Spectroscopic/Bond Property Relationship in Group 11 Dihydrides via Relativistic Four-Component Methods

Group 11 dihydrides MH2– (M = Cu, Ag, Au, Rg) have been much less studied than the corresponding MH compounds, despite having potentially several interesting applications in chemical research. In this work, their main spectroscopic constants (bond lengths, dissociation energies, and force constants) have been evaluated by means of highly accurate relativistic four-component coupled cluster (4c-CCSD(T)) calculations in combination with large basis sets. Periodic trends have been quantitatively explained by the charge-displacement/natural orbitals for chemical valence (CD-NOCV) analysis based on the four-component relativistic Dirac–Kohn–Sham method, which allows a consistent picture of the nature of the M–H bond to be obtained on going down the periodic table in terms of Dewar–Chatt–Duncanson bonding components. A strong ligand-to-metal donation drives the M–H bond and it is responsible for the heterolytic (HM···H–) dissociation energies to increase monotonically from Cu to Rg, with RgH2– showing the strongest and most covalent M–H bond. The “V”-shaped trend observed for the bond lengths, dissociation energies, and stretching frequencies can be explained in terms of relativistic effects and, in particular, of the relativistically enhanced sd hybridization occurring at the metal, which affects the metal–ligand distances in heavy transition-metal complexes. The sd hybridization is very small for Cu and Ag, whereas it becomes increasingly important for Au and Rg, being responsible for the increasing covalent character of the bond, the sizable contraction of the Au–H and Rg–H bonds, and the observed trend. This work rationalizes the spectroscopic/bond property relationship in group 11 dihydrides within highly accurate relativistic quantum chemistry methods, paving the way for their applications in chemical bond investigations involving heavy and superheavy elements.


Methodological details of the 4c-CCSD(T) geometry optimizations
Geometries have been optimized at the 4c-CCSD(T) level by performing the numerical first derivatives of the energy with respect to the nuclear coordinates, by considering several conditions: i) the geometries are expected to be linear from previous calculations [1][2][3][4] ; ii) the two M-H bonds are considered to be equal, on the basis of the complexes linear centrosymmetric symmetry (D ∞h ).
According to these conditions, we could optimize the MH 2 complexes by only accounting for one degree of freedom (the M-H bond) in an iterative approach. At each step, three single-point calculations were performed and a second order Newton's interpolation procedure was used for obtaining the minimum energy and the corresponding bond length. The steps have been repeated until convergence between the two bond lengths obtained after two consecutive steps was reached (10 -4 Å). Using this approach, it was possible to optimize the geometries with a computationally demanding state-of-the-art method in a reasonable amount of time.

Methodological details of the 4c-CCSD(T) vibrational frequencies calculations
The Hessian matrix has been calculated considering that, since in this class of linear triatomic centrosymmetric molecules the stretching and bending modes are not coupled, we could neglect any motion along the x and y axis and considering motion only along the z-axis. In this way, the Hessian is simplified and becomes a 3x3 matrix.
The elements of the Hessian have been calculated via numerical derivatives. The most convenient approach in our case is the central difference approximation. In this way first derivatives can be expressed as follows: where h is the displacement from the equilibrium x 0 value; f(x 0 +h) and f(x 0 -h) represent the function evaluated at both forward and backward displacements and εh 2 represents the error due to the approximation. This is a more convenient way of evaluating derivatives with respect to backward of forward difference approximation, since the error scales as h 2 .
In our case, the diagonal elements of the Hessian can be simply expressed as:

