Thermochemistry of neutral and anionic sulfur fluorides SF_n (n = 1–6): Revisited with theoretical G4/W1/W2 composite methods and the roles of metastable conformer of S$F4−$ anion

Sulfur fluoride (SF_n) compounds are involved in many important chemical applications as insulating dielectrics,¹ plasma etchants,² precursors of fluorine chemistry,³ and stratospheric chemicals.⁴ Therefore, reliable information on the thermochemical properties of these compounds, such as their electron affinity (EA) and bond dissociation enthalpy (⁠$D298K0$⁠), is an indispensable prerequisite for further related experimental and theoretical studies.

Studies conducted on the properties of SF_n species with n = 1–6 have a long history, as summarized four decades ago.⁵ Systematic theoretical studies^6,7 by density-functional-theory (DFT) methods were soon augmented with a more elaborate method, i.e., coupled-cluster single–double–triple [CCSD(T)], which includes extrapolation to the complete basis-set (CBS) limits.⁸ Intensive and elaborate experimental studies of the thermochemical properties of SF_n systems were actively conducted by numerous research groups during the 1980s and 1990s, and all of the results up to early 2000 were well summarized and assessed by Miller and co-workers;⁹ they compared the experimental results with theoretical data generated through composite procedures employing quantum chemical computations on the basis of the Gaussian-2 and Gaussian-3 (G2/G3) theories.^10,11 Miller and co-workers estimated the accuracy of their results by the G3 method within 0.1 eV for sulfur fluorides excluding SF₆.⁹ A few further theoretical studies with other variants of more elaborate theoretical methods were conducted thereafter^12,13 but were limited to either the simplest system, i.e., sulfur monofluoride SF,¹² or the neutral systems of SF_n only.¹³ 

Several noticeable discrepancies between the finest theoretical and experimental results still remain unexplained until now. For example, the discrepancy between theory and experiment is as large as approximately 0.7 eV for the bond dissociation of a neutral F atom from neutral and anionic SF₅ systems and a little over 0.4 eV for the bond dissociation of neutral SF₃ and anionic SF₄ systems. After the studies employing CCSD(T)-CBS computations⁸ and the G3 composite method,⁹ no further full-scale systematic and extensive studies at a higher level of theory have been conducted for the entire range of SF_n species with n = 1–6. A noticeable exception is the very extensive studies of the $SF6−$ anion.^14–16 The difference between the results obtained by the G3 method and the corresponding value from the CCSD(T)-CBS method is larger than 0.1 eV in many cases for the EA and $D298K0$⁠, and the theoretical limit of neither model can be regarded as converging within ±4 kJ/mol of chemical accuracy. Considering the ever-increasing accuracy of experimental techniques and high-level theoretical methods, the current theoretical and experimental data on the thermochemical properties of SF_n should be resolved or further explained properly for not only academic studies but also industrial applications of sulfur fluoride compounds.

To augment the previous theoretical studies^6,8,9,12,13 and provide additional insight into explaining the discrepancy between theory and experiment and the reasons underlying the large error bound of the experimental results, we carried out additional quantum chemical calculations on the thermochemical parameters of SF_n systems by applying the Gaussian-4 (G4)¹⁷ and Weizmann-1 and Weizmann-2 (W1/W2)^18–20 composite procedures as theoretical methods in the present work. The present detailed investigations of the potential energy surface of SF_n systems reveal that the existence of a metastable structure of the $SF4−$ anion and its roles should not be neglected in studying not only the $SF4−$ anion but also other related neutral and anionic SF_n chemicals. As discussed below, several previous discrepancies between the theoretical and experimental results can be resolved or explained, and the reasons underlying the large uncertainties of several experimental results can also be understood by alluding to the existence of the conformer of the $SF4−$ anion.

