Alistar Ottochian

Carmela Morgillo

Ilaria Ciofini

Michael J Frisch

Giovanni Scalmani
email: giovanni@gaussian.com

Carlo Adamo
email: carlo.adamo@chimie-paristech.fr

Double Hybrids and Time-Dependent Density Functional Theory: an Implementation and Benchmark on Charge Transfer Excited States

In this paper we present the implementation and benchmarking of a Time Dependent Density Functional Theory approach in conjunction with Double Hybrid (DH) functionals. We focused on the analysis of their performance for through space charge-transfer (CT) excitations which are well known to be very problematic for commonly used functionals, such as global hybrids.

Two different families of functionals were compared, each of them containing pure, hybrid and double-hybrid functionals. The results obtained show that, beside the robustness of the implementation, these functionals provide results with an accuracy comparable to that of adjusted range-separated functionals, with the relevant difference that for DHs no parameter is tuned on specific compounds thus making them more appealing for a general use. Furthermore, the algorithm described and implemented is characterized by the same computational cost scaling as that of the ground state algorithm employed for MP2 and double hybrid DFAs.

1.Introduction

Approaches rooted in Density Functional Theory (DFT) nowadays provide valuable insights for the understanding of a wide variety of chemical problems as witnessed by their widespread use in the last two decades [1][2][3] . Their success mainly relies on a very favorable cost to accuracy ratio enabling the description of ground state properties of relatively large molecular systems.

Nonetheless the accuracy reached using DFT approaches strongly depends on the nature of the density functional approximation (DFA) used and, in particular, on the expression of the contribution to exchange and correlation energies 1,2 . For the time being, one of the most sophisticated and accurate DFAs for ground state properties are represented by the so called Double Hybrid (DH) functionals casting in the DFA expression both Hartree-Fock (HF) exchange and a second order perturbative contribution computed with Kohn-Sham orbitals to the correlation [4][5][6] . These functionals thus introduce an explicit dependence of the energy expression on both occupied and virtual orbitals. Several papers in literature, following the seminal works of Truhlar 7 and Grimme 6 , have demonstrated their rigorous theoretical foundation [4][5][6][7][8][9][10][11][12] .

If these functionals have proven to be extremely performant for many ground state properties (such as thermochemistry or the description of weak interacting systems [13][14] ) nonetheless their largest limitation is the computational cost associated to the evaluation of their non-local correlation contribution. Indeed, even if scaling techniques may reduce the computational effort without altering their accuracy, DHs, with a formal scaling of at best, still remain expensive when compared to 'standard' global or range separated hybrids DFAs.

This situation is even more involved when considering the use of DHs for the treatment of excited state. Grimme and Neese proposed in 2007 15 a Time Dependent formulation in the context of the Tamm-Dancoff approximation for DHs resting on a standard linear response formulation corrected by a second-order perturbative term derived from the Head-Gordon's CIS(D) approach 16 (hereafter TD(A)-DHFT). This correction to excitation energy is scaled by the weight of the second order (MP2) correlation term in the ground state energy expression.

Using this formalism several authors have applied DHs to compute vertical excitation energies of organic molecules up to medium size [17][18][19][20] . Recently, scaling techniques 21,22 have been applied to both increase the TD(A)-DHFT accuracy and reduce associated computational cost (for instance, by neglecting spin opposite terms) 18,23 .

Since the results obtained are extremely encouraging in term of accuracy, in the present paper we present the implementation and benchmarking of TD(A)-DFT approaches focusing on the analysis of their performance for the description of through space charge-transfer (CT) excitations which are well known to be very problematic to be described using standard global hybrid approaches [24][25][26][27][28][29] .

Two different families of functionals will be compared: those derived imposing the fulfillment of some theoretical constraints, and those based on a more empirical fitting approach. For each of these families the performance of pure (GGA), hybrid (global and range separated hybrids) or double-hybrid functionals will be firstly assessed on standard benchmark and next analyzed using a set of charge transfer dimers, recently investigated by Baer and collaborators, as test case 28 .

The paper is structured as follows: after a brief description of the CIS(D) correction in the context of the Time Dependent implementation of DH functionals realized in the Gaussian code (section 2) and of the overall computational details (section 3), the implementation is validated is section 4 using two standard benchmark sets (hereafter named as Thiel and RLex80-EX7-0) before the results obtained on the charge transfer systems are discussed. In this case both results obtained in the gas phase and solution -here represented by a polarizable continuum model -will be discussed.

Overall the results we obtain show how DHs can be safely used to correctly describe CT excitations of medium sized molecules at still affordable costs both in the gas phase and in solution.

Theory and implementation

The description of electronic excited states can be approached in several different ways.

Conceptually the easiest approach is based on the Configuration Interaction method truncated to include only singly excited determinants (CIS) [START_REF] Szabo | Modern Quantum Chemistry: Introduction to Advanced Electronic Structure Theory[END_REF] . In this case the excited states are found by diagonalizing the Hamiltonian matrix that spans only the subspace generated by singly excited determinants. In its extreme simplicity CIS method lacks an explicit treatment of the electronic correlation of the ground (reference) determinant. This is often recovered, at a much greater computational cost, by including higher order excited determinants, such as in the case of the CISD method where doubly excited determinants are included. The CIS method can thus be considered the entry level approach for the treatment of excited states, just as the HF method is the basic model to study the ground state electronic structure. Following this line of development, the same considerations that lead to the use of second order Møller-Plesset perturbation theory (MP2), to include correlation effects at the HF ground state level, can be applied to the CIS method. Indeed, an MP2 correction of CIS states was firstly proposed by Foresman et al. [START_REF] Foresman | [END_REF] .

According to the resulting CIS-MP2 method, the excited state energies depend on two kinds of perturbations applied on the ground state: the static electron-electron correlation operators

T n (where n classifies the number of electrons considered in the excitation) and the time dependent ones U n . Each one of the U n -s should be considered as the first order developing term of the related static operator T n . For example, at the second order in perturbation theory these operators assume the form:

(

(

) 2
where V is the perturbation potential due to the electron correlation, a and b are the amplitudes relative to T and U operators respectively, Δ is an energy difference between HF orbital energies and, as usual, the i,j,.. indicate occupied orbitals whereas a,b,.. stand for virtual ones. In its first formulation the excited energy E CIS with the MP2 correction is then written as

Where the Φ CIS excited state mixes with states that came from the spaces of the double (U 2 Φ 0 ) and triple (U 3 Φ 0 ) excitations trough the perturbation to rise the direct and indirect terms, respectively. The indirect terms indeed describe triple excitation determinantal states that are created by the MP2 perturbation (a double excitation) of a single excited state. These are the terms responsible for the lack of self-consistency and the main computational burden, with an algorithmic complexity of O(N 6 ).

An appealing approximation of the indirect terms in Eq.( 3), based on the comparison of CISD and CCSD expansion, involves replacing the U 3 operator with the product T 2 U 1 , i.e. assuming that the MP2 amplitudes of pairs of electrons not involved in the excitations remain unchanged. The resulting CIS(D) method 16 and has the following total energy expression (4) then the (D)-corrected transition energy ω CIS (D) has to be computed with respect to the MP2 energy of the ground state as . In Eq.(3) and Eq.( 4) the quantities and are defined as

(5) (6)
The computational complexity of the CIS(D) method is reduced to O(N 5 ) with respect to CIS-MP2, but more importantly size-consistency is recovered.

