Accurate Quantum Chemical Reaction Energies for Lithium-Mediated Electrolyte Decomposition and Evaluation of Density Functional Approximations

Abstract

An important concern related to the performance of Li-ion batteries is the formation of a solid electrolyte interphase on the surface of the anode. This film is formed from the decomposition of electrolytes and can have important effects on the stability and performance. Here, we evaluate the decomposition pathway of ethylene carbonate and related organic electrolyte molecules using a series of density functional approximations and correlated wave function (WF) methods, including the coupled-cluster theory with single, double, and perturbative triple excitations [CCSD(T)] and auxiliary-field quantum Monte Carlo (AFQMC). We find that the transition state barrier associated with ring opening varies widely across different functionals, ranging from 3.01 to 17.15 kcal/mol, which can be compared to the value of 12.84 kcal/mol predicted by CCSD(T). This large variation underscores the importance of benchmarking against accurate WF methods. A performance comparison of all of the density functionals used in this study reveals that the M06-2X-D3 (a meta-hybrid GGA), CAM-B3LYP-D3 (a range-separated hybrid), and B2GP-PLYP-D3 (a double hybrid) perform the best, with average errors of about 1.50–1.60 kcal/mol compared to CCSD(T). We also compared the performance of the WF methods that are more scalable than CCSD(T), finding that DLPNO–CCSD(T) and phaseless AFQMC with a DFT trial wave function exhibit average errors of 1.38 and 1.74 kcal/mol, respectively.

Graphical Abstract

1. INTRODUCTION

Lithium-ion batteries have been the gold standard and ubiquitous driving force behind almost all electronic devices for over two decades.^1,2 A typical lithium-ion battery consists of a graphite anode, a transition metal oxide cathode, and an electrically insulating and ionically conducting nonaqueous organic electrolyte that acts as an ionic path between the two.^3,4 One key to understanding a battery’s electrochemistry is its electrode/electrolyte interface, which controls critical aspects of performance and is thus a target for improvement.^5,6 The decomposition of organic electrolyte molecules at the electrode surface yields a passivation layer called the solid electrolyte interphase (SEI).⁷ Understanding the SEI layer formation, composition, and growth over multiple time and length scales is essential for the design of next-generation safe and performant lithium-ion batteries.^8,9

The most common electrolytes are mixtures of alkyl carbonates such as ethylene carbonate (EC) or propylene carbonate (PC) and lithium salts such as LiAsF₆ or LiPF₆.¹⁰ Both components undergo chemical reactions that contribute to the composition of the SEI. The choice of an electrolyte is partially based on its compatibility with the electrode materials used, such that the electrolyte should either be electrochemically stable or form a stable and well-understood SEI at the electrode surface,⁶ necessitating a thorough understanding of the electrolyte stability and decomposition pathways. However, it is experimentally challenging to directly capture reactive processes leading to the formation of the SEI.^11,12

Alternatively, atomistic simulations, including electronic structure theory and molecular dynamics, have the potential to contribute to a fundamental understanding of the electrolyte decomposition pathways responsible for the SEI.^13-19 Although density functional theory (DFT) has become a standard approach due to the steep computational scaling of more accurate wave function (WF) methods that explicitly describe electron correlation, the results can vary depending on the density functional approximation used and the system of interest. Therefore, the selection of a particular density functional approximation for a specific application requires rigorous benchmarking studies.^20-24 In this work, we carry out such a study for the problem of organic electrolyte decomposition in the presence of lithium.