Initial studies to find possible metastable structures were carried out by applying the B3LYP functional in hybrid density-functional-theory (DFT)²¹ in conjunction with the 6-31G(2df,p)²² and cc-pvTZ+d²³ basis sets, mainly because the B3LYP/6-31G(2df,p) and B3LYP/cc-pvTZ+d methods are used in the geometry optimization step of the G4 and W1 procedures, respectively.^17,18 Our careful examinations at the middle stages of all geometry optimizations of the neutral and anionic SF_n (n = 1–6) systems ended up in virtually the same conclusions for the SF₂ and SF₃⁻ anions, as discussed earlier,^6,9 but produced somewhat different results for the metastable conformational isomer of the SF₄⁻ anion with C_2V symmetry. An earlier theoretical study already acknowledged the structure of the SF₄⁻ anion with C_2V symmetry⁶ but did not pay any further attention to any possible implication of this structure. All the subsequent theoretical and experimental studies neglected the possibility of any involvement of the metastable structure with C_2V symmetry and considered the role of the global minimum structure of the SF₄⁻ anion with C_4V symmetry only. In these studies, the metastable structure may have been considered an artifact of computational methods, or it was expected that the barrier height from the metastable state to the global minimum would be low enough, and thus, any significant role of the metastable structure was neglected. To check for possible artifacts in the results of the B3LYP calculations, the details of the metastable structure were examined again by applying the wB97XD functional²⁴ with the 6-311+G(3df) basis set²⁵ and the many-body second-order perturbation theory (MP2) method²⁶ with the cc-pvQZ+d basis set.²⁷ The coupled-cluster singles and doubles with noniterative triples, CCSD(T), method²⁸ with the frozen-core molecular orbital approximation²⁹ implemented in the ACES-II suite of the program package³⁰ was also applied herein. The structure of the transition state (TS) between the metastable state and the global minimum and the barrier height of the TS were also studied, confirming that the observation of the new conformer is not an artifact of the computational methods. The barrier height is also not negligible. The bond length, bond angles, and harmonic frequencies of the three configurations (the metastable, the global minimum, and TS structures) are given in Table I, and the important features are depicted in Fig. 1.

The thermochemical properties of SF_n with n = 1–6, including the new conformer of $SF4−$⁠, were then calculated by the G4 method implemented in the Gaussian-16 (G16) suite of the program package.³¹ As demonstrated in Sec. III, the results obtained with the G4 method do not match closely enough with the corresponding results from CCSD(T)-CBS reported in the previous work,⁸ and there is no assurance that the results obtained by the G4 method were sufficiently converged. For further verification, therefore, the W1 methodology implemented in G16 was applied again herein to all systems of SF_n. Both the W1U and W1BD variants¹⁹ of W1 theory implemented in G16 were used in our computations. The differences between the results obtained with the two variants of W1 theory, however, are very small in our cases, and only the results obtained with the W1BD variant are included in the results below. Because the size of the basis sets used in the geometry optimization step of the G4 and W1 methods is large enough, the problems caused by different geometries in the computation of the zero-point-energy (ZPE) and higher-level correlation energies were not encountered in our calculations with the G4 and W1 methodologies, even for the $SF6−$ and $SF2−$ anions, as discussed in earlier theoretical studies employing the G2/G3 methods.^6,9 Except for the fact that the reference geometry and vibration frequencies of anionic $SFn−$ were calculated with not B3LYP/cc-pvTZ+d but B3LYP/Aug-cc-pvTZ+d for better description of anionic systems, all of the remaining computational steps for extrapolation to the one-electron basis-set limit and electron correlations and possible relativistic effects were carried out through the standard procedure of the W1 methods as implemented in G16.³¹ Only one technical point deserves to be mentioned here, which is that the “opt = z-matrix” keyword has to be given in addition to the “G4” or the “W1BD” keyword when the metastable structure of the SF₄ anion is treated. Otherwise, the straightforward use of just the “G4” or the “W1BD” keyword alone produces results that do not correspond to the metastable state, but to the global minimum structure, partially because the barrier height between the two isomeric nuclear configurations is underestimated to be as small as ∼13 kJ/mol by the DFT method in the present case.

