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

are thoroughly understood, then deviations from linearity in the LFER are more easily understood, and the rationalizations for such deviations are more soundly based.

Steric effects have been recognized as important in organic chemistry since 1872 (6). However, it was not until the 1950s that a meaningful quantification of this steric effect appeared (7). Taft defined a steric parameter, ES, as the average relative rate of acid-catalyzed ester hydrolysis:

where k is the observed rate for the acid-catalyzed ester hydrolysis and k0 is the rate of methyl ester hydrolysis. Original values of ES were averaged over four kinetic measurements. Since then, there have been several changes to the experiments upon which the ES parameter is based, but the essential nature of the parameter has remained unchanged (1). There are several advantages to a steric measure based on kinetic data (1).

To successfully correlate thermodynamic or kinetic parameters with stereoelectronic effects, there must be a clean separation of the electronic from the steric effects (Eq. 1). Since there is no necessary reason for an experimentally based steric parameter to be free of electronic effects, workers have turned to applications of computational chemistry to achieve a quantitative measure of pure steric influence of a ligand.

In the early 1970s molecular mechanics was used to define the steric energy of a molecule (8). Molecular mechanics steric energies are the sum of all energies that cause a molecule to distort from an ideal, strain-free geometry. Thus, steric energy was defined as the sum of all bonded and nonbonded energies within a molecule. In 1992 Brown noted that the nonbonded interactions that express the steric requirements of a ligand are the nonbonded repulsive interactions between ligand and the binding site within a complex (9). Therefore, Brown defined a new parameter, the ligand repulsive energy, ER, as a quantitative measure of the steric influence of a ligand.

Traditionally, the steric influence of a ligand in organometallic chemistry is in some way related to the physical size of that ligand. Hence, the cone and solid-angle methodologies have been used as quantitative measures of steric effects in organometallic chemistry (1–3).

We divide this chapter into two parts: the use of molecular mechanics in defining the steric requirement of a ligand and the use of mathematical models to quantify steric size. In the first part of the chapter, we present the original definitions of steric energy and Brown’s extensions to formulate the ligand repulsive energy parameter. In the second part, we present Tolman’s cone-angle methodology and its modifications leading to various ligand profiles. Also, we present

the solid-angle methodology and its use in profiles. Finally, we conclude this chapter by presenting steric measures based upon Boltzmann-weighted averaging over the conformational space of the ligand.

2. STERIC EFFECTS IN MOLECULAR MECHANICS

2.1. Allinger’s Molecular Mechanics Programs

In the original molecular mechanics work, a steric energy, E, for a molecule was defined as the sum of the potentials for bond stretch, Es, angle bend, Eθ, torsional strain, Eω, nonbonded interactions, EvdW, and other terms, such as Urey–Bradley terms, cross-interaction terms, and electrostatic terms, (8).

In the original force fields, Allinger and others used Hooke’s-law harmonic potentials for a diagonal force field (Eq. 4). If l0, θ0, and ω0 are the strain-free bond distances, angles, and torsion angles and fl,i, fθ,k and fω,m are the relevant force constants, then the potential function for the molecule is given by

12 m fω,m(ωm ω0m)2

Because of their small magnitudes, all off-diagonal force constants are ignored. The Urey–Bradley force field (10) takes into account 1–3 nonbonded interactions, Eq. (5). If f, f ′, and f ″ are harmonic force constants, then the Urey–Bradley potential is given by

The original MM2 force field (represented by Eq. 4) was modified in 1989 (11–13). The MM3 force field employed a more refined set of functions to model structural, thermodynamic, and spectroscopic properties of molecules (for example, Eqs. 6–8; k and V represent force constants).

FIGURE 1 Plot of potential energy versus interatomic distance for a diatomic molecule. The solid curve is plotted with the Lennard–Jones potential (Eq. 11) and the dashed curve with the Buckingham potential (Eq. 12).

In Eq. (10), ε is the potential well depth (Fig. 1). The Lennard–Jones potential can be used as the nonbonded potential because it contains both an attractive (if n 6) and a repulsive part. In theory, n 6. But when n 12, the potential arranges to the simple Lennard–Jones 6/12 potential often used in molecular mechanics:

Alternatively, the Buckingham potential can be used instead of the Lennard– Jones potential:

In order to get the steepness of the potential wells plotted from Eqs. (11) and (12) to agree, α is usually set to 12 or 12.5 (11,13,15,16). In MM3 and MM4 the Buckingham potential is favored as the van der Waals term.

Although the sum of all potential energies was used to define the steric energy of a molecule, Brown found a very poor correlation between this quantity and kinetic or thermodynamic parameters known to be under steric control (9). Thus, Brown defined a new quantitative measure of the steric influence of a ligand, called the ligand repulsive energy, ER.

