9 Molecular Orbitals in Chemical Bonding

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9 Molecular Orbitals in Chemical Bonding OUTLINE 9-1 Molecular Orbitals 9-2 Molecular Orbital Energy Level Diagrams 9-3 Bond Order and Bond Stability 9-4 Homonuclear…
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9 Molecular Orbitals in Chemical Bonding OUTLINE 9-1 Molecular Orbitals 9-2 Molecular Orbital Energy Level Diagrams 9-3 Bond Order and Bond Stability 9-4 Homonuclear Diatomic Molecules 9-5 Heteronuclear Diatomic Molecules 9-6 Delocalization and the Shapes of Molecular Orbitals OBJECTIVES After you have finished studying this chapter, you should be able to ã Describe the basic concepts of molecular orbital theory ã Relate the shapes and overlap of atomic orbitals to the shapes and energies of the resulting molecular orbitals ã Distinguish among bonding, antibonding, and nonbonding orbitals ã Apply the Aufbau Principle to find molecular orbital descriptions for homonuclear diatomic molecules and ions ã Apply the Aufbau Principle to find molecular orbital descriptions for heteronuclear diatomic molecules and ions with small ⌬(EN) values ã Find the bond order in diatomic molecules and ions ã Relate bond order to bond stability ã Use the MO concept of delocalization for molecules in which valence bond theory would postulate resonance A computer representation of one of the ␲ molecular orbitals of benzene. W W e have described bonding and molecular geometry in terms of valence bond theory. In valence bond theory, we postulate that bonds result from the sharing of electrons in overlapping orbitals of different atoms. These orbitals may be pure atomic orbitals or hybridized atomic orbitals of individual atoms. We describe electrons in overlapping orbitals of different atoms as being localized in the bonds between the two atoms involved. We then use hybridization to help account for the geometry of a molecule. In molecular orbital theory, we postulate that the combination of atomic orbitals on different atoms forms molecular orbitals (MOs), so that electrons in them belong to the molecule as a whole. Valence bond and molecular orbital theories are alternative descriptions of chemical bonding. They have strengths and weaknesses, so they are complementary. Valence bond In some polyatomic molecules, a molecular orbital may extend over only a fraction of the molecule. 354 CHAPTER 9: Molecular Orbitals in Chemical Bonding Polyatomic ions such as CO32⫺, SO42⫺, and NH4⫹ can be described by the molecular orbital approach. theory is descriptively attractive, and it lends itself well to visualization. Molecular orbital (MO) theory gives better descriptions of electron cloud distributions, bond energies, and magnetic properties, but its results are not as easy to visualize. The valence bond picture of bonding in the O2 molecule involves a double bond. O O This shows no unpaired electrons, so it predicts that O2 is diamagnetic. Experiments show, however, that O2 is paramagnetic; therefore, it has unpaired electrons. Thus, the valence bond structure is inconsistent with experiment and cannot be accepted as a description of the bonding. Molecular orbital theory accounts for the fact that O2 has two unpaired electrons. This ability of MO theory to explain the paramagnetism of O2 gave it credibility as a major theory of bonding. We shall develop some of the ideas of MO theory and apply them to some molecules and polyatomic ions. 9-1 MOLECULAR ORBITALS An early triumph of molecular orbital theory was its ability to account for the observed paramagnetism of oxygen, O2. According to earlier theories, O2 was expected to be diamagnetic, that is, to have only paired electrons. See the Saunders Interactive General Chemistry CD-ROM, Screen 10.9, Molecular Orbital Theory. We learned in Chapter 5 that each solution to the Schrödinger equation, called a wave function, represents an atomic orbital. The mathematical pictures of hybrid orbitals in valence bond theory can be generated by combining the wave functions that describe two or more atomic orbitals on a single atom. Similarly, combining wave functions that describe atomic orbitals on separate atoms generates mathematical descriptions of molecular orbitals. An orbital has physical meaning only when we square its wave function to describe the electron density. Thus, the overall sign on the wave function that describes an atomic orbital is not important, but when we combine two orbitals, the signs of the wave functions are important. When waves are combined, they may interact either constructively or destructively (Figure 9-1). Likewise, when two atomic orbitals overlap, they can be in phase (added) or out of phase (subtracted). When they overlap in phase, constructive interaction occurs in the region between the nuclei, and a bonding orbital is produced. The energy of the bonding orbital is always lower (more stable) than the energies of the combining orbitals. When they overlap out of phase, destructive interaction reduces the probability of finding electrons in the region between the nuclei, and an antibonding orbital is produced. This is higher in energy (less stable) than the original atomic orbitals. The overlap of two atomic orbitals always produces two MOs: one bonding and one antibonding. We can illustrate this basic principle by considering the combination of the 1s atomic orbitals on two different atoms (Figure 9-2). When these orbitals are occupied by electrons, the shapes of the orbitals are plots of electron density. These plots show the regions in molecules where the probabilities of finding electrons are the greatest. In the bonding orbital, the two 1s orbitals have reinforced each other in the region between the two nuclei by in-phase overlap, or addition of their electron waves. In the antibonding orbital, they have canceled each other in this region by out-of-phase overlap, or subtraction of their electron waves. We designate both molecular orbitals as sigma (␴) molecular orbitals (which indicates that they are cylindrically symmetrical about the internuclear axis). We indicate with subscripts the atomic orbitals that have been combined. The star (★) denotes an antibonding orbital. Thus, two 1s orbitals produce a ␴1s (read “sigma-1s”) bonding orbital and a ␴★ 1s (read “sigma-1s-star”) antibonding orbital. The right-hand side of Figure 9-2 shows the relative energy levels of these orbitals. All sigma 9-1 Molecular Orbitals ⫹ ⫹ ⫺ Figure 9-1 An illustration of constructive and destructive interference of waves. (a) If the two identical waves shown at the left are added, they interfere constructively to produce the more intense wave at the right. (b) Conversely, if they are subtracted, it is as if the phases (signs) of one wave were reversed and added to the first wave. This causes destructive interference, resulting in the wave at the right with zero amplitude; that is, a straight line. ⫺ ⫹ ⫺ ⫹ ⫹ ⫺ ⫺ (a) In-phase overlap (add) 355 (b) Out-of-phase overlap (subtract) antibonding orbitals have nodal planes bisecting the internuclear axis. A node, or nodal plane, is a region in which the probability of finding electrons is zero. Another way of viewing the relative stabilities of these orbitals follows. In a bonding molecular orbital, the electron density is high between the two atoms, where it stabilizes the arrangement by exerting a strong attraction for both nuclei. By contrast, an antibonding orbital has a node (a region of zero electron density) between the nuclei; this allows for a strong net repulsion between the nuclei, which makes the arrangement less stable. Electrons are more stable (have lower energy) in bonding molecular orbitals than in the individual atoms. Placing electrons in antibonding orbitals, on the other hand, requires an increase in their energy, which makes them less stable than in the individual atoms. Nodal plane and 1s 1s In-ph overl ase ap (a dd) Atomic orbital 1s ★ σ1s (antibonding) σ1s (bonding) Atomic orbitals ★ σ1s (antibonding) Energy ase ct) a of-ph Out- ap (subtr l over Molecular orbitals Molecular orbitals Figure 9-2 Molecular orbital (MO) diagram for the combination of the 1s atomic orbitals on two identical atoms (at the left) to form two MOs. One is a bonding orbital, ