1 Introduction

The existence and energetics of aryl cations has long been a topic of interest among the organic chemistry community [1–6]. In particular, the nature of the ground state multiplicity for the parent and substituted phenyl cations has attracted considerable attention. The parent system is known to have two low-energy minima which corresponds to the 1A1 and 3B1 states, with the intersection between the lowest energy singlet and triplet hypersurfaces lying close to the triplet geometry and energy [7]. Coupled with non-trivial spin–orbit coupling between these two states near the crossing point [8], the triplet phenyl cation is a short-lived intermediate that rapidly decays [7] to the ground singlet state 102 kJ/mol below the triplet minimum [9]. The phenyl cation has been isolated and characterized in argon [10, 11] and LiCl matrices [12] and in the gas phase [13], and indirectly detected in aqueous solution with a lifetime of <500 ps [14].

The closed-shell singlet state of aryl cations has six π-electrons in the ring and an empty in-plane σ-like orbital on the dicoordinate carbon atom carrying the formal positive charge. In the triplet state, the σ orbital contains an unpaired electron and the ring has only five π-electrons [15]. Where substituents are present, the singlet state is stabilized by σ-donors in the positional order efficacy ortho > meta > para and by π-donors in the order para > ortho > meta, while triplet states are stabilized by π-donors in the order para ~ ortho > meta [16]. The differing spin states display varying chemoselectivity in their reactivity with nucleophiles [17, 18]. Singlet phenyl cations react rather unspecifically; triplets prefer unsaturated functional groups (π-nucleophiles) over those with lone pairs (n nucleophiles) [19]. The phenyl cation can also be stabilized via hyperconjugation with high-lying strained carbon–carbon bonds [20].

To better understand and predict experimental behavior, a range of theoretical investigations have considered the magnitude and direction of the singlet–triplet energy gap (ES−T) for the parent (reviewed in Ref. [9]) and substituted aryl cations. However, the prior work has generally been conducted using Hartree–Fock (HF) and density functional theory (DFT) methods, and these methods are known to significantly underestimate the ES−T of organic compounds. Consequently, the use of higher-level (e.g., composite) theoretical methods and/or more modern density functionals is required in order to achieve both qualitative and quantitative ES−T predictivity (see, Refs. [9, 21–27] and references therein). As a result, in the current study we undertook a broad examination of the parent and substituted and benzannelated aryl cations with the high-level G4(MP2) and G4 methods, approaches which should yield ES−T estimates at or near thermochemical accuracy.

2 Computational details

Composite method calculations were conducted at the G4(MP2) [28], G4 [29], and W1BD [30] levels of theory as employed in Gaussian 09 [31]. Additional calculations were performed at various semiempirical, Hartree–Fock, density functional, Moller–Plesset perturbation, and composite method levels of theory with a range of basis sets. Full reference details for these other methods are provided in our previous work [9], with the exception of the MN12L [32], MN12SX [33], M11L [34], SOGGA11X [35], APF and APFD [36], N12SX [33], and HISSbPBE [37] density functionals and the Def2TZV, Def2TZVP, Def2TZVPP, and QZVP basis sets [38, 39] which are cited herein. Dispersion corrections were applied using the D2 [40] and D3 (with Becke–Johnson damping) [41] versions of Grimme’s dispersion approaches. All calculations were conducted either in the gas phase (1 atm.) or solution phase (1 M) at 298.15 K. Solution phase calculations employed the SMD [42], IEF-PCM [43, 44], and C-PCM [45, 46] solvation models. Geometries were visualized using Gabedit 2.4.7 [47] and Avogadro 1.1.1 [48]. Geometry optimizations for all compounds were conducted in both the singlet and triplet state and converged absent imaginary frequencies.

3 Results and discussion

Our studies began with a comprehensive investigation into the effects of model chemistry on the calculated adiabatic singlet–triplet excitation energy (AES−T) of the parent phenyl cation. In prior work [9], we investigated well-to-well singlet–triplet excitation energies (WWES−T) using single point calculations on the B3LYP/TZVP optimized geometry (i.e., WWES−T at the x/TZVP//B3LYP/TZVP level of theory, where x is the model chemistry; semiempirical calculations were at the x//B3LYP/TZVP level). The values reported in Table 1 herein are full geometry optimizations and frequency calculations at the x/TZVP level (or at the semiempirical level), thereby constituting AES−T estimates.

