Electrochemical oxidation stability of anions for modern battery electrolytes: a CBS and DFT study

Abstract

The electrochemical stability vs. oxidation is a crucial property of anions in order to be suitable as components in lithium-ion batteries. Here the applicability of a number of computational approaches and methods to assess this property, employing a wide selection of DFT functionals, has been studied using the CCSD(T)/CBS method as the reference. In all, the vertical anion oxidation potential, ΔE_v, is a fair way to calculate the stability vs. oxidation, however, a functional of at least hybrid quality is recommended. In addition, the chemical hardness, η, is identified as a novel approach to calculate the stability vs. oxidation.

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1 Introduction

Electrolytes are a key component of batteries and their electrochemical stability windows (ESWs); the potential range wherein none of its components breaks down due to oxidation or reduction, currently limit the capabilities of Li-ion batteries (LIBs).¹ This is due to the high voltages possible for LIBs, often above 4 V. As there is a strong desire to produce even higher voltage cathodes,² to allow for even higher energy density LIBs – a very high oxidation stability (preferably >5 V vs. Li⁺/Li) is aimed at for future electrolytes. For the complementary/competing sodium-ion battery technology,³ currently seeing a rapid increase in attention, the overall development is foreseen to take the same direction. In order to reduce trial-and-error efforts, tools to predict electrochemical stability from chemical structure alone are of large interest. Generally, one way is by in silico screening, which allows for rapid evaluation of candidates for future synthesis.

Indeed, the ESWs of electrolyte solvents has been tackled computationally previously.^4–6 This is in part due to the large importance of the solid electrolyte interphase (SEI) passivation layer on the LIB anode, formed by (controlled) electrolyte degradation.⁷ As the solvent tends to be reduced, the reduction limit of the ESW has had focus on the solvent, with extensive studies to understand the SEI formation.^8–11 While the reduction limit of the anions has been studied in special solvents such as ionic liquids,^12–14 the anions are not in general reduced in LIB (or SIB) electrolytes. The creation of a SEI, partially by anion reduction, is a vast subject outside the scope of this paper.

In contrast, it is the (lithium) salt anion that often sets the oxidation limit, e.g. for ClO₄⁻ it is 6.1 V vs. Li⁺/Li, while it is 6.8 V for PF₆⁻ in the same solvent.¹⁵ In contrast, the solvents used are known to have a fair stability vs. oxidation, e.g. the standard blend of ethylene and dimethyl carbonate (EC/DMC) was found to have a limit of 6.7 V vs. Li⁺/Li.⁴ The stability of the anions versus oxidation are therefore often reported whenever novel salts for LIBs or SIBs are reported, either as experimental¹⁶ or computed¹⁷ values.

The experimentally measured ESWs have some variance due to the large number of factors affecting the final value, such as the sweep rate, solvent(s), and electrode(s) used.^4,15,18,19 Thus reference values for the ESWs, e.g. the intrinsic anion redox potentials, properties solely of the anions and independent of the electrode/solvent chemistry and physics, are highly warranted.

For the anion oxidation stability, a common approach is to calculate the HOMO energy,²⁰E_HOMO, due to the correlation between E_HOMO and the oxidation stability (based on the Koopmans theorem²¹). This approach can for example utilize Hartree–Fock (HF) methods with a small to medium sized basis set (e.g. 6-31G* or 6-311+G*) and is applicable to a vast number of compounds as it is computationally inexpensive.

To increase the accuracy, electron correlation must be taken in to account, such as with density functional theory (DFT). Unfortunately, the Koopmans theorem is often not valid within the DFT formalism, due to the poor quality of the orbitals,²² despite their physical relevance.^23–25 Thus an alternative approach, using the energy difference from a vertical excitation, is often used, either within the Frank–Condon approximation^26,27 (ΔE_v) or allowing also for geometry relaxation of the excited state²⁸ to obtain the ionization potential (IP). Both approaches are illustrated in Fig. 1. As hybrid (or better) functionals are recommended for anions,²² the computational cost for such calculations is higher than for a pure HF calculation, as this approach at least involves an extra single-point calculation and significantly higher if geometry optimization of the excited state is performed. Allowing for geometry relaxation, however, also allows for a thermodynamic approach.^28,29 If double-hybrid functionals are used, the computational cost increases even more.

