Computational Spectroscopy of the Cr–Cr Bond in Coordination Complexes

We report the accurate computational vibrational analysis of the Cr–Cr bond in dichromium complexes using second-order multireference complete active space methods (CASPT2), allowing direct comparison with experimental spectroscopic data both to facilitate interpreting the low-energy region of the spectra and to provide insights into the nature of the bonds themselves. Recent technological development by the authors has realized such computation for the first time. Accurate simulation of the vibrational structure of these compounds has been hampered by their notorious multiconfigurational electronic structure that yields bond distances that do not correlate with bond order. Some measured Cr–Cr vibrational stretching modes, ν(Cr2), have suggested weaker bonding, even for so-called ultrashort Cr–Cr bonds, while others are in line with the bond distance. Here we optimize the geometries and compute ν(Cr2) with CASPT2 for three well-characterized complexes, Cr2(O2CCH3)4(H2O)2, Cr2(mhp)4 , and Cr2(dmp)4. We obtain CASPT2 harmonic ν(Cr2) modes in good agreement with experiment at 282 cm−1 for Cr2(mhp)4 and 353 cm−1 for Cr2(dmp)4, compute 50Cr and 54Cr isotope shifts, and demonstrate that the use of the so-called IPEA shift leads to improved Cr–Cr distances. Additionally, normal mode sampling was used to estimate anharmonicity along ν(Cr2) leading to an anharmonic mode of 272 cm−1 for Cr2(mhp)4 and 333 cm−1 for Cr2(dmp)4.


Introduction
The concept of multiple bonding between metals dates back to the late 1920s with the first structural confirmation in 1964. ? Over the last 60 years, multiple bonds between a wide variety of transition metals have been assigned bond orders ranging from single to sextuple. ? ? ? Nowadays, synthetic chemists seek out new metal-metal bonds for as catalysts, building blocks in metal-organic frameworks, photosensitizers, and molecular conductors. ? ? ? ? ? Group 6 metalmetal bonds have been broadly studied for their interesting bonding and unique photophysical properties. ? ? Of these, Cr-Cr multiple bonds are notable because of their rich electronic structure and unique spectroscopic properties. Starting with Cotton's work in the 1970s, a variety of dichromium complexes have been synthesized; characterization by diffraction has shown that the bond distances range from 1.7 to 2.3 Å. ? ? ? ? ? ? ? ? ? Although in the early days formal bond orders were estimated based on distances, computational work using multiconfigurational methods has established that the electronic structure of these complexes is multiconfigurational due to the small energy splitting between the σ, π, and δ orbitals. The concept of an effective bond order (EBO), that accounts for partial occupation of low-lying antibonding orbitals, has been used to quantify the nature of the overall Cr-Cr bond that is composed of multiple partial bonds. ? ? Furthermore, these computations highlight that the multiconfigurational electronic structure of Cr 2 bonds is more nuanced compared to Mo 2 and W 2 analogues. ? As a result, the Cr-Cr bond order often does not correlate with bond distance. ? ? ? ? Note that for some particularly short Cr-Cr distances, density functional theory (DFT) has been used successfully since the electronic structure can be represented by fully occupied bonding orbitals. ?
The lack of correlation between bond distance and bond order motivated a number of vibrational studies of these compounds; diffraction studies alone cannot be used to predict bonding or, in turn, vibrational force constants. Da Re et al. emphasize the unique challenges in dichromium complexes by comparing the empirical force constants and bond distances for first and second row transition metals. For example, Mo(II)-Mo(II) bond distances empirically correlate to their vibrational frequencies, as expected, with force constants increasing monotonically with decreas-ing bond distance. ? Cr(II)-Cr(II) bonds deviate from the norm and have multiple modes in the low-frequency region of the spectra involving the Cr centers, making peak assignment challenging. The measured force constants in two complexes with ultrashort Cr-Cr bonds, Cr 2 (mhp) 4 and Cr 2 (dmp) 4 , drew our interest (mhp = deprotonated 6-methyl-2-hydroxypyridine and dmp = 2,6dimethoxyphenyl). In the former, the Cr-Cr stretching mode, ν(Cr 2 ), was first assigned at 556 cm −1 by Cotton et al. but was later revised to be at 340 (or 400) cm −1 by Manning and Trogler. ? ?
