Computational assessment on the Tolman cone angles for P-ligands

The Tolman cone angle ( θ ), the par excellence descriptor of the steric measure of a phosphine, has been recomputed for a set of 119 P-ligands, including simple phosphanes and phosphites, as well as bulky biaryl species often employed in catalytic processes. The computed cone angles are obtained from three different transition metal coordination environments: linear [AuCl(P)] ( θ L ), tetrahedral [Ni(CO) 3 (P)] ( θ T ) and octahedral [IrCl 3 (CO) 2 (P)] ( θ O ), allowing to observe the steric behavior of the ligand when increasing the steric hindrance around the metal center. The computed cone angles have been extracted from the lowest-energy conformer geometry obtained with a combined MM/DFT methodology. A conformational screening is done using MM, which allows us to identify the lowest energy structure of each ligand in each coordination environment. These low energy conformers are subsequently reoptimized at DFT theory level, from which the cone angle value can be extracted. The computed cone angles are compared with the original Tolman cone angles, and with other steric parameters such as solid angles ( Θ ), percent buried volumes (%V bur ), and angular symmetric deformation coordinate (S4’). This new set of values correlates with the phosphine ligand dissociation enthalpies in titanocene complexes of general formula [Ti(2,4-C 7 H 11 ) 2 (PR 3 )], and with reaction barriers in the Suzuki-Miyaura reaction between [Pd-PR 3 ] and bromobenzene, proving that this newly proposed set of cone angles can be employed to establish linear correlations between different experimental and calculated properties for systems in which the phosphine ligands play a significant role.


Introduction
Phosphines (PR 3 ) are among the most important and widely employed ligands in coordination chemistry. Known since 1870, 1 these compounds have some advantages over ammine ligands, such as their enhanced solubility in organic solvents and their compatibility with metals in multiple oxidation states. These two features have made metal-phosphine complexes very useful in the homogenous catalysis field. 2 Important chemical processes, including olefin hydrogenation (Wilkinson's catalyst), 3 olefin metathesis (Grubbs' catalyst), 4 or a wide range of palladium-catalyzed coupling reactions use metal-phosphine complexes. 5 Also, the tetrahedral nature of an sp 3 phosphorus atom with different substituents leads to a P-stereogenic center, and several transition metal complexes bearing such ligands have been used in enantioselective catalytic reactions. 6 It is precisely this high degree of functionalization, which allows controlling both electronic and steric properties of the phosphine ligand, what makes them highly effective in several chemical reactions. One of the first approaches to quantify the steric properties of the phosphines was done by Chadwick A. Tolman, when he proposed the Tolman cone angle (θ) as a measure of the steric bulk of the phoshphine ligand. 7 The Tolman cone angle is one of the most employed parameter for measuring the size of a phosphine ligand. This parameter is defined as the apex angle of a cone with origin at the metal center with spreading edges along the van der Waals spheres of the outermost atoms ( Figure 1). 8 Although its wide acceptance and constant use, this parameter has been flawed since its creation in the late 1970s. Originally, the Tolman cone angle was developed for symmetric monodentate phosphine ligands bound to a nickel center in a tetrahedral arrangement. The Ni-P distance was fixed to 2.28 Å, which is an average distance obtained from crystal structures, and the cone angle was measured using a physical space-filling model and a specialized ruler. In the case of asymmetric phosphine ligands the cone angle can be estimated by averaging the three angles between the phosphorus substituents: θ = 1/3 (θ 1 + θ 2 + θ 3 ).  13 and the percent buried volume descriptor (%V bur ). 14 The Sterimol descriptors, 15 which were originally not developed as a steric measure, have been successfully employed to account for ligand bulkiness in quantitative structure−activity relationships in drug design and catalytic processes. 16 Alternatively, other procedures, mostly related to computational chemistry methods, have been developed to derive new values for the Tolman cone angles. The usage of DFT-optimized structures as source for obtaining cone angles has been carried out by different research groups, and new methodologies such as AARON 17 and Solid-G, 9a, 18 have been reported. Much in the same way, other options to generate θ values consist of mapping the average local ionization energy of ligands 19 or using data generated from a molecular mechanics approach. 20 Due to nowadays computer power, which allows for accurate yet fast calculations on lots of molecules, a systematic and accurate approach to computing Tolman cone angles can be envisioned. To this end we have developed force fields able to screen for a great number of P-ligand (P) conformations in tetrahedral [Ni(CO) 3 (P)] complexes. The best candidates, this is, those with the lowest relative energies, have been identified, and a DFT geometry optimization was performed on them. These two procedures are carried out without any structural constraint i.e. the Ni-P distance is no longer kept fixed at a value of 2.28 Å. From the optimized geometry, the Tolman cone angle in a tetrahedral coordination environment (θ T ) is extracted with the FindConeAngle tool developed by Allen and coworkers, 21 currently implemented for Mathematica. 22 This method is fast and reliable, and allows the determination of the Tolman cone angle in any coordination environment without any of the assumptions made in the original cone angle development. The same research group has also reported a parallel procedure to obtain the exact solid angles (Θ) 11a for a wide range of mono-and polydentate ligands. 23 Additionally, and since the cone angle should be responsive to the coordination environment of the ligand, the methodology described above has been extended to compute the phosphine cone angles in linear [AuCl(P)] (θ L ) and octahedral [IrCl 3 (CO) 2 (P)] (θ O ) complexes ( Figure 2). The relative ligand arrangement in the latter complex corresponds to the one that minimizes the trans influence of the substituents on the iridium atom and consequently produces the lowest energy substitutional isomer. This set of structures includes the prototypical PR 3 , PX 3 , P(OR) 3 , plus some of their combinations, and also the most representative ligands in catalytic processes, including many of the Buchwald biaryl ligands, 24 for which the cone angle has never been determined. Finally, the computed Tolman cone angles will be compared to a) previously reported values, b) other typical P-ligand steric measures found in literature such as solid angles (Θ), 11a the LKB steric He 8 descriptor 12 and the angular symmetric deformation coordinate (S 4 '), 13 and c) recomputed percent buried volumes (%V bur ), 14 obtained from the lowest energy conformers found herein.

