Читать книгу Organic Mechanisms - Xiaoping Sun - Страница 30

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].

(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) in the line connecting the two nuclei, giving rise to the formation of the bonding MO (σ_2p = 2p_z,A + 2p_z,B) and antibonding MO (σ_2p^* = 2p_z,A − 2p_z,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 (π_p) and antibonding (π_p^*) MOs (Fig. 1.10a). They are formed by constructive and destructive sideway orbital overlaps, respectively: π_p = p₁ + p₂ (fused lobes due to a positive linear combination—constructive orbital overlap) and π_p^* = 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 (π_p) and antibonding (π_p^*) MOs is different (Fig. 1.10b). Usually, the p orbital in the more electronegative atom has greater contribution to the bonding MO (π_p), and the p orbital in the less electronegative atom has greater contribution to the antibonding MO (π_p^*). In the C=O π bond, the bonding (π_p) and antibonding (π_p^*) 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 π_p, the p orbital in oxygen (more electronegative) makes a greater contribution than does the p orbital in carbon (less electronegative). For the formation of π_p^*, 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 π_p MO is responsible for the formation of a π bond, and antibonding orbital π_p^* is responsible for dissociation of the π bond.

When more than two p orbitals overlap sideways, it results in the formation of a conjugate π bond. Similar to the separate π bonds (formed by sideway overlap of two p orbitals), a conjugate π bond consists of series of MOs formed by linear combinations of the contributing p orbitals. The number of constituent MOs is equal to the number of contributing p orbitals. For example, the conjugate π‐bond of allyl radical (CH₂=CHCH₂˙), formed by sideway overlap of three p orbitals in the carbon atoms, consists of the following three MOs (Fig. 1.11a):

The conjugate π‐bond of 1,3‐butadiene (CH₂=CHCH=CH₂), formed by sideway overlap of four p orbitals in the carbon atoms, consists of the following four MOs (Fig. 1.11b):

In each of the molecules, since all the p orbitals are from carbon atoms, their contributions to each of the MOs are equal.