orbitals.
Figure 1.8 illustrates how an H2 molecule is formed from two H atoms. When two H atoms approach to one another, their 1s orbitals (1sA and 1sB) overlap giving two molecular orbitals σ1s and σ1s* through the following linear combinations of 1sA and 1sB [2].
FIGURE 1.8 Formation of the hydrogen molecule (H2) from two hydrogen (H) atoms.
Since 1sA and 1sB are identical, their contributions to each of the MOs (σ1s and σ1s*) should be equal. Therefore, we have c1 = c2 = c (>0) and c1′ = c2′ = c′ (>0).
In order to normalize the molecular orbital σ1s, the following integral must have the value unity
where dτ is the volume factor.
Therefore,
Since the wavefunction of the 1s orbital is normalized, we have
The term S = ∫(1sA1sB)dτ is referred to as the overlap integral. Therefore, we have
Similarly, by normalizing σ1s*, we can obtain
Therefore, we have
(1.60)
(1.61)
The overlap integral S is determined by the internuclear distance. At equilibrium H─H bond distance, the electron density of σ1s in the midregion of the bond is maximum, while the electron density of σ1s* in the midregion of the bond is zero. Therefore, σ1s is called bonding molecular orbital. It is formed by constructive interaction (overlap) of two atomic orbitals and is responsible for the formation of the H─H σ bond. σ1s* is called antibonding molecular orbital. It is formed by destructive interaction (overlap) of two atomic orbitals and is responsible for dissociation of the H─H bond. Since each of the 1s orbitals makes the same contribution to the bonding σ1s and antibonding σ1s* MOs, the coefficients 1/[2(1 + S)]1/2 and 1/[2(1 − S)]1/2 are often omitted when writing the LCAOs. Therefore, the bonding and antibonding MOs in H2 can be simply written as σ1s = 1sA + 1sB and σ1s* = 1sA − 1sB.
The diatomic halogen X2 (X = F, Cl, Br, or I) molecules are among fundamental main group molecules. Bonding in these molecules, usually represented by F2, is described in Figure 1.9 using MO theory. As two F atoms come together, the two single electrons (in pz orbitals) interact resulting in constructive and destructive orbital overlaps (LCAOs) in the line connecting the two nuclei, giving rise to the formation of the bonding MO (σ2p = 2pz,A + 2pz,B) and antibonding MO (σ2p* = 2pz,A − 2pz,B), respectively.
In the category of nonmetallic main group elements, a π bond is formed by overlap of two p orbitals (LCAOs) in sideways (Fig. 1.10). If both p orbitals are identical (such as p orbitals in a C=C π bond), each of the p orbitals has the same contribution to the bonding (πp) and antibonding (πp*) MOs (Fig. 1.10a). They are formed by constructive and destructive sideway orbital overlaps, respectively: πp = p1 + p2 (fused lobes due to a positive linear combination—constructive orbital overlap) and πp* = p1 − p2 (separated lobes due to a negative linear combination—destructive orbital overlap). If the two p orbitals are from atoms of different elements (such as p orbitals in a C=O π bond), the contribution of each p orbital to the bonding (πp) and antibonding (πp*) MOs is different (Fig. 1.10b). Usually, the p orbital in the more electronegative atom has greater contribution to the bonding MO (πp), and the p orbital in the less electronegative atom has greater contribution to the antibonding MO (πp*). In the C=O π bond, the bonding (πp) and antibonding (πp*) MOs can be expressed as
FIGURE 1.9 Formation of the fluorine molecule (F2) from two fluorine (F) atoms.
FIGURE 1.10 Formation of (a) the C=C π bond from two equivalent p orbitals and (b) the C=O π bond from two nonequivalent p orbitals.
The above equations show that for the formation of πp, the p orbital in oxygen (more electronegative) makes a greater contribution than does the p orbital in carbon (less electronegative). For the formation of πp*, the p orbital in carbon (less electronegative) makes a greater contribution than does the p orbital in oxygen (more electronegative). In each case, the bonding πp MO is responsible for the formation of a π bond, and antibonding orbital πp* is responsible for dissociation of the