During the twenty five years since the introduction of the CIS(D) method in 1994, time-

dependent Density Functional Theory (TD-DFT) has emerged as the method of choice for the study of excited states, by providing a convenient compromise between accuracy and computational cost. In its full form, TD-DFT is a method related to the random phase approximation (RPA), and as such it involves both single excitations and deexcitations.

However, following the Tamm-Dancoff approximation, TD(A)-DFT is equivalent to perform a CIS calculation starting from the Kohn-Sham ground state determinant, involving only single excitations. On the other hand, while TD(A)-DFT was being recognized as the method of choice for the study of excited states, Grimme 6 proposed an approach to apply the usual MP2 perturbative correction to a Kohn-Sham ground state determinant. This idea led to a new class of density functional approximations (DFA) termed double hybrids, which are characterized by the energy expression (7) where the x and the c subscripts indicate exchange and correlation contributions, respectively, while E HF is the usual Hartree-Fock exchange energy and E c MP2 is the correlation energy correction according to the MP2 approach. The addition of this last contribution to the DFT energy expression increases the computational cost to O(N 5 ) but allows for an enhanced accuracy of the description of the electronic ground state, especially in those cases known to be problematic for standard DFAs 13,14 .

The combination of double hybrid DFA and a second order perturbative correction equivalent to CIS(D) can be readily achieved in the context of TD(A)-DFT, while for TD-DFT the most practical approach still involves just the use of the excitation single amplitudes in the (D) correction 23 . This can be done essentially introducing the correction to the contribution stemming from the DFA part (Ω TDA-GH in Eq.( 8)) for the same amount the MP2 is involved in the ground state energy of Eq.( 7) end obtaining that: (8) Also in this context the computational scaling and the performance of double hybrids can be improved if a separate scaling for the same-spin (SS) and the opposite-spin (OS) contributions to the corrections is applied 23,32 . Similarly, different scaling factors for the SS and SO terms has also been proposed and used to tune excited state calculations using the CIS(D) correction 32 . Moreover, an additional parameter λ can be introduced, which scales the TD(A)-DFT transition energy in the denominator of the second term on the right-hand side of equation ( 4) 23 . In the present work any scaling of the U 1 and the T 2 terms or λ values other than one, were employed only for purpose of validating our implementation against results published in the literature. To the best of the authors' understanding the (D) correction to TD(A)-DFT calculations using double-hybrid DFAs, has been until now implemented only in codes that make systematic use of the resolution-of-the-identity (RI) approximation 33 such as ORCA 34 , and Q-Chem 35 . Despite the nonnegligible saving in computational cost, the use of the RI approximation relies on the assumption that the auxiliary expansion basis set being used is complete. However, "there is no guarantee that the approximation becomes uniformly better as the […] completeness of the basis increases or that error in the approximation is known a priori" 33 .

The novelty of this work consists in our choice to evaluate the (D) correction within the code that implements the MP2 semi-direct algorithm, which is used in most practical cases by the Gaussian program [START_REF] Frisch | Gaussian Development Version[END_REF] to compute . Such algorithm, originally described in reference 37, does not rely in the RI approximation. One important feature of our implementation is the ability to compute the (D) correction to a list of N states at a cost of about (N+1) MP2 calculation. All required quantities are assembled from AO two electron integrals which are generated on the-fly in a direct fashion. In the first pass, the ground state DH-DFA energy is computed and at the same time we can also evaluate the contribution to the last term on the right hand side of Eq.( 4) arising from the whole terms that contribute to the (D) correction for all the N states under consideration. This can be explained rewriting this last term inserting Eq.( 6) in Eq.( 4) and recognizing the similarity with Eq.( 12), ( 13) and ( 14) in reference 38 in the following manner: (9) where the quantities and are defined as (10) and they are evaluated using the existing MP2 gradient code 38 . This allows to save a lot of computations because, for example we do not need to form the two-electron integrals with three virtual indexes during the evaluation of . To continue, the last term in Eq.( 9) involves a Fock matrix including Coulomb and (full) exchange contributions only, (11) which is produced in a direct fashion from the single amplitudes transformed to the AO basis.

This same term is evaluated for all the N states as the ground state is computed as usual by multiplying the double amplitudes with the fully transformed integrals in the MO basis.

Following this first pass, N more MP2-like energy calculations are performed to compute instead the second term on the right hand side of Eq.( 4) for each of the excited state under consideration. In fact, it can be shown that the required terms are double-like amplitudes and can be written as (12) where the two-electron integrals in each term, are transformed on three indexes using the regular MO coefficients, while the remaining index, marked with the tilde, is transformed using a second set of MO coefficients pre-contracted with the single amplitudes, .

This means that the difference between a regular two-electron integral and the tilded ones in Eq.( 12) resides only in one of the four quarter transformation where, instead of transforming the AO to one of the MO, we are transforming to one of the excitation orbitals usually a combination of occupied and virtual MO. Bearing in mind that an excitation orbital belongs to the same space spanned by the regular MO it is straightforward to employ the same algorithm used to compute the energy to evaluate the second term on the right hand side of Eq.( 4).

It is only necessary to recognize that we form the amplitudes instead of the usual double amplitudes, and that we have include the TD(A)-DFT transition energy of the excited state in the denominator. The resulting algorithm is then a generalization of the one initially proposed in reference 37. Explicitly, in the first quarter transformation a second kind of partially transformed integrals is produced in addition to the , while at the end of the second quarter transformation three kinds of half-transformed integrals are made available, namely , and . The third and fourth quarter transformations is then run two times on to produce two sets of fully transformed integrals and . On the other hand the usual second half transformations is applied to and to produce and , respectively. At this point, all the basics tilded integrals are available and, as the regular MP2 energy code does, it is possible to build routinely their anti-symmetrized combinations appearing in Eq.( 12), to calculate the amplitudes. These amplitudes are then added up with their relative weight depending on orbital energy difference to form the second term on the right hand side of Eq.(4).

The algorithm described above to calculate the (D) correction on one excited state is characterized by the same computational cost scaling as that of the ground state algorithm employed for MP2 and double hybrid DFAs. Then a linear increase in cost is observed as a function of the number of excited states which are considered to be corrected.

3.Computational details

All the calculations were performed with a modified version of the Gaussian development code [START_REF] Frisch | Gaussian Development Version[END_REF] . Ground state structural optimization were performed at the MP2 39 /6-31G(d) 40,41 level of theory for all molecules belonging to the Thiel 42 and RLex80-EX7-0 20,43 subsets.

Optimized structures for the Baer set (obtained at the B3LYP 44 level with cc-pVDZ and augcc-pVDZ 45 ) were taken from the reference 28.

Vertical singlet excited states were computed at the TD-DFT level using different exchangecorrelation functionals namely: BLYP 46,47 , B3LYP 44,48 , CAM-B3LYP 49 , PBE 50 , PBE0 51,52 , LC-PBE 53 ; the B2-PLYP 6 , PBE0-DH 12 , and PBE-QIDH 11 in the case of the Thiel and RLex80-EX7-0 sets. The TZVP basis 54 was used both for the Thiel and the RLex80-EX7-0 set. For selected molecules of this latter the def2-TZVP triple- basis set 55 used in the original publication of Bremond et al. 20 was also tested and the corresponding results are reported in SI. In the case of the Baer Set, PBE, PBE0, LC-PBE, and PBE-QIDH functionals were considered using the cc-pVDZ and aug-cc-pVDZ basis sets. All calculations involving DH functionals were performed at the TDA level to allow comparison with previous work.

