Selection of Low-Dimensional 3 ‑ D Geometric Descriptors for Accurate Enantioselectivity Prediction

: This study focuses on building enantioselectivity models using only a few intuitively meaningful descriptors based on the “ buried volume ” idea. Appropriate dissection of the sphere is used to calculate the buried volume of quadrants and octants, which we name % V BQ and % V BO . Propene polymerization catalysis to isotactic polypropylene (iPP) and 1,1 ′ -bis-2-naphthol (BINOL)- phosphoric acid-catalyzed thiol addition to N -acyl imines are used to illustrate the approach. For iPP, only a single steric descriptor derived from the comparison of hindrance in di ﬀ erently occupied octants ( Δ % V BO ) is needed, and electronic e ﬀ ects are unimportant. Moreover, the model (mean absolute deviation, MAD, 0.12 kcal/ mol) works for more than a single catalyst class, allowing in silico catalyst design. For thiol addition, the best performance is achieved by comparison of hindrance in octants, and one steric descriptor is needed ( Δ % V BO ) in addition to an electronic descriptor, the natural population analysis (NPA) charge on the P atom. In both cases, key ingredients are (a) the use of properly chosen “ scanning regions ” ( e.g. , octants or quadrants) and (b) the availability of highly accurate experimental data sets. The low dimensionality of descriptor space and their obvious intuitive meaning naturally provide guidelines for further catalyst optimization.


■ INTRODUCTION
Asymmetric catalysis, where selective access to one of two possible mirror-image products (enantiomers) is desired, is a central theme in synthetic organic chemistry 1 and polymer chemistry. 2,3Catalyst optimization for asymmetric reactions relies heavily on tuning of steric factors.Despite the simplicity of the relevant interactions, design is even nowadays still dominated by empirical optimization.
The aim of computer-aided catalyst design 4−6 is to speed up the optimization process and deliver optimized catalytic performance within an expedited timeframe.Direct prediction of catalyst selectivity by quantum mechanical methods is complicated by the high level of accuracy needed: fractions of a kcal/mol.This is an accuracy that even the most sophisticated methods cannot deliver. 7n this context, statistical modeling provides, in principle, an attractive alternative.−20 Several reasons make such an approach attractive: no mechanistic insight is needed, candidate catalyst structures can be characterized by "simple" descriptors that quantify their steric and electronic properties, and machine learning methods have the capacity to identify patterns in large sets of data that are incomprehensible to experimental practitioners and to generate models based on experimental data that can then be used to predict the performance of untested candidates. 11owever, such approaches need rather large databases (in the order of thousands or more data points) for training and validation.Modern high-throughput experimentation (HTE) approaches 21−29 allow, in favorable cases, fast construction of such databases.8][19][20]30 Descriptor design also is not necessarily straightforward. Focing on steric effects, over the years, different approaches have been developed to measure steric bulk: interference values, 31 A-values, 32 Charton, 33 and Taft 34 parameters, Tolman's cone angle, 35 AREA (θ), 36 Sterimol parameters, 37 comparative molecular field analysis (CoMFA), 38 percent buried volume (%V Bur ), 5 and average steric occupancy (ASO).11,39 The very fact that so many approaches have been reported testifies to the notion that "simple descriptor" does not mean "simple in design".