The Cr-Cr bond in Cr 2 (dmp) 4 is even shorter, but the ambiguity surrounding the vibrational mode remains. An estimation based on the aggregate isotope shift would lead to the assignment at 650 cm −1 ; however, an alternative assignment, based on the average of three low-frequency bands assigned to include motion in the Cr-Cr internal coordinate, would put ν(Cr 2 ) closer to 365 cm −1 . ? In both complexes, the larger value is consistent with the force constant that would be predicted from a simple harmonic approximation based on the bond distance. The lower values would suggest a significantly weaker bond than one would expect from a formally quadruple bond. Figure 1: The dichromium complexes studied in this work. mhp = deprotonated 6-methyl-2hydroxypyridine and dmp = 2,6-dimethoxyphenyl. Cr in orange, N in blue, O in red, C in gray, and H in white.
Vibrational analysis of compounds consisting of a few dozen atoms is routinely performed by DFT. However, as mentioned earlier, the electronic structure of dichromium complexes is highly multiconfigurational, which DFT cannot describe properly. ? For the computation of bond length, several procedures have been proposed to empirically correct DFT simulations. For example, DFT geometry optimizations can be performed keeping the Cr-Cr bond distance fixed at the experimen-tal value with subsequent analysis of the electronic structure with CASPT2. ? Alternatively, a series of constrained DFT optimizations can be performed to determine the Cr-Cr bond distance pointwise with CASPT2 yielding good agreement with experiment. ? ? ? ? These hybrid approaches, however, are not applicable to the computation of the vibrational spectra. Moreover, full CASPT2 geometry optimization for complexes with transition metals had only been performed on small molecules, and even then not routinely. ? ? ?
Here, we directly apply the CASPT2 electronic structure method to the prediction of vibrational spectra of Cr-Cr complexes, in which the vibrational frequencies and normal modes are computed from the second derivative of CASPT2 energies. Full geometry optimizations with the CASPT2 method and subsequent harmonic vibrational analysis are made possible by utilizing the recently developed CASPT2 analytical gradient program and extending it to compute second-order energy derivatives by means of finite difference of the first derivatives. Anharmonic effects are known to be important from experimental work on these complexes; ? therefore, the anharmonicity of the Cr-Cr harmonic stretching mode is studied.

Computational Details
Software Program for CASPT2 vibrational spectra The CASPT2 second-order energy derivatives are computed by the finite difference of the firstorder derivatives calculated using the analytical nuclear gradient program recently developed by the authors. ? ? We note in passing that the use of the analytical gradient program is essential in realizing the CASPT2 vibrational analysis because it reduces the computational cost by two-to three-orders of magnitude for the molecules in this study. An embarrassingly parallel algorithm has been included that allows the user to select the number of MPI processes required for each gradient calculation and distribute the 6N displacements among different tasks. The Hessian is symmetrized, mass weighted, and the translational and rotational degrees of freedom are set to zero using the usual projection (see Ref. ? for a concise summary). The normal modes are obtained by diagonalizing the projected mass-weighted Hessian yielding the normal modes (eigenvectors), and the corresponding harmonic vibrational frequencies can be calculated from the eigenvalues. The infrared intensity is also computed from the transition dipole moments. Raman intensities are not yet implemented.
In some of the calculations herein, we used Partial Harmonic Vibrational Analysis (PHVA) as described first by Head and coworkers ? ? ? ? ? and later by Li and Jensen ? to reduce the computational cost. In PHVA, a subblock of the Hessian matrix is diagonalized to obtained vibrational frequencies within that portion of the molecule. This scheme is particularly useful in this study because the Cr-Cr stretching mode is localized, allowing us to skip some of the displacements, especially those associated with bulky ligands, that do not couple directly or indirectly with the Cr-Cr stretching mode. The selection scheme of the atoms to be displaced is detailed below.

Simulation Parameters
CASPT2 geometry optimizations and vibrational frequencies have been performed as implemented in the bagel program package. ? An (8e, 8o) active space is used consisting of the 3d σ, π, and δ bonding and antibonding orbitals, resulting from the interaction of the 3d 4 atomic configurations of the two Cr(II) centers ( Figure ?? and ??, Figure S1).  Table S5 for calculations using other basis sets. A real shift of 0.3 a.u. was used. ? The computation of harmonic vibrational frequencies with both the Full Harmonic Vibrational Approximation (FHVA) and PHVA was performed. Complete active space self-consistent field (CASSCF) geometry optimizations were conducted for comparison. For Cr 2 (mhp) 4 and Cr 2 (dmp) 4 , CASPT2 geometry optimizations of the ground state followed by PHVA were performed using a basis set with cc-pVTZ on Cr and the first coordination sphere while cc-pVDZ was used on all other atom types. A real shift of 0.4 a.u. was used.
