Benchmarking the Fluxional Processes of Organometallic Piano-Stool Complexes

The correlation consistent Composite Approach for transition metals (ccCA-TM) and density functional theory (DFT) computations have been applied to investigate the fluxional mechanisms of cyclooctatetraene tricarbonyl chromium ((COT)Cr(CO)3) and 1,3,5,7-tetramethylcyclooctatetraene tricarbonyl chromium, molybdenum, and tungsten ((TMCOT)M(CO)3 (M = Cr, Mo, and W)) complexes. The geometries of (COT)Cr(CO)3 were fully characterized with the PBEPBE, PBE0, B3LYP, and B97-1 functionals with various basis set/ECP combinations, while all investigated (TMCOT)M(CO)3 complexes were fully characterized with the PBEPBE, PBE0, and B3LYP methods. The energetics of the fluxional dynamics of (COT)Cr(CO)3 were examined using the correlation consistent Composite Approach for transition metals (ccCA-TM) to provide reliable energy benchmarks for corresponding DFT results. The PBE0/BS1 results are in semiquantitative agreement with the ccCA-TM results. Various transition states were identified for the fluxional processes of (COT)Cr(CO)3. The PBEPBE/BS1 energetics indicate that the 1,2-shift is the lowest energy fluxional process, while the B3LYP/BS1 energetics (where BS1 = H, C, O: 6-31G(d′); M: mod-LANL2DZ(f)-ECP) indicate the 1,3-shift having a lower electronic energy of activation than the 1,2-shift by 2.9 kcal mol−1. Notably, PBE0/BS1 describes the (CO)3 rotation to be the lowest energy process, followed by the 1,3-shift. Six transition states have been identified in the fluxional processes of each of the (TMCOT)M(CO)3 complexes (except for (TMCOT)W(CO)3), two of which are 1,2-shift transition states. The lowest-energy fluxional process of each (TMCOT)M(CO)3 complex (computed with the PBE0 functional) has a ΔG‡ of 12.6, 12.8, and 13.2 kcal mol−1 for Cr, Mo, and W complexes, respectively. Good agreement was observed between the experimental and computed 1H-NMR and 13C-NMR chemical shifts for (TMCOT)Cr(CO)3 and (TMCOT)Mo(CO)3 at three different temperature regimes, with coalescence of chemically equivalent groups at higher temperatures.


Introduction
Fluxional molecules are dynamic compounds in which magnetically or chemically distinct groups can readily interchange positions. The stereochemically fluid nature of such molecules at room temperature is illustrated by the fluxional shifts that they undergo [1][2][3]. These systems perform a significant role in increasing enantioselectivity in asymmetric synthesis [4][5][6][7][8][9][10]. CpRu((R)-BINOP-F)(H 2 O)][SbF 6 ] has been used as a catalyst in the Diels-Alder reaction of methacrolein and cyclopentadiene to produce a [4+2] cycloadduct with enantioselectivity of 92% ee (exo) [9]. During this reaction, the catalyst was shown to exhibit a fluxional pendular motion of the BINOP-F ligand, thereby creating chemically equivalent environments about the two phosphorus substituents.
Density functional theory (DFT) is a useful tool used to characterize the details in the fluxional behavior of various complexes [11]. Previous studies incorporating DFT on fluxional systems range from biochemical applications, such as Cu (II)· · · GlyHisLys peptide Density functional theory (DFT) is a useful tool used to characterize the details in the fluxional behavior of various complexes [11]. Previous studies incorporating DFT on fluxional systems range from biochemical applications such as Cu (II)⋯GlyHisLys peptide binding [12] to understanding fluxionally chiral dimethylaminopyridine catalysts [4,10]. Haptotropic rearrangement processes in sandwich-type complexes have also been investigated by DFT approaches [13][14][15]. Similarly, DFT has been used to gain insight into the fluxionality of various ligands, such as phosphines, in transition metal complexes by way of simulated NMR spectroscopy [16,17].
Although DFT has been used to characterize the energetics of fluxional processes, composite approaches have yet to be utilized to calibrate DFT results on these systems. The correlation consistent Composite Approach for transition metals (ccCA-TM) has been previously utilized to benchmark energetics of transition metal complexes [18][19][20][21][22][23][24][25], and it has been shown to have a mean absolute deviation (MAD) from experiment of 3.0 kcal mol −1 ("transition-metal chemical accuracy"). The ccCA-TM methodology was utilized in this study of cyclooctatetraene chromium tricarbonyl ((COT)Cr(CO)3) to provide reliable energies for which to compare DFT results.
