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

rotation about the Cr–L bond, all L–Cr–CObasal torsion force constants were set to zero (see Tables 2 and 4). Finally, the basal CO groups of the Cr(CO)5 fragment provide a relatively rigid structure from which the ligand is repelled.

To test the general applicability of ER as a measure of the steric influence of a ligand, Brown computed ligand repulsive energies with a fragment of very different geometry: CpRh(CO) (22).* Ligand repulsive energy values generated from these two different fragments are highly correlated (r 0.95; Ref. 22). Subsequent to this work, all other reported ligand repulsive energies have been computed with the Cr(CO)5 fragments in addition to other fragments, e.g., CH3 and CH2COOH (21). In all cases, ER is highly correlated with E′R. Thus, ligand repulsive energy is a robust measure of the steric influence of a ligand in any prototypical environment. Brown has demonstrated that ER can be used in linear free-energy relationships, giving generally superior correlation coefficients than the other steric measures traditionally used in organometallic chemistry (discussed in Sec. 3) (9).

3. MATHEMATICAL MODELING OF STERIC EFFECTS

3.1. Tolman’s Cone Angle

Although not computational in nature, the cone angle, θ, is historically important as a steric measure in organometallic chemistry (28). Tolman reasoned that when a ligand binds to a metal, it will do so by adopting the least sterically demanding conformation to minimize any steric stress (28). To quantify the physical size of a ligand, Tolman built a CPK model of the ligand with a metal ligand distance typical of a Ni(CO)3L complex. A right circular cone was placed around the ligand and the interior angle of the cone, θ, measured using a protractor (Fig. 2). Cone angles have been reported for P-, As- N-, and C-donor ligands (1–3).

In the late 1980s, workers replaced the CPK models with idealized structures generated from simple molecular modeling packages (1–3). (In all cases, no attempt was made to rigorously energy-minimize the structures.) Interior linear angles, equivalent to Tolman’s cone angle, were calculated using measurement tools in the same molecular modeling package.

Recently, Coville and coworkers published a computational measure of cone angles (29). To exemplify the algorithm used, consider a PR3 ligand attached to a metal. Cone angles were calculated as follows:

1.The ligand was divided into groups, with each group bonded to the donor atom, P. For example, for PHMePh the groups are H, Me, and Ph.

*Cp η5–C5H5

FIGURE 2 Measurement of the Tolman cone angle, θ. A protractor is used to measure the angle between the straight edge and the block.

2.For each atom in the group, a vector was defined from the metal to the center of that atom.

3.The angle ax (Fig. 3) was defined as the angle between the M–P bond axis and the vector defined in step 2 (from the metal to the center of the atom).

4.The semivertex angle for the atom, αx (Fig. 3), was defined as the angle between the vector from the metal to the center of the atom (defined in step 2) and the vector tangential to the van der Waals radius of the atom taken from the metal (Fig. 3).

5.The cone angle for that atom, γX, was as given by the following:

6.Since the largest cone angle for a ligand in a particular conformation

was required, the vector resulting in the largest group cone angle, γi, was the one used for that group (i.e., γi is the largest of the γX values for a given group).

FIGURE 3 Definition of the angles ax and αx used to define the group cone angle, γX.

7.Finally, the Tolman cone angle, θ, for a ligand with n groups, each with group cone angle γi, was the average of all n group cone angles:

This algorithm is part of a program, Steric, published by Coville’s group. At the time of writing, Steric was available by ftp to hobbes.gh.wits.ac.za in /pub/steric

(login as anonymous and use a full e-mail address as password). (Alternatively, the site may be accessed through: ftp://hobbes.gh.wits.ac.za/pub/steric/)

The simplicity of the cone-angle methodology has made it a very popular measure of steric size in organometallic chemistry and, recently, organic chemistry (1). In addition, Tolman’s original choice of conformation for most of the ligands has resulted in a series of cone angles that are applicable to widely different reactions (2). However, any possible attraction between the ligand and its environment was ignored because the ligand is always placed in the conformation that gives rise to the smallest θ. Brown’s ligand repulsive energies, by contrast, takes into account all attractions between the ligand and prototypical fragment prior to isolating the pure repulsive part of the van der Waals potential to define the steric influence of the ligand (‘‘Calculation Algorithm’’ in Sec. 2.2).

Cone angles also ignore the finite spatial influence of a ligand on its molecular environment (30–33). For example, it is conceivable for a stable complex to form in which two ligands occupy adjacent coordination sites on a metal even though their cones overlap. One possible solution to the problem of interligand meshing is to generate a steric profile for each ligand.

