Cundari Th.R. -- Computational Organometallic Chemistry-0824704789

3.4.Solid-Angle Profiles

Radial Profiles

In Section 3.3, the solid angle was defined as the surface area of the ligand projected onto the inside of a sphere (Eq. 17). White and Coville reasoned that if the sphere originates at the metal in an organometallic complex and is allowed to grow out toward the ligand, then a partial solid angle can be calculated at each radius of the growing sphere (48). The resulting plot of solid angle as a function of radial distance is called a solid-angle radial profile. (Solid-angle radial profiles are conceptually similar to cone-angle radial profiles—see Figure 5—but are easier to implement. The easier implementation arises because the solid angle is defined by the projection of the ligand onto the inside of a sphere. As the radial profile is computed, the sphere onto which the ligand is projected simply grows from the metal.)

Equation (17) reveals that the solid angle is inversely proportional to the square of distance. This means that the radial profile for a sphere is not symmetrical (Fig. 5). The shape of a radial profile can provide information about the relative orientation of atoms within a ligand (or, by extension, between ligands in an organometallic complex). For example, consider a sphere A placed at some distance from a metal. Now, place a second sphere, B, different distances from A, as illustrated in Fig. 10. The shape of the radial profile generated from the metal is different for each of the geometrical arrangements of the two spheres. The differences in appearance of the radial profiles have enabled Coville and coworkers to examine possible stearic interactions between adjacent ligands (Fig. 11) (31).

Perhaps more powerful than the solid-angle radial profile is the quantification of the amount of overlap between two adjacent ligands.

Angles of Overlap

If two ligands are arranged around a metal so that the van der Waals radii of their atoms overlap, then a severe stearic strain should develop. Quantification of the amount of overlap is useful in predicting whether a given geometrical arrangement of ligands could exist. For example, suppose there is a nominal amount of overlap between two adjacent ligands. Then it is possible for some bonds to stretch, angles to bend, etc. to relieve the stress caused by the overlap. As the amount of overlap increases, the complex should become less and less stable. Coville and coworkers introduced a quantification of interatom overlap using both solid-angle and linear or vertex-angle concepts (49).

Using the solid-angle concept, there are two different types of overlap: overlap as a consequence of projecting the atoms onto a sphere, and physical overlap between the van der Waals radii of the atoms (Fig. 12). Only the latter is related to steric congestion. By using the solid-angle radial profile methodol-

FIGURE 10 Solid-angle radial profiles for two spheres, A and B, in different

the metal, sphere B is placed at 1.5 A, 2.2 A, 2.8 A, and 3.6 A (top to bottom)

from the metal. Notice that the shapes of the profiles, and the maximum solid angles, vary significantly as a function of the geometrical arrangement of the spheres. (From Ref. 48.)

ogy, all nonbonded overlap is eliminated from the computation. As the sphere grows from the metal in the solid-angle radial profile, the solid angle of only those atoms that intersect with the sphere is calculated. Thus, the possibility of overlap resulting from projecting the entire ligand onto a sphere is eliminated. For example, consider the case of the carbonyl ligand as viewed from a metal.

FIGURE 11 Use of solid-angle radial profiles. (a) Cis methyl groups give rise to the profiles indicated in (b). (b) The point marked X indicates the radial distance at which maximum steric interaction occurs. (From Ref. 48.) (c) Radial profiles for the cyclopentadienyl and P(OMe)3 ligands in [(η5–C5H4–p– C6H4Me)Fe(CO){P(OMe)3}I]. Nuclear magnetic resonance spectroscopy reveals steric interaction between the ligands as indicated. (From Ref. 31.)

There is no physical overlap between the C and O atoms, yet in the projection of the ligand from the metal, the carbon atom eclipses oxygen. In the radial profile (Fig. 10), this overlap is eliminated. By using the solid-angle methodology, a quantitative measure of the amount of overlap can be attained (49).

The solid angle of overlap, Λ, is calculated using the original solid-angle algorithm (Sec. 3.3) as follows:

FIGURE 12 Different types of overlap encountered in the solid-angle measure.

(a) In the process of projection, overlap that does not appear in the ligand appears in the projection. This overlap is called nonbonded overlap. (b) When two atoms physically overlap, the overlap in the projection is called bonded overlap. (From Ref. 48.)

1.A right circular cone is placed around two atoms.

2.The solid angle of the two intersecting atoms is calculated using the algorithm presented in Section 3.3.

3.The difference between the solid angle of each of the atoms (repre-

sented as spheres, obtained using Eq. 20) and the solid angle of the bonded atoms is the solid angle of overlap, Λ.

A semiquantitative measure of the amount of overlap between ligands can be attained using the semivertex angle of overlap, λ (49). In Figure 13, if the semivertex angle of atom A is α, for atom B is β, and the A-M-B bond angle is χ, then the vertex angle of overlap, λ, is

It should be noted that the vertex angle of overlap, λ, is not additive, which limits its general utility. The solid angle of overlap, Λ, has been used to rationalize metal–ligand bond lengths in Cr(CO)5L complexes, the Mn–Re bond length in MnRe(CO)10, and the conformational preferences in substituted cyclopentadienyl complexes of Mo and Ru (32,33,43,49).