Specifically, we study the reductive decomposition of the organic electrolytes: ethylene carbonate (EC), propylene carbonate (PC), and fluoroethylene carbonate (FEC). The reaction pathways studied here are similar to those investigated previously by Balbuena and co-workers²⁵ using DFT. We perform accurate benchmark calculations using a variety of high-level WF methods and then evaluate a range of DFT functionals; our work should be useful for the future development of accurate force field parameters for large-scale atomistic simulations of battery electrochemistry.^26,27 In addition to our evaluation of a range of DFT functionals, we also use EC decomposition to compare three accurate WF methods capable of serving as a benchmark: coupled-cluster theory with single, double, and perturbative triple excitations [CCSD(T)],²⁸ its low-scaling variant via domain-based local pair natural orbitals [DLPNO–CCSD(T)],²⁹ and phaseless auxiliary-field quantum Monte Carlo (ph-AFQMC) using a Kohn–Sham determinant trial wave function. We find that the three methods yield similar reaction energetics (within a few kcal/mol), and this internal consistency, especially between very different methods like coupled-cluster theory and AFQMC, supports the conclusion that all three give accurate results for this class of reactions. Consequently, we utilize the least expensive approach, DLPNO–CCSD(T), in studying the remaining reactions (involving PC and FEC electrolyte solvent components). It should be noted, however, that if the desired precision was chemical accuracy (1 kcal/mol), then further improvement of the reference results would be necessary.

2. METHODS

All geometry optimizations are performed at the ωB97X-D/def2-TZVP(-f) level of theory using Jaguar v11.4.³⁰ The transition state (TS) of the EC decomposition pathway is obtained using the automatic transition state (AutoTS) search approach, as detailed in ref 31. The transition states of the substituted ethylene carbonates (FEC and PC) are initially optimized with the eigenvector-following method using ωB97X-D/6-31G* as a guess, followed by frequency analysis to verify the TS. Finally, these preoptimized TS geometries are refined with the eigenvector-following method utilizing ωB97X-D/def2-TZVP(-f). These geometries are used for all subsequent energy evaluations.

DFT single-point energy calculations were performed with Jaguar v11.4 and Orca v5.0 using the def2-TZVP(-f) basis set without any further basis set extrapolation. We selected a wide range of DFT functionals belonging to different rungs of the Jacob’s ladder, including generalized gradient approximation (GGA), meta-GGA, hybrid, meta-hybrid-GGA, range-separated hybrid, and double hybrid.³² Many of the functionals were combined with various treatments of dispersion, including D3,³³ D3BJ,³⁴ and VV10.³⁵ The full list of density functional approximations can be found in Section 3. In total, we tested 17 different density functional approximations.

Canonical CCSD and CCSD(T) calculations were performed with PySCF v1.7.^36-38 DLPNO–CCSD and DLPNO–CCSD(T) calculations were performed with ORCA 5.0³⁹ using “TightPNO” settings and semicanonical perturbative triple corrections (i.e., without the improved iterative “(T1)” correction⁴⁰). All CC calculations were performed using a spin-unrestricted Hartree–Fock (HF) reference determinant and the core orbitals were frozen. All AFQMC calculations were carried out using single determinant trial wave functions, as detailed in ref 41. The trial wave functions (UB3LYP) and integrals for AFQMC were obtained with PySCF v2.0. Similar to previous studies,⁴² the imaginary time step for the AFQMC propagation was 0.005 Ha¹⁻, and the total propagation time was 200 Ha¹⁻, employing the “comb” algorithm for population control every 20 steps.⁴³ We used 3312 total walkers, which were initialized with a spin-restricted HF determinant.

Calculations were performed with the aug-cc-pVDZ, TZ, and QZ basis sets. The HF energy was extrapolated to the CBS limit with an exponential form and the correlation energy was extrapolated with an X^−3.05 form,⁴⁴ using the TZ and QZ results (X = 2, 3, 4 for DZ, TZ, QZ, respectively). As one exception, the reaction involving the formation of 8 was too large to treat at the QZ level with CC theory, so extrapolation was performed with an X^−2.51 form using only DZ and TZ results.⁴⁴

In this work, we focus on the purely electronic energies, i.e., we neglect vibrational corrections, and all calculations are performed in the gas phase. This approach facilitates a direct comparison of different levels of electronic structure theory but of course is not meant to be compared to experimental values. Although we expect small vibrational corrections on the order of 1–2 kcal/mol, the solvation energy corrections are expected to be very large. Nonetheless, we expect our conclusions concerning the performance of different levels of theory to hold, even when the solvent is treated explicitly or implicitly.