Fig. 1 Illustration of the different approaches used to calculate the anion oxidation potential using DFT; the vertical excitation energy (ΔE_v) and the ionization potential (IP).

Yet another approach is to study the chemical hardness, η – a concept that arises naturally from conceptual DFT:³⁰

which is the resistance of the electronic chemical potential, μ, to the changing of the number of electrons. To our knowledge, this is only the second attempt to utilize this concept for the anion ESW.³¹

Here the first systematic study involving CCSD(T), MP2, and DFT calculations with a series of basis sets is presented – employed to obtain reference values for the intrinsic anion oxidation potentials at the complete basis set (CBS) limit. This enables a thorough assessment of the applicability and performance of the various methods available to calculate intrinsic oxidation potentials of anions.^26–28

2 Computational methodology

A number of anions that all at some point have been considered for LIB electrolytes were selected to be studied in detail (a subset of those previously studied by us³²); BF₄⁻, PF₆⁻, ClO₄⁻, TFSI ([(CF₃SO₂)₂N]⁻), BOB ([B(C₂O₄)₂]⁻), TDI (C₅N₄CF₃⁻), Tf (CF₃SO₃⁻), N₅⁻, N₅C₂⁻, N₅C₄⁻ (two isomers), N₅C₆⁻ (two isomers), Im(BF₃)₂⁻ and N₅C₁₀⁻. A number of small anions were also added, as a control set. Additionally, some anions were found to be too large for the CBS method, and only partial results are therefore listed in the ESI.†

Single point energies were calculated with each of the following DFT functionals (with the 6-311+G* basis set): B3LYP,^33–35 BMK,³⁶ PBE0,³⁷ TPSS,³⁸ TPSSh,³⁹ the M06 suite (M06, M06-2X, M06-L),⁴⁰ VSXC,^41,42 B2PLYP,⁴³ and mPW2PLYP.⁴⁴ The B3LYP/6-311+G* equilibrium geometry was used for the anions and for the oxidized and reduced species.

The CCSD(T) calculations with the aug-cc-pVXZ (X = T, Q, 5) basis sets used an extrapolation to the CBS limit:⁴⁵

E(n) = E^CBS + An⁻³ (2)

where n is the cardinality of the basis set used, i.e. n = 3 for aug-cc-pVTZ. This formula, chosen for its simplicity and performance,⁴⁶ was used to extrapolate the HF, MP2 and CCSD(T) energies. A separate extrapolation for the correlated part of the energy and the HF energy proved to have no significant effect on the results.

Due to the O(N⁷) scaling of CCSD(T), the rise in computational cost and time is staggering‡ – making the full CCSD(T)/CBS treatment not feasible at present for all of the anions. Therefore, a mixed CBS method was employed, wherein a correction term is added to a MP2/CBS value (also extrapolated using eqn (2)). The correction term, ΔCCSD(T), was calculated with MP2/6-31G* and CCSD(T)/6-31G* (a small but reasonable basis set for our purpose⁴⁷):

ΔCCSD(T) = E^6-31G*_CCSD(T) − E^6-31G*_MP2 (3)

The CCSD(T)/CBS values are then:

E^CBS_CCSD(T) = E^CBS_MP2 + ΔCCSD(T) (4)

Values obtained with this method are forthcoming designated as ΔCBS.

A finite difference scheme was used for the calculation of the chemical hardness:

where E_N is the energy of the anion, and E_N+1 and E_N−1 are the energies of the reduced and oxidized anion species, respectively. The reduced anion species, E_N+1, could be considered an unphysical approximation, however, this is countered by the experimental observation of such species.¹²

E _HOMO values were extracted from the HF/6-311+G* calculations.

The vertical anion oxidation potentials, ΔE_v, were calculated using single point energies (ΔE_v = E_N−1 − E_N). As the experimental results in general are given vs. Li⁺/Li, the computed ΔE_v are all shifted by 1.46 V.⁸ The Gaussian 09⁴⁸ program was used for all calculations and the Int = ExactBasisTransform keyword was used for the CBS calculations in order to ensure that no basis functions that contribute to the energy were pruned away during basis transformations.