Cyclooctatetraene tricarbonyl d 4 complexes ((COT)M(CO)3) are fluxional molecules that contain an η 6 -bound COT ligand that results in a "piano-stool" conformation ( Figure  1A) [26]. In order to be comprehensive and specific, we provide, in Figure 1, an explicit accounting of the possible shifts in COT-type complexes. The lowest energy geometry for the (COT)M(CO)3 molecule is a piano-stool structure (A in Figure 1). These complexes can undergo a variety of fluxional shifts in which metal-COT carbon interactions are disrupted and then bound on a new carbon of the COT ligand. These processes can be denoted as a 1,n-shift (n = 2, 3, 4, 5), where n represents the carbon on the ring to which the reference bond on the ring has moved. For example, a 1,2-shift indicates a bonding rearrangement from the parent configuration ([1-6]-η 6 geometry, i in Figure 1) to another η 6 configuration ([2-7]-η 6 geometry, ii in Figure 1). Therefore, a 1,3-shift would result in the parent [1-6]-η 6 geometry rearranging to the [3-8]-η 6 geometry (iii in Figure 1).  Historical studies demonstrate that variable temperature NMR (VT-NMR) is a pivotal tool in understanding fluxional behavior [27]. For example, VT 1 H-NMR spectra provide insight into the energetics of the valency tautomerism of COT ligands about the metal center of (COT)M(CO) 3 (M = Cr, Mo) (∆G ‡ = 15.4 kcal mol −1 (k ≈ 25 s −1 ) at 20 • C for (COT)Cr(CO) 3 and ∆G ‡ = 14.8 kcal mol −1 (k ≈ 25 s −1 ) at 10 • C for (COT)Mo(CO) 3 [28]. The pioneering work of Cotton and coworkers proposed mechanisms for these low energy rearrangement processes [1,[26][27][28][29][30]. Whitesides and Budnik successfully studied the Mo derivative ten years later to arrive at similar conclusions [31]. Spin-saturation [32] and 2D-EXSY [33] have been used to understand fluxional processes. In addition, Lawless and Marynick's study provided insights from semiempirical computations into the ring rearrangement processes for (COT)Cr(CO) 3 [34]. In this study, we apply the robust ccCA-TM approach alongside DFT to provide insight into the ring-rearrangement processes of (COT)Cr(CO) 3 . Various ring-rearrangement pathways, including 1,2-, 1,3-, 1,4-, and 1,5-shifts, and (CO) 3 rotation about the metal center, will be considered in the fluxional processes of these complexes. Herein, we report a study of the energetics of (COT)Cr(CO) 3 and (TMCOT)M(CO) 3 (M = Cr, Mo, W), and an analysis of VT-NMR spectra for (TMCOT)Cr(CO) 3 and (TMCOT)Mo(CO) 3 . Figure 2 is the potential energy surface for the ring rearrangement processes of (COT)Cr(CO) 3 at various levels of theory, in which one minimum energy structure (I) and five transition states (TS-1 to TS-5) were identified. ∆E e ‡ values for the following transition states are given at the PBEPBE/BS1, B3LYP/BS1, and PBE0/BS1 levels of theory (see Methodology for basis set definitions). Additionally, given are the ∆E e ‡ values derived using ccCA-TM. TS-1 is C s -symmetric, where the COT ligand is η 6 -bound to Cr. This transition state represents the 120 • rotation of the three carbonyl groups about the metal center. TS-2 (1,3-shift) is a C s -symmetric complex where the COT ligand is η 4 -bound to Cr. TS-3 (1,2-shift) is C s -symmetric structure such that the COT ligand is η 5 -bound to Cr. TS-4 (1,5-shift) is a C s -symmetric complex, which possesses an η 4 -bound COT ligand. The highest-energy fluxional transition state, TS-5 (1,4-shift), is a C s -symmetric complex, where the COT ligand is η 4 -bound to Cr. Good agreement was observed in the relative ∆E e ‡ values computed using PBE0/BS1 and those derived using ccCA-TM. Tabulated bond lengths, ∆G ‡ , ∆E e ‡ , and ∆∆E e ‡ (relative to ccCA-TM) values computed at each level of theory are given in the Supporting Information (Table S1, Table S2, Table S3, and Table S4, respectively). Method and basis set testing was performed for each (I, TS-1-TS-5).

Results
Additionally, identified were two additional geometries, structure 2 and TS-6 (see Figure S4). Structure 2 is a local minimum (∆E e = 24.3 kcal mol −1 using the PBE0/BS1 level of theory) connected to I by way of TS-6 (∆E e ‡ = 25.3 kcal mol −1 ). Because 2 is higher in energy than I, it was not considered in the fluxional processes of (COT)Cr(CO) 3 .