The results of the computations employing W1 theory show that in the anionic SF_n species, some bond lengths are longer than the usual S–F bonds and a few vibrational frequencies are notably small. This suggests that the potential energy surfaces of the systems are very shallow, and the effect of the electron-correlation level and the size of the basis sets may be larger than the usual cases. Thus, the results obtained with both G4 and W1 methodologies may not be sufficiently reliable because the reference geometry for treating the electron-correlation effect at a higher level of theory is optimized by the B3LYP/6-31G(2df,p) and the B3LYP/cc-pvTZ+d method, respectively. To check the possibility of further improving the theoretically calculated results, the geometrical structures and thermochemical properties (EA and $D298K0$⁠) were calculated again by applying the W2 theory, which is even one-step higher in the hierarchy of composite computational procedures. The reference geometry was optimized by the CCSD(T)/cc-pvQZ+d method in the W2 methodology. A series of computation steps comprising W2 theory were carried out using the MOLPRO suite of programs,³² with a few marginal modifications as follows. First, the geometry was optimized and the harmonic frequencies were subsequently calculated by the B3LYP/cc-pvTZ+d method, and the zero-point-energy (ZPE) correction and the thermal energy contribution to the enthalpy at 298 K were obtained with a scaling factor of 0.985 for practical reasons, as suggested elsewhere.²⁰ The reference geometry for the remaining steps of the W2 procedure, however, was determined again by the CCSD(T) method³⁰ with the cc-pvQZ+d basis set.²⁷ The core electrons corresponding to the 1s atomic orbital (AO) of the F atom and the 1s, 2s, and 2p AOs of the S atom were all frozen²⁹ in the post-HF steps of the CCSD(T) method.²⁸ All other subsequent steps were carried out as per original W2 theory,¹⁸ except for the determination of the HF-limits, which were obtained by the two-point extrapolation method suggested elsewhere.²⁰ The magnitude of the spin–orbit coupling (SOC) effect in the F (²P), S (³P), S⁻ (²P), SF (²Π), and SF₂⁻ (²Π_u) systems was calculated by the MRCI method³³ implemented in MOLPRO.³² All electrons, except for the two 1s-electrons of the sulfur atom, were included in the MRCI calculations with the cc-pvQZ+d basis set for the SOC of the F, S, S⁻, SF, and SF₂⁻ systems. The same magnitudes of the SOC were also used in the calculation of the final energy using the W1 procedure.

The bond length, bond angles, and harmonic frequencies of all the neutral and anionic SF_n (n = 1–6) species calculated using B3LYP/(cc-pvTZ+d for neutrals and Aug-cc-pvTZ+d for anions) and CCSD(T)/cc-pvQZ+d in the W1 and W2 procedures, respectively, are shown in Tables S-1 and S-2 of the supplementary material. The components used in the computations of the final total energies in the W1 and W2 procedures are also summarized in the supplementary material; see Tables S-3 and S-4, respectively.

The total energies at 0 K (E₀ = E_e + ZPE), including the zero-point-energy (ZPE) correction, and the enthalpies at 298 K (⁠$H298K0$⁠) (including the thermal energy contributions calculated by the G4, W1, and W2 methods) are all summarized in Table II. The adiabatic electron affinity (EA) was calculated from the difference between the corresponding total energies by applying Eq. (1), whereas the bond dissociation enthalpies (⁠$D298K0$⁠) of the three dissociation pathways, defined in Eqs. (2)–(4) as in a previous work,⁹ were obtained using the related values of $H298K0$ given in Table II,

Note that $D298K0$ (SF_n-1–F⁻) and $D298K0$ (F–$SFn−1−$⁠) correspond to the dissociation of a F⁻ anion and a neutral F atom, respectively, from anionic $SFn−$⁠, whereas $D298K0$ (SF_n-1–F) indicates the dissociation of a neutral F atom from neutral SF_n. Although the bond dissociation energies at 0 K are not included here, they can be calculated with ease by combining the corresponding magnitudes of E₀(0 K) in Table II. The magnitudes of the EA and $D298K0$ calculated using different theoretical methods in the present work (i.e., the G4/W1/W2 methods) are collated in Table III, with previous comparable theoretical results.^6,8,9,12,13 The experimental values^5,34–41 summarized by Miller et al.⁹ are also included for comparison and discussion.

The electronic states of the chemical species, as well as the symmetry of their molecular geometries, are shown in the first column of Table II. Additional quantum chemical calculations indicate that the other possible electronically excited states, including different spin-multiplicities, do not affect the results and discussions of the present work. The main features of the molecular geometries of neutral and anionic SF_n (n = 1–6) are well illustrated in Figs. 1–6 of the previous theoretical work with several combinations of DFT functionals and basis sets.⁷ The present results obtained by the B3LYP/cc-pvTZ+d method are basically the same as those in Ref. 7, except for marginal differences in the magnitude. Herein, we refrain from discussing the small differences in the magnitudes of the geometrical parameters because the small differences have little impact on the thermochemical properties in the present work. A new metastable conformational isomer of the $SF4−$ anion, however, is a notable exception.

