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Conformational analysis of hydrocarbon geometries

We have demonstrated that eFF obtains correct ground state geometries for the simple hydrocarbons of Figure 4.4 and the constrained hydrocarbons of Figure 4.5. We ask now whether eFF can differentiate between multiple conformers of the same hydrocarbon. This is a tricky task, since we must now (1) have the correct energetics of bending valence electrons away from a tetrahedral arrangement, and (2) accurately describe the magnitude of steric repulsions between electrons on different atoms. In traditional force fields, these interactions are handled via separate angle, dihedral, and noncovalent interaction terms, but in eFF they should arise out of a proper consideration of electrostatics and the Pauli principle.

We start by looking at the cyclic hydrocarbons cyclopropane, cyclobutane, cyclopentane, and cyclohexane (Figure 4.7). The bonding in cyclopropane is known to involve curved bonds, a compromise between the geometrical requirements of the molecule and the hybridization of orbitals on carbon. In eFF, we see that the bonding electrons lie outside the perimeter of a line drawn connecting the carbons, with an angle between bonding electrons of $ \mathrm{98^{o}}$; valence bond calculations [25] show similarly curved bonds with an interorbital angle of $ \mathrm{110^{o}}$. Curved bonds appear naturally in eFF as a consequence of the repulsion between the three carbon-carbon bonding electron pairs.

Figure 4.7: eFF reproduces curved bonds of cyclopropane, and pucker of five and six membered rings.

Continuing on to larger rings, it is known that cyclobutane and cyclopentane attempt to make carbon tetrahedral, but instead of forming curved bonds like cyclopropane, they ``pucker'' so that the nuclei do not all lie in the same plane. Our force field reproduces this pucker in cyclopentane (dihedral $ \mathrm{21.5^{o}}$ versus $ \mathrm{33.2^{o}}$ B3LYP/6-311g**), but not in cyclobutane (dihedral $ \mathrm{0.3^{o}}$ versus $ \mathrm{18.0^{o}}$ B3LYP). In cyclobutane the difference in energy between puckered and planar conformations is known to be small ($ \sim$1.5 kcal/mol B3LYP), making the electron force field's error reasonable.

In cyclohexane, there are two conformers -- chair and twist-boat -- with a more significant energy difference of 6.3 kcal/mol (B3LYP, Figure 5.5). We obtain with eFF an energy difference of 4.7 kcal/mol and dihedral angles that compare well to known values ( $ \mathrm{57.7^{o}}$ versus $ \mathrm{56.6^{o}}$ B3LYP for the chair, and $ 34.0^{o}$ versus $ 32.3^{o}$ B3LYP for the twist-boat). The agreement of cyclohexane energies and geometries with B3LYP values is remarkable, considering that it involves a balance between the barrier of twisting about each carbon-carbon bond, and the steric repulsion between axial hydrogens. To test whether it is a fortuitous agreement, or the sum of reliable quantities, we examine the conformational preferences of some simpler systems (Figure 4.8).

Table 4.3: Energy differences between conformers examined. $ \mathrm {^{*}}$Gauche butane is not a local minimum, and is constrained at $ \mathrm {60^{o}}$.
      $ \Delta E$ (kcal/mol)
system energy of relative to eFF exact
ethane eclipsed staggered 2.1 2.7
butane gauche trans 1.6* 0.9
cyclohexane twist-boat chair 4.7 6.3
1,3-dimethyl-cyclohexane ax-ax eq-eq 5.8 5.9
  ax-eq eq-eq 2.7 2.1
decalin cis trans 12.1 3.2
2-pentene major minor 5.5 4.6

Figure 4.8: eFF reproduces steric repulsions within alkanes

In ethane, we find the energy difference between eclipsed and staggered conformations -- known to be a consequence of Pauli repulsion between C-H bond electrons -- to be slightly low (2.1 versus 2.7 kcal/mol B3LYP). In butane, we find the difference between gauche and trans forms, which arises from the repulsion between methyl groups, to be slightly high (1.6 versus 0.9 kcal/mol B3LYP). This difference is not surprising given our previous observation that carbon-carbon bonds repel carbon-hydrogen bonds more than they should. The combination of high methyl-methyl repulsions and low barriers for hydrogen eclipsing causes gauche butane to not be a local minimum structure, but to optimize directly to trans butane; the energy difference given is for a fixed dihedral angle of $ \mathrm {60^{o}}$.

In substituted cyclohexanes, axial substituents can become equatorial and vice versa through a chair flip. The stability of a cyclohexane conformer is particularly affected by repulsions between axial substituents, since they are close to each other (2.66 $ \mathrm{\AA}$ B3LYP) and oriented in the same direction. To quantify the magnitude of these 1,3-diaxial interactions, we consider the relative energetics of axial-axial, axial-equatorial, and equatorial-equatorial 1,3-dimethylcyclohexane. The axial-equatorial dimethyl and axial-axial dimethyl geometries adopt an overly-twisted geometry, a consequence of the overly large repulsion between axial methyl and axial hydrogen. Nonetheless -- and this should be viewed as an accident -- the energy differences between these conformers closely matches the exact values (5.8 versus 5.9 kcal/mol B3LYP, and 2.7 versus 2.1 kcal/mol B3LYP). To emphasize this point, we examine cis versus trans decalin, two hydrocarbon conformers that also differ in the number of interactions between axial substituents, but are more rigid and cannot relax so readily. In this case the energy difference is larger than the exact value (12.1 versus 3.2 kcal/mol B3LYP).

When a single bond is connected to a substituted double bond, allylic strain can restrict the rotation about the single bond [26]; this effect is used in designing reagents to perform highly selective diastereoselective enolations [27]. To compute the magnitude of allylic 1,3-strain, we consider two conformers of 2-pentene, noting that the minor form is substantially destabilized by the repulsion between methyl groups (Figure 4.9). eFF estimates the energy difference between conformers to be slightly higher than the exact value (5.5 versus 4.6 kcal/mol B3LYP). In this case, the too-high repulsion between methyl groups in eFF is balanced by the too-long double bond to give a value that agrees well with the known value.

Figure 4.9: eFF reproduces allylic strain interaction

We find that the electron force field gives remarkably good estimations of the energy differences between hydrocarbon conformers. In some cases, this is due to the force field parameters being balanced at a point such known biases of the force field, such as overly repulsive methyl groups, are compensated for by other biases in the force field, such as overly flexible carbon-hydrogen bonds. Overall, it is encouraging that the simple eFF functions can describe subtle conformational preferences of hydrocarbons as well as coarse properties like bond formation and atom hydridization. This is itself noteworthy, considering the number of terms and parameters in a conventional force field [1] devoted solely to the task of computing preferred bond lengths, angles, and torsions within molecules.

next up previous contents
Next: Methyl cation, radical, and Up: Validation against ground state Previous: Carbon forms multiple bonds,   Contents
Julius 2008-04-29