next up previous contents
Next: Carbocation rearrangements Up: Validation against ground state Previous: Methyl cation, radical, and   Contents

Homolytic versus heterolytic bond cleavage

We have previously discussed the energetics of breaking hydrogen molecule into hydrogen radicals. Since we have computed the energy of methyl radical and the energy of ethane, we can compute the analogous energetics of breaking ethane into two methyl radicals. We find that while the bond dissociation energy of $ \mathrm {H_{2}}$ is underestimated by eFF (67.2 kcal/mol versus 104.2 kcal/mol exact), the bond dissociation energy of the carbon-carbon bond in ethane is overestimated (163.5 kcal/mol versus 89.7 kcal/mol exact).

The error in homolytic bond dissociation energies arises from differences in how well the eFF wavefunctions represent the true electron density in the molecule, versus separated fragments. For hydrogen molecule, the true electron density is a doubly peaked atom-centered function, which eFF describes as a singly peaked bond-centered function. Hydrogen atom in contrast is represented well, because in both eFF and in the true case, the electron density has a maximum at the nucleus.

In the ethane carbon-carbon bond, the errors in basis representation take the opposite form. Carbon-carbon sigma bonds have an electron density that is concentrated in the region between the nuclei; hence the eFF bond-centered representation is a good one. In comparison, the methyl radical is poorly represented because, as we have seen in the last section, eFF does not have the proper $ p$ functions to describe the radical electron. Hence the relative error is in the opposite direction as in the $ \mathrm {H_{2}}$ case, and we find that $ \mathrm {H_{2}}$ is underbound while ethane is overbound.

In carbon-hydrogen bonds, the basis representation errors of the molecule versus the dissociated fragments cancel, and the bond dissociation energy is near the correct values (methane 119.9 kcal/mol versus 104.8 kcal/mol exact).

eFF can describe heterolytic bond dissociation as well, where electron pairs split asymmetrically, so that one species is left with two electrons while another is left with none at all. A common example is protonation, and we find that the species $ \mathrm{HeH^{+}}$ has a bond dissociation energy near the exact value (44.1 kcal/mol versus 47.1 kcal/mol exact). This excellent agreement comes about because the electron pair of $ \mathrm{HeH^{+}}$ is mainly centered on the helium, making the singly-peaked eFF density a good approximation to the true electron density (Figure 4.11).

The protonation energy of ammonia is too high (446.0 kcal/mol versus 207.0 kcal/mol), an indicator that eFF does not currently compute the energetics of molecules containing lone pairs correctly.


Table 4.4: Hydrocarbon and protonated species bond dissociation energies.
    $ \Delta E$ (kcal/mol)
relative to energy of eFF exact
$ \mathrm{H-H}$ $ \mathrm{2 H \cdot}$ 67.2 104.2
       
$ \mathrm{H_{3}C-H}$ $ \mathrm{H_{3}C + H \cdot}$ 111.9 104.8
$ \mathrm{(CH_{3})_{3}C-H}$ $ \mathrm{(CH_{3})_{3}C + H \cdot}$ 108.2 95.2
       
$ \mathrm{H_{3}C-CH_{3}}$ $ \mathrm{2 H_{3}C \cdot}$ 163.5 89.7
$ \mathrm{(CH_{3})_{3}C-CH_{3}}$ $ \mathrm{(CH_{3})_{3}C \cdot + \cdot CH_{3}}$ 121.4 86.0
       
$ \mathrm{HeH^{+}}$ $ \mathrm{He + H^{+}}$ 44.1 47.1
$ \mathrm{NH_{4}^{+}}$ $ \mathrm{NH_{3} + H^{+}}$ 446.0 207.0


Figure 4.11: Protonation of helium and ammonia


next up previous contents
Next: Carbocation rearrangements Up: Validation against ground state Previous: Methyl cation, radical, and   Contents
Julius 2008-04-29