Solvent effects (methylene chloride) included using an implicit solvation model as implemented in the Gaussian code (C-PCM 56 ).

4.Results and discussion

In order to validate our implementation, we firstly computed the excitation energies of the molecules belonging to the Thiel set for which DHs results can be found in literature 17 and more recently ameliorated by the use of scaling techniques of spin-component and spinopposite (SCS/SOS) of electron-pair contributions to the nonlocal correlation components 23 .

The set contains the 28 molecules reported in SI classed, following the original paper of Thiel, in 4 groups: 1) unsaturated hydrocarbons (7 molecules); 2) aromatic hydrocarbons and heterocycles (11 molecules); 3) aldehydes, ketones and amides (6 molecules), and 4) nucleobases (4 molecules). Detailed data (that are computed transition energies) for each group can be found in Supporting Information while the resulting Mean Absolute Deviations (MADs) are reported in Table 1.

As expected, the obtained results are completely consistent to the previously reported data 17 for what concerns the B2-PLYP functional with a computed MAD of 0.24 eV to be compared with the 0.22 eV result reported in literature, thus validating our current implementation.

Comparing the two families of functionals, that are those resting on the Becke exchange and LYP correlation with those mixing PBE exchange and correlation, the same general trend can be observed. Introducing exact exchange (B3LYP and PBE0) enhances the performances with respect to parent GGAs (BLYP and PBE) especially in the case of Group 3 and Group 4 molecules. This systematic improvement of performance is not observed when going from these global hybrids (B3LYP and PBE0) to the corresponding range separated hybrids (CAM-B3LYP and LC-PBE) for which the agreement with the reference data is generally worse with few exceptions being represented by CAM-B3LYP performance for group 1 and group 3 molecules. Indeed, in the case of the PBE family the range separated LC-PBE shows a larger total MAD with respect to the corresponding global hybrid PBE0 (0.63 eV with respect to 0.30 eV).

On the other hand, all double hybrid functionals represent at best only a slight improvement with respect to global hybrid with a total MAD reducing from 0.35 eV to 0.26 eV (from B3LYP to B2-PLYP). Of note in the case of the PBE family both the PBE-QIDH and PBE0-DH performance is slightly worse (0.40 eV) than that of PBE0 (0.30 eV) essentially due to the extremely good agreement with the reference data observed for the PBE0 functional for the molecules belonging to group 2 and group 4 with a MAD of 0.25 eV and 0.18 eV, respectively.

All these general trends are in agreement with previously reported investigations 17,20 . Worth of discussion are indeed the results labelled as "PBE-QIH" in Table 1. These data correspond to the MADs associated to excitation energies computed at TD(A)-DFT level using the PBE-QIDH functional but not including the doubles corrections. Comparison between the PBE-QIH and PBE-QIDH MADs thus allow to evaluate directly the magnitude of these corrections. Clearly due to the large percentage of HF exchange, the PBE-QIH functional overestimates the transition energies providing very large MAD independently of the group of molecules considered with MADs ranging from 0.84 eV for group 1 to 1.01 eV for group 4.

Indeed, in this case the corrections ( ) are very large ranging from 0.15 eV to roughly 0.70 eV, thus highlighting the importance of the correct treatment of perturbative contributions in DH functionals.

To further test our implementation the seven molecules belonging to the RLex80-EX7-0 set were also considered. This benchmark set, depicted in Figure 1, was recently specifically developed to enable the fast evaluation of the performance of exchange correlation functionals for excited states. 20 Data reported in Table 2 are obtained using the TZVP basis set for all functionals but the results and conclusions are comparable to those obtained using the 6-31+G(d) basis set for GGA and hybrid functionals and the def2-TZVP basis for DH functionals as suggested in the original paper , reported for selected systems in SI.

Compared to Thiel set, the RLex80-EX7-0 includes also transitions with a sizable intramolecular Charge Transfer character (CT) such as that of the molecule 74 (Figure 1), a push-pull chromophore, or of the molecule 75.

Overall the same trends already observed for the Thiel set can be noted ( acting as an acceptor. For these molecules accurate experimental data both in the gas phase and/or in solution can be found in literature 58,59 . Of note we consider here only the effect of the functional on excitation energies, the geometry of each dimer being kept fixed at the B3LYP-optimized one as in the original paper of Baer 28 .

First the results obtained for the Ar-TCNE dimers (Ar=benzene, toluene, o-xylene and napthatalene) will be considered since for these systems experimental data both in the gasphase and in methylene chloride solution are available 58 .

As reported in Table 3, for all dimers the first transition corresponds, as expected, to a HOMO-LUMO excitation, the HOMO being centered on the aryl donor and the LUMO on the TCNE acceptor. Analogously, for all systems also the second transition (of HOMO-1 to LUMO character) corresponds to an intermolecular  Ar - TCNE CT excitation. These two transitions are generally non-degenerate their difference in transition energies increasing with the size of the aromatic systems up to a value of 0.8 eV in the case of the naphthalene. This trend is general roughly independently of the functional considered and of the presence or absence of the solvent. Starting from the third electronic transition, different functionals provide different ordering of the electronic transitions, pure functional predicting essentially a CT excitation of n Ar - TCNE character while global, range-separated and double hybrids are predicting a local (and intense)  TCNE - TCNE excitation. In the case of the naphthalene-TCNE system all functionals predict a third intermolecular CT excitation of  Ar - TCNE character.

In order to evaluate the error in computed transition energies, since the reference data are experimental, two different MADs have been reported in Table 3. The first (MAD f ) considering always the first CT excitation with the largest oscillator strength and the second one (MAD 1st ) considering always the first computed transition independently of the associated intensity. The transition energies considered in the first case are reported in bold face in Table 3. Of note the computed oscillator strengths are, in agreement with the experimental data, relatively small and -using the basis set here considered, i.e. cc-pVDZ-not necessarily converged as already noted by Baer et al. 28 Nonetheless the use of the larger aug-cc-pVDZ does not significantly change the general trend here reported (see SI for corresponding data).

Independently of the criteria used to compute the MAD, both in the gas phase and in solution, the smallest errors are obtained using the PBE-QIDH functional while extremely large errors are computed at the GGA (PBE) or global hybrid (PBE0) level. DHs are better performing also with respect to range-separated hybrids (LC-PBE). Of note the agreement with the experimental data increases in going from the gas-phase to solution data. This is due essentially to error cancellation since transition energies are basically underestimated at all levels of theory as it is the effect of the solvent, which is experimentally also red-shifting the transition energies. As consequence the underestimation of solvent effect compensates the underestimation in computed transition energies allowing to afford extremely low values for the computed MAD in solution in the case of PBE-QIDH (0.16-0.23 eV).

Overall from the analysis of our data we can conclude that DHs represent a good alternative to adjusted range separated hybrids for the treatment of intramolecular charge transfer excitations provided that the double correction is correctly included. Indeed excitation energies of 3.8 eV, 3.4 eV, 3.0 eV and 2.7 eV were reported for Aryl-TCNE (Aryl = benzene, toluene, o-xylene, naphthalene, respectively) by Baer and collaborators 28 using the BNL functional and of 3.9 eV, by Truhlar and collaborators, using the revM11 functional in the case of the benzene-TCNE system. 57 To further confirm these finding, the set of substituted anthracene-TCNE dimers (Table 4) by

Baer was also considered. In this case experimental data are only available in solution 59 .