The TMCOT ligand in the X-ray crystal structure of (TMCOT)Cr(CO) 3 reported by Cotton and coworkers is η 6 -bound to Cr. An overlay of experimental and PBE0/BS1 optimized structures is shown in Figure 3.
The XRD structure (CSD entry: TMCOCR) [29] matches well with the optimized geometry of the lowest energy structure (II) for (TMCOT)Cr(CO) 3 (RMSD = 0.038 Å for the 19 heavy atoms using PBE0/BS1 optimized structure). The X-ray crystal structures for the Mo and W complexes have not been reported. However, the computed structures for these derivatives each contain a η 6 -bound TMCOT ligand and appear to be very similar to the lowest-energy structure computed for the Cr complex (RMSD = 0.146 Å for Mo, 1.181 Å for W relative to (TMCOT)Cr(CO) 3 for 19 heavy atoms; RMSD calculated using PBE0/BS1 optimized structures). Additionally, identified were two additional geometries, structure 2 and TS-6 (see Figure S4). Structure 2 is a local minimum (ΔEe = 24.3 kcal mol −1 using the PBE0/BS1 level of theory) connected to I by way of TS-6 (ΔEe ‡ = 25.3 kcal mol −1 ). Because 2 is higher in energy than I, it was not considered in the fluxional processes of (COT)Cr(CO)3.
The TMCOT ligand in the X-ray crystal structure of (TMCOT)Cr(CO)3 reported by Cotton and coworkers is η 6 -bound to Cr. An overlay of experimental and PBE0/BS1 optimized structures is shown in Figure 3.  The XRD structure (CSD entry: TMCOCR) [29] matches well with the optimized geometry of the lowest energy structure (II) for (TMCOT)Cr(CO)3 (RMSD = 0.038 Å for the 19 heavy atoms using PBE0/BS1 optimized structure). The X-ray crystal structures for the Mo and W complexes have not been reported. However, the computed structures for these derivatives each contain a η 6 -bound TMCOT ligand and appear to be very similar to the lowest-energy structure computed for the Cr complex (RMSD = 0.146 Å for Mo, 1.181 Å for W relative to (TMCOT)Cr(CO)3 for 19 heavy atoms; RMSD calculated using PBE0/BS1 optimized structures).
The potential energy surface for the various rearrangements processes for The potential energy surface for the various rearrangements processes for (TMCOT)Cr(CO) 3 is given in Figure 4. The lowest-energy geometry is represented by structure II. ∆E e ‡ values for the described transition states are given at the: PBEPBE/BS1, B3LYP/BS1, and PBE0/BS1 levels of theory for (TMCOT)Cr(CO) 3 . TS-A (CO-rotation) is C 1 -symmetric and represents a 120 • (CO) 3 rotation about Cr, where the TMCOT ligand remains η 6 -bound to the Cr. The , is a C s -symmetric structure in which TMCOT is η 5 -bound to Cr. In TS-B, the mirror plane passes through two of the opposing C-H units in the TMCOT ligand. TS-C (1,3-shift) is C 1 -symmetric, in which the TMCOT ligand is η 4 -bound to the Cr atom. TS-D (1,5-shift) is a C 1 -symmetric structure in which TMCOT is η 4 -bound to the metal. TS-E (1,2-shift-b) is a C s symmetric structure in which TMCOT is η 5 -bound to the Cr center and represents a second 1,2-shift. The mirror plane in this structure passes through the opposing C-CH 3 units, in contrast to the opposing C-H units as seen in TS-A. TS-F (1,4-shift) represents the transition state in which the complex is C 1 -symmetric with an η 4 -bound TMCOT ligand.   Table 1 and Table S3, respectively. An additional 1,4-shift transition state (TS-G was located computationally for (TMCOT)W(CO)3 ( Figure S3).   Table 1 and Table S3, respectively. An additional 1,4-shift transition state (TS-G) was located computationally for (TMCOT)W(CO) 3 ( Figure S3). Table 1. Computed relative electronic energies and free energies (in parentheses) for (TMCOT)M(CO) 3 (M = Cr, Mo, W) fluxional transition states with DFT/BS1 (all energies reported in kcal mol −1 ).