2.2.Brown’s Ligand Repulsive Energy

Definition of Ligand Repulsive Energy

Brown reasoned that the steric influence of a ligand is more complicated than simply the physical size of that ligand (9). The steric effect of a ligand should optimally be considered to arise from the nonbonded repulsion between the ligand and its molecular environment. The geometry of an organometallic complex is determined by electronic as well as steric effects. To isolate the steric part, Brown reasoned that nonbonded repulsive interactions need to be considered. The attractive part of the van der Waals potential is derived from dispersion forces between nonbonded atoms in a molecule (Eq. 9). Since the dispersion forces affect electronic energy levels (17), inclusion of the attractive part of the van der Waals potential in a pure steric measure is inappropriate (9). These conclusions also support not using the total steric energy (defined in Eq. 3) as a quantitative measure of a ligand’s steric requirement.

Since we are interested only in the steric influence exerted by the ligand on its environment, the van der Waals repulsion within the ligand needs to be excluded. Consider, for example, the calculation of the steric influence of a ligand, L, attached to a prototypical organometallic group such as Cr(CO)5. (The choice of this prototypical fragment will be discussed in the upcoming section on ‘‘Generality of the Approach.’’) To exclude intraligand repulsion, Brown calculated the van der Waals repulsive energy, EvdW,R, as a function of the Cr–L bond distance:

As the ligand moves toward the fragment in a direction perpendicular to the basal plane of the carbonyl groups, all intraligand repulsion is held constant. Thus, only the repulsion between ligand and organometallic group to which it is bound, Cr(CO)5, will change when the Cr–L bond distance is varied. The slope of the plot of van der Waals repulsive energy versus distance gives dimensions of energy over distance. To give dimensions of energy, Brown scaled this slope by

the equilibrium metal–ligand distance re. Thus, ligand repulsive energy, ER, is defined as the slope of the plot of van der Waals repulsive energy versus distance scaled by re (the negative sign in the following equation ensures that the sign of the ligand repulsive energy is positive):

∂EvdW,R

ER re ∂ (14) r

To date, ligand repulsive energies have been computed for a variety of different P-, As- (9), N-(18), O-, and S-donor ligands (19), η2-coordinated olefins (20), and alkyl groups (21). In addition to the Cr(CO)5 fragment, Brown and others have computed ligand repulsive energies, ER′ , using [(η5–C5H5)Rh(CO)] (19,20,22), CH3, and CH2COOH (21) fragments.*

Calculation Algorithm

In the original papers, ligand repulsive energies were computed manually (9,18– 22). Recently, we developed a program, ERCODE, to compute ligand repulsive energies using the methodology developed by Brown (23). Ligand repulsive energies are computed as follows.

1.The Cr(CO)5L complex is built using molecular modeling software, and energy-minimized using a modified MMP2 (24–26) or the universal force field (27). (These modifications are listed in Tables 1–4.)

2.A conformational search is carried out to determine the best representation of the lowest-energy structure. Typically, 2000 conformers are generated using a Monte Carlo algorithm in which the torsion angles of all rotatable bonds are simultaneously varied by randomly different amounts (23).

3.The lowest-energy structure found in step 2 is energy-minimized using

˚ tight termination criteria (typically of 0.0100 kcal/mol A).

4.The van der Waals function is changed from the Buckingham potential (Eq. 12) to the pure repulsive form (Eq. 13).

5.With all other internal coordinates frozen, the metal–ligand bond is varied by small amounts. Typically, seven distances are used: one is

˚

the equilibrium distance, re, three are shorter than re (each by 0.01 A),

˚

and three are longer than re (each by 0.01 A).

6.A plot of van der Waals repulsive energy, EvdW,R, versus distance, r, is constructed and the slope calculated. In practice, this plot is linear over

*The label ER has been reserved for the Cr(CO)5 fragment. Ligand repulsive energies computed with other fragments are called ER′ (fragment).

46 White

TABLE 1 Bond-Stretching Parameters Added to the MMP2 Force Field to Enable Modeling of Cr(CO)5L Complexes

Source: Refs. 18–20, 25, and 26.

the small range of the distances by which the metal–ligand bond distance is varied.

7.The negative of the slope of the EvdW,R versus r plot from step 6 is multiplied by re to give ER (Eq. 14).

Generality of the Approach

Brown chose the Cr(CO)5 fragment for several practical reasons. The vibrational spectra of Cr(CO)6 are well known, and force constants could be extracted (25,26), which allowed for the parameterization of the MMP2 force field. In addition, several crystal structures of Cr(CO)5L (L P-donor ligand) were available, so computed and observed structures could be compared. In addition to parameterization arguments, Brown used the Cr(CO)5 fragment for geometrical reasons. The Cr(CO)5 fragment has a fourfold axis of symmetry, which, in the Cr(CO)5L complex, is collinear with the pseudo rotational axis of the ligand (threefold in the case of P-, As-, N-, and C-donor ligands, and twofold in the case of O- and S-donor ligands and olefins). These collinear axes of symmetry simplify the parameterization of the torsional part of the force field: to allow free

TABLE 2 Bond-Angle Deformation Parameters Added to the MMP2 Force Field to Enable Modeling of Cr(CO)5L Complexes

Source: Refs. 18–20, 25, and 26.