Contrary to Baer previous work 28 but analogously to what reported in Table 3, transition energies were thus directly computed in solution using a polarizable continuum model. Of note BNL energies in solution taken from Baer work are instead estimated subtracting 0.32 eV from gas-phase computed data, the 0.32 eV shift being estimated as an average experimental gas-phase to solvent shift 59 on the basis also of a previous theoretical work 60 .

The general trend observed are, as expected, in line with those of the previously discussed Ar-TCNE systems with a large/slight underestimation of transition energies provided by the GGA/global hybrid functional (PBE/PBE0) and an overestimation of transition energy provided by the range-separated LC-PBE. The results obtained with the PBE-QIDH functional are indeed in very good agreement with the experimental data in solution but generally all red-shifted with the prediction obtained at BNL level of theory in the gas-phase (see Table S5). These data are consistent with the previously pointed out error compensation between the underestimation of gas to solvent shift and transition energies.

Overall when considering the overall performance of PBE-QIDH, this functional not only shows a very small MAD with respect to solution data (0.14 eV), comparable to that of the BNL functional (0.09 eV), but, more relevantly is able to correctly recover the ordering of computed energies as a function of the substituent with the only exception of the inversion between the 9-methyl and 9, 10 dimethyl substituent. Furthermore it should be underlined that, contrary to adjusted range separated functionals, in double hybrids no parameter is tuned on a specific class of compounds to reproduce its excited state properties making them more appealing for a general use.

5.Conclusions

The results reported in this paper show how a Double Hybrids DFAs can be an effective tool to predict excited states energies of molecular compounds of relatively large size. Indeed, the algorithm described and implemented is characterized by the same computational cost scaling as that of the ground state algorithm employed for MP2 and double hybrid DFAs, although a linear increase in cost is observed as a function of the number of states considered. From the point of view of the performance we showed that the use of double hybrids allows for a better description of through space charge-transfer excitations which are well known to be very problematic to be described with commonly used functionals, such as global hybrids.

In particular, our results show, beside the robustness of the implementation, that double hybrids enable to obtain results comparable to that of adjusted range separated functionals, with the relevant difference that for DHs no parameter is tuned on a specific class of compounds to reproduce its excited state properties thus making them more appealing for a general use.

Figure captions

Figure 1 Molecules of the RLex80-EX7-0 set 20 considered in this work

Figure 1

 1 Figure 1

Table 2

 2 To test the performance of DH functionals in the prediction of through-space CT excitations, the set of molecules recently proposed by Baer and collaborators28 was considered in the case of the PBE family of functionals (PBE, PBE0, LC-PBE and PBE-QIDH). This set contains a

	number of dimers composed by an aromatic molecule (benzene, toluene, o-xylene,
	naphthalene and substituted anthracene) acting as a donor, and tetracyanoethylene (TCNE)

). If global hybrids (B3LYP and PBE0) still represent the best cost to accuracy compromise, double hybrids, and particularly B2-PLYP, are extremely well performing with total MAD ranging from 0.11 eV (B2-PLYP) to 0.19 and 0.21 eV for PBE-QIDH and PBE0-DH, respectively. Of note all three DH functionals perform very well for molecules 74 and 75, that are those presenting transitions of partial CT character. For molecule 74 the effect of the inclusion of the perturbative correction is particularly large, as can be seen comparing the MAD computed for the PBE-QIH (1.01 eV) and PBE-QIDH (0.49 eV). Indeed, in such a case the inclusion of a larger percentage of HF exchange in the PBE-QIH it is not sufficient to recover the correct transition energy. Of note PBE0 is providing the smallest error and B3LYP is also showing an error of only 0.14 eV.

Nonetheless if both B3LYP and PBE0 provide very accurate intramolecular CT excitation energies in the case of molecule 74 and 75, it is well known that these global hybrid functionals fail to reproduce through space CT excitations.

Table 1 .

 1 Computed Mean Absolute Deviation (MAD, in eV) for all functionals for the Thiel benchmark set. For groups definition refer to Figure1. PBE-QIH results are obtained from PBE-QIDH TD(A)-DFT calculations not including the doubles correction (that is the term in eq. 8).

	Functional			MAD (eV)
		Group 1	Group 2	Group 3	Group 4	Total
	BLYP	0.72	0.39	0.70	0.86	0.61
	B3LYP	0.60	0.24	0.36	0.29	0.31
	CAM-B3LYP	0.54	0.29	0.44	0.26	0.35
	B2-PLYP a	0.31	0.29	0.21	0.16	0.26
	PBE	0.78	0.38	0.68	0.86	0.62
	PBE0	0.60	0.25	0.38	0.18	0.30
	LC-PBE	0.63	0.52	0.62	0.57	0.63
	PBE-QIH a	0.54	0.84	0.77	1.01	0.88
	PBE-QIDH a	0.39	0.46	0.29	0.34	0.40
	PBE0-DH a	0.25	0.47	0.26	0.33	0.40

a Within the Tamm-Dancoff approximation.

Table 2 .

 2 Relative and Mean Absolute Deviations (MAD, in eV) computed using the TZVP basis set for the molecules belonging to the RLex80-EX7-0 data set (figure1). The reference CC2 values, taken from reference 43, are also reported.

		01 Mol	08 Mol	33 Mol	41 Mol	52 Mol	74 Mol	75 Mol	MAD
	BLYP	0.57	0.3	1.06	0.98	0.34	0.72	1.16	0.73
	B3LYP	0.15	0.06	0.27	0.71	0.08	0.14	0.52	0.28
	CAM-B3LYP	0.00	0.19	0.15	0.42	0.2	0.41	0.2	0.22
	B2-PLYP a	0.09	0.04	0.02	0.27	0.01	0.08	0.23	0.11
	PBE	0.53	0.3	1.04	0.94	0.33	0.71	1.15	0.71
	PBE0	0.05	0.01	0.1	0.62	0.01	0	0.37	0.17
	LC-PBE	-0.17	0.43	0.44	0.11	0.5	0.88	0.65	0.45
	PBE-QIH a	-0.32	0.74	0.72	0.1	0.64	1.01	0.49	0.57
	PBE-QIDH a	0.02	0.21	0.29	0.07	0.19	0.45	0.12	0.19
	PBE0-DH a	0.06	0.28	0.25	0.24	0.22	0.35	0.08	0.21
	CC2	5.41	2.24	4.06	4.77	2.65	3.71	3.37
	a Within the Tamm-Dancoff approximation

Table 3 .

 3 Computed and experimental excitation energies (eV) for Ar-TCNE systems. For the sake of clarity, only the three lowest electronic transitions are reported. In bold transitions that have been considered to compute the MAD f (see text for definition). Highlighted in yellow intermolecular  Ar - TCNE CT transitions; in green intermolecular n Ar - TCNE CT transitions; in pink local  TCNE - TCNE excitations and in grey local  Ar - Ar excitations. Within the Tamm-Dancoff approximation. b) from reference 58.

	gas phase

Table 4 .

 4 Computed and experimental excitation energies (in eV) for a series of substituted anthracene-TCNE systems.

	Substituent	PBE	PBE0 LC-PBE PBE-QIH a PBE-QIDH a BNL b	exp. c
	none	1.38	1.47	2.47	2.09	1.58	1.82	1.73
	9-cyano	0.65	1.02	3.12	2.42	1.90	2.03	2.01
	9-chloro	1.16	1.27	2.56	2.08	1.58	1.82	1.74
	9-carbomethoxy	1.06	1.22	2.66	2.15	1.64	1.84	1.84
	9-methyl	1.26	1.34	2.31	1.93	1.42	1.71	1.55
	9-nitro	0.92	1.43	2.68	2.50	1.98	2.12	2.03
	9,10-dimethyl	1.52	1.59	2.24	2.03	1.53	1.77	1.44
	9-formyl	0.85	1.26	2.72	2.20	1.7	1.95	1.9
	9-formyl 10-chloro	0.83	1.25	2.83	2.30	1.79	1.96	1.96
	MAD	0.63	0.49	0.81 5	0.39 5	0.14	0.09

a) Within the Tamm-Dancoff approximation; b) From reference 59. c) From reference 28.