3 Results & discussion

The analysis of the results starts with a comparison between the CBS and the ΔCBS methods for small anions, followed by a further analysis of ΔCBS values of larger anions. Thereafter, we compare the performance of different functionals. Each section treats and discusses the HOMO, ΔE_v, and η approaches.

3.1 CBS vs. ΔCBS

Only a portion of the anions were small enough to allow CCSD(T) calculations with the aug-cc-pV5Z basis sets. Therefore, most anions had to resort to the ΔCBS measure rather than CBS. As MP2 energies are included in the results of the CCSD(T) calculations, the anions in Table 1 can also function as an internal benchmark for the ΔCBS method. In general, the values obtained from the ΔCBS method are close to the values from the CCSD(T)/CBS method. This holds for both the ΔE_v and η approaches, showing the general validity of the ΔCBS method. For BF₄⁻, however, the ΔCBS values are 10% and 6% lower than the CBS values, giving a possible upper bound for possible errors. The BF₄⁻ values are close to revised experimental value of 6.2 V vs. Li⁺/Li.

Table 1 Anions with full CCSD(T)/CBS results. All values are given in V vs. Li⁺/Li, except η and E_HOMO (eV). E_ox is experimental values

Anion	ΔEv	ΔEv	η	η	E HOMO	E ox
	CCSD(T)/CBS	ΔCBS	CCSD(T)/CBS	ΔCBS	HF/6-311+G*	Experimental
Br^−	2.12	2.10	4.37	4.36	−3.79	4.1^49
F^−	1.96	1.99	6.02	6.07	−4.08	5.9^49
Cl^−	2.21	2.25	4.89	4.89	−4.83	4.4^49
HS^−	0.89	0.93	2.93	2.92	−2.56	—
HF2^−	4.93	4.97	5.40	5.44	−7.73	—
NO3^−	2.72	2.74	4.40	4.30	−6.33	—
OCN^−	2.33	2.35	4.06	4.07	−4.35	—
SCN^−	2.16	2.22	3.64	3.66	−3.84	—
BF4^−	7.03	6.35	6.42	6.06	−10.45	6.2;^28 6.6, 8.3^15
BrO3^−	3.44	3.54	4.48	4.55	−6.04	—
ClO4^−	4.34	4.36	5.06	5.06	−7.89	6.0;^28 6.1, 7.0^15
HCO2^−	2.80	2.85	3.80	3.81	−5.11	—
BH4^−	3.21	3.23	3.90	3.85	−5.19	—

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Surprisingly, neither the CBS nor the ΔCBS ΔE_v values for the halides are in the expected order of F⁻ > Cl⁻ > Br⁻, instead Cl⁻ > Br⁻ > F⁻ is observed. Comparing the E_ox values with the ΔE_v values for the halides, shows that the ΔE_v value is drastically lower. Similarly, the experimental E_ox value of ClO₄⁻ of 6.0 V vs. Li⁺/Li is far higher than expected from the CBS calculations, with a ΔE_v value of 4.34 V vs. Li⁺/Li. It seems as if the atomic anions have a problem with a proper radical description affecting this measure.

The hardness, η, has the expected ordering for the halides, furthermore, the hardness values are on average closer to the E_ox than the ΔE_v values. For example, BF₄⁻ has a hardness that is well within the range of the E_ox values, despite having the highest difference between η values calculated by CBS and ΔCBS. BF₄⁻ is also the only anion with a higher ΔE_v than η. ClO₄⁻ is the only anion for which all E_ox values are larger than the hardness.

The E_HOMO values have the same issues for the halides as the ΔE_v values, i.e. wrong ordering. Unfortunately, it is not the same ordering, i.e. Cl⁻ > F⁻ > Br⁻. Looking at the E_ox values, the E_HOMO value for BF₄⁻ is very large, −10.45 eV, far larger than expected. It is not clear what causes BF₄⁻ to stand out for all measures.

3.2 ΔCBS

In Table 2 we show the ΔCBS values only for the larger anions for which CBS was not viable. A comparison vs. experimental data show that the ΔE_v approach tends to overestimate the oxidation potentials. This includes the BOB (by 1.63 V) and MOB (by 1.09 V) anions – with BOB correctly described to be more stable. Furthermore, some values are very close to the measured E_ox with Im(BF₃)₂⁻ as the stellar example (only 0.07 V difference). For some other anions, a definite comparison is not possible, due to the large variations in E_ox values. These include TFSI, where the ΔE_v value is close to some of the E_ox values. Both AsF₆⁻ and PF₆⁻ have ΔE_v values close to the E_ox maximum values.