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(2) Instead, in the case of non-diagonal elements, we need to perform subsequent first derivatives for calculating the gradients (G i + and G i -) at first and then the element of the Hessian matrix: - A critical choice for a good approximation is in the value of h. A too small value of h may cause problems in the sense that the difference between the function evaluated at backwards and forward displacement may be smaller than the precision of the calculations, therefore causing the final values to oscillate and therefore not being accurate. Instead, a too big value of h may cause problems concerning a too big error in the estimation of the derivative and therefore a bad approximation of the latter.
With the aim of understanding which the optimal value of h for introducing the smallest possible error was, we calculated the elements of the Hessian for CuH 2 with different values of h ranging from 0.005 to 0.05. Then, rearranging Eqs. (2) and (3), we could find two expressions that, via linear or polynomial regression, could allow us to estimate the error we introduce in the approximation and therefore the real force constant.
For diagonal elements, rearranging Eq. (2) we obtain: This means that in both cases we should be able, by plotting the k ii apparent values (which are the ones we determine through our calculations) versus h 2 , by the means of a linear regression, we should be able to interpolate them to a straight line, whose intercept corresponds to the real force constant.
In the case of non-diagonal elements, manipulating the expressions for the gradients and the final derivative, we obtain the following expression: This means that for k 12 (and k 13 , for which one obtains the same equations), one can estimate the error due to the approximation by plotting the k 12 apparent values versus h and interpolate the data with a second-order polynomial.

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Below are shown the plots for the four constants and the k ij real values, comparing the results with the ones where the smallest value of h is used (i.e. 0.005 Å).

