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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].

(1.58)

(1.59)

      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 (1s_A and 1s_B) overlap giving two molecular orbitals σ_1s and σ_1s^* through the following linear combinations of 1s_A and 1s_B [2].

FIGURE 1.8 Formation of the hydrogen molecule (H₂) from two hydrogen (H) atoms.

      Since 1s_A and 1s_B are identical, their contributions to each of the MOs (σ_1s and σ_1s^*) should be equal. Therefore, we have c₁ = c₂ = c (>0) and c₁′ = c₂′ = c′ (>0).

      In order to normalize the molecular orbital σ_1s, 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 = ∫(1s_A1s_B)dτ is referred to as the overlap integral. Therefore, we have

      Similarly, by normalizing σ_1s^*, 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 σ_1s in the midregion of the bond is maximum, while the electron density of σ_1s^* in the midregion of the bond is zero. Therefore, σ_1s 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. σ_1s^* 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 σ_1s and antibonding σ_1s^* 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 σ_1s = 1s_A + 1s_B and σ_1s^* = 1s_A − 1s_B.

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 p_z orbitals) interact resulting in constructive and destructive orbital overlaps (LCAOs)

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