CoMFA, %V Bur , and ASO are voxel approaches: a scanning region is defined, and points within it are assigned a numerical value based on their being occupied by the catalyst (ASO takes catalyst flexibility into account by merging values over lowenergy conformations).Selected combinations of voxels can then be used as black-box descriptors (ASO, CoMFA).This is the approach followed in the entirely statistical protocol developed by Denmark, 11,39 in which the emphasis is on machine learning.Alternatively, voxel values can be summed over well-defined scanning regions of space (e.g., sphere, quadrants, octants), resulting in one or a few descriptors, which in favorable cases can be intuitively connected to the underlying reaction mechanism and thus emphasize human learning.One could argue that for the massive statistical approach, the choice of scanning regions is hardly relevant, but if the results are to be combined in just a few descriptors, the choice of regions becomes crucial.They should include all of the space thatif occupiedaffects enantioselectivity, but not any space that would not make a difference.Asymmetry in space occupation between given regions can then be correlated to enantioselectivity.−43 For quantitative analysis within the %V Bur framework, a logical starting point is dissecting the whole coordination sphere, properly oriented, into quadrants (Figure 1).Differences in %V Bur between more and less hindered quadrants can then be used as a descriptor (indicated here by Δ%V BQ ).An alternative approach could instead focus only on the hemisphere where the reaction occurs, i.e., the "front-half", with the "back-half" being considered irrelevant. 44Dissection of the properly oriented hemisphere would then lead to octants, and differences in %V Bur between more and less hindered octants can then be used as a descriptor (Δ%V BO ).−46 Using the octants approach (initially called quadrants of the hemisphere) and literature polymerization data, some of us showed that this enabled qualitative prediction of enantioselectivity (R 2 between experimental and fitted selectivities equal to 0.74) in propene polymerization promoted by a series of metallocene and non-metallocene catalysts. 44Of course, this low R 2 would prevent developing any model having predicting capability.It should be noted that we here use the term qualitative to indicate models that predict rough trends (mean absolute deviation, MAD, ≥ 1 kcal/mol) and quantitative for models good enough to use for catalyst optimization (MAD ≈ 0.2 kcal/mol).
Octant models (or quadrants of a half-sphere) have been also used recently by the Denmark group in the analysis of asymmetric cyclopropanation, hydrogenation, and thiol addition. 47Differently, some of us very recently showed that quadrants of the full sphere can also be used for quantitative prediction of isotactic polypropylene (iPP) enantioselectivity (R 2 between experimental and fitted data 0.92) of rigid C 2symmetric indenyl-based ansa-zirconocenes 2,30,63,64 , even allowing moderate extrapolation. 30However, the step from qualitative to quantitative prediction required an accurate HTE database, a re-benchmarking of the sphere radius used for scanning, and the deletion of some substituents in the ligand backbone and on the outer edges of the catalytic pocket (believed to be irrelevant to enantioselectivity, yet still impacting calculated steric bulk in the scanning sphere). 18,20hile the latter study allowed us to reach accurate correlation between experimental and fitted values, manipulating the structure of the catalyst to delete irrelevant parts is not a practical approach.
Given this background, the present work aims to test the performance of the quadrant and octant versions of the Δ% V Bur descriptor in two cases of enantioselective catalysis.The first is isoselectivity in propene polymerization to iPP, using a sizable data set obtained through modern HTE methods; we will show that the octant Δ%V BO descriptor works across (some) catalyst classes (metallocenes and salans).This will validate the octant model and will clarify that the qualitative performance achieved previously by some of us was essentially due to inconsistency in the experimental data set used for training, as it was assembled using data from different research groups. 44ecognizing that iPP catalysis might be a special case, we also reanalyzed 1,1′-bis-2-naphthol (BINOL)-derived chiral phosphoric acid-catalyzed thiol addition to N-acyl imines and will show that a single-conformer method, together with the octant Δ%V BO descriptor, can be used successfully.For the Δ%V BQ and Δ%V BO are calculated as the difference between %V Bur of the more hindered and of the less hindered quadrants (or octants), as defined in (a).sake of completeness, in both cases, we will also discuss the performance of the quadrant Δ%V BQ descriptor, showing its potentials and limitations.

■ RESULTS AND DISCUSSION
Stereoselectivity in Olefin Polymerization with C 2 -Symmetric Catalysts.The origins of stereoselectivity induction in olefin polymerization (Cossee−Arlman mechanism, 48−51 Corradini model for enantioselectivity 52−54 ) are well known.Furthermore, it is well established that stereoselectivity is only depending on steric factors provided certain experimental conditions are met (see the SI for a short discussion). 3,55,56Unsurprisingly, therefore, a long history of predictive modeling both via direct quantum mechanics (QM) prediction 57−61 and structure-performance-type statistical approaches 16,18,20,30,44,46,62 exists.However, olefin polymerization is also extremely sensitive to variations in the polymerization conditions, and only recently, HTE has enabled researchers to build catalyst performance databases of sufficient size to allow the generation of quantitative models, employing virtually identical polymerization conditions.. [16][17][18][19][20]29 In the present case, we looked at the performance of 50 catalyst precursors in propene polymerization, belonging to four different catalyst families: 28 ansa-zirconocenes (A1− A28), 16,18 10 bis(phenolate-amine) catalysts (salan-type, S1− S10), 66,67 11 bis(phenolate-ether) catalysts ([OOOO]-type, O1−O11), 68,69 and one Mitsui-type catalyst (M1, 65a Scheme 1). Diffent from earlier attempts, our HTE workflow allowed us to assemble a coherent database with results generated under rigorously identical conditions (T p = 60 °C, p propene = 6.6 bar, toluene solvent) where contributions from chain epimerization 3,55,56 are negligible.Data for the ansa-zirconocenes was previously published by us and taken from refs 16 and 18.Data for the post-metallocenes was generated for the express purpose of this paper (see the Experimental Section and SI Table S1).
Quadrants and Octants.Figure 2A illustrates the performance of the plain quadrant model (Δ%V BQ ) on the C 2symmetric ansa-metallocene set of medium to high (σ < 0.9995) stereoselectivity.To cover the influential 4,4′-position substituents, a scanning sphere radius of 5.0 Å was used. 16An R 2 of 0.73 is achieved, i.e., the model is only qualitatively correct and lacks quantitative predictive power.More concerning is that R 2 drops to 0.43 if the four catalysts with low enantioselectivity (ΔΔG enantio ‡ < 3.5 kcal/mol) are excluded.
The metallocene model can be applied to other catalyst classes, and we tested its performance for the salan catalysts S1−S10 (Table S2) and the Mitsui-type catalyst M1 (i.e., using the slope and intercept of the metallocene model to predict the performance of catalysts of the other two catalyst classes).A severe underestimation of enantioselectivity can be noted (in terms of ΔΔG enantio ‡ , MAD = 1.69 kcal/mol) for all but the single nonstereoselective salan catalyst (S6) in the set, while an Scheme 1. Precatalysts Employed for Predictive Modeling of Stereoselectivity in Propene Polymerization even more dramatic deviation is observed for M1 (4.5 kcal/ mol, overestimation).These poor results are due to the quadrant model scanning steric bulk in regions that are irrelevant to catalyst performance.As mentioned above, the deletion of selected substituents before calculating Δ%V BQ offers an avenue to circumvent this problem.Figure 2B illustrates the performance of our published quantitative model for stereoselectivity of the ansa-zirconocene set, showing an R 2 of 0.92, still using a scanning sphere of 5.0 Å.However, the selection of atoms for deletion is backbone-dependent and can only be unambiguously defined within a catalyst class: the model cannot be used as is for other catalyst classes such as salan or [OOOO] catalysts.
This result is unsurprising when looking at Figure 3.For ansa-(bis)indenyl zirconocenes, the distribution of steric bulk in the catalyst backbone situated in the back hemisphere (i.e., opposite to the active pocket) aligns nearly perfectly with the occupied and formally empty quadrants of the front hemisphere.
On the contrary, for salan catalysts, most of the steric bulk in the back hemisphere is situated in regions formally associated with the empty front quadrants.Therefore, in the evaluation of Δ%V BQ , some of the separation of steric bulk found in the front hemisphere is canceled by contributions found in the back hemisphere that are not actually relevant to enantioselectivity.Since this type of compensation will be different for different catalyst classes, a global model applicable to multiple classes and based on a single quadrant descriptor cannot be expected to work.An octant model fares much better.Figure 4 shows the correlation of predicted with experimentally observed enantioselectivity for the metallocene set.The R 2 on the metallocene subset is 0.85, i.e., a significant improvement is observed compared to the quadrant model with no deletions, using the same sphere size of 5.0 Å (variation of the scanning sphere did not improve R 2 ).
Moreover, salan catalysts (and the lone Mitsui-type catalyst) now fit perfectly in the correlation derived from metallocenes.The regression equation changes only minimally.The R 2 for the combined model is 0.95.Leave-one-out cross-validation (LOOCV) analysis results in a Q 2 value of 0.93.
Using the slope and intercept of the metallocene model to predict the performance of the other two catalyst classes, the predicted performance of post-metallocenes S1−S10 and M1 shows a MAD from the experiment of only 0.20 kcal/mol  Occupied quadrants gray, "empty" quadrants white.The bulk of the ligand framework in the back hemisphere of a typical ansa-zirconocene (A2, left) is concentrated in the same quadrants as the important 4,4'-Ph substituents (front hemisphere).For the generic o-tBu-substituted salan (post-metallocene, smaller ortho-substituent chosen for better visualization), the ortho-substituent defines the contents of the "occupied" quadrant, but the bulk of the ligand framework in the back hemisphere is concentrated in the formally "empty" quadrants.When quadrants are used to scan the catalyst, this cancels some of the separation of steric bulk in the evaluation of Δ%V BQ .(Table S3), which drops further to 0.10 kcal/mol when the highly enantioselective catalysts S1 and S8 are excluded (σ = 0.9998, large experimental error bars result from σ approaching 1).Interestingly, even the four Hf-based salans (S7−S10) included in the salan data set fit the model (derived from a different catalyst class and from a different central metal) perfectly.Ansa-hafnocene performance is dominated by chain epimerization at 60 °C. 62Consequently, it is impossible to include them in the model.However, post-metallocenes are less prone to the series of β-hydrogen eliminations that need to occur for chain epimerization.
As indicated by these results, a well-chosen single steric descriptor can quantitatively describe enantioselectivity of the active pocket across different catalyst classes (metallocenes and post-metallocenes of salan and Mitsui type).Interestingly, this indicates that while salans are more flexible in a general sense, 65b they are essentially similarly "inflexible" compared to ansa-zirconocenes at the olefin insertion state (TS) stage.The ortho-carbons bearing bulky substituents in octahedral postmetallocenes are located in the back hemisphere, while the 4position carbon in (bis)indenyl ansa-zirconocenes serving the same role is already located in the front hemisphere (to prove the point, a salan with an ortho-H substituent yields atactic PP, 66 while Me 2 Si(2-Me-4-H-indenyl) 2 ZrCl 2 yields iPP 2 ).Therefore, larger substituents at the ortho-carbon of salan catalysts are required to achieve high enantioselectivity than are needed for the 4-position carbon of (bis)indenyl ansazirconocenes (Figure 5).
Extension to [OOOO]-Type Catalysts.While the ansazirconocene-derived regression equation can be used to predict the performance of salans and Mitsui-type catalysts, the extension to [OOOO]-type catalysts O1−O11 fails and large deviations (MAD 1.50 kcal/mol) from experimental enantioselectivities are observed (Table S4).However, the observed [OOOO] data appear to be systematically offset relative to the predicted values (Figure 6, blue datapoints, the blue trendline serves as a guide to the eye).For the [OOOO] subset O1− O11 on its own, R 2 is 0.92 and the LOOCV Q 2 is 0.89 (Figure 6, red trendline).The slopes of the models for the metallocene + salan data set and the [OOOO] data set, respectively, are nearly identical, and the systematic deviation is indicated by a different intercept.
Most likely, this bias can be traced to an increased (and unfavorable) flexibility of the backbone, even at the TS stage (Figure 7).Catalysts O1−O11 have rather flexible backbone linker C 2 , C 3 , or C 4 groups, as recently demonstrated by Cuthbert et al. 68,69 In a broader sense, this suggests that modeling of the active pocket via a single three-dimensional (3D) geometric descriptor determined from a dichloride precursor and a global model can only be expected to work when relatively stiff metal complexes are targeted, i.e., when the precursor resembles the active site.For flexible catalysts, a truly global model must fail.However, the presence of backbone flexibility does not prevent building a catalyst class specific model from a precatalyst structure, as indicated by the nearly unchanged slope for the [OOOO] catalysts compared to the metallocene and salan set in Figure 6.
Stereoselectivity in Thiol Addition to N-Acyl Imines.While accurate description of enantioselectivity in a specific reaction class using the Δ%V BO descriptor is certainly encouraging, it is not obvious that this approach would work for other reactions: iPP might be a special case.Therefore, we also tested the Δ%V BQ and Δ%V BO models for an example of organocatalysis, reanalyzing HTE data collected by Denmark for thiol addition to acyl imines (Scheme 2; catalysts P1−P19, 52−70 in ref 11).The Denmark group tested both ASO and quadrant-based descriptors for this reaction. 11,39,47They also reported that electronic effects are significant and included a descriptor based on the electrostatic potential.Multivariate linear regression using ASO and electrostatic potential resulted in excellent correlation of the experimental enantioselectivities, with an R 2 of 0.95 and a MAD on ΔΔG enantio, exp ‡ equal to 0.16 kcal/mol only.
Herein, we resorted to test the capability of Δ%V BQ and Δ% V BO descriptors analogous to the approach employed in iPP modeling.Like Denmark, we find that electronic effects are

Scheme 2. Precatalysts Employed for Predictive Modeling of Enantioselectivity in Thiol Addition to N-Acyl Imines
important, and we use the natural population analysis (NPA) charge on P (q P ) to describe this.We initially performed a radius scan to calculate Δ%V BQ and Δ%V BO at varying sphere radius and settled on a radius of 3.0 Å, associated with the largest R 2 value.
A quadrant model comprising only one electronic parameter (q P ) and one steric descriptor (Δ%V BQ ) performs moderately well, with R 2 = 0.87 and Q 2 = 0.78 (see SI Figure S1 for the regression plot), resulting in reasonable correlation and predictive power.This is a better performance compared to that observed in the case of propene polymerization discussed above.The quadrant model improves slightly with the inclusion of a second steric parameter: the occupancy of the most hindered quadrant, here denoted by %V BQ .This gave a reasonable three-descriptor model, with an R 2 value of 0.92, though it resulted in no improvement in LOOCV accuracy, Q 2 = 0.79 (Figure 8A).
For the octants case, the two-descriptor model, using only Δ%V BO as a steric parameter along with one electronic descriptor, yields R 2 = 0.94 and Q 2 = 0.90, which is a clearly better predictive model than its quadrant counterpart (Figure 8B).This is consistent with the case of propene polymerization discussed above (see the SI for the regression plot).In this case, inclusion of %V BO for the highly occupied octant, resulting in a three-descriptor model, produced no improvement, with R 2 = 0.94 and Q 2 = 0.89 comparable to the twodescriptor model (Figure S2).
Overall, the multivariate linear regression equations derived from quadrant and octant models, as displayed in regression plots in Figure 8, show good performance, with the twodescriptor octants model giving the best performance, close to the R 2 of 0.95 achieved by Denmark with the conformationally dependent ASO model.Furthermore, the two-descriptor octant model also has an appreciable predictive power for this class of organocatalysts, with a MAD value of 0.16 kcal/ mol, again consistent with the value achieved by Denmark and co-workers.As for the quadrant models, the reasonable accuracy they achieve can be easily rationalized, considering that all of the precatalysts of Scheme 2 have the same BINOL linker, thus resulting in similar occupancy in the back hemisphere over the whole set of systems.This is at odds with the case of catalysts for iPP production discussed above, where the different linkers yield different occupancies in the back hemisphere, resulting in poor performance of the quadrant-based model.
To further confirm the necessity of using an electronic descriptor, already highlighted by Denmark and co-authors, 11 a regression analysis was also carried out by excluding the q P descriptor, creating a steric-only octant model based on Δ% V BO .The performance of this model drops to R 2 = 0.68 and Q 2 = 0.59, emphasizing the significance of electronic effects for this specific class.Finally, using a steric-only octant model based on both Δ%V BO and %V BO yields R 2 = 0.83 and Q 2 = 0.76, still clearly worse than the model based on Δ%V BO and the NPA charge on P.

■ CONCLUSIONS
The two examples discussed above demonstrate that it is possible to develop truly predictive models based on simple, intuitively meaningful descriptors.The %V Bur approach appears to be a good starting point, leading to the Δ%V BQ and Δ%V BO descriptors, but care should be taken when selecting regions for descriptor evaluation (e.g., quadrants vs octants, choice of radius).6][17][18][19][20]29,47,62 The above work on enantioselectivity in iPP catalysis indicates that a quantitative model relying on a single octantbased steric descriptor, Δ%V BO , can accurately correlate the experimental values, whereas the quadrant-based steric descriptor, Δ%V BQ , only offers mediocre performance. Moreoverthe octant-based model can be based on, and is valid for, more than a single catalyst class, indicating the transferability of the model.Conformational averaging does not appear to be needed in this case.The use of a low-dimensional model for iPP leads to particularly simple guidelines for systematic catalyst improvement: fill the full quadrants/octants further and keep the empty regions empty.The work on enantioselectivity in thiol addition to N-acyl imines catalyzed by BINOL-based phosphoric acids indicates that it is possible to build useful models for thiol addition based on a quadrant steric descriptor, Δ%V BQ , together with one electronic descriptor.Replacing the quadrant descriptor by the analogous octant-based descriptor Δ%V BO results in clearly superior performance, comparable to that achieved via comprehensive sampling of conformational space by Denmark.
