Tetrahedral carbon forms bonds to other carbons and hydrogen

Optimizing atoms and molecules with eFF, we observe that (1) opposite spin electrons pair, (2) for atoms larger than helium, electrons separate into core electrons that are nucleus centered and valence electrons that are larger than the core electrons, and (3) valence electrons pack like hard spheres, with a maximum of four electron pairs around each core (``octet rule''). It is apparent that the basic rules of Lewis bonding and hybridization arise as a natural result of balancing kinetic energy, electrostatic potential, and Pauli repulsion.

When carbon has a full octet of electrons, they arrange themselves into a tetrahedral $\mathrm{sp^{3}}$ packing. Methane is stable, and its valence electrons are centered at $\sim$80% of the distance from the core center to the proton, reflecting the greater electronegativity of carbon over hydrogen (Table 4.1, Figure 4.4).

Table 4.1: Geometries of primary, secondary, and tertiary-substituted carbon

	$\mathrm{d_{CC} (\AA)}$			$\mathrm{d_{CH} (\AA)}$			angle (degrees)
	eFF	exact		eFF	exact		eFF	exact
$\mathrm{CH_{4}}$				1.143	1.094		109.5	109.5
$\mathrm{CH_{3}(CH_{3})}$	1.501	1.536		1.173	1.091		110.8	110.9
$\mathrm{CH_{2}(CH_{2})_{2}}$	1.513	1.526		1.229	1.096		107.9	109.5
$\mathrm{CH(CH_{3})_{3}}$	1.529	1.525		1.424	1.108		101.8	109.4
$\mathrm{C(CH_{3})_{4}}$	1.573	1.534

Figure 4.4: eFF geometries of simple substituted hydrocarbons

Ethane is stable as well, with sigma-bond electrons centered at the bond midpoint, as required by symmetry. The C-C bonding electrons do not overlap significantly with the nucleus, unlike the bonding electrons in $\mathrm {H_{2}}$ This difference is due to the Pauli repulsion between the sigma electrons and the $\mathrm{1s^{2}}$ cores of the carbons in ethane; protons do not have such $\mathrm{1s^{2}}$ cores. Thus carbon-carbon bonds are longer than either carbon-hydrogen or hydrogen-hydrogen bonds. The Pauli function parameters were adjusted so that the carbon-hydrogen and carbon-carbon bond lengths of methane and ethane were close to known values [23].

In eFF, carbon-carbon bonds have slightly smaller electrons than carbon-hydrogen bonds, which causes them to repel each other more strongly than they should. This imbalance causes distortions away from an ideal tetrahedral geometry in secondary and tertiary carbons; for example, isobutane has a too-small HCC angle ( $\mathrm{101.8^{o}}$ instead of $\mathrm{109.4^{o}}$ exact), and a too-long carbon-hydrogen bond length (1.424 $\mathrm{\AA}$ vs 1.108 $\mathrm{\AA}$).

Carbon-hydrogen bonds have lengths in eFF that are too variable, but we will see later that their dissociation energies are less variable than their distance variation would suggest. Also, the carbon-carbon bond lengths are relatively fixed with respect to different substitution, as they should be. These observations suggest we would do well to focus on geometries with a core carbon skeleton and outwardly oriented hydrogens, where too-long C-H bonds would not clash.

Many organic molecules of interest, as well as bulk and surface diamond, fall into this category. Figure 4.5 shows that eFF can describe a variety of bridged, fused-cyclic, and strained carbon skeletons, with largely correct carbon-carbon distances. The worst discrepancies in bond distances involve quaternary carbons in compounds like $\mathrm{^{t}Bu-^{t}Bu}$ (1.708 $\mathrm{\AA}$ vs 1.592 $\mathrm{\AA}$ exact) and diamond (1.681 $\mathrm{\AA}$ vs 1.545 $\mathrm{\AA}$ exact).

Figure 4.5: eFF geometries of larger hydrocarbons, bond lengths in Angstroms