The existence of the metastable structure of the SF₄⁻ anion with C_2V symmetry was already mentioned in an earlier report,⁶ but only the global minimum structure with C_4V symmetry was considered in theoretical and experimental studies of SF_n thereafter. The barrier height between the metastable conformer and its global minimum was probably considered small enough for the role of the metastable structure to be neglected. We, however, have found that the new metastable stationary structure of the $SF4−$ anion can exist separately with a notable energy barrier. Not only the bending modes but also the asymmetric bond stretch of the F^ShotSF^Opp moiety is required to overcome the barrier height, as shown in Fig. 1. F^Shot stands for the F atom with the shortest S–F bond length, whereas F^Opp is the F atom at the opposite side of F^Shot. The geometrical structure of the metastable isomer, determined by both the B3LYP/cc-pvTZ+d and wB97XD/6-311+G(3df) methods, corresponds to the C_S symmetry because the bond lengths are R(S–F^Shot) = 1.60 Å–1.63 Å and R(S–F^Opp) = 1.98 Å–1.95 Å from the two DFT calculations, as shown in the middle part of Table I. The two bond lengths, however, were the same, R(S–F^Shot) = R(S–F^Opp) = 1.7075 Å, when the MP2/cc-pvQZ+d method was applied. More extensive computations by applying the CCSD(T)/cc-pvQZ+d method resulted in marginal differences between the two bond lengths: R(S–F^Shot) = 1.7112 Å and R(S–F^Opp) = 1.7129 Å. The correct symmetry of the metastable isomer would be the C_2V point group, but the artifactual symmetry breaking hidden here results in a distorted structure with C_S symmetry, as observed in other systems such as NO₃ and O₄⁺.^42,43 We refrain from a more detailed study of the symmetry breaking problem of the metastable structure, as discussed in previous works,^44,45 partially because the energy difference caused by the artifactual symmetry breaking in the metastable structure of the $SF4−$ anion is approximately ∼2 kJ/mol according to our preliminary study using the CCSD(T)/cc-pvQZ+d method. To emphasize the possibility of the occurrence of the artifactual symmetry breaking, the symmetry of this new metastable isomer is represented by C_S-2V hereinafter.

From calculations using the B3LYP/cc-pvTZ+d method, the energy of the new metastable structure with C_S-2V symmetry is 50 kJ/mol higher than that of the global minimum structure, and the magnitude changes to 44 kJ/mol, 43 kJ/mol, and 39 kJ/mol when computed using the wB97XD/6-311+G(3df), MP2/cc-pvQZ+d, and CCSD(T)/cc-pvQZ+d methods, respectively, with a final value of 42 kJ/mol from the W2 composite method.

The lowest-energy transition-state (TS) structure connecting the two isomers clearly has C_S symmetry, as shown in Fig. 1. The F^Opp atom in the metastable structure with C_S-2V symmetry now becomes the F^Long atom in the TS structure with C_S symmetry. We carried out the “following of the intrinsic-reaction-coordinate (IRC-following)” using the only one imaginary frequency of the TS structure and confirmed that the TS connects the metastable and the global minimum structures correctly. The barrier height from the metastable structure to the TS calculated by the B3LYP/cc-pvTZ+d method is ∼13 kJ/mol and was almost the same using the wB97XD/6-311+G(3df) method. The barrier height, however, became notably higher, i.e., 37 kJ/mol and 27 kJ/mol, when the MP2/cc-pvQZ+d and CCSD(T)/Aug-cc-pvQZ+d//MP2/cc-pvQZ+d methods were, respectively, applied. Regardless of the exact magnitude of the barrier height, it is clear that the barrier is large enough to make the two structures with C_S-2V and C_4V symmetry distinguishable conformational isomers, i.e., conformers. The two conformers can play somewhat different roles in the determination of not only the thermochemical properties but also the kinetics and dynamical behaviors of the $SF4−$ anion. The existence of the two isomers will become more evident in the discussions on the thermochemical properties below.

The electron affinity (EA) and bond dissociation enthalpies (⁠$D298K0$⁠) of three dissociation channels, calculated by Eqs. (1)–(4) using the quantities given in Table II, are given in Table III. Though not included in Table III, the EAs of the F and S atoms can be easily calculated from the related E₀(0 K) values in Table II. The values of EA(F) obtained with W1 and W2 are 3.405 eV and 3.403 eV, respectively, which are close enough to the most reliable experimental value of 3.401 189 ± 0.000 003 eV.⁴⁶ The values of EA(S) obtained by the W1 and W2 methods are 2.082 eV and 2.070 eV, respectively, which are also sufficiently close to the most reliable experimental value of 2.077 103 ± 0.000 001 eV.⁴⁷ The errors in the values of EA(F) and EA(S) determined by the W2 method are less than ±0.01 eV. The main goal of the present work is to produce thermochemical properties within the chemical accuracy, ±4 kJ/mol ≈ ±0.04 eV, and the results obtained with the W2 method seem sufficiently reliable here.