Computational details
As mentioned above, one of the problems that we face when trying to compute phosphine cone angles is the wide range of configurations that they can adopt. This is particularly critical in bulkier ligands with aliphatic groups, in which the flexibility may lead to several possible conformations, thus leading to multiple values for the corresponding cone angle. To perform an optimal screening of all possible configurations, and select among them the lowest energy ones, we performed a molecular mechanics (MM) screening using a generic linear dicoordinated [AuCl(P)] molecule, where P corresponds to all the P-ligands in Table 1. Using the DL_POLY Classic software for Molecular Mechanics, 25 a 1 ns trajectory in the NVT ensemble at 300 K has been done for all [AuCl(P)] systems (see SI for force field details). The corresponding outputs from the simulation have been later analyzed in order to extract the 10 lowest energy configurations across the trajectory. The lowest energy geometry has been used as a starting geometry the Density Function Calculations (DFT). This procedure was repeated for the tetrahedral [Ni(CO) 3

Results and discussion
The The Buchwald and other bulky biaryl P-ligands (Table 1, entries 36-60, structures shown in Table S1) show a similar behavior, which is often invoked to rationalize the reaction mechanism of catalytic homogeneous processes. These ligands present two different -and characteristic-most likely conformations depending of the orientation of the secondary aryl ring, which may be pointing away (open conformation) or towards (close conformation) the metal center, establishing a metal-arene interaction ( Figure 3). Table 1. Original (θ) and computed ligand cone angles (in degrees) in different coordination environments:   (Table S2). In all cases, the lowest-energy structures correspond to those displaying the close conformation, in which the shortest contact between the gold atom and the dangling group of the biaryl substituent (usually a carbon atom) is found at distances ranging from 3.10 to 3.40 Å.
These distances are, in most cases, shorter than the sum of van der Waals radii of the  As may be observed (Table 1) (Table 2).   The octahedral computed cone angle (θ O ) has been compared ( Figure 6) with other available phosphine steric parameters such as the solid angles (Θ), 11a the LKB-P steric descriptor He 8 , 12b-d the angular symmetric deformation coordinate (S 4 '), 13 and the percent buried volumes (%V bur , Figure 7). 14 A complete list of values for these quantities can be found in the ESI (Table S3)  The monodentate phosphine Ligand Knowledge Base (LKB-P) contains the steric descriptor He 8 . 12a, 12b This descriptor is calculated as the interaction energy between a P-ligand and a ring of 8 He atoms, which remain fixed in a regular distribution on a circle of radius 2.5 Å. As in the Tolman cone angle, the distance between the P atom and the centroid of the He 8 ring is kept frozen at 2.28 Å. This setup aims to mimic the interaction between the P-ligand and other cis-coordinated groups in an octahedral complex. One of the cons of employing He 8 is that its interpretation is not immediate and sometimes is difficult to think of an energy value as a steric measure of a ligand. As may be observed the correlation between θ O and He 8 is not good, e.g., ligands such as P(Np) 3 (#8) and P t BuPh 2 (#109) have practically the same θ O value while the for the three %V bur quantities can be found in Table S3. The correlation between the computed θ and %V bur are moderately good in all cases; Figure 7 shows the best correlation, obtained between θ L and %V bur-L (correlations between the tetrahedral and octahedral analogs can be found in Figure S1). This correlation clearly indicates a positive relationship between θ L and %V bur-L, as should be expected. However, a deviation can be clearly appreciated for the smaller ligands, mainly those with θ L < 130°, for which larger %V bur values than expected are found. Those ligands belong mostly to the PX 3 class, where X is an electronegative substituent (F, Br, Cl, I or CF 3 ).