Overall, this work has demonstrated that sectioning the original %V Bur descriptor into different regions of the first coordination sphere around the active center can lead to a variety of models that are highly accurate in fitting known data and can predict the experimental behavior of systems out of the training data set with good accuracy.Of course, the selection of the most appropriate model depends on the specific case in hand.An apparent feature of both discussed catalytic reactions is that the precursor sufficiently resembles the active site.On the other hand, if the precatalyst geometry is not representative of the actual catalyst at the transition state of the selectivity-determining step, these models, as any regression model, will probably fail.Finally, one cannot expect models to be valid across systems where the real reaction mechanism changes (or where computed structures "misbehave").
The more one knows about a reaction, the easier it will be to develop "intuitive descriptors".In this respect, polyolefins are somewhat of a special case because so much is already known about geometries and energetics.Still, there are important and well-studied out there that might conceivably benefit from this type of modeling.As demonstrated with the BINOLbased phosphoric acids, enantioselectivity predictions employing low-dimensional intuitive descriptors also work for other C 2 -symmetric systems.Conceivably, the approach should be easily amendable to predict the enantioselectivity of C 2symmetric catalysts outside of the chosen two examples, provided accurate experimental data is available.Extension to C 1 -symmetry needs to be tested.Catalyst chemoselectivity, regioselectivity, or molar mass capability are obvious further catalyst performance parameters of interest, and we are actively engaged in extending the intuitive descriptors approach, within the world of olefin polymerization and beyond.

■ EXPERIMENTAL SECTION
Catalyst Synthesis.synthesis of the precatalysts is detailed in the Supporting information.
The polymers were characterized by (a) high-temperature gel permeation chromatography (GPC) with a Freeslate Rapid-GPC setup and (b) quantitative 13 C NMR with a Bruker DRX 400 spectrometer equipped with a hightemperature cryoprobe (for 5 mm OD tubes) and a preheated robotic sample changer.All results are averages of dublicate polymerization experiments.More details can be found in SI Table S1.
Computational Details.Precatalyst Optimization: Olefin Polymerization.A protocol was chosen that was previously tested extensively for olefin polymerization-related problems, including modeling. 16,18The structures of metal dichlorides were fully optimized using the Gaussian16 software package 75 in combination with the OPTIMIZE routine of Baker 76,77 and the BOpt software package. 78Following the protocol proposed in ref 79, all precatalysts were optimized at the TPSSTPSS 80 / cc-pVDZ(-PP) 81−83 level of theory, using a small core pseudopotential on Zr. 84,85 Justifying the accuracy of the calculated geometries, the protocol has been successfully used, in combination with M06-2X 86 single point energy corrections, to address several polymerization-related problems: absolute barrier heights for propagation, 87 comonomer reactivity ratios, 88,89 metal−carbon bond strengths under polymerization conditions, 90−92 electronic and steric tuning of molar mass capability, 93 and modeling. 16,18The density fitting approximation (resolution of identity, RI) 94−97 and standard Gaussian16 quality settings were used.All structures represent true minima (as indicated by the absence of imaginary frequencies) or transition states (exactly one imaginary frequency, corresponding to the reaction coordinate).
BINOL-PA Optimization: Thiol Addition.The structures of all precursors were fully optimized using the Gaussian16 software package, 75 at the PBE0/TZVP level of theory.Standard Gaussian16 quality and convergence settings were used.All structures represent true minima (as indicated by the absence of imaginary frequencies).Manual conformational search was performed to identify the lowest energy conformation of the systems.