Acknowledgments

I.C and C.M acknowledge funding from the European Research Council (ERC) under the European Union's Horizon 2020 research and innovation programme (CoG STRIGES grant agreement No 648558). I.C. and C.A. acknowledge the French National Research Agency, ANR ("E-StorIc" project : ANR-14-CE05-0002).


Dr Pierre-Adrien Payard

Dr Yu-Ting Zheng

Dr Wen-Juan Zhou
email: wenjuan.zhou@solvay.com

Lhoussain Khrouz

Laurent Bonneviot

Dr Raphael Wischert

Dr Laurence Grimaud

Dr Marc Pera-Titus
email: marc.pera-titus-ext@solvay.com

Iron Triflate Salts as Highly Active Catalysts for the Solvent-Free Oxidation of Cyclohexane

Keywords: Iron Triflate, Oxidation, KA oil, C-H activation, EPR, DFT, Cyclohexane, Homogeneous Catalysis

Among a series of iron salts, iron triflates revealed as highly active catalysts for the oxidation of cyclohexane by t-butyl hydroperoxide into cyclohexanol and cyclohexanone with initial turnover frequencies higher than 10,000 h -1 . The structure of the iron complexes under reaction conditions was studied by combining electron paramagnetic resonance (EPR) spectroscopy and DFT calculations. The coordination of the catalytic iron center readily evolved in the presence of the reaction products, leading ultimately to its deactivation. Iron and organic superoxo intermediates were identified as plausible active species allowing to rationalize the high activity of iron ligated by highly-delocalized counter-anions.

Introduction

The oxidation of cyclohexane (CyH) to cyclohexanol (CyOH) and cyclohexanone (CyONE), also known as KA oil, is a well-known industrial process for the production of adipic acid (AA). [START_REF] Musser | Cyclohexanol and Cyclohexanone[END_REF] This process comprises two consecutive oxidation steps (Scheme 1): (i) non-catalytic autoxidation of CyH by O2 yielding cyclohexyl hydroperoxide (CyHP), and (ii) deperoxidation of CyHP to KA oil catalyzed by a transition metal, typically a homogeneous cobalt catalyst. [START_REF] Cavani | Radical Chain Oxidations[END_REF] The process is conducted at very low CyH conversion (~5%) with a CyHP selectivity typically of 60-70% in the first step. [START_REF] Hermans | [END_REF] The industrial challenge is to increase the catalytic activity for KA formation while keeping the CyHP concentration as low as possible using a clean and safe route. In this view, a fast and selective catalytic pathway using a non-toxic metal would be desirable. ! Scheme 1. Industrial process for the oxidation to CyH to KA oil.

High-valent iron-oxo complexes (e.g., cytochrome P450) are known to efficiently catalyze oxidation reactions in living systems. [4] Iron-oxo systems have been extended to non-heme bio-inspired catalysts based on iron centers, [5] and to mononuclear iron complexes. [6] The most studied and simplest non-heme iron system relies on the so-called Fenton reagent consisting of an aqueous solution of hydrogen peroxide and an iron salt. [7] This reagent has been widely used in the total oxidation of organic pollutants in liquid waste treatment, as it generates very active and non-selective OH° radicals. [8] The search for high selectivity in partial oxidation reactions has led to the development of alternative catalytic systems comprising highly oxidized iron-oxo intermediates (IV or V). [9] To stabilize high-valent iron intermediates, a ligand is generally used. [10] A well-known application of iron complexes for CyH oxidation relies on the pioneering work of Barton, using an iron catalyst combined with metallic zinc and oxygen as oxidant. [11] This chemistry was improved by replacing Fe(II)/Zn 0 /O2 by Fe(II)/ superoxide. Using an analogous system based on Fe(III)/TBHP ( T B H P = t -b u t y l h y d r o p e r o x i d e ) b a s e d o n (tris(trimethylacetate))iron(III) [Fe(tma)3] and (tris (1,1,1,2,2,3,3heptafluoro-7,7-dimethyl-4,6-octanodionate)iron(III) [Fe(fod)3], Schuchardt et al reported a selectivity of 91% at 11% conversion for CyH oxidation. [12] However, the product (AA) deactivated the active iron centers.

Recently, iron triflates, i.e. Fe II (OTf)2 and Fe III (OTf)3, have been reported as active catalysts in low-coordinating solvents for a variety of Lewis-acid catalyzed reactions. [13] Typical examples include imination of sulfoxides, [14] aziridination of enol silyl ethers, [15] hydroaddition to unsaturated C-C bonds, [16] etherification and transetherification of alcohols, [17] and glycosylation of sugar derivatives. [18] Despite the potential of iron triflates for C-H activation in C-N, [19] C-C, [20] and C-X (X = F, Cl, I) [21] coupling reactions, few reports are available on oxidation reactions. [5b,22] Herein we report an unprecedented high catalytic activity of Fe III (OTf)3 for the oxidation of neat CyH towards CyOH and CyONE (KA oil) using TBHP as oxidant (Scheme 2). Fe III (OTf)3 behaved as precursor of the active iron species for CyH oxidation. We provide detailed insight into the structure of the catalyst under reaction conditions by combining electron paramagnetic resonance (EPR) and DFT calculations. [a] Reaction conditions: 80 °C, CyH (4.0 g) containing TBHP (7.5 wt%, 3.3 mmol), 0.14 mmol catalyst (10 ppm Fe), 60 min. [b] OTf: SO2CF3, NTf2: N(SO2CF3)2, TPP: tetraphenylporphyrine, OTs: OSO2Tol [a] Reaction conditions: 80 °C, CyH (4.0 g) containing TBHP (7.5 wt%), 60 min Table 3. CyH and TBHP conversion, KA selectivity and yield, A/K molar ratio and TON as a function of the temperature [a] [a] Reaction conditions: 80 °C, CyH (4.0 g) containing TBHP (7.5 wt%), 60 min, 0.14 mM Fe III (OTf)3 (10 ppm Fe), 60 min

Results and Discussion

Definition of the catalyst performance metrics

The catalytic performance of the different Fe complexes was primarily characterized in terms of the TBHP conversion, KA selectivity and KA yield with respect to TBHP, as well as by the turnover frequency at time t = 0 (TOF0) and the turnover number (TON) at 60 min. [START_REF]the kinetic curve plotting the KA yield at time t = 0, divided by the number of Fe centers[END_REF] More details on the reaction metrics can be found in the SI (Experimental section, section 1.3).