The CASPT2 energy was converged to 1 × 10 −8 and the forces were converged to 3 × 10 −4 . The def2-TZVPP-JKFIT basis was used in all calculations for density fitting ? and the so-called IPEA shift was not employed since it is not yet implemented for CASPT2 nuclear gradients. ? In PHVA, the Cr atoms and the first coordination sphere were displaced.
A scan of the potential energy surface (PES) was then performed by displacing the geometry along the harmonic normal mode (Q i ) prior to mass weighting in Cartesian increments of 0.25 for DFT geometry optimizations and harmonic vibrational frequencies were computed as implemented in the Turbomole program package. ? The M06 functional ? was used with the def2-TZVP basis set. ? Singlet and triplet spin states were considered. Attempts to optimize the geometry with a broken-symmetry singlet routinely relax to the closed-shell singlet.
The effect of the basis set is small, and PHVA yields similar performance but reduces the total computational cost significantly ( reported by a large margin and the Cr 2 (dmp) 4 complex, when measured, was expected to lead to an even higher value. This value is consistent with an empirically assigned force constant based on a simple diatomic oscillator approximation employing the measured Cr-Cr distance. However, the vibronic features in the absorption and emission spectra as well as combined infrared and Raman analysis performed by Manning and Trogler ruled out this assignment suggesting that ν(Cr 2 ) lies at a much smaller wavenumber, either 340 or 400 cm −1 . ? They specifically note that either of the two Raman bands could be the Cr-Cr stretching mode but could not obtain solution spectra to further solidify the assignment. Since a vibronic progression at 320 cm −1 , assigned to the so-called δ → δ * excited state, was also observed, Manning and Trogler suggested that the 340 cm −1 peak was more likely to be the Cr-Cr stretching mode. This was based on the observation of a reduction in ν(M 2 ) by 10-15% in the excited state for quadruple bonded M 2 systems.
The CASPT2 harmonic Cr-Cr stretching mode is at 282 cm −1 ; however, modes at 266, 267, 359, and 360 cm −1 consist of coupled metal-metal and metal-ligand stretching motions. We can compare ν(Cr 2 ) with experiment directly but note that the average of all modes involving significant Cr-Cr motion would give nearly the same value, 299 cm −1 , supporting Manning and Trogler's assignment of 340 cm −1 . Furthermore, these results reinforce previous conclusions based on the vibrational spectrum that the formally Cr 2 quadruple bond in this complex is much weaker, consistent with the electronic structure (vida supra). Although, to our knowledge, isotopic labeling experiments have not been performed for Cr 2 (mhp) 4 , ν(Cr 2 ) shifts to 287 cm −1 for 50 Cr and 277 cm −1 for 54 Cr. Similar sized shifts are observed for the modes consisting of combinations of metalmetal and metal-ligand motions (Table S6).
Additionally, ν(Cr 2 ) was computed for Cr 2 (dmp) 4 and the predominate Cr-Cr stretch is at 353 cm −1 . The mode is the largest of the three complexes, consistent with having the shortest bond distance and highest bond order, but remains significantly smaller than empirical estimates. The CASPT2 value for ν(Cr 2 ) is in good agreement with the estimate of this mode at 365 cm −1 based on averaging three bands observed in experiment. Specifically, three bands appear in experiments using the naturally occurring isotope (∼87% 52 Cr) at 345, 363, and 387 cm −1 (labeled as bands a, b, and c; Table ??) and were assigned to involve Cr motions due to their shifts upon isotope substitution. ? Since several CASPT2 normal modes contributions from Cr-Cr, Cr-C and Cr-N stretching motions, further analysis including 50 Cr and 54 Cr isotopes was performed (Table S7).
Of the normal modes computed by CASPT2 with mixed metal-metal and metal-ligand stretching motions, the two at 310 and 324 cm −1 have the largest shifts upon isotope substitution (Table S6).
We tentatively assign these two modes along with the relatively unmixed ν(Cr 2 ) mode as bands a, b, and c ( Table ??). The CASPT2 bands are shifted to lower wavenumbers at 310, 324, and 353 cm −1 but the spacing between peaks is in good agreement with experiment. This is likely due in part to the overestimation of the calculated bond distance.
