Molecular Orbital Theory (LCAO) | Chemistry Optional Notes for UPSC PDF Download

Molecular Orbitals Involving Only ns Atomic Orbitals

• We begin our discussion of molecular orbitals with the simplest molecule, H₂, formed from two isolated hydrogen atoms, each with a 1s¹ electron configuration. As discussed previously, electrons can behave like waves. In the molecular orbital approach, the overlapping atomic orbitals are described by mathematical equations called wave functions. The 1s atomic orbitals on the two hydrogen atoms interact to form two new molecular orbitals, one produced by taking the sum of the two H 1s wave functions, and the other produced by taking their difference:
  (11.5.1)
• The molecular orbitals created from Equation  11.5.1 are called linear combinations of atomic orbitals (LCAOs) Molecular orbitals created from the sum and the difference of two wave functions (atomic orbitals). A molecule must have as many molecular orbitals as there are atomic orbitals.
• Adding two atomic orbitals corresponds to constructive interference between two waves, thus reinforcing their intensity; the internuclear electron probability density is increased. The molecular orbital corresponding to the sum of the two H 1s orbitals is called a σ1s combination (pronounced “sigma one ess”) (part (a) and part (b) in Figure  11.5.1). In a sigma (σ) orbital, A bonding molecular orbital in which the electron density along the internuclear axis and between the nuclei has cylindrical symmetry, the electron density along the internuclear axis and between the nuclei has cylindrical symmetry; that is, all cross-sections perpendicular to the internuclear axis are circles. The subscript 1s denotes the atomic orbitals from which the molecular orbital was derived: The ≈ sign is used rather than an = sign because we are ignoring certain constants that are not important to our argument.

Figure 11.5.1: Molecular Orbitals for the H₂ Molecule. (a) This diagram shows the formation of a bonding σ1s molecular orbital for H₂ as the sum of the wave functions (Ψ) of two H1s atomic orbitals. (b) This plot of the square of the wave function (Ψ²) for the bonding molecular orbital illustrates the increased electron probability density between the two hydrogen nuclei. (Recall that the probability density is proportional to the square of the wave function.) (c) This diagram shows the formation of an antibonding   molecular orbital for H₂ as the difference of the wave functions (Ψ) of two H 1s atomic orbitals. (d) This plot of the square of the wave function (Ψ²) for the antibonding molecular orbital illustrates the node corresponding to zero electron probability density between the two hydrogen nuclei.

σ_1s ≈ 1s (A) + 1s (B) (11.5.2)

Conversely, subtracting one atomic orbital from another corresponds to destructive interference between two waves, which reduces their intensity and causes a decrease in the internuclear electron probability density (part (c) and part (d) in Figure  11.5.1). The resulting pattern contains a node where the electron density is zero. The molecular orbital corresponding to the difference is called   (“sigma one ess star”). In a sigma star (σ*) orbital An antibonding molecular orbital in which there is a region of zero electron probability (a nodal plane) perpendicular to the internuclear axis., there is a region of zero electron probability, a nodal plane, perpendicular to the internuclear axis:

 ≈ 1s(A)−1s (B)  (11.5.3)

• A molecule must have as many molecular orbitals as there are atomic orbitals.

The electron density in the σ_1s molecular orbital is greatest between the two positively charged nuclei, and the resulting electron–nucleus electrostatic attractions reduce repulsions between the nuclei. Thus the σ_1s orbital represents a bonding molecular orbital. A molecular orbital that forms when atomic orbitals or orbital lobes with the same sign interact to give increased electron probability between the nuclei due to constructive reinforcement of the wave functions. In contrast, electrons in the orbital are generally found in the space outside the internuclear region. Because this allows the positively charged nuclei to repel one another, the   orbital is an antibonding molecular orbital (a molecular orbital that forms when atomic orbitals or orbital lobes of opposite sign interact to give decreased electron probability between the nuclei due to destructive reinforcement of the wave functions).

• Antibonding orbitals contain a node perpendicular to the internuclear axis; bonding orbitals do not

Energy-Level Diagrams

Because electrons in the σ1s orbital interact simultaneously with both nuclei, they have a lower energy than electrons that interact with only one nucleus. This means that the σ1s molecular orbital has a lower energy than either of the hydrogen 1s atomic orbitals. Conversely, electrons in the  orbital interact with only one hydrogen nucleus at a time. In addition, they are farther away from the nucleus than they were in the parent hydrogen 1s atomic orbitals. Consequently, themolecular orbital has a higher energy than either of the hydrogen 1s atomic orbitals. The σ1s (bonding) molecular orbital is stabilized relative to the 1s atomic orbitals, and the (antibonding) molecular orbital is destabilized. The relative energy levels of these orbitals are shown in the energy-level diagram (a schematic drawing that compares the energies of the molecular orbitals (bonding, antibonding, and nonbonding) with the energies of the parent atomic orbitals) in Figure 11.5.2

Figure  11.5.2: Molecular Orbital Energy-Level Diagram for H₂. The two available electrons (one from each H atom) in this diagram fill the bonding σ1s molecular orbital. Because the energy of the σ1s molecular orbital is lower than that of the two H 1s atomic orbitals, the H₂ molecule is more stable (at a lower energy) than the two isolated H atoms.

• A bonding molecular orbital is always lower in energy (more stable) than the component atomic orbitals, whereas an antibonding molecular orbital is always higher in energy (less stable).

To describe the bonding in a homonuclear diatomic molecule (a molecule that consists of two atoms of the same element) such as H₂, we use molecular orbitals; that is, for a molecule in which two identical atoms interact, we insert the total number of valence electrons into the energy-level diagram (Figure  11.5.2). We fill the orbitals according to the Pauli principle and Hund’s rule: each orbital can accommodate a maximum of two electrons with opposite spins, and the orbitals are filled in order of increasing energy. Because each H atom contributes one valence electron, the resulting two electrons are exactly enough to fill the σ_1s bonding molecular orbital. The two electrons enter an orbital whose energy is lower than that of the parent atomic orbitals, so the H₂ molecule is more stable than the two isolated hydrogen atoms. Thus molecular orbital theory correctly predicts that H₂ is a stable molecule. Because bonds form when electrons are concentrated in the space between nuclei, this approach is also consistent with our earlier discussion of electron-pair bonds.

Bond Order in Molecular Orbital Theory

• In the Lewis electron structures, the number of electron pairs holding two atoms together was called the bond order. In the molecular orbital approach, bond order One-half the net number of bonding electrons in a molecule. is defined as one-half the net number of bonding electrons: (11.5.4)

• To calculate the bond order of H₂, we see from Figure  11.5.2 that the σ_1s (bonding) molecular orbital contains two electrons, while the (antibonding) molecular orbital is empty. The bond order of H₂ is therefore
   (11.5.5)
• This result corresponds to the single covalent bond predicted by Lewis dot symbols. Thus molecular orbital theory and the Lewis electron-pair approach agree that a single bond containing two electrons has a bond order of 1. Double and triple bonds contain four or six electrons, respectively, and correspond to bond orders of 2 and 3.
• We can use energy-level diagrams such as the one in Figure 11.5.2 to describe the bonding in other pairs of atoms and ions where n = 1, such as the H₂⁺ ion, the He₂⁺ ion, and the He₂ molecule. Again, we fill the lowest-energy molecular orbitals first while being sure not to violate the Pauli principle or Hund’s rule.

Figure  11.5.3: Molecular Orbital Energy-Level Diagrams for Diatomic Molecules with Only 1s Atomic Orbitals. (a) The H₂⁺ ion, (b) the He₂⁺ ion, and (c) the He₂ molecule are shown here.

• Part (a) in Figure 11.5.3 shows the energy-level diagram for the H₂⁺ ion, which contains two protons and only one electron. The single electron occupies the σ_1s bonding molecular orbital, giving a (σ_1s)¹ electron configuration. The number of electrons in an orbital is indicated by a superscript. In this case, the bond order is (1-0)/2 = 1/2 Because the bond order is greater than zero, the H₂⁺ ion should be more stable than an isolated H atom and a proton. We can therefore use a molecular orbital energy-level diagram and the calculated bond order to predict the relative stability of species such as H₂⁺. With a bond order of only 1/2 the bond in H₂⁺ should be weaker than in the H₂ molecule, and the H–H bond should be longer. As shown in Table  11.5.1, these predictions agree with the experimental data.
• Part (b) in Figure 11.5.3 is the molecular orbital energy-level diagram for He₂⁺. This ion has a total of three valence electrons. Because the first two electrons completely fill the σ_1s molecular orbital, the Pauli principle states that the third electron must be in the antibonding orbital, giving a  (σ_1s)²(σ⋆_1s)¹  electron configuration. This electron configuration gives a bond order of (2-1)/2 = 1/2. As with H₂⁺, the He₂⁺ ion should be stable, but the He–He bond should be weaker and longer than in H₂. In fact, the He₂⁺ ion can be prepared, and its properties are consistent with our predictions (Table 11.5.1).

• Finally, we examine the He2 molecule, formed from two He atoms with 1s² electron configurations. Part (c) in Figure  11.5.3 is the molecular orbital energy-level diagram for He₂. With a total of four valence electrons, both the σ_1s bonding and  antibonding orbitals must contain two electrons. This gives a  (σ_1s)²(σ⋆_1s)¹ electron configuration, with a predicted bond order of (2 − 2) ÷ 2 = 0, which indicates that the He2 molecule has no net bond and is not a stable species. Experiments show that the He₂ molecule is actually less stable than two isolated He atoms due to unfavorable electron–electron and nucleus–nucleus interactions.
• In molecular orbital theory, electrons in antibonding orbitals effectively cancel the stabilization resulting from electrons in bonding orbitals. Consequently, any system that has equal numbers of bonding and antibonding electrons will have a bond order of 0, and it is predicted to be unstable and therefore not to exist in nature. In contrast to Lewis electron structures and the valence bond approach, molecular orbital theory is able to accommodate systems with an odd number of electrons, such as the H₂⁺ ion.

In contrast to Lewis electron structures and the valence bond approach, molecular orbital theory can accommodate systems with an odd number of electrons.

Solved Examples

Example 1: Use a molecular orbital energy-level diagram, such as those in Figure  11.5.3, to predict the bond order in the He₂²⁺ ion. Is this a stable species?
Given: chemical species
Asked for: molecular orbital energy-level diagram, bond order, and stability
Strategy:
(a) Combine the two He valence atomic orbitals to produce bonding and antibonding molecular orbitals. Draw the molecular orbital energy-level diagram for the system.
(b) Determine the total number of valence electrons in the He₂²⁺ ion. Fill the molecular orbitals in the energy-level diagram beginning with the orbital with the lowest energy. Be sure to obey the Pauli principle and Hund’s rule while doing so.
(c) Calculate the bond order and predict whether the species is stable.
Ans: (a) Two He 1s atomic orbitals combine to give two molecular orbitals: a σ_1s bonding orbital at lower energy than the atomic orbitals and a antibonding orbital at higher energy. The bonding in any diatomic molecule with two He atoms can be described using the following molecular orbital diagram:

(b) The He₂²⁺ ion has only two valence electrons (two from each He atom minus two for the +2 charge). We can also view He₂²⁺ as being formed from two He⁺ ions, each of which has a single valence electron in the 1s atomic orbital. We can now fill the molecular orbital diagram:
The two electrons occupy the lowest-energy molecular orbital, which is the bonding (σ_1s) orbital, giving a (σ_1s)² electron configuration. To avoid violating the Pauli principle, the electron spins must be paired. (c) So the bond order is
He₂²⁺ is therefore predicted to contain a single He–He bond. Thus it should be a stable species.