3. RESULTS AND DISCUSSION

We study in detail the reductive decomposition of ethylene carbonate (EC) in the presence of a lithium ion, and we closely follow the mechanism identified in ref 25. We also study the same reaction for propylene carbonate (PC) and fluoroethylene carbonate (FEC) as a confirmation of the generality of our conclusions.

3.1. Reductive Decomposition of Ethylene Carbonate.

Figure 1 illustrates the steps of the reductive decomposition of EC.²⁵ Step 1 involves the strong binding of the EC molecule 1 with a Li⁺ ion to form the ion-pair intermediate 2. This ion-pair intermediate 2 is reduced in step 2 to form the neutral adduct 3; the energy change for this step is large and negative due to the strong electron affinity induced by the Li⁺ ion (in the absence of Li⁺, the addition of an electron to gas-phase EC requires about +8 kcal/mol).²⁵ The adduct 3 subsequently undergoes homolytic ring cleavage, passing through the transition state 4 in step 3 to form the intermediate radical 5 in step 4. A nonplanar pathway in steps 2 and 3 is also possible, as shown in Figure S1. Once the radical 5 is formed, there are several possible termination pathways. Here, we study two such pathways, as illustrated in steps 5 and 6, which correspond to pathways C and B, respectively, as detailed in ref 25. The first is a reductive decomposition to form a lithium carbonate anion 6 and ethylene 7. The second is a dimerization of the radical 5 to form ethylene 7 and lithium ethylene dicarbonate 8. We note that both lithium carbonate and lithium ethylene dicarbonate have been implicated as major components of the SEI, but there are some conflicting observations.^45-47

Figure 1.

Reductive decomposition of EC in the presence of a lithium ion calculated by using three different levels of theory. Relative energies with respect to the initial species are given in kcal/mol.

Using the electronic structure methods described in Section 2, we calculate the electronic energy differences of all six steps. A T1 diagnostic test⁴⁸ on the CC calculations confirms that all species are qualitatively single-reference, supporting the application of DFT, CC, and AFQMC with a single determinant trial. Using CCSD(T) as the reference, we then calculate the mean absolute deviation (MAD) over these six steps for each method considered. We find the performance of different methods to be similar for both pathways (Tables S1-S4, SI). Therefore, we consider the pathway involving planar transition while calculating the MAD across different methods. The detailed reaction energy profile predicted by PBE-D3, CAM-B3LYP-D3, and CCSD(T) is shown in Figure 1, and the MAD from all methods is presented in Figure 2.

Figure 2.

Mean absolute deviation with respect to CCSD(T) of the stepwise energy differences for the reductive decomposition of EC, as shown in Figure 1. Hatched bars show functionals with added dispersion.

First, we consider the performance of the wave function methods. Among those we tested, DLPNO–CCSD(T) has the smallest MAD of 1.38 kcal/mol, where the largest deviations of 2.70 and 3.01 kcal/mol are observed for the reactions involving the transition state (steps 3 and 4), which is consistent with other reports of overestimations of transition state barriers by DLPNO–CCSD(T).⁴⁹ Without steps 3 and 4, the MAD of DLPNO–CCSD(T) is reduced to 0.64 kcal/mol, which is excellent in light of its significantly lower computational cost compared to canonical CCSD(T). To enhance the accuracy of DLPNO–CCSD(T) for barrier heights, we calculated the reaction energies for steps 3 and 4 using different TcutPNO thresholds, as illustrated in Figure S8. As expected, the error with respect to canonical CCSD(T) decreases with an increasingly tight threshold, as illustrated in Figure S8. The smallest deviations, i.e., 1.94 kcal/mol for step 3 and 0.23 kcal/ mol for step 4 are obtained using the 6/7 TcutPNO extrapolation with “T1” corrections. Notably, this observation aligns with the findings reported by Bistoni and co-workers,⁵⁰ where improved accuracy was demonstrated through the use of a 6/7 TcutPNO extrapolation in comparison to the default TightPNO settings. The MADs obtained for CCSD, DLPNO–CCSD, and AFQMC are 1.73, 2.61, and 1.74 kcal/mol, respectively, where AFQMC has an average statistical uncertainty of ±1 kcal/mol. The highest deviations obtained for AFQMC and CCSD are for steps 4 and 5. As shown in Figures 3 and 4, all other wave function methods overestimate barrier heights compared to CCSD(T), which is opposite to the behavior displayed by most density functionals. Overall, the wave function methods are reasonably consistent, justifying their usage as a benchmark, and the good performance of DLPNO–CCSD(T) supports its application to larger molecular systems than accessible with canonical CCSD(T).