Table 1 Gas phase standard state (298.15 K, 1 atm.) AES−T and singlet/triplet C6–C1–C2 angles of the parent phenyl cation at the x/TZVP level of theory (where ‘x’ denotes the model chemistry employed; e.g., “MP2/TZVP”)
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The AES−T estimates for the phenyl cation range widely over 186 kJ/mol depending on model chemistry, from −41.6 kJ/mol at the HF/TZVP level up to +144.0 kJ/mol at the MP2/TZVP level. Hartree–Fock is known to give lower energies for triplets, which have smaller correlation energies [49]. Almost all DFT methods cluster between 76 and 97 kJ/mol. The G4 AES−T, which has been previously validated [9] against experimental data with no systematic bias nor absolute deviations greater than 6.5 kJ/mol, is thereby taken as the benchmark method for comparison. There are only two density functionals (e.g., B1B95, M11L) within ±~4 kJ/mol from the G4 benchmark, demonstrating the unlikelihood of achieving accurate AES−T estimates via almost all DFT methods, including the more modern functionals. In addition, semiempirical methods are poor at predicting AES−T of these types of compounds, negating their potential utility for large organic structures where the low computational cost of semiempirical approaches is very attractive. The less expensive Gaussian-n composite methods [e.g., G4(MP2), G3, G3(MP2B3), G3(MP2), and G3(B3)] offer comparable accuracy to the G4 levels and may offer promise where G4 calculations are prohibitively time-consuming.

The broad AES−T range among the various methods is not due to large differences in the geometry of the singlet or triplet phenyl cation between computational approaches. The C6–C1–C2 angle for the singlet state phenyl cation averages 147.6° over all methods with a standard deviation (SD) of only 1.2°. The corresponding triplet state C6–C1–C2 angle averages 127.8° with even less method-dependent variation (SD = 0.5°). Basis set effects (Table 2) on the phenyl cation AES−T are much smaller than model chemistry impacts, but are still non-negligible. Depending on the basis set chosen, AES−T may vary by upwards of 13 kJ/mol within a specific model chemistry. Dispersion correction impacts are minimal, resulting in—at most—a few tenths of a kJ/mol change in the AES−T (Table 3). Similarly, there are collectively only modest (i.e., several kJ/mol or less) effects of different solvation models (SMD, IEF-PCM, and C-PCM) and different polar protic through apolar aprotic solvents on the calculated AES−T of the phenyl cation (Table 4).

Table 2 Gas phase standard state (298.15 K, 1 atm.) AES−T of the parent phenyl cation using various Pople, Dunning, and Ahlrichs basis sets with the HF, PBE0, wB97XD, MN12SX, and MP2 model chemistries
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Table 3 Gas phase standard state (298.15 K, 1 atm.) AES−T of the parent phenyl cation at the x/TZVP level of theory with and without dispersion corrections for representative model chemistries
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Table 4 Standard state AES−T of the parent phenyl cation at the G4(MP2) and G4 levels of theory in various solvents (298.15 K, 1 M) using the SMD, IEF-PCM, and C-PCM solvation models
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G4(MP2) and G4 calculations were used to probe the effects when a range of electron-withdrawing and releasing substituents were placed in the ortho-, meta-, and para-positions relative to the phenyl cation (Table 5). For the hydroxy, thiol, methoxy, and thiomethoxy substituents, both anti- and syn- conformations are possible (Fig. 1). In all cases, the ortho–anti conformation AES−T is higher (generally by several kJ/mol) than the corresponding ortho–syn conformation AES−T, but these effects are minor relative to the difference between substituents and near the error of the computational methods. Earlier work [16] using low-level STO-3G calculations reported a stable higher energy orthogonal geometry for the amino substituent in the ortho-, meta-, and para-positions. We were unable to reproduce these findings at the G4(MP2) and G4 levels. All substituents assumed a planar configuration with respect to the phenyl moiety regardless of position on the ring or starting conformation.