This assignment is further supported by the calculated isotope shifts. The predicted shift due to substituting 54 Cr with 50 Cr for a simple diatomic oscillator with a mode of 340 cm −1 was estimated to be ∼13 cm −1 . ? The experimentally observed shifts are 10.1, 9.4, and 5.6 cm −1 for peaks a, b, and c, respectively. CASPT2 computed values are in excellent agreement at 9.3, 10.3, and 12.2 cm −1 , with the largest deviation for ν(Cr 2 ) where the effect of an elongated Cr-Cr bond distance and anharmonicity of the mode is expected to be greatest. The aggregate Cr isotope shift from the three bands was 25.1 cm −1 in experiment and is 31.9 cm −1 with CASPT2.
Finally, we note that the largest complex, Cr 2 (dmp) 4 , contains 78 atoms and the calculation was performed on 768 CPUs. The PHVA calculation took 5 days to complete involving 60 gradient calculations required for Hessian evaluation. FHVA would have required 468 gradient calculations.

Anharmonicity along the Cr-Cr Stretching Mode
Although the agreement between experiment and CASPT2 offers clear improvement over DFT, the bond distances are longer than what is observed for typical agreement for single-reference transitional metal complexes between DFT and experiment. Moreover, low frequency Cr-Cr stretching modes are known to exhibit significant anharmonicity. ? ? ? ? ? ? Therefore, a scan of the PES along the harmonic normal mode was performed at the same level of theory as in the harmonic vibrational analysis and by including the so-called IPEA shift in the CASPT2 zeroth-order Hamiltonian. For Cr 2 (mhp) 4 and Cr 2 (dmp) 4 , the structure was displaced along the harmonic mode associated with the Cr-Cr stretching motion, Q i , in the positive and negative direction up to 3Q i . The resulting potential was fit to the fourth order polynomial, to obtain cubic and quartic anharmonic corrections. This approach has been used in a series of The inclusion of anharmonic corrections shifts the vibrational mode to a lower value compared to the harmonic mode by 10 and 20 cm −1 , respectively. The CASPT2 minima obtained from the PES scan of Cr 2 (mhp) 4 without the IPEA shift is the same as that obtained by geometry optimization, 1.989 Å. This is true using either the Dunning or the ANO-RCC basis sets. However, the minima shifted to 1.887 Å when the IPEA shift is employed, in excellent agreement with the experimental value of 1.889(1) Å (Table ??  Conclusions CASPT2 harmonic vibrational analysis can provide insight into the complex low-frequency region of experimental vibrational spectra for large dichromium complexes. The calculated geometries and vibrational modes are consistent with empirical Cr 2 bond distances and frequencies. Specifically, the CASPT2 harmonic Cr-Cr stretching mode is predicted to lie at 196/266, 282, and 353 cm −1 for [Cr 2 (O 2 CCH 3 ) 4 ](H 2 O) 2 , Cr 2 (mhp) 4 , and Cr 2 (dmp) 4 , respectively. The bonding in the three complexes is weaker than their formal assignments based on bond distance and consistent with lower frequency vibrational modes. The complexity of the potential energy surface around the Cr centers manifests in this rich low-energy spectrum, in particular for Cr 2 (mhp) 4 and Cr 2 (dmp) 4 .
The modes between 250 and 400 cm −1 include normal modes that couple metal-metal and metalligand motions. While not all of these modes are confirmed to be Raman active, those that shift upon isotope substitution are consistent with the observed experimental bands, albeit systematically shifted to lower values. The PES calculations along the ν(Cr 2 ) normal mode and observation of improved Cr-Cr bond distances with the inclusion of the IPEA shift support the hypothesis by Da Re et al. that there is a distorted Cr-Cr potential in these Cr(II) 2 complexes similar to that inferred for Cr(0) 2 contributing to the vibrational behavior of the Cr-Cr complexes. ? Finally, the ν(Cr 2 ) mode is anharmonic and including cubic and quartic corrections shifts the vibrational mode to lower values by 10 to 20 cm −1 . Nevertheless, questions remain in describing this complicated low frequency region, in particular the nature of peaks assigned to an excited state transitions observed in the spectra, assumed to arise from the so-called δ → δ * transition. These results emphasize that the interpretation of vibrational spectra for complexes with strong metal-metal and metal-ligand interactions is challenging requiring insights from both experiment and theory.
CASPT2 vibrational analysis is anticipated to be a valuable tool in furthering understanding of multiconfigurational bonds in larger complexes with metal-metal bonds.