Table 2 Anions with ΔCBS results for ΔE_v and η. All values are given in V vs. Li⁺/Li, except η and E_HOMO which are in eV

Anion	ΔEv	η	E HOMO	E ox
				Experimental
N5^−	4.54	5.80	−6.10	—
NCN2^−	2.86	4.44	−4.82	—
AsF6^−	8.38	7.02	−12.13	6.5;^28 6.8, 8.6^15
PF6^−	8.57	7.21	−11.67	6.3;^28 6.8, 8.4^15
C(CN)3^−	2.78	3.81	−4.60	—
CH3COO^−	1.48	2.99	−5.10	—
N5C2^−	4.27	4.62	−5.70	—
Tf	5.31	5.04	−7.50	5.0, 5.9;^28 6.0, 7.0^15
FSI	5.34	5.24	−8.66	>4.5^50
N5C4a^−	4.32	5.51	−5.78	>4.0^51,52
N5C4b^−	4.05	5.01	−5.55	—
N5C6a^−	4.45	5.02	−6.04	—
N5C6b^−	4.08	5.37	−5.49	—
BOB	6.13	5.30	−8.81	4.5^53
N5C8^−	4.20	4.47	−5.80	—
TDI	4.02	5.21	−5.54	4.80^54
TFSI	6.12	5.62	−8.60	5.3, 6.1;^28 6.3, 7.1;^15 5.40^55
Im(BF3)2^−	4.92	4.53	−6.32	4.85^16
MOB	5.49	5.23	−8.72	4.4^56
N5C10^−	4.59	—	−6.24	—
B(CH3)4^−	3.01	3.57	−5.34	2.9^15

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Overall, the hardness has qualitatively similar predictive power, as the ΔE_v values. In general, the η values are close to the E_ox values. Starting with the anions with single E_ox values, the η values are slightly higher, however, the difference is not as large as for the ΔE_v values. BOB and MOB have the largest differences, both 0.8 V, between E_ox and η. The η values of TFSI, Tf, PF₆⁻ and AsF₆⁻ are all within the range of reported E_ox values. Looking at the relative stability of BOB and MOB, we note that the difference between the ΔE_v values (0.6 V) is far higher than the η values (0.1 eV), the latter being closer to the E_ox difference.

The E_HOMO values show an unexpected ordering of oxidative stability; the third most stable anion, is the BOB anion (−8.81 eV), more stable than TFSI (−8.60 eV) and TDI (−5.54 eV), despite both having higher E_ox values than BOB. It is also notable that E_HOMO is the only measure that indicates AsF₆⁻ to be more stable than PF₆⁻, as obtained by the maximum E_ox values.

3.3 Functional dependency

The analysis of the DFT functional performance is discussed first for ΔE_v and subsequently for η and grouped by functional type.

The ΔE_v values from DFT and HF in Table 3 in general underestimate ΔE_v – using ΔCBS as reference – with an average deviation negative for HF and all functionals (except M06-2X, +0.01 V), as seen in Fig. 2.

Table 3 Vertical oxidation potential (ΔE_v) of anions from selected DFT functionals and HF. ΔCBS values given for comparison. All values in V vs. Li⁺/Li. The average and standard deviations are comparative to the ΔCBS values

