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Carbon-carbon single and multiple bonds

In the new eFF, carbon-carbon single bonds have the bond-centered closed-shell form as in the old eFF; but double and triple bonds now exist as banana bonds instead of sigma-pi bonds (Figure 5.12). Without any particular optimization of parameters, we find that carbon-carbon bond lengths for ethane, and ethylene, and acetylene are within 0.01 $ \mathrm{\AA}$ of the exact values (Figure 5.13). Carbon-hydrogen bond lengths are slightly longer than they should be by $ \mathrm{\approx 0.05\ \AA}$.

Figure 5.12: Electron arrangements in carbon-carbon single, double, and triple bonds.

Figure 5.13: Bond dissociation energies and geometry parameters of ethane, ethylene, and acetylene.

As in the old eFF, carbon-carbon single and double bonds are too strong (for ethane, 140 kcal/mol BDE versus 90 kcal/mol BDE exact; for ethylene, 211 kcal/mol BDE versus 170 kcal/mol BDE exact). Carbon-hydrogen bonds are too weak (for ethane, 76 kcal/mol BDE versus 100 kcal/mol BDE exact; for ethylene, 82 kcal/mol BDE versus 113 kcal/mol BDE exact). The differences in CC and CH bond dissociation energies between ethane, and ethylene, and acetylene however are close to the exact values (with the exception of the CC triple bond energy), suggesting that the energy differences stem from a systematic bias in the energetics of CC versus CH versus radical electrons.

Ethane has a barrier to twisting of 1.6 kcal/mol as it passes from a staggered to an eclipsed form (versus 3.0 kcal/mol exact [6]). As we see later, intermediate range steric repulsions are systematically underestimated in our version of eFF. Twisting ethylene causes the banana bond electrons to separate; at $ \mathrm{90^{o}}$, the up spin electrons align along one axis while the down spin electrons align along a $ \mathrm{90^{o}}$ axis. Twisted ethylene is higher in energy than planar ethylene by nearly 15 kcal/mol (versus 65 kcal/mol exact [7]).

Several problem cases that plagued the old eFF are now handled well with the new eFF (Figure 5.14). Previously, it was found that carbon-hydrogen bond lengths for secondary and tertiary carbons were unreasonably high, reaching 1.424 $ \mathrm{\AA}$ in isobutane. With the new eFF, the isobutane CH bond distance is now 1.137 $ \mathrm{\AA}$, near the exact value of 1.108 $ \mathrm{\AA}$.

Figure 5.14: Improved geometries for old eFF ``problem hydrocarbons''.

Carbon-carbon bonds are found to be more rigid as well. Previously, $ \mathrm{^{t}Bu-^{t}Bu}$ had a central carbon-carbon bond distance of 1.708 $ \mathrm{\AA}$; with the new eFF, it is now a more correct 1.519 $ \mathrm{\AA}$ (versus an exact value of 1.592 $ \mathrm{\AA}$). Diamond now has a CC bond distance of 1.510 $ \mathrm{\AA}$ versus an exact value of 1.545 $ \mathrm{\AA}$. Geometries such as 1,3-dimethylcyclohexane (axial-axial) and cis-fused decalin no longer display inappropriate twisting or bending; they in fact superimpose nearly exactly onto B3LYP- optimized structures. It appears as though the attractive three-body potential between valence electrons and nuclei is enforcing more reasonable limits on the range of valence electron motions. We quantify these observations more precisely by measuring the bond lengths and angles of a range of simple substituted alkanes and alkenes (Figures 5.15 and 5.16), and comparing them to exact values.

Figure 5.15: Key geometric parameters of substituted alkanes and alkenes.

Figure 5.16: Comparison of old versus new eFF geometric parameters for simple substituted alkanes and alkenes.

We find that carbon-hydrogen and carbon-carbon bond lengths are now closer to the correct values. Under the old eFF, carbon-carbon double bonds were nearly the same length as single bonds but under the new eFF are the correct 1.33 $ \mathrm{\AA}$ length. Carbon-hydrogen bonds are no longer overly flexible. We do find some less satisfying consequences, however: carbon-carbon single bonds are too rigid, and carbon-carbon double bonds actually shrink with increasing substitution instead of expanding. Bond angles also show more ``scatter'' from the exact values. Two possibilities exist: either the bonds themselves are too rigid, or steric effects between adjacent alkyl groups are too small. Our later results tend to support the latter hypothesis, although we have not ruled out the former possibility.