Figure 3.

Signed deviation with respect to CCSD(T) for the transition state barrier for ring opening (step 3). Hatched bars show functionals with added dispersion.

Figure 4.

Signed deviation with respect to CCSD(T) for step 4. Hatched bars show functionals with added dispersion.

Turning to the performance of DFT, we see that PBE-D3, arguably the most popular GGA functional in battery modeling, has a large MAD of 6.83 kcal/mol. The meta-GGAs M06-L, SCAN, and TPSS exhibit MADs between 5.57 and 6.39 kcal/mol. These results raise concerns about the reliability of commonly used functionals for applications related to SEI modeling. Hybrids perform slightly better, with MADs between 4 and 5 kcal/mol, with the lowest MAD of 4.18 kcal/mol for B3LYP-D3. The meta-hybrid functional M06-2X-D3 performs best with a MAD of 1.48 kcal/mol, which is comparable to DLPNO–CCSD(T). However, the other meta-hybrid SCAN0 exhibits a relatively large MAD of 4.59 kcal/mol, with the largest deviation of 11.26 kcal/mol obtained for step 5 (reductive decomposition). The range-separated hybrid functionals also perform well, especially when dispersion corrections are included. The lowest MAD is obtained for CAM-B3LYP-D3 (1.65 kcal/mol). The MAD of other leading range-separated hybrid functionals, such as ωB97X-D3BJ and ωB97X-V, is 2.40 and 1.88 kcal/mol. The double-hybrid functionals give similar average errors ranging from 1.49 to 2.70 kcal/mol, with B2GP-PLYP(-D3) having the lowest MAD of less than 2 kcal/mol. Based on all of these results we can identify dispersion-corrected M06-2X (meta-hybrid GGA), CAM-B3LYP and ωB97X (range-separated hybrids), and B2-GPLYP (double hybrid) as among the best-performing functionals.

In the SI (S-1.2), we provide errors for each step of the decomposition pathway from all methods, which provides a more detailed look into their performance. Here, we focus on their performance for the critical step 3, i.e., the transition state barrier for ring opening, which determines the kinetics of the decomposition. Figure 3 shows the absolute deviation of all methods with respect to CCSD(T) for this step. The deviation ranges from 0.002–9.84 kcal/mol, with B2-GPLYP-D3 (0.02 kcal/mol), B2-GPLYP (0.002 kcal/mol), CAM-B3LYP-D3 (0.45 kcal/mol), and M06-2X-D3 (0.63 kcal/mol) agreeing well with the reference value. While ωB97X-V has a mean error of 1.88 kcal/mol when averaged over all reaction steps, it exhibits a significant error of 3.74 kcal/mol for this critical step. Importantly, we find that some of the most common functionals used for materials simulations resulted in large deviations. For example, PBE-D3, SCAN, TPSS, and B3LYP-D3 underestimate the barrier height by more than 6 kcal/mol. Similar behavior is seen for step 4, which is to be expected because the magnitude of its energy difference is the barrier height for the reversed (ring-closing) reaction. As shown in Figure 4, the error for this step closely tracks that of the ring-opening barrier height (step 3), but with an opposite sign. The observed challenge in describing the energy differences of these steps is presumably due to the self-interaction error and associated stabilization of the delocalized electronic structure of transition state geometry 4. Although we observe an improvement by over 7 kcal/mol for PBE0-D3 compared to PBE-D3 for these steps, attributable to its reduced self-interaction error, such improvement is not consistent across all steps of the reaction. For example, in steps 5 and 6 (Figure S4a,b), the errors obtained for PBE0-D3 are larger than those for PBE-D3. Thus, it is difficult to rank PBE0-D3 and PBE-D3 for this reaction pathway.