Anion	ΔCBS	B2PLYP	B3LYP	HF	M06-L	M06-2X	mPW2PLYP	PBE0	TPSSh	VSXC
Br^−	2.10	1.91	2.13	1.02	1.85	2.03	1.93	2.05	2.06	2.02
Cl^−	1.99	2.02	2.26	1.02	2.26	2.23	2.02	2.17	2.15	2.20
F^−	2.25	1.84	2.03	−0.21	1.70	1.71	1.80	1.76	1.78	2.01
HS^−	0.93	0.62	0.87	−0.33	0.74	0.84	0.63	0.77	0.76	0.81
HF2^−	4.97	4.72	4.60	4.94	4.05	5.12	4.81	4.56	4.22	4.09
NO3^−	2.74	2.30	2.64	3.31	2.19	3.22	2.45	2.63	2.31	2.11
OCN^−	2.35	2.20	2.23	0.95	2.05	2.37	2.21	2.16	2.05	2.00
SCN^−	2.22	1.96	2.08	0.95	1.95	2.19	1.98	2.05	1.99	1.95
BF4^−	6.35	6.43	5.98	6.05	5.36	6.85	6.58	6.03	5.52	5.31
BrO3^−	3.54	3.19	3.31	2.95	2.89	3.61	3.27	3.28	3.05	2.97
ClO4^−	4.36	4.20	4.20	4.08	3.80	4.74	4.31	4.21	3.88	3.72
HCO2^−	2.85	2.14	2.31	2.13	1.91	2.57	2.23	2.24	2.03	1.94
BH4^−	3.23	3.00	3.11	2.27	2.80	3.10	3.02	3.01	2.96	2.70
N5^−	4.54	4.15	4.26	4.72	3.81	4.72	4.25	4.26	3.95	3.74
N(CN)2^−	2.86	2.67	2.65	1.37	2.49	2.88	2.68	2.62	2.47	2.38
AsF6^−	8.38	7.22	7.00	9.16	6.26	8.24	7.46	7.11	6.47	6.19
PF6^−	8.57	6.95	6.66	8.94	5.96	7.85	7.16	6.78	6.16	5.93
C(CN)3^−	2.78	2.53	2.60	1.27	2.45	2.80	2.55	2.58	2.44	2.34
CH3COO^−	1.48	2.06	2.20	0.72	1.83	2.28	2.14	2.16	1.91	1.82
N5C2^−	4.27	4.10	4.08	2.48	3.94	4.28	4.11	4.06	3.93	3.85
Tf	5.31	4.90	3.97	5.39	3.60	5.41	5.02	4.96	3.69	3.60
FSI	5.34	5.10	5.08	4.92	4.65	5.67	5.20	5.11	4.76	4.59
N5C4a^−	4.32	4.10	4.02	2.23	3.86	4.26	4.11	4.01	3.87	3.78
N5C4b^−	4.05	3.86	3.87	2.58	3.71	4.10	3.88	3.86	3.72	3.62
N5C6a^−	4.45	4.19	4.19	2.85	4.04	4.44	4.21	4.19	4.05	3.94
N5C6b^−	4.08	3.78	3.77	2.27	3.61	4.02	3.80	3.77	3.63	3.52
BOB	6.13	4.80	4.77	5.70	4.08	5.67	4.99	4.85	4.32	4.06
N5C8^−	4.20	3.95	3.97	2.70	3.81	4.23	3.97	3.98	3.83	3.71
TDI	4.02	3.80	3.79	2.32	3.60	4.05	3.82	3.77	3.64	3.57
TFSI	6.12	5.39	5.40	4.74	4.46	5.79	5.45	5.37	4.59	4.50
Im(BF3)2^−	4.92	4.76	4.81	3.47	4.68	4.99	4.77	4.77	4.63	4.66
MOB	5.49	5.06	4.92	5.56	4.24	5.60	5.18	4.99	4.47	4.21
N5C10^−	4.59	4.26	4.26	2.95	4.09	4.55	4.28	4.28	4.11	3.99
B(CH3)4^−	3.01	2.65	2.55	2.08	2.36	2.84	2.69	2.52	2.39	2.21
Avg. dev.		−0.35	−0.36	−0.92	−0.70	0.01	−0.29	−0.35	−0.62	−0.73
Std. dev.		0.39	0.49	0.81	0.65	0.29	0.36	0.42	0.60	0.65

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Fig. 2 The deviation, δΔE_v of the ΔE_v value from the ΔCBS reference value for each of the methods. Average and standard deviations are also shown.

Starting with HF, one notable issue is the values for F⁻ and HS⁻, both negative (vs. Li⁺/Li). In practice this means that HF predicts that these anions would reduce Li⁺. HF also has the highest average and standard deviations. Notably, the Hückel anions (N₅C_2n⁻, excluding N₅⁻) have far lower values than the ΔCBS reference, with the difference ranging from 1.47 V to 2.09 V. The highly symmetric and quite special anion N₅⁻ is, however, slightly higher in energy than the reference, by 0.18 V. Other notable anions are AsF₆⁻ and PF₆⁻, higher by 0.78 V and 0.38 V, respectively.

