Organic Mechanisms. Xiaoping Sun
(c) Hydrolysis, (d) SN2 reaction, and (e) Dissociation.
1.8 THE MOLECULAR ORBITAL THEORY
1.8.1 Formation of Molecular Orbitals from Atomic Orbitals
The microscopic particles such as electrons possess dual properties, which are particle‐like behavior and wave‐like behavior. The latter can be quantitatively characterized by the wavefunction (ψ), which is a function of the space coordinates (x, y, z in three dimensions) of an electron. The one‐electron wavefunction in an atom is called atomic orbital (AO). The square of a wavefunction (ψ2) is the probability of finding an electron (also called electron density). The atomic orbitals in the valence shells of the atoms of main group elements include s and p orbitals. Their shapes in the three‐dimensional space are illustrated in Figure 1.7.
FIGURE 1.7 The shapes of the s and p orbitals in the three‐dimensional space.
Studying the behavior of fundamental particles in chemistry must eventually go beyond the classical laws. It requires that chemical bonding in molecules be explained as a superposition phenomenon of electron wavefunctions. When two atoms (such as hydrogen atoms) approach each other, their valence electrons will start interacting. This makes the wavefunctions (atomic orbitals) of the interacting atoms superimpose (overlap). Mathematically, such a superposition phenomenon (also called orbital overlap) can be expressed in terms of the linear combinations of atomic orbitals (LCAOs) leading to a set of new wavefunctions in a molecule, called molecular orbitals (MOs) which are shown in Equations 1.58 and 1.59 [2].
ψ1 and ψ2 represent atomic orbitals of the two approaching atoms 1 and 2, respectively. c11, c12, c21, and c22 are constants (positive, zero, or negative). Φ1 and Φ2 are the resulting molecular orbitals from linear combinations of ψ1 and ψ2. By the nature, the molecular orbitals are one‐electron wavefunctions. However, they can approximately characterize the behavior of electrons in a many‐electron molecule. In principle, the number of molecular orbitals formed is equal to the number of participating atomic orbitals which overlap in a molecule. In other words, the participating atomic orbitals can combine linearly in different ways. The number of LCAOs is equal to the number of the atomic 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)