Table 5 Gas phase standard state (298.15 K, 1 atm.) AES−T of substituted phenyl cations at the G4(MP2) and G4 levels of theory
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Fig. 1
figure 1

Structures of the syn- and anti-conformations for the hydroxy (X=O, R=H), thiol (X=S, R=H), methoxy (X=O, R=CH3), and thiomethoxy (X=S, R=CH3) substituted phenyl cations

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Previous investigations [15, 16, 50, 51] at lower levels of theory predicted that meta-substituted amino- and thiomethoxy-phenyl cations—as well as the para-substituted methoxy-, thio-, and hydroxy-phenyl cations—would all be triplet ground states, whereas the G4(MP2) and G4 calculations suggest these will be singlet ground states. As Bondarchuk and Minaev [52] (B3LYP/6-31G(d,p)) and Lazzaroni et al. [50] (UB3LYP/6-31G(d)) have previously shown, the singlet o-nitrophenyl cation cannot likely exist in the gas phase since calculations lead to a rearranged structure. At the G4(MP2) and G4 levels, we are able to reproduce these findings, obtaining a ring-opened species via intramolecular oxygen transfer to the carbocation from starting geometries where the nitro group was in-plane and orthogonal to the aryl moiety.

Electron-donating substituents on the phenyl cation lower the AES−T, following a general pattern that correlates significantly (p < 0.05)—but having less than desirable predictivity—with the corresponding Hammett [53] σm and σp (Fig. 2a) and σ +p (Fig. 2b) substituent constants. Other researchers have observed this via theoretical studies [15, 16]. Cox et al. [54] experimentally observed a triplet ground state for the p-dimethylaminophenyl cation. At the G4 level, we calculate AES−T of −30.4, 18.0, and −36.6 kJ/mol for the ortho-, meta-, and para-substituted dimethylaminophenyl cation derivatives, in support of the experimental claims for a para-substituted triplet ground state.

Fig. 2
figure 2

Correlations between the gas phase standard state (298.15 K, 1 atm.) AES−T of substituted phenyl cations from Table 5 calculated at the G4(MP2) level and the corresponding Hammett (a) σm and σp and (b) σ +p substituent constants

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In a series of experimental studies on substituted phenyl cations, Ambroz et al. [55–57] reported that the 3-methoxy, 4-methoxy, 2,4-dimethoxy, 3,5-dimethoxy, 3,4,5-trimethoxy, 2,4,6-trimethoxy, 3,4-dichloro, 2,4,5-trichloro, and 2,4,6-tribromo substituted phenyl cations were likely singlet ground states, whereas the 2,4,5-trimethoxy and 3,5-dichloro-5-amino substituted phenyl cations were probably ground state triplets. At the G4(MP2) level, our calculations support the assignment of the 3,5-dichloro-5-amino (AES−T = −32.4 kJ/mol) and 2,4,5-trimethoxy (AES−T = −34.3 kJ/mol) derivatives as ground state triplets, as well as the 3,5-dimethoxy (AES−T = 43.6 kJ/mol), 2,4,6-trimethoxy (AES−T = 6.4 kJ/mol), 3,4-dichloro (AES−T = 38.4 kJ/mol), and 2,4,6-tribromo (AES−T = 9.1 kJ/mol) derivatives as ground state singlets. On the other hand, G4(MP2) calculations suggest the 2,4-dimethoxy (AES−T = −33.8 kJ/mol), 3,4,5-trimethoxy (AES−T = −27.1 kJ/mol), and 2,4,5-trichloro (AES−T = −4.5 kJ/mol) derivatives will be ground state triplets rather than singlets, although the computational error for the 2,4,5-trichlorophenyl cation AES−T leaves room for potential singlet–triplet isoenergicity.

To investigate the potential generalizability of these structure–property substituent trends, the AES−T of various silylenes was calculated via the G4 and W1BD methods (Table 6). At these levels of theory, we obtain AES−T which are in reasonable agreement with previously published [58] CCSD(T)/6-311++G(d,p)//QCISD/6-31G(d) [G4 mean signed deviation (MSD) = 7.5, mean absolute deviation (MAD) = 7.5, and root mean squared deviation (RMSD) = 8.4; W1BD MSD = 5.0, MAD = 5.0, and RMSD = 5.4; all values in kJ/mol] and B3LYP/AUG-cc-pVTZ//B3LYP/6-31+G(d) (G4 MSD = 15.5, MAD = 15.5, and RMSD = 15.9; W1BD MSD = 13.4, MAD = 1.4, and RMSD = 13.8; all values in kJ/mol) ES−T results. However, both the CCSD(T)/6-311++G(d,p)//QCISD/6-31G(d) and B3LYP/AUG-cc-pVTZ//B3LYP/6-31+G(d) methods offer systematically lower ES−T than the G4 and W1BD values by between about 4–20 kJ/mol (averaging about 4–8 and 13–17 kJ/mol negative deviations, respectively) depending on the compound/method combination.