Example 2: Use a molecular orbital energy-level diagram to predict the valence-electron configuration and bond order of the H₂²⁻ ion. Is this a stable species?
Ans: H₂²⁻ has a valence electron configuration of  (σ_1s)²()²  with a bond order of 0. It is therefore predicted to be unstable.

• So far, our discussion of molecular orbitals has been confined to the interaction of valence orbitals, which tend to lie farthest from the nucleus. When two atoms are close enough for their valence orbitals to overlap significantly, the filled inner electron shells are largely unperturbed; hence they do not need to be considered in a molecular orbital scheme. Also, when the inner orbitals are completely filled, they contain exactly enough electrons to completely fill both the bonding and antibonding molecular orbitals that arise from their interaction. Thus the interaction of filled shells always gives a bond order of 0, so filled shells are not a factor when predicting the stability of a species. This means that we can focus our attention on the molecular orbitals derived from valence atomic orbitals.
• A molecular orbital diagram that can be applied to any homonuclear diatomic molecule with two identical alkali metal atoms (Li₂ and Cs₂, for example) is shown in part (a) in Figure 11.5.4, where M represents the metal atom. Only two energy levels are important for describing the valence electron molecular orbitals of these species: a σns bonding molecular orbital and a σ*_ns antibonding molecular orbital. Because each alkali metal (M) has an ns¹ valence electron configuration, the M₂ molecule has two valence electrons that fill the σns bonding orbital. As a result, a bond order of 1 is predicted for all homonuclear diatomic species formed from the alkali metals (Li₂, Na₂, K₂, Rb₂, and Cs₂). The general features of these M₂ diagrams are identical to the diagram for the H₂ molecule in Figure 11.5.2. Experimentally, all are found to be stable in the gas phase, and some are even stable in solution.

Figure 11.5.4: Molecular Orbital Energy-Level Diagrams for Alkali Metal and Alkaline Earth Metal Diatomic (M₂) Molecules. (a) For alkali metal diatomic molecules, the two valence electrons are enough to fill the σ_ns (bonding) level, giving a bond order of 1. (b) For alkaline earth metal diatomic molecules, the four valence electrons fill both the σ_ns (bonding) and the σ_ns* (nonbonding) levels, leading to a predicted bond order of 0.

Similarly, the molecular orbital diagrams for homonuclear diatomic compounds of the alkaline earth metals (such as Be₂), in which each metal atom has an ns² valence electron configuration, resemble the diagram for the He₂ molecule in part (c) in Figure  11.5.3 As shown in part (b) in Figure  11.5.4 , this is indeed the case. All the homonuclear alkaline earth diatomic molecules have four valence electrons, which fill both the σns bonding orbital and the σ_ns* antibonding orbital and give a bond order of 0. Thus Be₂, Mg₂, Ca₂, Sr₂, and Ba₂ are all expected to be unstable, in agreement with experimental data. In the solid state, however, all the alkali metals and the alkaline earth metals exist as extended lattices held together by metallic bonding. At low temperatures,  Be₂ is stable.

Solved Examples

Example 1: Use a qualitative molecular orbital energy-level diagram to predict the valence electron configuration, bond order, and likely existence of the Na₂⁻ ion.
Given: chemical species
Asked for: molecular orbital energy-level diagram, valence electron configuration, bond order, and stability
Strategy:
(a) Combine the two sodium valence atomic orbitals to produce bonding and antibonding molecular orbitals. Draw the molecular orbital energy-level diagram for this system.
(b) Determine the total number of valence electrons in the Na₂⁻ ion. Fill the molecular orbitals in the energy-level diagram beginning with the orbital with the lowest energy. Be sure to obey the Pauli principle and Hund’s rule while doing so.
(c) Calculate the bond order and predict whether the species is stable.
Ans: (a) Because sodium has a [Ne]3s¹ electron configuration, the molecular orbital energy-level diagram is qualitatively identical to the diagram for the interaction of two 1s atomic orbitals.
(b) The Na₂⁻ ion has a total of three valence electrons (one from each Na atom and one for the negative charge), resulting in a filled σ_3s molecular orbital, a half-filled σ_3s* and a  (σ_3s)²(σ^⋆_3s)¹ electron configuration.
(c) The bond order is (2-1)÷2 = 1/2 With a fractional bond order, we predict that the Na₂⁻ ion exists but is highly reactive.

Example 2: Use a qualitative molecular orbital energy-level diagram to predict the valence electron configuration, bond order, and likely existence of the Ca₂⁺ ion.
Ans: Ca₂⁺ has a  (σ_4s)²(σ⋆_4s)¹ electron configurations and a bond order of 1/2 and should exist.