This completes our analysis of the reductive decomposition of EC, but before moving on, we note that exhaustive benchmarking against CCSD(T) is computationally expensive, especially for increasingly large molecules. DLPNO–CCSD(T) provides a cheaper alternative to serve as a benchmark, especially given its good agreement with canonical CCSD(T). To test whether our conclusions are changed, we repeated the same analysis using DLPNO–CCSD(T) as the reference, and the results are shown in Figure 5. Overall, we observe very similar trends, although the MAD of almost all functionals is increased by about 0.5–1.4 kcal/mol due to errors introduced by the DLPNO approximations. For example, the 2.70 kcal/ mol overestimation of the barrier height (step 3) by DLPNO–CCSD(T) changes the apparent accuracy of one of the best-performing functionals, M06-2X-D3, from 1.48 to 2.85 kcal/mol. This effect needs to be kept in mind when benchmarking DFT functionals against affordable DLPNO–CCSD(T) calculations.

Figure 5.

MAD of DFT functionals and wave function methods, with respect to DLPNO–CCSD(T). Hatched bars show functionals with added dispersion.

To test the transferability of our conclusions above with regard to functional performance, we extended our study to two electrolyte solvent molecules with substitution, −CH₃ and −F (PC and FEC). Given the comparatively lower cost and good performance of DLPNO-CCSD(T) with respect to CCSD(T), we selected DLPNO–CCSD(T) as the reference to compare the performance of different density functionals. Our detailed numerical results are given in the SI (S-2 and S-3), but, unsurprisingly, the findings are very similar to those for EC, reinforcing our previous conclusions. Specifically, B3LYP-D3, M06-2X-D3, CAM-B3LYP-D3, and the double hybrids are the best-performing functionals within their respective rungs.

4. CONCLUSIONS

In this work, we benchmarked different density functionals against accurate correlated WF methods for lithium-mediated electrolyte decomposition. In addition, we compared the agreement among different correlated methods. Overall, DLPNO–CCSD(T) shows good agreement with canonical CCSD(T) for the reaction energies. When the reactions involving the transition state are excluded, the MAD is 0.64 kcal/mol. Moreover, AFQMC also shows good agreement with a MAD of 1.74 kcal/mol and an average statistical uncertainty of ±1 kcal/mol. Considering the accuracy and cost of different correlated methods, we conclude that DLPNO–CCSD(T) is an excellent benchmark method for reaction energies of complex systems relevant to battery electrochemistry, i.e., reactions involving organic molecules and Li atoms. For barrier heights, improved results with deviations lower than 2 kcal/mol are obtained using the TcutPNO 6/7 extrapolation with “T1” corrections.

A performance comparison of all of the density functional approximations used in this study reveals that CAM-B3LYP-D3, M06-2X-D3, and double hybrids perform very well for this system. Perhaps, most importantly, the DFT transition state barrier heights vary from 3.01 to 17.15 kcal/mol, while the CCSD(T) barrier height is 12.84 kcal/mol. GGAs, in particular, are especially prone to a significant underestimation of the barrier height by up to 10 kcal/mol. Therefore, we urge caution when interpreting the results of GGA-based molecular dynamics simulations of electrolyte decomposition and SEI formation. However, despite the good performance of meta-GGAs, range-separated hybrids, and double hybrids observed in this molecular study, we note that these families of functionals are limited in their applicability to periodic metals.⁵¹ The development of accurate and broadly applicable electronic structure methods for surface chemistry is an important task for the community, especially toward the development of accurate force fields for reactive molecular dynamics simulations.