With more complex alkene geometries, eFF-optimized structures superimpose well onto B3LYP optimized geometries (Figure 5.17). This includes cyclic alkenes with conjugated and non-conjugated double bonds, such as cycloheptene (2.36 $ \mathrm{\AA}$ RMSD), 1,3-cycloheptadiene (0.16 $ \mathrm{\AA}$ RMSD), and even the anti-aromatic cyclooctatetraene (3.29 $ \mathrm{\AA}$ RMSD), where the out-of-plane nature of the molecule is captured. More rigid structures such as dimethylfulvene (1.79 $ \mathrm{\AA}$ RMSD) and dicyclopentadiene (0.22 $ \mathrm{\AA}$ RMSD) match B3LYP structures even more closely.

Figure 5.17: Optimized eFF and B3LYP alkenes superimposed, with root-mean-squared deviations (RMSD) given in parenthesis (angstroms).

Work remains to improve the eFF description of multiple bonding. Although bond dissociation energies are reasonable, eFF greatly overestimates the energy gained from turning double bonds into single bonds. As a test case, we consider the [4+2] cycloaddition between 1,3-butadiene and ethylene to produce cyclohexene, whose driving force is the conversion of two double bonds into two single bonds. eFF estimates the reaction energy to be -235 kcal/mol, significantly greater than the B3LYP value of -36 kcal/mol; we see a similar result for the [2+2] addition of two ethylenes to form cyclobutane.

Triple bonds are even more unstable relative to lower-order bonds; they are higher in energy than diradical double bonds by 42 kcal/mol in acetylene and 54 kcal/mol in dimethylacetylene. It appears that adjacent banana bond electrons in eFF repel each other too much. eFF has been parameterized to correctly describe the orthogonality of valence electrons around a single nucleus; perhaps further modifications are needed to transfer those energy expressions to the case of orthogonal electrons within a multiple bond. The too-strong repulsion may also originate from our electron anti-coalescence function, which aggressively keeps electrons of similar size separated.

Figure 5.18: eFF overestimates the energy gained from turning double bounds into single bonds; prototype $ [2+2]$ and $ [4+2]$ cycloadditions are shown, eFF (B3LYP) energy differences in kcal/mol.

Figure 5.19: eFF alkynes are unstable relative to alkene diradicals. eFF (B3LYP) distances in angstroms.

We have made no efforts to include conjugation or resonance effects in the eFF model, so it is no surprise that benzene exists as 1,3,5-cyclohexatriene in our description, with distinct single and double bonds (with lengths of 1.505 $ \mathrm{\AA}$ and 1.309 $ \mathrm{\AA}$, respectively). In fact, diradical 1,4-cyclohexadiene -- which would be a resonance structure of benzene -- exists in eFF as a separate local minimum geometry that is less stable than 1,3,5-cyclohexatriene by 12 kcal/mol. Also, propene and propane have nearly identical CH bond dissociation energies on their alkyl end (75 and 76 kcal/mol respectively), indicating that the allyl radical has no special stability.

Figure 5.20: eFF shows no resonance stabilization of conjugated double bonds.

All in all, we have made a significant advance over the previous eFF in describing hydrocarbons. Complex structures containing single and double bonds now match closely with B3LYP optimized geometries. Double and triple bonds are now the correct length and size; because double bonds now contain compact banana electrons rather than overly diffuse sigma-pi electrons, structures such as cyclooctatetraene have the correct geometry. Carbon-carbon and carbon-hydrogen bond dissociation energies increase by the correct amount as unsaturation increases (with triple bonds being an exception). It is apparent that the same Pauli function that stabilizes lone pairs on neon also serves to stabilize banana bonds in double and triple bonds; and it is remarkable that without further parameterization that the correct lengths of multiple bonds and the geometries of complex molecules, are reproduced well. Further work remains to obtain better isodemic reaction energies, stabilize triple bonds, and include conjugation effects. Nonetheless, we have validated a basic point -- exchange attraction is useful not only for obtaining correct ionization potentials for high spin atoms, but also for properly describing multiple bonds.

next up previous contents
Next: Heteroatoms single and multiple Up: Results and discussion Previous: Atom hydrides   Contents
Julius 2008-04-29