Molecular Orbitals Formed from ns and np Atomic Orbitals

• Atomic orbitals other than ns orbitals can also interact to form molecular orbitals. Because individual p, d, and f orbitals are not spherically symmetrical, however, we need to define a coordinate system so we know which lobes are interacting in three-dimensional space. Recall that for each np subshell, for example, there are np_x, np_y, and np_z orbitals. All have the same energy and are therefore degenerate, but they have different spatial orientations.
  σ_npz = np_z(A)−np_z(B) (11.5.6)
• Just as with ns orbitals, we can form molecular orbitals from np orbitals by taking their mathematical sum and difference. When two positive lobes with the appropriate spatial orientation overlap, as illustrated for two np_z atomic orbitals in part (a) in Figure 11.5.5, it is the mathematical difference of their wave functions that results in constructive interference, which in turn increases the electron probability density between the two atoms. The difference therefore corresponds to a molecular orbital called a σ_npz  bonding molecular orbital because, just as with the σ orbitals discussed previously, it is symmetrical about the internuclear axis (in this case, the z-axis):
  (11.5.7)

The other possible combination of the two np_z orbitals is the mathematical sum:
(11.5.8)

In this combination, shown in part (b) in Figure 11.5.5, the positive lobe of one npz atomic orbital overlaps the negative lobe of the other, leading to destructive interference of the two waves and creating a node between the two atoms. Hence this is an antibonding molecular orbital. Because it, too, is symmetrical about the internuclear axis, this molecular orbital is called a σnp_z = np_z(A)−np_z(B) antibonding molecular orbital. Whenever orbitals combine, the bonding combination is always lower in energy (more stable) than the atomic orbitals from which it was derived, and the antibonding combination is higher in energy (less stable).

Figure 11.5.5: Formation of Molecular Orbitals from np_z Atomic Orbitals on Adjacent Atoms. (a) By convention, in a linear molecule or ion, the z-axis always corresponds to the internuclear axis, with +z to the right. As a result, the signs of the lobes of the np_z atomic orbitals on the two atoms alternate − + − +, from left to right. In this case, the σ (bonding) molecular orbital corresponds to the mathematical difference, in which the overlap of lobes with the same sign results in increased probability density between the nuclei. (b) In contrast, the σ* (antibonding) molecular orbital corresponds to the mathematical sum, in which the overlap of lobes with opposite signs results in a nodal plane of zero probability density perpendicular to the internuclear axis.

• Overlap of atomic orbital lobes with the same sign produces a bonding molecular orbital, regardless of whether it corresponds to the sum or the difference of the atomic orbitals. The remaining p orbitals on each of the two atoms, np_x and np_y, do not point directly toward each other. Instead, they are perpendicular to the internuclear axis. If we arbitrarily label the axes as shown in Figure  11.5.6, we see that we have two pairs of np orbitals: the two np_x orbitals lying in the plane of the page, and two npy orbitals perpendicular to the plane. 
• Although these two pairs are equivalent in energy, the np_x orbital on one atom can interact with only the np_x orbital on the other, and the npy orbital on one atom can interact with only the npy on the other. These interactions are side-to-side rather than the head-to-head interactions characteristic of σ orbitals. Each pair of overlapping atomic orbitals again forms two molecular orbitals: one corresponds to the arithmetic sum of the two atomic orbitals and one to the difference. 
• The sum of these side-to-side interactions increases the electron probability in the region above and below a line connecting the nuclei, so it is a bonding molecular orbital that is called a pi (π) orbital (a bonding molecular orbital formed from the side-to-side interactions of two or more parallel np atomic orbitals). The difference results in the overlap of orbital lobes with opposite signs, which produces a nodal plane perpendicular to the internuclear axis; hence it is an antibonding molecular orbital, called a pi star (π*) orbital An antibonding molecular orbital formed from the difference of the side-to-side interactions of two or more parallel np atomic orbitals, creating a nodal plane perpendicular to the internuclear axis..
   (11.5.9)
   (11.5.10)
• The two np_y orbitals can also combine using side-to-side interactions to produce a bonding π_npy molecular orbital and an antibonding π^⋆_npy molecular orbital. Because the np_x and np_y atomic orbitals interact in the same way (side-to-side) and have the same energy, the π_npx and π_npy molecular orbitals are a degenerate pair, as are the π^⋆_npx and π^⋆_npy molecular orbitals.

Figure 11.5.6: Formation of π Molecular Orbitals from npx and npy Atomic Orbitals on Adjacent Atoms.(a) Because the signs of the lobes of both the npx and the npy atomic orbitals on adjacent atoms are the same, in both cases the mathematical sum corresponds to a π (bonding) molecular orbital. (b) In contrast, in both cases, the mathematical difference corresponds to a π* (antibonding) molecular orbital, with a nodal plane of zero probability density perpendicular to the internuclear axis.

• Figure 11.5.7 is an energy-level diagram that can be applied to two identical interacting atoms that have three np atomic orbitals each. There are six degenerate p atomic orbitals (three from each atom) that combine to form six molecular orbitals, three bonding and three antibonding. The bonding molecular orbitals are lower in energy than the atomic orbitals because of the increased stability associated with the formation of a bond. Conversely, the antibonding molecular orbitals are higher in energy, as shown. 
• The energy difference between the σ and σ* molecular orbitals is significantly greater than the difference between the two π and π* sets. The reason for this is that the atomic orbital overlap and thus the strength of the interaction are greater for a σ bond than a π bond, which means that the σ molecular orbital is more stable (lower in energy) than the π molecular orbitals.

Figure 11.5.7: The Relative Energies of the σ and π Molecular Orbitals Derived from npx, npy, and npz Orbitals on Identical Adjacent Atoms. Because the two npz orbitals point directly at each other, their orbital overlap is greater, so the difference in energy between the σ and σ* molecular orbitals is greater than the energy difference between the π and π* orbitals.

• Although many combinations of atomic orbitals form molecular orbitals, we will discuss only one other interaction: an ns atomic orbital on one atom with an np_z atomic orbital on another. As shown in Figure 11.5.8, the sum of the two atomic wave functions (ns + np_z) produces a σ bonding molecular orbital. Their difference (ns − np_z) produces a σ* antibonding molecular orbital, which has a nodal plane of zero probability density perpendicular to the internuclear axis.