3.2. Cone-Angle Profiles

Ligand Profiles

Cone-angle profiles were first introduced in 1977 by Alyea and Ferguson (34– 36), Farrar and Payne (37), and Smith and Oliver (38). (Ligand profiles have also been used to define cone angles (39–42). This will be discussed more fully in Section 3.5.) In all cases, crystal structure data were used, hydrogen atoms

˚

(of radius 1.20 A) were added and some sort of profile plotted. Usually, the semivertex angle, θ/2, versus rotational angle, φ, was plotted in either Cartesian or polar coordinates (Fig. 4). These plots, called ligand profiles, were inspected for potential steric interaction between adjacent ligands. Workers have found ligand profiles useful in understanding coordination numbers of particularly bulky, conformationally flexible ligands, such as tricyclohexylphosphine, PCy3. For example, Ferguson, Alyea, and coworkers found that the conformation of PCy3 in [Hg(OAc)2PCy3]2 was significantly different from its conformation in

ligand exerts a different steric demand on its environment. For example, consider a simple sphere representing the ligand placed at some distance from the metal (Fig. 5). A second sphere of variable radius, r1 to rn, is allowed to grow from the metal. At rf , the sphere growing from the metal firsts encounters the ligand (the ligand is also a sphere in this case). The growing sphere reaches a maximum at rmax when it intersects a cone enveloping the ligand (Fig. 5). (This is the point at which the Tolman cone angle would be defined.) At the center of the ligand sphere, the growing sphere has radius labeled rcen (Fig. 5). At each radius, r1 to rn, the cone angle is calculated using Steric (see Sec. 3.1). The plot of cone angle, θ, versus the radius of the growing sphere, d, is called the cone-angle radial profile, CARP (Fig. 5).

Suppose at radius d the ligand has cone angle θCARP. Then a 3D plot can be generated using a circle of radius θCARP at each d (Fig. 6). Finally, Coville and coworkers generated a more meaningful 3D profile by plotting θCARP versus d versus φCARP (φ is as defined in Fig. 4), as shown for PH3 in Figure 6.

Analyses of cone-angle radial profiles have been used to correlate multinuclear NMR data with spatial regions of steric overlap (43). Cone-angle radial profiles have the potential to be used to predict relative stabilities of cis and trans isomers and to predict coordination numbers of bulky ligands. However, coneangle profiles are very sensitive to the conformation of a ligand.

3.3. Solid Angles

A different approach for taking into account interligand meshing is to mathematically remove all unoccupied space from the calculated ligand size. In other words, generate a computation of only the amount of space occupied by the atoms in a ligand (Fig. 7), called the solid angle, Ω (44)

The first attempt to apply solid angles to the calculation of steric sizes of ligands was by Immirzi and Musco, who placed a right circular cone around the ligand and calculated the solid angle of that cone (45). A solid angle can be thought of as the surface area of the shadow of the projection of a ligand into the inside of a unit sphere (Fig. 8). This means that the solid angle can provide shape-related quantification of the amount of space occupied by the ligand (Fig. 7). Unfortunately, by placing a cone around the ligand and calculating a solid cone angle, presumably because of the difficulty in obtaining an analytical solution to Eq. (17), the values reported by Immirzi and Musco are equivalent in concept to the cone angle.

In the 1980s and early 1990s several workers presented numerical solutions to Eq. (17) (1). In 1993, White et al. presented an analytical solution to the defining equation for a solid angle (46):

(a)

FIGURE 6 Three-dimensional cone-angle radial profiles. (a) A circle of radius θCARP is plotted at each distance, d (see Fig. 5). (b) At each distance d, a surface of θCARP versus φ (the M–P rotational angle; see Fig. 4) is plotted. (From Ref. 29.)

(In Eq. 17, r is the position vector of an element of surface with respect to an origin and r is the magnitude of r.) Since the solid angle is shape-dependent, it is very sensitive to the conformation of the ligand. In the limit of free rotation about the metal–ligand bond, which implies limited interligand meshing, the ligand occupies the same amount of space as a cone placed around it, so the solidangle concept reduces to that of the cone angle.

For simplicity, let us consider the solution to Eq. (17) for a sphere of radius

(b)

rA a distance dA from the origin enclosed in a right cone of semivertex angle α. The element of solid angle can be written as

FIGURE 7 Difference between a cone angle and a solid angle. Cone angles completely envelop the ligand and include some unused space in the measure. Solid angles include only the space occupied by the ligand in the measure. (From Ref. 48.)

The key to solving Eq. (17) analytically is not attempting to work with an entire ligand at once. Thus, White et al. (46) considered only one pair of atoms in the ligand at a time, calculating the solid angle for that pair. As seen in Figure 9, some single atoms (spheres) will be counted too often. The solid angles for these atoms can be subtracted using the solid angle for a sphere (Eq. 20). To calculate the solid angle for a pair of atoms, a right circular cone was placed around them. The atoms (spheres) projected onto the base of the cone as ellipses. The solid angle of the cone can be calculated easily (45), and the solid angle of the unused space can be removed by integrating over the perimeter of the ellipses. The mathematical details of the integration are outside the scope of this chapter (approximately 46 equations form the basis of the algorithm), but can be obtained from Ref. 46.

As the spheres are projected, a certain amount of overlap between the ellipses occurs as a consequence of the projection. The original algorithm dealt only with overlap between two ellipses (46). It is possible that higher orders of overlap could result, and these have been removed in subsequent, numerical versions of the algorithm. (47)