Table 6 Gas phase standard state (298.15 K, 1 atm.) AES−T of various silylenes at the G4 and W1BD levels of theory
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The H2C=Si, H2Si=Si, HN=Si, (H3C)HSi=Si, and (H3C)2Si=Si silylenes are consistently predicted to clearly be ground state singlets using the G4, W1BD (with the exception of (H3C)2Si=Si, for which the calculation cost was too expensive), CCSD(T)/6-311++G(d,p)//QCISD/6-31G(d), and B3LYP/AUG-cc-pVTZ//B3LYP/6-31+G(d) methods, analogous to the prediction of clear triplet ground states for (H2N)HSi=Si, (H2N)2Si=Si, (HO)2Si=Si, and F2Si=Si using all four levels of theory (with the exception of (H2N)2Si=Si, for which W1BD calculations were too computationally expensive). However, there is disagreement as to the ground state multiplicity for (HO)HSi=Si and FHSi=Si using the different theoretical methods. The CCSD(T)/6-311++G(d,p)//QCISD/6-31G(d) and B3LYP/AUG-cc-pVTZ//B3LYP/6-31+G(d) methods predict (HO)HSi=Si will be a ground state triplet, whereas the G4 and W1BD methods predict either a slightly energetically favored ground state singlet (G4) or an energetic degeneracy between the two multiplicities (W1BD). For FHSi=Si, both the G4 and W1BD methods predict a clear ground state singlet, whereas the CCSD(T)/6-311++G(d,p)//QCISD/6-31G(d) method predicts effective energetic degeneracy, and the B3LYP/AUG-cc-pVTZ//B3LYP/6-31+G(d) method predicts a clear ground state triplet. Overall, the phenyl cations and silylenes display analogous structure–property substituent patterns, with an AES−T ordering trend of parent > methyl > fluoro > hydroxy > amino and multiple substitutions further lowering the AES−T below the mono-substituted compound.

In contrast to the unsubstituted phenyl and naphthyl cations, which will clearly be ground state singlets [9], the 1-, 2-, and 9-anthracenyl cations will be ground state triplets, having AES−T of −29.4, −24.8, and −40.4 kJ/mol, respectively (Table 7). Laali et al. [51] reported a ES−T gap of −56 kJ/mol at the B3LYP/6-311+G(d)//B3LYP/6-311+G(d) level for the 9-anthracenyl cation. Our value is about 16 kJ/mol higher, but in qualitative agreement.

Table 7 Gas phase standard state (298.15 K, 1 atm.) AES−T of the phenyl, naphthyl, and anthracenyl cations at the G4(MP2) and G4 levels of theory
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The nature and magnitude of the singlet–triplet transition for phenyl cations has important practical implications. For example, correlations between ES−T gaps for unsaturated compounds and yields of the corresponding Meerwein reaction products have been reported [59], thereby offering mechanistic insights that may be extended to other systems. In addition, the current findings support prior theoretical work explaining why aryl cations have differing ground state multiplicities depending on the type and location of substituents. The singlet ground state and two (3B1, 3A2) low-lying triplet excited states are known to have different geometric and electronic structures [49]. For the singlet, the positive charge resides in the σ system; however, for the triplets the cation is delocalized throughout the π system. With π-acceptor substituents (including the parent phenyl cation), singlet ground states occur. As substituents become increasingly π-donating, the triplet ground state becomes increasingly favorable. In general, it can be summarized that singlet phenyl cations are best stabilized by σ-donating substituents in the positional order ortho > meta > para, with π-donors being effective in the expected resonance-based ordering pattern para > ortho > meta. On the other hand, π-donors also stabilize the triplet state in the order para ~ ortho > meta, and for strong π-donors such as the amino moiety, the relative stabilization of the triplet far exceeds that of the singlet, resulting in a ground state triplet [16]. An examination of the aryl cation ES−T substituent trends presented herein at the G4 level shows excellent agreement with this theoretical framework.