Figure 11.5.8: Formation of Molecular Orbitals from an ns Atomic Orbital on One Atom and an npz Atomic Orbital on an Adjacent Atom.(a) The mathematical sum results in a σ (bonding) molecular orbital, with increased probability density between the nuclei. (b) The mathematical difference results in a σ* (antibonding) molecular orbital, with a nodal plane of zero probability density perpendicular to the internuclear axis.

Second Row Diatomic Molecules

If we combine the splitting schemes for the 2s and 2p orbitals, we can predict bond order in all of the diatomic molecules and ions composed of elements in the first complete row of the periodic table. Remember that only the valence orbitals of the atoms need be considered; as we saw in the cases of lithium hydride and dilithium, the inner orbitals remain tightly bound and retain their localized atomic character. We now describe examples of systems involving period 2 homonuclear diatomic molecules, such as N₂, O₂, and F₂. When we draw a molecular orbital diagram for a molecule, there are four key points to remember:

• The number of molecular orbitals produced is the same as the number of atomic orbitals used to create them (the law of conservation of orbitals).
• As the overlap between two atomic orbitals increases, the difference in energy between the resulting bonding and antibonding molecular orbitals increases.
• When two atomic orbitals combine to form a pair of molecular orbitals, the bonding molecular orbital is stabilized about as much as the antibonding molecular orbital is destabilized.
• The interaction between atomic orbitals is greatest when they have the same energy.

The number of molecular orbitals is always equal to the total number of atomic orbitals we started with.

• We illustrate how to use these points by constructing a molecular orbital energy-level diagram for F₂. We use the diagram in part (a) in Figure  11.5.9; the n = 1 orbitals (σ_1s and σ_1s*) are located well below those of the n = 2 level and are not shown. As illustrated in the diagram, the σ_2s and σ_2s* molecular orbitals are much lower in energy than the molecular orbitals derived from the 2p atomic orbitals because of the large difference in energy between the 2s and 2p atomic orbitals of fluorine.
• The lowest-energy molecular orbital derived from the three 2p orbitals on each F is  σ_2pz  and the next most stable are the two degenerate orbitals,  π_2px and  π_2py . For each bonding orbital in the diagram, there is an antibonding orbital, and the antibonding orbital is destabilized by about as much as the corresponding bonding orbital is stabilized. As a result, the  σ^⋆_2pz orbital is higher in energy than either of the degenerate  π^⋆_2px and  π^⋆_2py  orbitals. We can now fill the orbitals, beginning with the one that is lowest in energy.
• Each fluorine has 7 valence electrons, so there are a total of 14 valence electrons in the F₂ molecule. Starting at the lowest energy level, the electrons are placed in the orbitals according to the Pauli principle and Hund’s rule. Two electrons each fill the σ_2s and σ_2s* orbitals, 2 fill the  σ_2pz orbital, 4 fill the two degenerate π orbitals, and 4 fill the two degenerate π* orbitals, for a total of 14 electrons. To determine what type of bonding the molecular orbital approach predicts F₂ to have, we must calculate the bond order. According to our diagram, there are 8 bonding electrons and 6 antibonding electrons, giving a bond order of (8 − 6) ÷ 2 = 1. Thus F₂ is predicted to have a stable F–F single bond, in agreement with experimental data.

Figure 11.5.9: Molecular Orbital Energy-Level Diagrams for Homonuclear Diatomic Molecules.(a) For F₂, with 14 valence electrons (7 from each F atom), all of the energy levels except the highest,  σ^⋆_2pz are filled. This diagram shows 8 electrons in bonding orbitals and 6 in antibonding orbitals, resulting in a bond order of 1. (b) For O₂, with 12 valence electrons (6 from each O atom), there are only 2 electrons to place in the  pair of orbitals. Hund’s rule dictates that one electron occupies each orbital, and their spins are parallel, giving the O₂ molecule two unpaired electrons. This diagram shows 8 electrons in bonding orbitals and 4 in antibonding orbitals, resulting in a predicted bond order of 2.