CyH oxidation using TBHP as oxidizing agent

Screening of Fe complexes

The catalytic activity of a series of Fe complexes was first investigated using CyH as reactant and TBHP as an oxidant (7.5 wt% or 0.65 mM in CyH, see SI for preparation, section 1.2) at 80 o C under reflux. For comparison, a porphyrin based Fe salt (Cl-Fe-TPP) was chosen as reference catalyst, and its reactivity was compared to that of Fe(II) and Fe(III) salts, i.e. Fe II (OTf)2, Fe III (OTf)3, Fe III (NTf2)3, Fe III (OTs)3 and Fe III Cl3, featuring in all cases a highly delocalized, poorly coordinating, counter anion. The products and reactants were analyzed by GC (see SI, section 1.3 for detailed method) after 60 min (Table 1). Among the different Fe complexes, the least soluble in CyH is Fe III (OTf)3 with a solubility of 0.59 mM and 0.24 mM at 20 o C and 80 o C, respectively (see SI, section 1.4 for solubility measurements). In light of these results, a Fe concentration of 0.14 mM (10 ppm Fe) was chosen to compare the catalytic activity of the complexes. The most active complexes for CyH oxidation also appear to be the most selective towards KA oil (entries 1-3) by minimizing undesired TBHP disproportionation to t-butyl alcohol and O2, and the formation of t BuO-O t Bu (Scheme 2). Note that t BuO-OCy was not detected, opposing previous studies encompassing CyH oxidation by TBHP using solvents (e.g,, acetonitrile) under dilute conditions. [START_REF] Leising | [END_REF] Among the different Fe complexes, Fe III (OTf)3 exhibits the highest TBHP conversion and KA yield with values of 73% and 25%, respectively. Also, the TON after 60 min is the highest with a value of ca. 1200. Surprisingly, both triflate and triflimide Fe salts perform much better than the porphyrin based Fe salt. The TON decreases in the order: Fe III (OTf)3 > Fe II (OTf)2 > Fe II (NTf2)3 > Cl-Fe III TPP > Fe III (OTs)3 > Fe III Cl3. Noteworthy, at the reaction conditions tested, Fe II (OTf)2 is slightly less active than Fe III (OTf)3. However, Fe III (OTf)3 shows a higher alcohol-toketone molar ratio (A/K) (6.4 vs. 3.3).

Effect of the Fe III (OTf)3 concentration

The catalytic properties of Fe III (OTf)3 were then explored at variable catalyst concentration in the range 0.014-1.95 mM (1-140 ppm Fe) (Table 2). Increasing the Fe concentration from 0.014 mM (1 ppm Fe) to 0.56 mM (40 ppm Fe) clearly promotes the CyH and TBHP conversion, but at the expense of the TON. This observation suggests that a high catalyst concentration (especially >0.14 mM or 10 ppm Fe) does not enhance the catalytic activity. The highest TON at 60 min (about 2600) is achieved at very low KA yield (5.7%) and at the lowest Fe III (OTf)3 concentration (0.014 mM). Besides, the A/K molar ratio declines at higher Fe concentration from 82 at 0.014 mM to 4.0 at 0.14 mM, suggesting that CyONE is a secondary oxidation product of CyH.

Effect of the reaction temperature

The effect of the temperature was studied in the range 40-100 o C using 0.14 mM Fe III (OTf)3 (10 ppm Fe) (Table 3). CyH oxidation is enhanced at higher temperature at the expense of disproportionation into t-butyl alcohol and O2, with KA oil being enriched in CyOH. In parallel, the A/K molar ratio decreases with the temperature from an initial value of 5.5 at 40 °C to 2.8-4.0 in the temperature range 45-80 o C.

Reactant and product evolution during the reaction

The reaction kinetics was measured at different temperatures during the first 20 min of reaction using Fe III (OTf)3 and Fe III (NTf2)3 as catalysts (Figure 1). For both catalysts, KA oil formation proceeds fast during the first 15 min with a decrease of the A/K molar ratio, but stabilizes after 10-20 min for temperatures higher than 50 °C. At first sight, this observation Complex [b] CyH conv (%) suggests catalyst deactivation, which could be caused either by the reaction products, or from by-products. Indeed, traces of AA were detected, which combined with water are known to deactivate Fe centers. [5c,12] To gain more insight into the underlying mechanism of Fe deactivation, CyOH, CyONE and AA were added to the reaction media in the beginning of the reaction. While CyOH or CyONE only exert a slight effect, AA strongly lowers the KA yield, thus confirming the deactivating role of AA (Table 4). Table 4. Evolution of the KA yield upon addition of CyOH, CyONE and AA before the reaction. [a] The concentration of products was adjusted to magnify the effect on the KA yield.

[a] Reaction conditions: 80 °C, 10 ppm Fe III (OTf)3, 4 g of CyH containing 7.5 wt% TBHP (3.3 mmol), 60 min.

TOF and activation energy for CyH oxidation

The reaction rate (i.e. TOF0) was measured at time t = 0 in the temperature range 45-70 o C for Fe III (OTf)3 and Fe III (NTf2)3. In both cases, very high TOFs of 14,000 h -1 and 31,000 h -1 , respectively, at 60 o C, are observed. The activation energy measured from the Arrhenius plots of ln(TOF0) vs. 1/T is 145 ± 12 kJ/mol for Fe III (OTf)3 and 115 ± 6 kJ/mol for Fe III (NTf2)3

(Figure 2). At all temperatures, the TOF0 is higher for Fe III (NTf2)3. However, higher KA yields and TONs at 60 min are obtained with Fe III (OTf)3 (Table 1, entries 1 and 3). This observation points out a faster deactivation of Fe III (NTf2)3 during the reaction despite its higher initial reaction rate.

EPR investigation of Fe(III) coordination

EPR is a sensitive technique for the analysis of Fe(III) species with unpaired electrons at very low concentration. In frozen solution, the g-tensor component of Fe(III) can be measured at low-and high-spin states providing information on the symmetry and coordination sphere of the species and their evolution during the reaction. In the case of high-spin Fe(III), one should also consider the contribution of the zero-field-splitting (ZFS), which can be assessed by the so-called the D tensor. This tensor includes two components, i.e. D and E, accounting for the intensity and anisotropy of the interaction. [START_REF] Hagen | Biomolecular EPR Spectroscopy[END_REF] Fe(II) ions, which are also paramagnetic are silent in conventional X-band (9.6 GHz) spectrometers.

Coordination sphere of Fe(III)

A CyH solution of Fe III (OTf)3 exhibits an EPR signal at g = 2.02 (Figure 3), which is also observed on the solid salt (Figure S1).

Since the triflate ion is usually described as a weak ligand, one could reasonably expect it to be in high-spin state (S = 5/2). Indeed, Fe III (OTf)3 and Fe III (NTf2)3 have been described as highspin complexes, as also confirmed by DFT calculations (see SI, Table S1). [START_REF] Cabrero-Antonino | [END_REF] In our case, the fit of the EPR spectrum of Fe III (OTf)3 using the Matlab ® routine Easyspin is consistent with a single weakly distorted high-spin Fe center (g = 2.012, D = 0.01 cm 1 , E/D = 0.1, see fitted spectrum in SI). [27] DFT calculations (see SI for Computational Details) yield a structure for Fe III (OTf)3 featuring three bidentate triflate anions interacting with the Fe center through two oxygen atoms (Figure 3). The metal-ligand interaction in Fe III (OTf)3 is mainly ionic as confirmed by NBO analysis: both Fe and O are strongly charged (+1.7 on Fe and -0.9 on O), and the spin is located mainly on Fe (spin density on Fe: 4.0, on O: 0.1). The values gx = gy = gz = 2.01 and D = 0.09 cm -1 were obtained using the EPR calculation module of ORCA, in good agreement with the experimental spectrum.