Discussion
Several DFT methods utilized for the (COT)Cr(CO) 3 system disagreed with the relative energy ordering of the fluxional processes obtained with the more rigorous ccCA-TM methodology. It is helpful to determine which methodology is in closest agreement with this composite approach. After testing several levels of theory, it was found that PBE0/BS1 and PBE0/BS3 computed ∆E e ‡ values do indeed qualitatively correlate with ccCA-TM. Consequently, PBE0/BS1 was extended to the larger, less symmetric (TMCOT)M(CO) 3 systems. Notably, ccCA-TM relative energetics computed using the weighted and nonweighted double-ζ basis sets for CCSD(T) and CCSD(T, FC1) single points were isoenergetic within 0.1 kcal mol −1 .
The ∆E e ‡ values for (COT)Cr(CO) 3 computed using the PBE0/BS1 and PBE0/BS3 levels of theory were both in agreement with the relative ordering presented by ccCA-TM. However, PBEPBE energetics computed with BS1, BS2, and BS3 each showed that the 1,2 and 1,3-shifts were nearly isoenergetic with an energy difference of 0.2 kcal mol −1 . B3LYP electronic energetics computed with BS1, BS3, and BS5 each showed that the 1,3-shift was the lowest energy structure, where the 1,2-shift was lower in energy than the COrotation. The electronic energies computed at the B97-1/BS2 and B97-1/BS3 levels of theory depicted that the 1,3-shift was the lowest energy process, followed by the CO-rotation and the 1,2-shift. This data is provided in Table S3.
Solvation of (TMCOT)M(CO) 3 in chloroform SMD raises the ∆G ‡ of most transition states, and slightly alters the relative ordering of the fluxional processes of each (TM-COT)M(CO) 3 complex (Table S7).
Simulated 1 H-NMR and 13 C-NMR spectra computed at the GIAO-PBE0/BS6//PBE0/BS1 level of theory ( Figures 5 and 6, respectively) show that coalescence of peaks is observed when higher temperature regimes are considered. When the temperature is increased, higher energy fluxional transition states are more easily accessible for (COT)Cr(CO) 3   The experimental and computed 1 H-NMR spectra of (TMCOT)Cr(CO) 3 illustrate that there are eight distinct peaks at −23 • C ( Figure 5A). In the computed gas-phase 1 H-NMR spectrum, the peaks for vinyl protons 8 and 2 have reversed assignments when compared to experimental results. Cotton and coworkers stated that there was no basis for assigning the peaks for the methyl groups in the low-temperature limit 1 H-NMR spectrum, so the experimental labeling is arbitrary [30]. The computed assignments for the methyl peaks are b, c, a, d (downfield to upfield) while the experimental assignments were c, d, a, b (downfield to upfield) ( Figure 5A).
Increasing the temperature to 46 • C leads to coalescence of equivalent vinyl protons 2/6 and the methyl protons represented by b/c and a/d, respectively, while vinyl protons 4 and 8 remain chemically distinct. This leads to five discernable peaks: three for the vinyl protons and two for the methyl protons ( Figure 5B).
The experimental VT 13 C-NMR spectrum of (TMCOT)Cr(CO) 3 shows twelve distinct peaks ( 13 C-NMR chemical shifts for CO groups were not reported) [26] at low temperature while the simulated spectrum shows fifteen distinct peaks ( Figure 6A). By increasing the temperature to an intermediate temperature, the peaks for carbons 4 and 8 of the TMCOT ligand remain chemically non-equivalent ( Figure 6B). However, coalescence of peaks is observed in carbons 1/7, 3/5, 2/6, b/c, and a/d, and carbons 9 The experimental and computed 1 H-NMR spectra of (TMCOT)Cr(CO)3 illustrate that there are eight distinct peaks at −23 °C ( Figure 5A). In the computed gas-phase 1 H-NMR spectrum, the peaks for vinyl protons 8 and 2 have reversed assignments when compared to experimental results. Cotton and coworkers stated that there was no basis for assigning the peaks for the methyl groups in the low-temperature limit 1 H-NMR spectrum, so the experimental labeling is arbitrary [30]. The computed assignments for the methyl peaks are b, c, a, d (downfield to upfield) while the experimental assignments were c, d, a, b (downfield to upfield) ( Figure 5A).
Increasing the temperature to 46 °C leads to coalescence of equivalent vinyl protons 2/6 and the methyl protons represented by b/c and a/d, respectively, while vinyl protons 4 and 8 remain chemically distinct. This leads to five discernable peaks: three for the vinyl protons and two for the methyl protons ( Figure 5B).