The uppermost part of Table III compares the magnitudes of the EA for the SF_n systems obtained herein with the corresponding values from other comparable previous theoretical studies,^8,9,12,13 as well as the reference experimental values summarized by Miller and co-workers.⁹ The improvement in the EA of SF, the simplest diatomic case, calculated using G3/G4 vs W1/W2 is not very significant, with a difference of only −0.03/−0.01 eV = 2.29 eV (W2) − 2.32/2.30 eV (G3/G4). The corresponding improvements for the other SF_n species, however, are more significant and much larger than the claimed error bound, ±0.041 eV, of the G3 method for the EAs;⁴⁸ the differences between the EA obtained with the previous G3/G4 methods and our final W2 method are as follows: −0.15/−0.09 eV (SF₂), −0.12/−0.06 eV (SF₃), −0.22/−0.19 eV (SF₄), −0.11/−0.03 eV (SF₅), and −0.21/−0.21 eV (SF₆). Note that all the changes correspond to an improvement relative to the corresponding reference experimental value, except in the case of SF₅ where the present calculation results in a stronger deviation from the experimental reference value, 4.23 ± 0.12 eV. The experimentally suggested value, however, is not determined by direct experimental measurements but is deduced by combining several observations related to the thermochemical cycle, Eq. (1), in Ref. 9.

For the SF system, the simplest case in the SF_n series, it was shown that the difference between the data obtained herein using W2, 2.29 eV, and the highly accurate experimental value, 2.285 ± 0.006 eV, can be treated by combining a series of computations more elaborately.¹² The elaborate procedure used in the reference is basically the same as that of our W2 computations, and the magnitude of the spin–orbit coupling (SOC) term for neutral SF is the only difference. We also note that very extensive computations with the W4lite theory³⁸ were used for the study of EA of SF₆.¹⁶ However, such further elaborate work for other SF_n species with n = 2–6 is not the focus of the present work, partially because the accuracy we are claiming in the present work is within 0.04 eV and partially because the accuracy of the experimentally suggested reference value of the other SF_n systems is not sufficiently high given the present situation. The W2 results for EA(SF), EA(SF₂), and EA(SF₄) obtained herein are basically the same as the previous results from CCSD(T) computation in conjunction with the complete basis-set limit (CBS) extrapolations.⁸ The changes in the EA calculated with CCSD(T)-CBS vs W2, however, are more significant in the other three cases: 3.10 vs 3.01 eV for SF₃, 4.08 vs 4.00 eV for SF₅, and 0.90 vs 1.00 eV for SF₆. The differences in the values obtained with W1 vs W2, however, were all less than 0.03 eV, and we believe that the error bound of the final W2 results obtained herein would all adequately be within ±0.04 eV, corresponding to a chemical accuracy of approximately 4 kJ/mol, not from the experimentally suggested value but from the ultimate theoretical limit of each system. Thus, we suggest the use of our results as a reference in future studies.

As a final discussion on the EA, it is noteworthy that the calculated EA for SF₄ to form the $SF4−$ anion with C_4V symmetry, 1.42 eV, is sufficiently close to the experimental value, 1.5 ± 0.2 eV, whereas the calculated EA for forming the metastable $SF4−$ anion with C_S-2V symmetry, 1.02 eV, differs notably from the reference experimental value. Figure 2 compares the shape of the highest occupied molecular orbital (HOMO) of the neutral and anionic SF₄ systems. The shape of the HOMO of neutral SF₄ is shown in Fig. 2(a), whereas the shapes of the singly occupied molecular orbital (SOMO) of the two isomeric structures of the $SF4−$ anion with C_S-2V and C_4V symmetry are shown in Figs. 2(b) and 2(c), respectively. Though the shape of the lowest unoccupied molecular orbital (LUMO) of the neutral species is not included in Fig. 2, it is qualitatively very close to that of the SOMO shown in Fig. 2(b). The shapes of the MO in Fig. 2 suggest that the initial stage of electron attachment to neutral SF₄ would be more similar to the metastable structure of the $SF4−$ anion with C_S-2V symmetry, and the global minimum of the anion with the C_4V structure may be attained after some delay to allow for internal motions, starting from the initial C_S-2V structure. The possibility of direct electron attachment from the opposite direction of the HOMO of neutral SF₄, combined with the bending (opening) motion of ∠F^ShotSF^Shot of neutral SF₄, to form the $SF4−$ structure with C_4V symmetry from the beginning cannot be neglected, but such detailed dynamics is out of the scope of the present work. Meanwhile, the shapes of the SOMO of the SF₄⁻ anion in the two isomeric conformations suggest that the isomerization here could be called “orbital isomerization” because the difference in the shape of the SOMO is the only feature distinguishing the two isomers. However, the soundness of using such a new term, “orbital isomerization,” is an open question.

The middle section of Table II shows the magnitude of the bond dissociation enthalpy (⁠$D298K0$⁠) of the two dissociation pathways for the anionic $SFn−$ species. Among the 12 cases for bond dissociation of the six $SFn−$ anions, the change in $D298K0$ calculated using G3 vs W2 was less than 0.03 eV for five cases but as high as 0.07 eV–0.12 eV for the other seven cases, which is much higher than the expected error bound of the G3 method, ±0.041 eV. Not only the difference in the result obtained herein using G4 vs the reported W2 computations but also that between the results obtained with the CCSD(T)-CBS limit⁸ and the present W2 computations was significant in several cases. Meanwhile, the differences in the results obtained herein by the W1 method vs the W2 computations were generally approximately 0.03 eV–0.04 eV, except for four cases of D°_{298 K}(F–SF₂⁻), D°_{298 K}(F–SF₃⁻), D°_{298 K}(F–SF₄⁻), and D°_{298 K}(SF₅–F⁻). We believe that the present W2 results are all within ±0.04 eV of the ultimate theoretical results, except for the four cases. The difference in the results obtained with W1 vs W2 for the four cases was as large as 0.12 eV, and we cannot be sure that convergence of theoretical limits was obtained in these cases.

The error bounds of the experimentally suggested reference values of D°_{298 K}(SF_n-1–F⁻) and D°_{298 K}(F–SF_n-1⁻) are so large that a detailed comparison of our final W2 results with the experimental values is not very meaningful. However, noticeable discrepancies still remain between our final W2 results and the corresponding experimental reference values, even after considering the large error bound of the experiments. Two of the following discrepancies are especially noticeable: 2.19[1.78] eV (W2) vs 1.84 ± 0.16 eV (expt.) for $D298K0$ (SF₃–F⁻) and 4.18[4.59] eV (W2) vs 5.0 ± 0.6 eV (expt.) for $D298K0$ (F–SF₄⁻). Other noticeable points are the large error bound of the reference values for $D298K0$ (F–SF₃⁻), 2.3 ± 0.7 eV, and for $D298K0$ (F–SF₂⁻), 4.4 ± 1.0 eV.

The existence of two conformational isomers of the $SF4−$ anion, with either C_4V or C_S-2V symmetry, provides additional insight into explaining the above discrepancies. The energy for the dissociation of one F⁻ ion from the $SF4−$ anion is calculated to be 2.19 eV or 1.78 eV depending on whether the initial structure of the $SF4−$ anion has C_4V or C_S-2V symmetry. The experimentally suggested reference value of 1.84 ± 0.16 eV suggests that the experimental results may correspond to the dissociation from the metastable structure with C_S-2V symmetry. The experimental conditions of previous studies probably do not correspond to the dissociation of an F⁻ ion from the global minimum structure of the SF₄⁻ anion but from the metastable isomeric structure of the SF₄⁻ anion. The metastable structure can be generated by the thermal excitation of the global minimum structure, or the dissociative electron attachment of neutral SF₄ may proceed through the metastable structure of the anion with C_S-2V symmetry obtained by the initial attachment of an electron to the LUMO of neutral SF₄ because the shape of the SOMO of the $SF4−$ anion with C_S-2V structure is very similar to that of the LUMO of neutral SF₄. Meanwhile, the energy for the dissociation of one neutral F atom from the $SF4−$ anion was calculated to be 2.55 eV or 2.14 eV, depending on the symmetry of the initial structure of the $SF4−$ anion. Both values are within the range of the error bound of the experimental reference value, 2.3 ± 0.7 eV, and the average magnitude of the two values, i.e., (2.55 + 2.14)/2 = 2.345, is sufficiently close to the center of the experimental values. The two dissociation pathways, from the C_4V and the C_S-2V configuration, may both have been involved under the actual experimental conditions. Such mixing of two isomers of the $SF4−$ anion may be the reason why the error bound of the experimental reference value is so large for this case of $D298K0$ (F–SF₃⁻), 2.3 ± 0.7 eV, as well as other reports. We, therefore, can anticipate that the kinetics of the dissociation of an F⁻ anion from $SF4−$ would be faster than that of the dissociation of a neutral F atom from $SF4−$⁠. The larger magnitude of $D298K0$ (F–SF₃⁻) than $D298K0$ (SF₃–F⁻) also matches with the anticipation. Such detailed features of kinetic and dynamical processes, however, cannot be assured at this point and need much more elaborate theoretical and experimental studies.

The dissociation of neutral or anionic F from the $SF5−$ anion may follow two different pathways because the F atom located along the C_4V axis of $SF5−$ could behave differently from the remaining four F atoms. In the case of dissociation to neutral SF₄ and anionic F⁻, the two pathways lead to the same thermochemical results because the dissociated neutral SF₄ has only one stable structure. In the dissociation toward a neutral F atom and $SF4−$ anion, however, the two channels result in different thermochemical situations because of the two different isomeric structures of the $SF4−$ anion. $D298K0$ (F–SF₄⁻) is calculated to be either 4.18 eV or 4.59 eV depending on whether the structure of the remaining $SF4−$ anion has either C_4V or C_S-2V symmetry, respectively. The experimental reference value, 5.0 ± 0.6 eV, suggests that the experimental results may correspond more closely to the dissociation pathway generating the metastable isomer of $SF4−$ with C_S-2V symmetry initially. The theoretical results obtained herein suggest that the previous experiments for the dissociation of both the $SF4−$ and $SF5−$ anions should be repeated or reanalyzed with a careful consideration of the roles of the two isomeric structures of the $SF4−$ anion separately.

As a final remark on the bond dissociation of anionic $SFn−$⁠, the present theoretical value for the dissociation of a neutral F atom from the $SF3−$ anion is 3.95 eV, based on the W2 method, whereas the corresponding experimental value is 4.4 ± 1.0 eV. The wide range of the error bound of the experimental value stems from the fact that the value is determined not by direct measurement but by combining the results of several experiments. We again believe that the result obtained by the present W2 computations is much more reliable because this theoretical value can be easily and directly calculated from the magnitudes of H(298 K) in Table II.

The lowest part of Table II shows the magnitude of the bond dissociation enthalpy (⁠$D298K0$⁠) of neutral SF_n species. A previous theoretical study employing the CCS(DTQ) method¹³ was limited to the dissociation of neutral SF_n systems and is included here for comparison. Notably, the results of the previous work¹³ are located between the corresponding results obtained by the W1 and W2 methods. The convergence from G3 to G4/W1/W2 cannot be said to be straightforward and steady, but the changes shown here suggest that the final values obtained by the W2 method approach sufficiently close to the ultimate theoretical result of each case. Notably, the difference in the values from the G3/G4 and W2 methods is very small, approximately 0.02 eV for SF, SF₂, SF₃, and SF₄. $D298K0$ calculated herein by the W2 procedure seems to be adequately within ±0.04 eV of the ultimate theoretical result for SF_n with n = 1–4. The differences in the values obtained with G3/G4 vs W2 for SF₅ and SF₆, however, are not small: −0.07 eV for SF₅ and −0.13 eV for SF₆. The difference in the calculated $D298K0$ (SF_n⁻–F) values from W1 vs CCS(DTQ) and W2, however, indicates stable convergence. Therefore, we argue that our final results for $D298K0$ (SF_n⁻–F) from the W2 method are all within the chemical accuracy of ±4 kJ/mol ≈ ±0.04 eV.

The experimentally suggested reference values of $D298K0$ (SF_n-1–F) for SF, SF₂, and SF₆ are close enough to the calculated magnitudes from the W2 computations: 3.52 ± 0.09 eV (expt.) vs 3.59 eV (W2) for SF, 3.98 ± 0.19 eV (expt.) vs 3.86 eV (W2) for SF₂, and 4.35 ± 0.10 eV (expt.) vs 4.45 eV (W2) for SF₆. However, noticeable discrepancies still remain between the values for SF₃, SF₄, and SF₅: 2.74 ± 0.31 eV (expt.) vs 2.36 eV (W2) for SF₃, 3.74 ± 0.34 eV (expt.) vs 4.10 eV (W2) for SF₄, and 2.30 ± 0.26 eV (expt.) vs 1.62 eV (W2) for SF₅. The difference between the experimental and theoretical values of $D298K0$ (SF_n-1–F) for SF₅ is especially large. As mentioned in an earlier work,⁹ the experimentally suggested reference values are the results of combining numerous experimental measurements (for example, 56 reactions for the negative ion chemistry of SF₄),⁴⁸ rather than being directly determined values. On the other hand, the theoretical values obtained by the W2 procedure are generated in a more direct and straightforward manner, and we expect the present results to be a very helpful guide in analyzing the previous experimental results or repeating the corresponding experiments.