Coordination sphere of Fe(III) in the presence of CyOH and CyONE

As pointed out above, CyONE and CyOH are potential ligands to Fe(III), but showing a contrasting effect on the catalytic activity: CyONE seems to promote deactivation, whereas CyOH enhances the activity. To gain more insight into the coordination ability of CyONE and CyOH on the Fe(III) centers, a dedicated EPR study was performed. The addition of CyONE to a CyH solution of Fe III (OTf)3 (10 mM) does not alter the EPR signal (Figure 4, Figure S2). This result is rather unexpected, since ketones should interact with Fe(III), generating a Lewis adduct. Indeed, the Gibbs free energies of coordination computed by DFT strongly support this idea (Table 5): the most stable adduct between Fe III (OTf ! ! 2) CyOH (10 equiv), ( 3) CyONa (1 equiv), ( 4) CyOH (10 equiv, after 1 night), and ( 5) CyONa (10 equiv).

Table 5. DFT-calculated enthalpy (ΔH, kJ mol -1 ) and Gibbs free energy at 25 °C (ΔG, kJ mol -1 ) of high spin complexes resulting from the coordination of CyOH and CyONE to Fe III (OTf)3.

Likewise, the addition of CyOH to Fe(OTf)3 does not modify the EPR spectrum upon addition up to 3 equiv (Figure 5-1). As for CyONE, DFT calculations confirm that simple coordination of CyOH is not expected to modify the signal to an important extent (Table 5, complexes 3-5

Species

ΔG [a] (kJ mol -1 ) ΔH [a] ( k J mol a] The enthalpy and Gibbs free energies of complex formation were calculated with respect to free Fe III (OTf)3, CyOH and CyONE. In the case of complexes 10 to 12, the formation of CyOH2,OTf ion pair was considered (ΔG = -6.8 kJ mol -1 , ΔH = -61.6 kJ mol -1 ), since triflic acid is less favorable. The formation of monoalkoxy Fe species, i.e. (Fe(OTf)3 + (n + 2) CyOH = Fe(OTf)2(CyO)(CyOH)n + CyOH2.OTf, with n = 0, 1 and 2, complexes 10-12), turns out to be thermodynamically favorable, while species featuring multiple alkoxy ligands such as Fe(OTf)(CyO)2(CyOH)n and Fe(CyO)3(CyOH)n, with n = 0, 1, 2 and 3 (Table S2), are less favorable. [START_REF]Triflic acid is a strong acid that is expected to protonate water (ΔG = -38.8 kJ mol -1 ), CyOH (ΔG = -44[END_REF] The latter complexes are all predicted to be high-spin species (see SI). The formation of iron-alkoxy species was experimentally confirmed by adding variable amounts of sodium cyclohexanate (CyONa) to Fe III (OTf)3 (Figure 5-3, Figure S5). Indeed, after addition of 1 equiv of CyONa, the low-field signal immediately emerges. However, under CyONa excess, only a single broad signal appears at around g = 2 (Figure 5-5). Interestingly, a similar evolution occurs when heating at 40 °C overnight a solution of Fe III (OTf)3 with CyOH (10 equiv) in CyH (Figure 54). Since the formation of low-spin complexes seems unlikely with only alkoxy ligands, which is confirmed by DFT (Table S1), this observation might be explained by the formation of Fe(III) clusters featuring CyO -as bridging ligand. In this view, the weak dipolar interaction between localized spin momenta would broaden the signal due to unresolved coupling. [START_REF] Guisado-Barrios | [END_REF] Indeed, these signals were successfully fitted using an increasing proportion of Gaussian part in the function (Figure S5,S6).

Catalyst ageing

The signal attributed to Fe clusters is also observed in the EPR spectrum measured after the reaction, when the catalyst is supposed to be deactivated (Figure S7). Besides the above mentioned broad signal, [START_REF]The sharp multiple between 3000 and 4000 G is due to Mn(II) impurity resulting from Mn(III) reduction after TBHP addition[END_REF] the spectrum can be decomposed into a mixture of high-spin complexes with high ZFS interaction (D = 0.6 cm -1 E/D = 0.33, and D = 0.7 cm -1 , E/D = 0.21). All the signals may be attributed to alkoxy or hydroxy Fe complexes. Overall, CyOH is expected to solubilize the salt, but in excess may inhibit the Fe centers.

Reaction intermediates and proposed mechanism

When TBHP is added to a solution of Fe III (OTf)3 in CyH, the signal ascribed to Fe(III) vanishes and only sharp signals at g = [2.024 2.009 2.007] and g = [2.031 2.011 2.003] are visible (Figure 6, Figure S8). These signals are characteristic of the superoxide radical O2°-, either free or stabilized on an EPR-silent Fe center (Fe II -OO°). [31] Surprisingly, no signal ascribed to Fe III (OOtBu) is observed. [5c] Radical trapping experiments were also carried out using 5,5-dimethyl-1-pyrroline N-oxide (DMPO) (Figure 7). [START_REF] Villamena | EPR Spin Trapping in Reactive Species Detection in Biology[END_REF] Under standard reaction conditions, a single radical is observed (Figure 7-1, S = 1/2, g = 2.004, aN = 13.25 G, aH = 9.98, 1.45 G), corresponding to the adduct formed between DMPO and t BuO° (or t BOO°). [START_REF] Haire | [END_REF] No traces of OH°, typical of Fenton-type systems, or t Bu° trapped by DMPO, are observed. The addition of CyONE (10 equiv) results in a similar signal, but less intense, while a second species appears (Figure 7-2, S = 1/2, g = 2.006, aN = 7 G, aH = 3.5 G, see also Figure S9, S10) due to DMPO oxidation. [34] This experiment reinforces the idea of Fe poisoning by CyONE.

With these results in hand and supported by the literature, we provide in the following lines a tentative list of reactions participating during CyH oxidation with the detected species highlighted in bold. For a critical review on Fe-peroxo catalytic mechanisms, see Barbusinsky et al. [35] The first reaction involves the deprotonation of TBHP to yield Fe-peroxo complexes (Eq 1-2) (X = OTf -, CyO -, OH -or t BuOO -). Note that the coordination sphere may be completed by additional neutral ligands such as CyOH, CyONE and H2O, which, for the sake of clarity, have been omitted.

(1) Fe III (OTf)X2 + t BuOOH + CyOH ⇔ Fe III (OO t Bu)X2 + CyOH2.OTf

(2) Fe II (OTf)X + t BuOOH + CyOH ⇔ Fe II (OO t Bu)X + CyOH2.OTf

These reactions (in equilibria) should be favored under TBHP excess, since one TfOH molecule is generated upon addition of CyOH to Fe III (OTf)3. However, the release of free TfOH proved to be thermodynamically unfavorable, as inferred by DFT. Hence, we propose instead the participation of a CyOH2.OTf adduct (X = OTf, n = 0, ΔG1 = -11 kJ mol -1 ). The Fe III (OO t Bu)X2 complex may further evolve by oxidation of Fe(III) to Fe(IV) generating t BuO° radicals (Eq 3), [36] matching the observation by spin trapping (vide supra). Homolytic cleavage of the C-O bond to form t Bu° (Eq 4-5) is energetically unfavorable (X = OTf, n = 0, ΔG4 = +336 kJ mol 1 ), [31b] and this radical was not experimentally detected. Accordingly, the formation of superoxo radicals by this pathway is unlikely. As an alternative, TBHP-assisted C-O cleavage could be driven by Fe III (OO t Bu)X2 complexes according to Eq 6 [31b] (3) Fe The as-generated ferryl-oxo species [i.e. (Fe IV =O)X2] (Eq 3) should be able to activate the C-H bond in CyH (Eq 7) and CyOH (Eq 8). [37] Besides, FeOO° could also oxidize CyH (Eq 9 and Eq 10). [31b] (7) (Fe IV =O)X2 + CyH → Fe II X2 + CyOH We were curious to figure out why Fe III (OTf)3 and Fe III (NTf2)3 salts are highly efficient for CyH oxidation. Assuming that C-H activation by a Fe(IV)=O moiety is the rate-limiting step of the reaction, it may be interesting to question the very high activity of these salts. In a recent study, Sautet and coworkers reported that the activation energy for C-H activation and the energy level of the acceptor frontier orbital of the initial Fe(IV)=O intermediate (σ* and π* for quintuplet and triplet state complexes, respectively) are linearly correlated. [38] Relying on this work, we computed the orbital energies for some plausible Fe(IV)=O intermediates featuring triflate ligands and for a porphyrin-ligated Fe(IV)=O as a reference (Table 6).