• We now turn to a molecular orbital description of the bonding in O₂. It so happens that the molecular orbital description of this molecule provided an explanation for a long-standing puzzle that could not be explained using other bonding models. To obtain the molecular orbital energy-level diagram for O₂, we need to place 12 valence electrons (6 from each O atom) in the energy-level diagram shown in part (b) in Figure  11.5.9. We again fill the orbitals according to Hund’s rule and the Pauli principle, beginning with the orbital that is lowest in energy. Two electrons each are needed to fill the σ_2s and σ_2s* orbitals, 2 more to fill the  σ_2pz orbital, and 4 to fill the degenerate  π^⋆_2px and  π^⋆_2py orbitals. According to Hund’s rule, the last 2 electrons must be placed in separate π* orbitals with their spins parallel, giving two unpaired electrons. This leads to a predicted bond order of (8 − 4) ÷ 2 = 2, which corresponds to a double bond, in agreement with experimental data (Table 4.5): the O–O bond length is 120.7 pm, and the bond energy is 498.4 kJ/mol at 298 K.
• None of the other bonding models can predict the presence of two unpaired electrons in O₂. Chemists had long wondered why, unlike most other substances, liquid O₂ is attracted into a magnetic field. As shown in Figure  11.5.10, it actually remains suspended between the poles of a magnet until the liquid boils away. The only way to explain this behavior was for O₂ to have unpaired electrons, making it paramagnetic, exactly as predicted by molecular orbital theory. This result was one of the earliest triumphs of molecular orbital theory over the other bonding approaches we have discussed.
• The magnetic properties of O₂ are not just a laboratory curiosity; they are absolutely crucial to the existence of life. Because Earth’s atmosphere contains 20% oxygen, all organic compounds, including those that compose our body tissues, should react rapidly with air to form H₂O, CO₂, and N₂ in an exothermic reaction. Fortunately for us, however, this reaction is very, very slow. The reason for the unexpected stability of organic compounds in an oxygen atmosphere is that virtually all organic compounds, as well as H₂O, CO₂, and N₂, have only paired electrons, whereas oxygen has two unpaired electrons. Thus the reaction of O₂ with organic compounds to give H₂O, CO₂, and N₂ would require that at least one of the electrons on O₂ change its spin during the reaction. This would require a large input of energy, an obstacle that chemists call a spin barrier. Consequently, reactions of this type are usually exceedingly slow. If they were not so slow, all organic substances, including this book and you, would disappear in a puff of smoke!
• For period 2 diatomic molecules to the left of N₂ in the periodic table, a slightly different molecular orbital energy-level diagram is needed because the  σ_2pz  molecular orbital is slightly higher in energy than the degenerate π^⋆_npx and π^⋆_npy orbitals. The difference in energy between the 2s and 2p atomic orbitals increases from Li₂ to F₂ due to increasing nuclear charge and poor screening of the 2s electrons by electrons in the 2p subshell. The bonding interaction between the 2s orbital on one atom and the 2pz orbital on the other is most important when the two orbitals have similar energies. This interaction decreases the energy of the σ_2s orbital and increases the energy of the  σ_2pz orbital. Thus for Li₂, Be₂, B₂, C₂, and N₂, the  σ₂pz orbital is higher in energy than the  σ_3pz orbitals, as shown in Figure  11.5.11 Experimentally, it is found that the energy gap between the ns and np atomic orbitals increases as the nuclear charge increases (Figure  11.5.11). Thus for example, the  σ_2pz molecular orbital is at a lower energy than the  π_2px,y  pair.

Figure 11.5.11: Molecular Orbital Energy-Level Diagrams for the Diatomic Molecules of the Period 2 Elements. Unlike earlier diagrams, only the molecular orbital energy levels for the molecules are shown here. For simplicity, the atomic orbital energy levels for the component atoms have been omitted. For Li₂ through N₂, the  σ_2pz orbital is higher in energy than the  π_2px,y orbitals. In contrast, the  σ₂pz orbital is lower in energy than the  π_2px,y orbitals for O₂ and F₂ due to the increase in the energy difference between the 2s and 2p atomic orbitals as the nuclear charge increases across the row.

Completing the diagram for N₂ in the same manner as demonstrated previously, we find that the 10 valence electrons result in 8 bonding electrons and 2 antibonding electrons, for a predicted bond order of 3, a triple bond. Experimental data show that the N–N bond is significantly shorter than the F–F bond (109.8 pm in N₂ versus 141.2 pm in F₂), and the bond energy is much greater for N₂ than for F₂ (945.3 kJ/mol versus 158.8 kJ/mol, respectively). Thus the N₂ bond is much shorter and stronger than the F₂ bond, consistent with what we would expect when comparing a triple bond with a single bond.

Solved Examples

Example 1: Use a qualitative molecular orbital energy-level diagram to predict the electron configuration, the bond order, and the number of unpaired electrons in S₂, a bright blue gas at high temperatures.
Given: chemical species
Asked for: molecular orbital energy-level diagram, bond order, and number of unpaired electrons
Strategy:
(a) Write the valence electron configuration of sulfur and determine the type of molecular orbitals formed in S₂. Predict the relative energies of the molecular orbitals based on how close in energy the valence atomic orbitals are to one another.
(b) Draw the molecular orbital energy-level diagram for this system and determine the total number of valence electrons in S₂.
(c) Fill the molecular orbitals in order of increasing energy, being sure to obey the Pauli principle and Hund’s rule.
(d) Calculate the bond order and describe the bonding.
Ans: (a) Sulfur has a [Ne]3s²3p⁴ valence electron configuration. To create a molecular orbital energy-level diagram similar to those in Figure  11.5.9 and Figure  11.5.11, we need to know how close in energy the 3s and 3p atomic orbitals are because their energy separation will determine whether the  π_3px,y or the  σ_3pz > molecular orbital is higher in energy. Because the ns–np energy gap increases as the nuclear charge increases (Figure  11.5.11), the  σ₃pz molecular orbital will be lower in energy than the  π₃px,y pair.

(b) The molecular orbital energy-level diagram is as follows:
Each sulfur atom contributes 6 valence electrons, for a total of 12 valence electrons.
(c) Ten valence electrons are used to fill the orbitals through  π_3px and  π_3py , leaving 2 electrons to occupy the degenerate  π^⋆_3px  and  π^⋆_3py pair. From Hund’s rule, the remaining 2 electrons must occupy these orbitals separately with their spins aligned. With the numbers of electrons written as superscripts, the electron configuration of
with 2 unpaired electrons. The bond order is (8 − 4) ÷ 2 = 2, so we predict an S=S double bond.