The experimental VT 13 C-NMR spectrum of (TMCOT)Cr(CO)3 shows twelve distinct peaks ( 13 C-NMR chemical shifts for CO groups were not reported) [26] at low temperature while the simulated spectrum shows fifteen distinct peaks ( Figure 6A). By increasing the temperature to an intermediate temperature, the peaks for carbons 4 and 8 of the TMCOT ligand remain chemically non-equivalent ( Figure 6B). However, coalescence of peaks is observed in carbons 1/7, 3/5, 2/6, b/c, and a/d, and carbons 9/10/11 (only observed computationally), thereby indicating a 1,2-shift within this temperature region. Due to the rotation rate of the three CO ligands about the Cr atom, all CO ligands are chemically equivalent. At high temperatures, coalescence is observed in each the olefin carbons, methyl carbons, and carbonyl carbons, respectively ( Figure 6C).
E ref (ccCA) represents energies at the MP2/aug-cc-pVXZ (X = D, T, Q) complete basis set (CBS) limit added to the HF/CBS energy. The HF CBS limit can be computed with a two-point extrapolation of HF energies with the aug-cc-pVTZ and aug-cc-pVQZ basis sets as seen in Equation (2).
∆E(CC) is a correction that stems from CCSD(T) to account for higher order dynamic correlation effects, as it is not adequately described using MP2. ∆E(CV) represents a basis set correction to the core-core and core-valence electron interactions in CCSD(T). ∆E(ZPE) is the result of zero-point energy and thermal corrections at 298.15 K, which uses harmonic vibrational frequencies scaled by 0.989. ∆E(SO) is the atomic spin-orbit coupling correction [84].
Magnetic shielding tensors for (TMCOT)Cr(CO) 3 and (TMCOT)Mo(CO) 3 were computed using the Gauge-Independent Atomic Orbital (GIAO) [85][86][87][88] method at the GIAO-PBE0/BS6//PBE0/BS1 level of theory. BS6 is described as the LANL08(f) [46,89] basis sets and corresponding ECP for Cr and Mo, LANL08(d) [48,89] with LANL2DZ ECP for Si, and the IGLO-II basis sets for H, C, and O [90]. The gas-phase computed chemical shift is represented by the difference between the absolute isotropic shielding constant for the reference atom (as computed in TMS) and the absolute isotropic shielding constant for the considered atom within the complex.
Simulated 1 H-NMR and 13 C-NMR spectra were obtained using an in-house Fortran program by convoluting the computed absolute isotropic shielding values and relative chemical shift with a Gaussian line shape and broadening of 0.025 ppm [91]. To account for peak averaging, the relative isotropic shielding values for chemically equivalent protons and carbons were considered (e.g., the relative isotropic shielding values for the three protons on a methyl group were considered, and the resulting peak was given at the mean chemical shift of these three peaks). To account for temperature-based peak averaging, the chemical shifts of protons and carbons that become chemically equivalent through fluxional processes at each temperature regime were averaged, thereby resulting in coalesced peaks.
Solvation effects on the fluxional processes have been considered by the using the SMD (chloroform) implicit solvation model via self-consistent reaction field (SCRF). Singlepoint solvation energy computations were performed on gas-phase optimized structures at the SMD-PBE0//PBE0/BS1 level of theory [92].

Conclusions
Density functional theory (DFT) was applied to investigate the fluxional processes in (COT)Cr(CO) 3 and (TMCOT)M(CO) 3 (M = Cr, Mo, and W). ccCA-TM energetics demonstrated that TS-2 (1,3-shift) is the lowest-energy fluxional process of (COT)Cr(CO) 3 (∆E e ‡ = 17.2 kcal mol −1 ). It was also discovered that the 1,2-shift represented by TS-B (also known as 1,2-shift-a) is the lowest-energy fluxional process for all three (TMCOT)M(CO) 3 complexes (∆G ‡ = 12.6 kcal mol −1 , 12.8 kcal mol −1 , and 13.2 kcal mol −1 for M = Cr, Mo, and W, respectively), which was reaffirmed by the analysis of the experimental and computed 1 H and 13 C-NMR chemical shifts. The computed free energy of activation for TS-B of each of these complexes is consistently slightly lower than the reported experimental results. Implicit solvation slightly alters the relative energies and ordering for the fluxional transition states of all (TMCOT)M(CO) 3 complexes. By increasing the temperature to the 112 • C, coalescence of the vinyl hydrogen and methyl peaks, respectively, is observed in the 1 H-NMR spectrum of (TMCOT)Cr(CO) 3 . Similarly, methyl, olefin, and carbonyl carbon peaks coalesce in the high temperature region of the 13 C-NMR spectrum of (TMCOT)Cr(CO) 3 .