Computational details of the DKS-CD analysis
The CD analysis has also been applied to quantify the rearrangement of the electron density in the MH 2 complexes at relativistic four-component Dirac-Kohn-Sham level 5 using BERTHA code [6][7][8][9] with the following computational set-up. The large component of the basis set for all the atoms was generated by uncontracting triple-ζ quality Dyall's basis sets [10][11][12][13] .The corresponding small component basis was generated using the restricted kinetic balance relation 14 . For the metal atoms (i.e. Cu, Ag, Au, Rg), a previously optimized auxiliary basis set for density fitting denoted as B20 was used 15 . For H atoms, accurate auxiliary basis sets were generated using a simple procedure starting from available DeMon 16 Coulomb fitting basis set. It is worth recalling that the Hermite Gaussian Type Functions (HGTFs) used as fitting functions are grouped together in sets sharing the same exponents (analogous scheme is adopted in the nonrelativistic DFT code DeMon) 16 Figure S1. Correlation between the square of the atomic number (Z 2 ) and the relativistic bond contraction (RBC) in MH complexes. R 2 =0.998.      Bottom: corresponding DKS-CD curves for the CuH 2 complex. Red dots indicate the position of the nuclei along the z axis. The vertical line marks the isodensity boundary. S15 Figure S7. Top: DKS-CD calculated isodensity surfaces of the total (Δρ) and of its first five NOCV components for the AgH 2 complex. The isovalue for the upper three surfaces (i.e. Δρ, Δρ 1 , Δρ 2 ) is ±0.001 e a 0 -3 , whereas for the three lower surfaces (i.e. Δρ 3 , Δρ 4 , Δρ 5 ) the isovalue is ±0.0001 e a 0 -3 . Blue regions indicate electron charge accumulation areas, whereas red regions indicate depletion areas.
Bottom: corresponding DKS-CD curves for the AgH 2 complex. Red dots indicate the position of the nuclei along the z axis. The vertical line marks the isodensity boundary. S16 Figure S8. Top: DKS-CD calculated isodensity surfaces of the total (Δρ) and of its first five NOCV components for the AuH 2 complex. The isovalue for the upper three surfaces (i.e. Δρ, Δρ 1 , Δρ 2 ) is ±0.001 e a 0 -3 , whereas for the three lower surfaces (i.e. Δρ 3 , Δρ 4 , Δρ 5 ) the isovalue is ±0.0001 e a 0 -3 . Blue regions indicate electron charge accumulation areas, whereas red regions indicate depletion areas.
Bottom: corresponding DKS-CD curves for the AuH 2 complex. Red dots indicate the position of the nuclei along the z axis. The vertical line marks the isodensity boundary. S17 Figure S9. Top: SR-CD calculated isodensity surfaces of the total (Δρ) and of its first five NOCV components for the CuH 2 complex (the isovalue is ±0.005 e a 0 -3 ). Blue regions indicate electron charge accumulation areas, whereas red regions indicate depletion areas. Bottom: corresponding SR-CD curves for the CuH 2 complex. Red dots indicate the position of the nuclei along the z axis. The vertical line marks the isodensity boundary. S18 Figure S10. Top: SR-CD calculated isodensity surfaces of the total (Δρ) and of its first five NOCV components for the AgH 2 complex (the isovalue is ±0.005 e a 0 -3 ). Blue regions indicateelectron charge accumulation areas, whereas red regions indicate depletion areas. Bottom: corresponding SR-CD curves for the AgH 2 complex. Red dots indicate the position of the nuclei along the z axis. The vertical line marks the isodensity boundary. S19 Figure S11. Top: SR-CD calculated isodensity surfaces of the total (Δρ) and of its first five NOCV components for the AuH 2 complex (the isovalue is ±0.005 e a 0 -3 ). Blue regions indicate electron charge accumulation areas, whereas red regions indicate depletion areas. Bottom: corresponding SR-CD curves for the AuH 2 complex. Red dots indicate the position of the nuclei along the z axis. The vertical line marks the isodensity boundary. S20 Figure S12. Top: SR-CD calculated isodensity surfaces of the total (Δρ) and of its first five NOCV components for the RgH 2 complex (the isovalue is ±0.005 e a 0 -3 ). Blue regions indicate electron charge accumulation areas, whereas red regions indicate depletion areas. Bottom: corresponding SR-CD curves for the RgH 2 complex. Red dots indicate the position of the nuclei along the z axis. The vertical line marks the isodensity boundary.  Figure S13. DKS-CD curves of the Δρ 1 component (red line) and of parent NOCV-pairs densities |Φ −1 | 2 (orange line) and |Φ +1 | 2 (green line) for the CuH 2 complex. The isodensity pictures (isovalue: ± 0.005 e a 0 -3 ) corresponding to the two parent densities have been also reported, where red and blue regions correspond to electron charge depletion and accumulation areas, respectively. Figure S14. DKS-CD curves of the Δρ 1 component (red line) and of parent NOCV-pairs densities |Φ −1 | 2 (orange line) and |Φ +1 | 2 (green line) for the AgH 2 complex. The isodensity pictures (isovalue: ± 0.005 e a 0 -3 ) corresponding to the two parent densities have been also reported, where red and blue regions correspond to electron charge depletion and accumulation areas, respectively. Figure S15. DKS-CD curves of the Δρ 1 component (red line) and of parent NOCV-pairs densities |Φ −1 | 2 (orange line) and |Φ +1 | 2 (green line) for the AuH 2 complex. The isodensity pictures (isovalue: ± 0.005 e a 0 -3 ) corresponding to the two parent densities have been also reported, where red and blue regions correspond to electron charge depletion and accumulation areas, respectively. Figure S16. DKS-CD curves of the Δρ 1 component (red line) and of parent NOCV-pairs densities |Φ −1 | 2 (orange line) and |Φ +1 | 2 (green line) for the RgH 2 complex. The isodensity pictures (isovalue: ± 0.005 e a 0 -3 ) corresponding to the two parent densities have been also reported, where red and blue regions correspond to electron charge depletion and accumulation areas, respectively. Table S6. Population of (n+1)s and nd orbitals of the metal atoms of the Group 11 anion dihydrides (n=4 for Cu, n=4 for Ag, n=5 for Au and n=6 for Rg) calculated by the means of non-relativistic (NR) and 4-component (4c) projection analysis.   Figure S17. Correlation between the populations of (n+1)s orbitals of the metal atom for Group 11 dihydrides (calculated at the 4-component level by the means of projection analysis) and the calculated extent of the sd hybridization (Δ HYBR ).