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Carbon forms multiple bonds, with a preference for $ \sigma -\pi $ bonding

When two electron pairs are squeezed into the space between the carbon nuclei of ethylene, they may avoid each other either by moving apart above and below the plane of the to form ``banana bonds''; or by having one electron form a node and become a pi bond that is orthogonal to the other electron pair, which forms a sigma bond. In the Hartree-Fock description, which operates on a basis of orthgonalized orbitals, the two pictures are equivalent, since they can be related to each other by a unitary transformation. Valence bond calculations performed without an orthogonalization constraint show similarly that the two models are nearly identical, with a slight preference toward banana bonding in ethylene (6.5 kcal/mol difference).

It is reasonable to expect that our force field would prefer banana bonding, since no provision has been made for electrons to attain $ p$ character. In the FSGO method, this lack of $ p$ functions has dire consequences [24]: electrons in multiple bonds coalesce into the same function, which in theory provides a p-like electron in the limit of infinitesimal separation, but in practice causes linear dependency problems, and makes the barrier to rotation of the ethylene pi bond negligible.

We find that contrary to expectation, our force field prefers a $ \sigma -\pi $ mode of bonding, but does so in a curious way: a sigma electron pair sits in between the carbons, then the electrons of the other electron pair split, so that an electron of one spin goes above the plane, and an electron of the other spin goes below the plane (Figure 4.6). This spin-polarized bond creates a diffuse effective $ p$ function; this mode of sigma-pi bonding is stabilized over equivalent banana bonding in eFF by 160 kcal/mol (Table 4.2).

Table 4.2: Energetics and geometries of double and triple bonds
    ethylene acetylene
Banana vs. $ \sigma\pi$ energy (kcal/mol)   -160.5 -183.4
(minus means banana preferred)      
Hydrogenation energy (kcal/mol) eff-$ \sigma\pi$ -141.6 -406.9
  exact -32.6 -41.7
CC bond length ($ \AA$) eff-$ \sigma\pi$ 1.517 1.383
  eff-banana 1.442 1.334
  exact 1.339 1.203
CH bond length ($ \AA$) eff-$ \sigma\pi$ 1.089 1.052
  eff-banana 1.125 1.064
  exact 1.086 1.063

Figure 4.6: Multiple bonds can split $ \sigma -\pi $ or form symmetric ``banana'' pairs.

Triple bonds display a similar preference (183 kcal/mol) for $ \sigma-\pi_{x}-\pi_{y}$ bonding over banana bonding where the bonding electron pairs arrange themselves into a triangle normal to the bond.

However, we find that eFF multiple bonds are too long (1.517 $ \mathrm{\AA}$ double bond versus 1.339 $ \mathrm{\AA}$ exact), too unstable (as shown from the hydrogenation energies), and too diffuse in the region above and below the plane, which can cause inappropriate steric clashes with molecular elements lying above and below pi bonds. The weakness of multiple bonds stands in contrast to strength of sigma bonds in eFF, which as we will see in later sections bind overly strongly (163.5 kcal/mol versus 89.7 kcal/mol exact).

It is promising that banana bonds show some stability, as well as a more reasonable length (1.442 $ \mathrm{\AA}$ double bond) within our scheme, and do not coalesce. It may be preferrable in future versions of eFF to construct the potentials so that banana bonds are more stable than $ \sigma -\pi $ bonds. It would be elegant if the same Pauli repulsion that separates valence electrons and gives carbon-carbon single bonds the correct length could also separate electrons in multiple bonds and give them the correct length and energy. In the meantime, we limit our applications to those involving saturated hydrocarbons.

next up previous contents
Next: Conformational analysis of hydrocarbon Up: Validation against ground state Previous: Tetrahedral carbon forms bonds   Contents
Julius 2008-04-29