Each of the functionals used have different issues, a few observations, grouped by functional complexity, are here highlighted.

Starting with the non-hybrid functionals, VSXC has the largest average deviation (−0.73 V) of the DFT functionals. It underestimates the ΔE_v of a few anions by more than 2 V: PF₆⁻ is underestimated by 2.65 V, AsF₆⁻ by 2.19 V, and BOB by 2.07 V. A few other anions are underestimated by more than 1 V: Tf, TFSI, MOB and BF₄⁻. Two anions have their ΔE_v overestimated compared to the reference; Cl⁻ (by 0.22 V) and CH₃COO⁻ (by 0.33 V). M06-L has the second highest average deviation (−0.70 V) of the DFT functionals. The problematic anions are the same as for VSXC with only slightly different values.

The hybrid functionals have better performance characteristics, however, B3LYP has the same problematic cases as VSXC and M06-L. Though, as the average deviation implies, the errors are smaller in magnitude; PF₆⁻ is underestimated by 1.91 V and AsF₆⁻ by 1.38 V and most others anions are below 1 V in difference. However, the overestimates of CH₃COO⁻ and Cl⁻ are somewhat large, 0.72 V and 0.28 V, respectively. PBE0 has similar performance characteristics as B3LYP – including the same problems with PF₆⁻, underestimated by 1.79 V, BOB by 1.28 V, and AsF₆⁻ by 1.27 V.

TPSSh performs somewhat worse than B3LYP. It is also the only hybrid functional that has any value that is underestimated by more than 2 V: PF₆⁻ by 2.41 V. The deviations for AsF6 and BOB are also large: 1.91 V and 1.81 V, respectively.

M06-2X has the distinction of having the lowest deviation of all of the functionals. However, it has a number of overestimates such as CH₃COO⁻ by 0.80 V, BF4 by 0.50 V, and NO₃⁻ by 0.48 V. Furthermore, the underestimates are low; PF₆⁻ 0.72 V, F⁻ 0.53 V, and BOB 0.46 V. It is also notable that AsF₆⁻ is underestimated by only 0.14 V. Interestingly, the BF₄⁻ value of 6.85 V is closer to the CBS value (7.03 V), than the ΔCBS value (6.35 V).

The two double hybrid functionals have very small deviations, as mPW2PLYP has the second lowest average deviation (−0.29 V) and only has two values that differ by more than 1 V: PF₆⁻ and BOB are underestimated by 1.41 V and 1.14 V, respectively. B2PLYP is similar to mPW2PLYP, however, it has a slightly higher deviation; −0.35 V. It also has three anions that differ by more than 1 V from the reference: PF₆⁻ is underestimated by 1.62 V, AsF₆⁻ by 1.16 V, and BOB by 1.33 V.

When the results from all the DFT functionals are taken together, a few points become apparent. First, a larger number of electronegative groups present in the anion structure, tends to cause worse results. Thus heavily fluorinated molecules in general have ΔE_v values underestimated by 1–2 V. This is readily apparent for the PF₆⁻ and AsF₆⁻ anions. This is also an issue for BOB, but to a less extent for the MOB variant, which has the less electronegative malonato moiety. This could be an issue for future ab initio work on novel electrolyte anions, as they tend to be themed around strongly electronegative groups.

Second, while DFT on the whole results in too low ΔE_v values, the ΔE_v value for CH₃COO⁻ is in general overestimated from 0.5 V to 0.8 V. The ΔE_v for Cl⁻ is also often overestimated.

Third, in general the performance of the DFT functionals follows the Jacob's ladder of DFT.⁵⁷ The hybrid functionals perform better than the local functionals, VSXC and M06-L, while the double-hybrid functionals, mPW2PLYP and B2PLYP, perform even better. There is one functional that performs far better, M06-2X, though that may be a fortunate cancellation of errors, as this functional is heavily parameterized.