These electron-poor ligands produce a relatively short Au-P distances and takes most parts of the ligand into the 3.5 Å sphere employed to derive the %V bur , which in the end produces a larger steric hindrance. The same behavior is also observed for the same ligands in the computed %V bur-T and %V bur-O descriptors. In contrast, the relationship between the phosphine dissociation enthalpies and the computed θ O parameter is quite good (Figure 8). The determination coefficient found is   Table S5.  (Table 4). As may be observed in Table 4, the major ligand effect on the oxidative addition barrier is related to the σ-donation ability, while the steric hindrance of the ligand has only a limited effect; the standardized regression coefficients of the HOMO energy and θ T are -1.03 and 0.09, respectively, indicating that the former is almost 11 times more important. This should not be surprising since the reaction takes place on a quite unhindered monoligated palladium complex. The negative sign of the E HOMO descriptor indicates that the stronger σ-donor ligands favor the Pd insertion in the C-Br bond, thus contributing in lowering the barrier for this reaction stage in agreement with previous reports. 45 The determination coefficient found for the oxidative addition multilinear regression is very high R 2 = 0.999, indicating an almost perfect match.
In the case of the transmetalation barrier a quite good multilinear regression model is also found (R 2 = 0.973) when employing the E LUMO as the electronic acceptance character of the phosphine ligand; the usage of E HOMO as electronic descriptor provided a poorer regression (R 2 = 0.743), and was consequently discarded. The transmetalation regression model shown in Table 4 states that the electronic factors are ca. 2 times more important than the sterics: β(E LUMO ) = 0.93 while β(θ T ) = 0.39. The sign of the θ T steric parameter is positive, which indicates that bulkier ligands should produce higher transmetalation barriers; this behavior could be related to the steric repulsion produced by the phenylboronate and the phosphine ligand in the process of ligand exchange. The sign of de E LUMO descriptor is also positive, indicating that the more π-acceptor phosphines will lower the transmetallation barrier, probably by stabilizing the additional electron density on the metal.
The multilinear regression model for the reductive elimination barrier shows also a good determination coefficient (R 2 = 0.963), which states that E HOMO and θ T can be employed as descriptors to model this reaction step. The standardized coefficients of 27 both descriptors are β(E HOMO ) = 1.04 and β(θ T ) = -0.63, which show that both factors are important in the reductive elimination. The sign θ T is negative, indicating that bulky phosphines will lower the reductive elimination barrier, probably by pushing the two phenyl groups closer and facilitating the biphenyl product formation. In contrast the sign of E HOMO is positive, which means that stronger σ-donor ligands will be more prone to delocalize electron density on the metal center, hence producing a higher reductive elimination barrier. This regression model indicates that bulky and electron-poor phosphines produce the lowest reductive elimination barriers, as reported previously. 46 In addition, the multilinear models shown above can be employed to predict the reaction barriers of the other ligands in Table 1, provided the electronic descriptors E HOMO and E LUMO can be found in the LKB-P database. Therefore, applying the multilinear regression models to the additional 64 available ligands allows generating electronic/steric ligand maps for each reaction barrier (Figure 9, see Table S6 for predicted barrier values), where the barrier height is projected in a bidimensional surface. It has to be noted that none of the predicted reaction barriers takes a negative value, not even in cases where the barriers had to be extrapolated.

Conclusions
The

Conflicts of interest
There are no conflicts to declare.