3.5. Boltzmann-Weighted Steric Measures

Thus far in this chapter all steric measures presented have been based on a single conformation of the ligand. In the case of ligand repulsive energies, attempts

FIGURE 13 Definition of the vertex angle of overlap. The semivertex angle of one sphere, α, is indicated by the solid lines and the semivertex angle of the other sphere, β, indicated by dashed lines. The vertex angle of overlap is given by λ. (Redrawn from Ref. 49.)

were made to find a good representation of the lowest energy conformation prior to computation of the steric parameter (9,18–23). Mosbo and coworkers were the first to recognize that it would be appropriate to use a Boltzmann weighting factor to average cone angles over the entire conformational space of the ligand (39).

Mosbo began by plotting half cone angles θ/2, versus M–P rotation axis, φ (Fig. 4), in 1° increments to obtain ligand profiles for each phosphines (see Sec. 3.2 for a description of ‘‘Ligand Profiles’’). Four different definitions of the cone angle were proposed using these ligand profiles:

1. θI: twice the maximum in the ligand profile:

θI 2 θ 2 (22)

max

2.θII: twice the maximum for each group in the ligand profile averaged over the three groups attached to P. This is equivalent to the Tolman cone angle defined in Sec. 3.1 (2):

3. θIII: twice the average maximum for each group in the ligand profile:

FIGURE 14 Classification of conformers of phosphine ligands used by Mosbo and coworkers. The metal is excluded from the molecular modeling calculations and is represented by a lone pair on phosphorus. (Redrawn from Ref. 39.)

4. θIV: twice the average of all 360 half cone angles:

When the ligand contains conformational degrees of freedom, each one angle, θI–θIV, was subjected to an energy-weighted averaging.

Several low-energy conformations for the phosphorus donor ligand were considered: trans, gauche-right, gauche-left, gauche-right gauche-left, trans gauche-right, and trans gauche-left (Fig. 14). For aryl-substituted phosphines, four additional conformers were considered: staggered-right, staggered-left, eclipsed-right, and eclipsed-left (Fig. 14). Symmetry-related conformers were considered degenerate. For ligands with conformational degrees of freedom, a Boltzmann-averaged cone angle, θ, was calculated:

In Eq. (26), θi is the cone angle of conformer i with mole fraction ni. Each mole fraction (Eq. 27) is obtained by calculating the heat of formation of the conformer using molecular modeling (discussed next):

the cone angle calculated using customized code. In the case of the MM2 calculations, a slightly modified procedure was used for ligands containing conjugated π systems in which SCF (self-consistent field) were implemented in the final step. Additional conformers were also found by a slightly modified conformational search strategy. Consider PPh3 as an example: One phenyl group was constrained and the dihedral driver of MM2 was used to construct a 2D-grid search of the conformational space of the other two rings (a 30° grid was used). For each of the minima found, the two varied phenyl rings were constrained and the ligand allowed to energy-minimize (thus generating a minimum for the third ring relative to the first two). Finally, all constraints were removed and the ligand was allowed to energy-minimize.

Cone angles generated from the preceding methodology, Eq. (22)–(25), were used in a linear free-energy relationship to rationalize cis:trans ratios in W(CO)4LL′ complexes (42). In addition, several regression analyses were performed on this data set comparing Tolman’s cone angle (2) to Mosbo’s cone angles (39–41) to Brown’s ER values (9,41).

White and coworkers also computed Boltzmann-weighted solid angles (see Sec. 3.3) by implementing Eq. (28) (ni is defined in Eq. (27) and Ω in Eq. 17) (50):

Instead of working with the isolated ligand, Cr(CO)5L complexes were built in a molecular modeling program and between 500 and 1000 conformers were generated per ligand. A random conformational search strategy was employed in which torsion angles for all rotatable bonds were simultaneously allowed to vary by randomly different amounts. (This is the same search strategy used to compute ER (21).) Each conformer was energy-minimized using the MMP2 force field with modifications by Brown and coworkers (see Tables 1–4 for modifications) (9,25,26). Finally, all 500–1000 conformers were submitted to Steric (see Sec. 3.1) for energy-weighted solid-angle calculation (Eq. 28). These workers found that the energy-weighted solid angle correlated better with other steric parameters (θ, ER) than the solid angle based on a single, low-energy conformer. In addition, the energy-weighted mean solid angle also performed better in linear free-energy relationships than the solid angle based only on the lowest-energy conformer (50).

4. SUMMARY

In essence there are three quantitative measures of steric size in organometallic chemistry: the cone angle, the solid angle, and the ligand repulsive energy. Cone angles have appeal because they are easy to visualize and because then perform well in linear free-energy relationships. However, cone angles may overstate the

physical size of the ligand and cannot take into account ligand meshing. Solid angles can drastically understate the physical size of a ligand, but can be usefully applied to the problem of ligand meshing. The most appealing quantification of true steric demand is the ligand repulsive energy. ER values are easy to measure and are robust. In addition, they are a sound quantitative measure of the steric influence of a ligand on its environment in the absence of electronic complications. The only current drawback of the measures is that current ER values are based on a single, low-energy conformation. At the time of writing, Brown and White were in the process of deriving a set of Boltzmann-weighted ligand repulsive energies.

ACKNOWLEDGMENTS

The author would like to thank Professors Theodore L. Brown, University of Illinois at Urbana-Champaign, and S. Bart Jones and Michael Messina, University of North Carolina at Wilmington, for helpful comments in preparing this manuscript.

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