The thermochemical properties of neutral and anionic systems of SF_n with n = 1–6 were studied by applying G4, W1, and W2 composite procedures of computational theories to augment previous theoretical results by the G2/G3 and the CCSD(T)-CBS methods,^6–9 with special attention to the characteristics of the metastable structure of the $SF4−$ anion. The energy of the metastable structure is 39 kJ/mol higher than that of the global minimum structure with C_4V symmetry, and the barrier height from the metastable state to the global minimum is calculated to be 27 kJ/mol by the CCSD(T)/Aug-cc-pvQZ+d//MP2/cc-pvQZ+d method. It is shown that the existence of the metastable $SF4−$ anion and its possible roles are very important in understanding the experimental results for neutral and anionic SF₄ and SF₅ systems, and many of the previous discrepancies that have persisted between theoretical and experimental studies for almost two decades can be explained better than ever before on the basis of the present theoretical results that consider the metastable isomer of the $SF4−$ anion.

We compared the results obtained by the G2, G3, G4, CCSD(T)-CBS, W1, CCS(DTQ), and W2 methods for the EA and D°_{298 K} of neutral and anionic SF_n. The results obtained by the G4 method are still notably different from the corresponding values obtained by the CCSD(T)-CBS or W1 methods in many cases. The results obtained by the W2 method are thought to have finally attained the proper convergence in the series of theoretical explorations. Notably, the variation between the calculated values from the G3 and W2 methods is much larger than 0.041 eV, the claimed accuracy of the G3 method, in many cases. The differences are especially large for hypervalent SF_n with n = 3–6. After detailed analyses of the differences in the magnitude of the calculated values, we claim that the final results generated herein by applying the W2 composite procedure are now adequately within the chemical accuracy, ±4 kJ/mol ≈ ±0.04 eV, in relation to the ultimate theoretical limits for all individual cases.

For a few cases, however, the differences between the final results obtained herein (W2) and the corresponding experimentally suggested reference values (expt.) are still notable, even when the experimental error bounds are considered: 2.36 eV (W2) vs 2.74 ± 0.31 eV (expt.) for D°_{298 K}(SF₂–F), 4.10 eV (W2) vs 3.74 ± 0.34 eV (expt.) for D°_{298 K}(SF₃–F), 1.62 eV (W2) vs 2.30 ± 0.26 eV (expt.) for D°_{298 K}(SF₄–F), and 4.00 eV vs 4.23 ± 0.12 eV for the EA of SF₅. As mentioned in earlier works, the experimentally suggested reference values are the results of combining numerous experimental measurements, rather than being directly determined values. Quantum chemical calculation, on the other hand, is more straightforward and direct and may be more reliable for determining the thermochemical properties in the present work. Though detailed analysis of the procedures for determining the experimental values was not attempted in the present work, we believe that the error bounds of the experimentally suggested reference values could be further refined by simply considering the involvement of two distinguishable isomers of the $SF4−$ anion.

In addition to the thermochemical properties, a few implications of the new metastable structure of the $SF4−$ anion on the ultrafast kinetics and dynamics of sulfur fluoride systems are briefly mentioned in the present paper. The implications stemming from the existence of two distinguishable isomeric structures of the $SF4−$ anion provide new insights not disclosed or mentioned before. The isomeric anions may have more delicate and complicated involvement in plasma processes with SF_n and other sulfur fluoride systems. Further theoretical and experimental studies on this matter are warranted.

See the supplementary material for the complete contents of the following: Table S-1. Bond lengths and angles by B3LYP/(Aug-)cc-pvTZ+d and CCSD(T)/cc-pvQZ+d of the W1 and W2 procedures, respectively. Harmonic frequencies (in cm⁻¹) are calculated by the B3LYP/(Aug-)cc-pvTZ+d method only. Table S-2. Cartesian coordinates of neutral and anionic SFn systems for n = 3–6 optimized by the CCSD(T)/cc-pvQZ+d method. Table S-3. W1BD energy with and without spin–orbit coupling. Table S-4. Component energies consisting of the final W2 energy.

This work was supported by the National Research Foundation of Korea (Grant No. NRF-2018-R1A2B6008396).

The data that support the findings of this study are available within its supplementary material.