Moving on to the chemical hardness measure, on average the DFT functionals overestimate the hardness (Table 4 and Fig. 3), as all average deviations are positive. Furthermore, with the large standard deviations, there will be outliers for each functional. Each functional must be inspected, following the same analysis layout as above. Starting with VSXC and M06-L, these have the same outliers, albeit differing slightly in magnitude. Both overestimate the F⁻ and HF₂⁻ by over 2 eV, and also have issues with CH₃COO⁻ and Cl⁻. The functionals in general underestimate the hardness of the Hückel anions (N₅C_2n, n = 2…), and BOB, by 0.5 eV or more. Furthermore, both functionals underestimate PF₆⁻ and AsF₆⁻, M06-L by 0.89 eV and 0.75 eV and VSXC by 1.11 eV and 1.20 eV, respectively. Interestingly, the average deviations for the two functionals are small, 0.13 eV for VSXC and 0.39 eV for M06-L.

Table 4 Hardness (η) of anions from selected DFT functionals and HF. ΔCBS values given for comparison. All values in eV. The average and standard deviations are comparative to ΔCBS values

Anion	ΔCBS	B2PLYP	B3LYP	HF	M06-L	M06-2X	mPW2PLYP	PBE0	TPSSh	VSXC
a N5C10^− has no reference value and is thus neglected in the statistics.
Br^−	4.36	4.92	4.93	4.69	4.86	4.88	4.94	4.90	4.94	5.06
Cl^−	4.89	5.75	5.75	5.50	5.83	5.74	5.76	5.71	5.73	5.43
F^−	6.07	8.47	8.42	7.83	8.33	8.30	8.46	8.30	8.31	8.58
HS^−	2.92	4.17	4.16	3.87	4.14	4.12	4.18	4.10	4.10	4.14
HF2^−	5.44	8.13	7.89	8.69	7.85	8.25	8.19	7.94	7.78	7.78
NO3^−	4.30	5.47	5.50	5.88	5.58	5.80	5.51	5.55	5.34	5.37
OCN^−	4.07	4.73	4.66	4.26	4.80	4.73	4.73	4.63	4.59	4.44
SCN^−	3.66	4.23	4.20	3.91	4.33	4.27	4.24	4.19	4.18	4.02
BF4^−	6.06	6.76	6.43	6.74	6.29	6.94	6.82	6.48	6.25	6.00
BrO3^−	4.55	4.87	4.86	4.83	4.85	5.08	4.91	4.84	4.76	4.65
ClO4^−	5.06	5.81	5.72	6.07	5.97	6.08	5.86	5.78	5.65	5.43
HCO2^−	3.81	5.10	4.77	4.86	4.74	4.95	5.14	4.73	4.64	4.52
BH4^−	3.85	5.10	5.07	4.88	5.04	4.96	5.11	5.01	5.01	4.75
N5^−	5.80	6.61	6.49	7.09	6.48	6.86	6.64	6.57	6.39	6.39
N(CN)2^−	4.44	4.51	4.43	4.05	4.65	4.56	4.52	4.43	4.39	4.11
AsF6^−	7.02	6.54	6.29	7.86	6.27	7.16	6.64	6.42	6.11	5.83
PF6^−	7.21	6.91	6.65	8.15	6.32	7.36	7.00	6.75	6.32	6.10
C(CN)3^−	3.81	4.19	4.16	3.75	4.39	4.28	4.20	4.17	4.12	3.81
CH3COO^−	2.99	4.29	4.28	3.71	4.31	4.35	4.33	4.26	4.16	4.04
N5C2^−	4.62	5.07	4.99	5.00	5.45	5.14	5.08	4.98	4.94	4.75
Tf	5.04	5.67	5.11	6.05	5.19	5.97	5.72	5.62	5.01	4.85
FSI	5.24	5.56	5.44	5.80	5.61	5.89	5.60	5.52	5.35	5.12
N5C4a^−	5.51	5.02	4.78	4.33	4.79	4.99	5.00	4.80	4.75	4.74
N5C4b^−	5.01	5.06	4.85	4.74	4.86	5.09	5.05	4.88	4.82	4.37
N5C6a^−	5.02	4.68	4.48	4.31	4.47	4.69	4.68	4.50	4.45	4.43
N5C6b^−	5.37	4.65	4.44	4.14	4.44	4.65	4.64	4.46	4.41	4.39
BOB	5.30	5.05	4.87	5.76	4.69	5.37	5.12	4.94	4.69	4.48
N5C8^−	4.47	4.32	4.11	3.95	4.09	4.33	4.31	4.13	4.08	4.05
TDI	5.21	4.74	4.54	4.26	4.54	4.74	4.73	4.55	4.51	4.49
TFSI	5.62	5.53	5.38	5.51	5.15	5.78	5.55	5.41	5.01	4.83
Im(BF3)2^−	4.53	5.06	4.99	4.57	5.39	5.17	5.05	4.98	4.94	5.02
MOB	5.23	5.23	4.98	5.75	4.78	5.39	5.27	5.06	4.81	4.58
N5C10^−	—^a	4.30	4.08	3.87	4.05	4.30	4.29	4.10	4.04	4.01
B(CH3)4^−	3.57	4.21	4.07	4.15	4.32	4.27	4.23	4.08	4.04	3.77
Avg. dev.		0.49	0.35	0.45	0.39	0.61	0.52	0.38	0.26	0.13
Std. dev.		0.78	0.82	0.89	0.86	0.74	0.78	0.79	0.83	0.90

