 48 49 50 51 52 53 54 55 56 57 58 59 60


Table 7. Performance of Density Functionals for the Relative Energy (kcal/mol) of Mid-

sized Li2Sn (n = 5 − 8) using plane wave/PAW calculations MSE

MUE

N12-SX

−1.18

1.21

SCAN

−1.09

1.24

HLE17

−0.20

1.37

GAM

−1.17

1.48

PBE

−2.13

2.18

BEEF

−2.10

2.23



TOC graphic


Page 36 of 36

Accurate Binding Energies for Lithium Polysulfides and Assessment of

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