Example 2: Use a qualitative molecular orbital energy-level diagram to predict the electron configuration, the bond order, and the number of unpaired electrons in the peroxide ion (O₂²⁻).
Ans:

bond order of 1;
no unpaired electrons

Molecular Orbitals for Heteronuclear Diatomic Molecules

Diatomic molecules with two different atoms are called heteronuclear diatomic molecules. When two nonidentical atoms interact to form a chemical bond, the interacting atomic orbitals do not have the same energy. If, for example, element B is more electronegative than element A (χ_B > χ_A), the net result is a “skewed” molecular orbital energy-level diagram, such as the one shown for a hypothetical A–B molecule in Figure  11.5.12. The atomic orbitals of element B are uniformly lower in energy than the corresponding atomic orbitals of element A because of the enhanced stability of the electrons in element B. The molecular orbitals are no longer symmetrical, and the energies of the bonding molecular orbitals are more similar to those of the atomic orbitals of B. Hence the electron density of bonding electrons is likely to be closer to the more electronegative atom. In this way, molecular orbital theory can describe a polar covalent bond.

Figure 11.5.12: Molecular Orbital Energy-Level Diagram for a Heteronuclear Diatomic Molecule AB, Where χB > χA. The bonding molecular orbitals are closer in energy to the atomic orbitals of the more electronegative B atom. Consequently, the electrons in the bonding orbitals are not shared equally between the two atoms. On average, they are closer to the B atom, resulting in a polar covalent bond.

• A molecular orbital energy-level diagram is always skewed toward the more electronegative atom.

An Odd Number of Valence Electrons: NO

• Nitric oxide (NO) is an example of a heteronuclear diatomic molecule. The reaction of O₂ with N₂ at high temperatures in internal combustion engines forms nitric oxide, which undergoes a complex reaction with O₂ to produce NO₂, which in turn is responsible for the brown color we associate with air pollution. Recently, however, nitric oxide has also been recognized to be a vital biological messenger involved in regulating blood pressure and long-term memory in mammals.
• Because NO has an odd number of valence electrons (5 from nitrogen and 6 from oxygen, for a total of 11), its bonding and properties cannot be successfully explained by either the Lewis electron-pair approach or valence bond theory. The molecular orbital energy-level diagram for NO (Figure  11.5.13) shows that the general pattern is similar to that for the O₂ molecule (Figure  11.5.11 ). Because 10 electrons are sufficient to fill all the bonding molecular orbitals derived from 2p atomic orbitals, the 11th electron must occupy one of the degenerate π* orbitals. The predicted bond order for NO is therefore (8-3) ÷ 2 = 2 1/2 . Experimental data, showing an N–O bond length of 115 pm and N–O bond energy of 631 kJ/mol, are consistent with this description. These values lie between those of the N₂ and O₂ molecules, which have triple and double bonds, respectively. As we stated earlier, molecular orbital theory can therefore explain the bonding in molecules with an odd number of electrons, such as NO, whereas Lewis electron structures cannot.

Figure 11.5.13: Molecular Orbital Energy-Level Diagram for NO . Because NO has 11 valence electrons, it is paramagnetic, with a single electron occupying the pair of orbitals.

• Note that electronic structure studies show the ground state configuration of  NO to be  in order of increasing energy. Hence, the  π_2px,y orbitals are lower in energy than the  σ_2pz orbital. This is because the NO molecule is near the transition of flipping energies levels observed in homonuclear diatomics where the sigma bond drops below the pi bond (Figure  11.5.11).
• Molecular orbital theory can also tell us something about the chemistry of NO . As indicated in the energy-level diagram in Figure  11.5.13, NO has a single electron in a relatively high-energy molecular orbital. We might therefore expect it to have similar reactivity as alkali metals such as Li and Na with their single valence electrons. In fact,  NO  is easily oxidized to the  NO⁺ cation, which is isoelectronic with  N₂ and has a bond order of 3, corresponding to an N ≡ O triple bond.

Nonbonding Molecular Orbitals

• Molecular orbital theory is also able to explain the presence of lone pairs of electrons. Consider, for example, the HCl molecule, whose Lewis electron structure has three lone pairs of electrons on the chlorine atom. Using the molecular orbital approach to describe the bonding in HCl, we can see from Figure  11.5.14 that the 1s orbital of atomic hydrogen is closest in energy to the 3p orbitals of chlorine. Consequently, the filled Cl 3s atomic orbital is not involved in bonding to any appreciable extent, and the only important interactions are those between the H 1s and Cl 3p orbitals. Of the three p orbitals, only one, designated as 3p_z, can interact with the H 1s orbital. 
• The 3px and 3py atomic orbitals have no net overlap with the 1s orbital on hydrogen, so they are not involved in bonding. Because the energies of the Cl 3s, 3p_x, and 3p_y orbitals do not change when HCl forms, they are called nonbonding molecular orbitals. A nonbonding molecular orbital occupied by a pair of electrons is the molecular orbital equivalent of a lone pair of electrons. By definition, electrons in nonbonding orbitals do not affect bond order, so they are not counted in the calculation of bond order. Thus the predicted bond order of HCl is (2 − 0) ÷ 2 = 1. Because the σ bonding molecular orbital is closer in energy to the Cl 3p_z than to the H 1s atomic orbital, the electrons in the σ orbital are concentrated closer to the chlorine atom than to hydrogen. A molecular orbital approach to bonding can therefore be used to describe the polarization of the H–Cl bond to give  H^δ+−−Cl^δ− .

Figure 11.5.14: Molecular Orbital Energy-Level Diagram for HCl. The hydrogen 1s atomic orbital interacts most strongly with the 3p_z orbital on chlorine, producing a bonding/antibonding pair of molecular orbitals. The other electrons on Cl are best viewed as nonbonding. As a result, only the bonding σ orbital is occupied by electrons, giving a bond order of 1.

Electrons in nonbonding molecular orbitals do not affect bond order.