Irrespective of the ligand, the quintuplet state is favored for triflate-containing complexes, except in the case of CyO -, for which both the quintuplet and triplet states are very close in energy (<1 kJ mol -1 ). In contrast, the triplet state is favored for the porphyrin-ligated Fe(IV)=O. Poorly ligated triflate Fe(IV)=O is the most electrophilic species with a very low σ* orbital at -5.4 eV. CyOH-and CyONE-ligated triflate Fe(IV)=O are predicted to be a few less active (-4.2 eV and -3.9 eV). This lower electrophilicity upon coordination may account for the rapid decrease of TON along the reaction (vide infra). Coordination of the CyO -ligand is expected to strongly increase the σ* level (-3.5 eV), which is in line with a potential catalyst deactivation. Finally, porphyrin-ligated Fe(IV)=O exhibits the highest frontier orbitals (π*: -5.6 eV, σ* = -3.3 eV), in agreement with the lower KA yield of Cl-Fe III TPP compared to that of Fe III (OTf)3 (Table 1). Thus, the remarkable activity of Fe III (OTf)3 for CyH oxidation may be ascribed to the higher electrophilicity of triflate-ligated Fe(IV)=O species, allowing a relatively low C-H activation barrier compared to other Fe-ligated catalysts. Fe II X2 may be re-oxidized to Fe(III) or Fe(IV) by O2 or TBHP according to Eq 11 and Eq 12 (11) Fe II X2 + O2 → Fe III X2O2° (12) Fe II (HOO t Bu)X2 → Fe III (O t Bu)X2 + OH° Table 6. Energy level of frontier orbitals (in eV) of S=1 and S=2 spin Fe(IV)=O complexes a However, the above reactions seem unlikely, since no traces of OH° or Fe III X2O2° were experimentally detected. [39] On the contrary, Eq 13 would allow the regeneration of (Fe IV =O)X2 (13) Fe II (HOO t Bu)X2 → (Fe IV =O)X2 + t BuOH Finally, mechanisms involving the formation of CyOOH and its disproportionation to CyOH and CyONE are unlikely, since the A/K molar ratio evolves during the reaction, excluding a termination reaction according to Eq 14 [5c] (14) 2 CyOOH → CyOH + CyONE

Conclusion

Highly electrophilic high-valent iron triflate salts proved to be of high interest for the C-H activation of cyclohexane. These salts selectively and efficiently catalyze the oxidation of cyclohexane to cyclohexanol and cyclohexone (KA oil) with very high turnover frequencies. Combined EPR spectroscopy with DFT calculations helped us to propose structures intermediates, as well as to find out possible sources of catalyst deactivation. [a] Fe(IV) oxo complexes can be on singlet, triplet or quintuplet states. Both triplet and quintuplet states were optimized for each complex. The energy of the acceptor orbitals (relevant to estimate the activation energy of the H-atom transfer) are indicated in eV. The energy of the π* and σ* orbitals are reported for the triplet (S = 1) and quintuplet state (S = 2), respectively, in agreement with the conclusions in ref [38]

Figure 1 .

 1 Figure 1. Time-evolution of KA yield and A/K molar ratio in CyH oxidation catalyzed by (A) Fe III (OTf)3 and (B) Fe III (NTf2)3 with TBHP as oxidant. Reaction conditions: 45-70 °C, 0.14 mM catalyst (10 ppm Fe), 4 g of CyH containing 7.5 wt% (3.3 mmol) TBHP, 0-20 min.

!Figure 2 .

 2 Figure 2. Arrhenius plots of TOF0 for Fe III (OTf)3 and Fe III (NTf2)3. TM refers to the mean temperature of the series (TM = 330.7 K). Reaction conditions as in Figure 1.

Figure 3 .

 3 Figure 3. Experimental (blue) and fitted (orange) EPR spectra at 110 K of Fe III (OTf)3 in CyH (10 mM).

Figure 4 .!Figure 5 .

 45 Figure 4. Experimental (blue) and fitted (red) EPR spectra at 110 K of Fe III (OTf)3 (10 mM) in CyH in the presence of CyONE (10 equiv).

  ). Conversely, in the presence of an excess of CyOH, a new signal appears at ca. 1500 G (Figure5-2, FigureS3, S4), while the solubility of the salt increases significantly. The fitting of the spectrum reveals the presence of at least three different high-spin species. The first species corresponds to that described in Figure5-1, whereas the remaining two species possess a slightly expanded g

[

Figure 6 .!Figure 7 .

 67 Figure 6. Experimental (blue) and fitted (red) EPR spectra at 110 K of Fe III (OTf)3 (10 mM) in a solution of TBHP in CyH (7.5 wt%).

( 8 )

 8 (Fe IV =O)X2 + CyOH → Fe II (H2O)X2 + CyONE (9) Fe III (OO°)X2 + CyH → (Fe IV =O)X2 + CyOH (10) Fe II (OO°)X + CyH → Fe II (O°)X + CyOH

Table 1 .

 1 Oxidation of CyH to KA oil catalyzed by Fe catalysts. CyH and TBHP conversion, KA selectivity and yield, A/K molar ratio and TOFs as a function of the catalyst at 0.14 mM Fe III (OTf)3 (10 ppm Fe)[a]

	H OOH	[Fe]	OH	O	OH
	+	80	°C	+	+
	[Fe]
		tBuOH + (tBuO) 2 by-products

! Scheme 2.

Table 2 .

 2 CyH and TBHP conversion, KA selectivity and yield, A/K molar ratio and TON as a function of the Fe III (OTf)3 concentration[a]

  III (OO t Bu)X2 → (Fe IV =O)X2 + t BuO° (4) Fe III (OO t Bu)X2 → Fe III (OO°)X2 + t Bu° (5) Fe II (OO t Bu)X → Fe II (OO°)X + t Bu° (6) Fe II (OO t Bu) + t BuOOH → Fe II (OO°) + t BuO° + t BuOH

Acknowledgements

The authors would like to express their gratitude to Solvay, CNRS, PSL, ENS Paris, ENS de Lyon and UPMC for funding. P.A. Payard is grateful to ENS Paris Saclay for a PhD grant. W-J. Zhou gratefully acknowledges the Shanghai Youth Top-Notch Talent development program. The authors would like to thank Ilaria Ciofini, Laurent Binet and Nadia Touati (EPR network RENARD, IT CNRS 3443) from ChimieParisTech for precious assistance and advice. The authors are also grateful to Dr. J. Lai (E2P2L) and Dr C. Michel (ENS Lyon) for fruitful discussion on triflate catalysis and computation of orbital energies of Fe(IV)=O intermediates, respectively.