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Fig. 3 The deviation of the hardness from the ΔCBS reference value for each of the methods. Average and standard deviations are also shown.

Looking at the hybrid functionals; B3LYP, M06-2X, PBE0, and TPSSh, all reveal the same pattern as for the VSXC and M06-L functionals, e.g. an overestimated hardness for F⁻ and HF₂⁻, in each case by ≥2 eV. However, the underestimates of the PF₆⁻ and AsF₆⁻ anions are much reduced for all the hybrid functionals (except TPSSh). M06-2X has even a slight overestimate of the PF₆⁻ and AsF₆⁻, by 0.16 eV and 0.14 eV, respectively.

The double hybrid functionals, B2PLYP and mPW2PLYP, have some of the largest average deviations of all the functional of Table 4, however, they have some of the smallest standard deviations, implying that they tend to overestimate the hardness. They also have many of the same outliers as the other functionals, in particular F⁻ and HF₂⁻, which are overestimated by ≥2.4 eV. Furthermore, Cl⁻ and CH₃COO⁻ continue to have some overestimation (≈0.8 eV and 1.3 eV, respectively). The Hückel anions (N₅C_2n, n = 2…), continue to have some underestimation. However, the magnitude of the underestimate for PF₆⁻ and AsF₆⁻ is smaller than for most of the hybrid functionals, with 0.30 eV and 0.38 eV for B2PLYP and 0.21 eV and 0.38 eV for mPW2PLYP, respectively.

Comparing the deviations from the reference values for both approaches, ΔE_v (Fig. 2) and η (Fig. 3), it must be noted that the magnitude of the deviations of η are far larger (note the figure scales). Furthermore, looking at Fig. 1 and 2 of the ESI,† show that while DFT can tackle ΔE_v calculations, the same is not clear for η, due to the low correlation with the ΔCBS values.

4 Conclusions

For the calculation of the stability vs. oxidation for the present set of anions, using the methodology of ΔE_v, a hybrid (or better yet, double hybrid) functional is recommended. Their resulting ΔE_v values are closer to the benchmark, surpassing the previously recommended VSXC functional of older works.^26,27 Out of the DFT functionals tested, M06-2X performs the best.

Using the E_HOMO measure to predict the stability vs. oxidation is debatable, due to the unexpected ordering of the anions' oxidative stability. Furthermore, neither the E_HOMO nor the ΔE_v approaches were able to predict the expected ordering of F⁻ > Cl⁻ > Br⁻.

Hardness is another interesting property to study in connection with oxidation potentials, though the origin of its strong correlation with E_ox is not clear. Out of the many DFT functionals tested, no strong general recommendation can be made, however, the double hybrids and M06-2X are candidates for the calculation of hardnesses.

Acknowledgements

We would like to thank the Swedish Energy Agency for funding via a VR/STEM grant and SNIC for the computational resources. Prof. Johan Scheers is thanked for his helpful comments and questions. Support to Patrik Johansson from Chalmers Areas of Advance Transport is gratefully acknowledged.

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Footnotes

† Electronic supplementary information (ESI) available: A table showing the extrapolation details of the CBS approach, the ΔE_v and η values for a number of additional anions and functionals. See DOI: 10.1039/c4cp04592k

‡ Illustrating this, moving from TZ (230 basis functions) to 5Z (635 basis functions) for BF₄⁻, increases the CPU time needed by a factor 100.

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