Solved Examples

Example 1: Use a “skewed” molecular orbital energy-level diagram like the one in Figure  11.5.12 to describe the bonding in the cyanide ion (CN−). What is the bond order?
Given: chemical species
Asked for: “skewed” molecular orbital energy-level diagram, bonding description, and bond order
Strategy:
(a) Calculate the total number of valence electrons in CN−. Then place these electrons in a molecular orbital energy-level diagram like Figure  11.5.12  in order of increasing energy. Be sure to obey the Pauli principle and Hund’s rule while doing so.
(b) Calculate the bond order and describe the bonding in CN−.
Ans: (a) The CN⁻ ion has a total of 10 valence electrons: 4 from C, 5 from N, and 1 for the −1 charge. Placing these electrons in an energy-level diagram like Figure  11.5.12 fills the five lowest-energy orbitals, as shown here:

Because χN > χC, the atomic orbitals of N (on the right) are lower in energy than those of C.
(b) The resulting valence electron configuration gives a predicted bond order of (8 − 2) ÷ 2 = 3, indicating that the CN− ion has a triple bond, analogous to that in N₂.

Example 2: Use a qualitative molecular orbital energy-level diagram to describe the bonding in the hypochlorite ion (OCl⁻). What is the bond order?
Ans: All molecular orbitals except the highest-energy σ* are filled, giving a bond order of 1.

Although the molecular orbital approach reveals a great deal about the bonding in a given molecule, the procedure quickly becomes computationally intensive for molecules of even moderate complexity. Furthermore, because the computed molecular orbitals extend over the entire molecule, they are often difficult to represent in a way that is easy to visualize. Therefore we do not use a pure molecular orbital approach to describe the bonding in molecules or ions with more than two atoms. Instead, we use a valence bond approach and a molecular orbital approach to explain, among other things, the concept of resonance, which cannot adequately be explained using other methods.

Summary

• Molecular orbital theory, a delocalized approach to bonding, can often explain a compound’s color, why a compound with unpaired electrons is stable, semiconductor behavior, and resonance, none of which can be explained using a localized approach.
• A molecular orbital (MO) is an allowed spatial distribution of electrons in a molecule that is associated with a particular orbital energy. Unlike an atomic orbital (AO), which is centered on a single atom, a molecular orbital extends over all the atoms in a molecule or ion. Hence the molecular orbital theory of bonding is a delocalized approach. Molecular orbitals are constructed using linear combinations of atomic orbitals (LCAOs), which are usually the mathematical sums and differences of wave functions that describe overlapping atomic orbitals. Atomic orbitals interact to form three types of molecular orbitals.
  • Orbitals or orbital lobes with the same sign interact to give increased electron probability along the plane of the internuclear axis because of constructive reinforcement of the wave functions. Consequently, electrons in such molecular orbitals help to hold the positively charged nuclei together. Such orbitals are bonding molecular orbitals, and they are always lower in energy than the parent atomic orbitals.
  • Orbitals or orbital lobes with opposite signs interact to give decreased electron probability density between the nuclei because of destructive interference of the wave functions. Consequently, electrons in such molecular orbitals are primarily located outside the internuclear region, leading to increased repulsions between the positively charged nuclei. These orbitals are called antibonding molecular orbitals, and they are always higher in energy than the parent atomic orbitals.
  • Some atomic orbitals interact only very weakly, and the resulting molecular orbitals give essentially no change in the electron probability density between the nuclei. Hence electrons in such orbitals have no effect on the bonding in a molecule or ion. These orbitals are nonbonding molecular orbitals, and they have approximately the same energy as the parent atomic orbitals.
• A completely bonding molecular orbital contains no nodes (regions of zero electron probability) perpendicular to the internuclear axis, whereas a completely antibonding molecular orbital contains at least one node perpendicular to the internuclear axis. A sigma (σ) orbital (bonding) or a sigma star (σ*) orbital (antibonding) is symmetrical about the internuclear axis. Hence all cross-sections perpendicular to that axis are circular. Both a pi (π) orbital (bonding) and a pi star (π*) orbital (antibonding) possess a nodal plane that contains the nuclei, with electron density localized on both sides of the plane.
• The energies of the molecular orbitals versus those of the parent atomic orbitals can be shown schematically in an energy-level diagram. The electron configuration of a molecule is shown by placing the correct number of electrons in the appropriate energy-level diagram, starting with the lowest-energy orbital and obeying the Pauli principle; that is, placing only two electrons with opposite spin in each orbital. From the completed energy-level diagram, we can calculate the bond order, defined as one-half the net number of bonding electrons. In bond orders, electrons in antibonding molecular orbitals cancel electrons in bonding molecular orbitals, while electrons in nonbonding orbitals have no effect and are not counted. Bond orders of 1, 2, and 3 correspond to single, double, and triple bonds, respectively. Molecules with predicted bond orders of 0 are generally less stable than the isolated atoms and do not normally exist.
• Molecular orbital energy-level diagrams for diatomic molecules can be created if the electron configuration of the parent atoms is known, following a few simple rules. Most important, the number of molecular orbitals in a molecule is the same as the number of atomic orbitals that interact. The difference between bonding and antibonding molecular orbital combinations is proportional to the overlap of the parent orbitals and decreases as the energy difference between the parent atomic orbitals increases. With such an approach, the electronic structures of virtually all commonly encountered homonuclear diatomic molecules, molecules with two identical atoms, can be understood. 
• The molecular orbital approach correctly predicts that the O₂ molecule has two unpaired electrons and hence is attracted into a magnetic field. In contrast, most substances have only paired electrons. A similar procedure can be applied to molecules with two dissimilar atoms, called heteronuclear diatomic molecules, using a molecular orbital energy-level diagram that is skewed or tilted toward the more electronegative element. Molecular orbital theory is able to describe the bonding in a molecule with an odd number of electrons such as NO and even to predict something about its chemistry.