Descriptor Determination.Descriptors were calculated based on the optimized geometries of metal dichloride precursors or BINOL-phosphoric acid catalysts, using the SambVca 2.1 standalone fortran code, 5 which can be downloaded from the SambVca 2.1 webserver, as well as with an equivalent standalone python script that facilitated batch processing (AutoSMaps). 98Probe spheres were centered on the metal or phosphorus atoms, with the positive direction of the z-axis along the axis passing through the metal and the midpoint of the Cl atoms for iPP polymerization catalysts and with the negative direction of the z-axis along P atom and the midpoint of the C−C BINOL bridge for thiol addition catalysts, and the xz plane defined by one of the Cl atoms (iPP catalysts) and by one of the C atoms of the C−C BINOL bridge for thiol addition catalysts.In both cases, the resulting coordinate system defines the front hemisphere as the one containing the Cl atoms (iPP) and P(O)OH fragment, respectively.The octants analyzed are situated in the front hemisphere (see Figure 1).
For the iPP example, optimization of the probe sphere radius was performed previously for the quadrant model; 16 a limited testing confirmed that 5.0 Å was also an optimal choice for the octant model.For the BINOL-phosphoric acids example, the sphere radius was fully optimized within the 3.0−8.0Å range, with a step size of 0.1 Å, and a final best radius of 3.0 Å for both the quadrant and octant models.For the iPP polymerization catalysts, hydrogen atoms were included, but the MCl 2 fragment was not.For the BINOLphosphoric acid catalysts, hydrogen atoms were not included, but the P(O)OH fragment was; the Δ%V BQ and Δ%V BO values reported are differences between the averages of the two most occupied quadrants/octants and theaverages of the two least occupied quadrants/octants.

■ ASSOCIATED CONTENT
* sı Supporting Information

Figure 1 .
Figure 1.(a) Definition of quadrants and octants of the scanning sphere.(b) Orientation of the catalysts in the scanning sphere.The projections of the most hindered quadrants (or octants) are colored.Δ%V BQ and Δ%V BO are calculated as the difference between %V Bur of the more hindered and of the less hindered quadrants (or octants), as defined in (a).

Figure 2 .
Figure 2. Correlation of predicted and experimentally observed stereoselectivity (as ΔΔG enantio ‡ ) using a quadrant model for ansazirconocenes A1−A28 and employing (A) original input with no deletions and (B) edited input where the SiR 2 bridge and the substituents in position 5, 6, and 7 are deleted.R sphere = 5.0 Å.

Figure 3 .
Figure3.Illustration of why a quadrant model cannot simultaneously describe different catalyst classes.Occupied quadrants gray, "empty" quadrants white.The bulk of the ligand framework in the back hemisphere of a typical ansa-zirconocene (A2, left) is concentrated in the same quadrants as the important 4,4'-Ph substituents (front hemisphere).For the generic o-tBu-substituted salan (post-metallocene, smaller ortho-substituent chosen for better visualization), the ortho-substituent defines the contents of the "occupied" quadrant, but the bulk of the ligand framework in the back hemisphere is concentrated in the formally "empty" quadrants.When quadrants are used to scan the catalyst, this cancels some of the separation of steric bulk in the evaluation of Δ%V BQ .

Figure 5 . 5 Figure 6 .
Figure 5. (A) Optimized transition-state geometries for stereoerror insertion in a basic metallocene (A2, left) and a generic o-tBusubstituted salan (right, smaller ortho-substituent chosen for better visualization).The arrows indicate non-bonded C−C distances where ligand bulk is closest to the methyl group of the inserting monomer: position 4 for metallocenes, the phenolate ortho position for salan.Blue arrow: shortest distance, red arrows: other short distances.(B) Topographic steric maps generated with SambVca 2.1. 5

Figure 7 .
Figure 7. Visualization of the backbone flexibility of catalyst O1.Faint dichloride precursor structure in background, overlayed with the TS backbone geometry.Clearly visible is the change in the dihedral angle between the backbone Ph and o-Ph substituent (A + B, indicated by red arrows) leading to reduced steric pressure in the occupied quadrants and the movement of backbone Ph toward the active pocket leading to increased steric pressure in the formally empty quadrants (C, indicated by violet arrow).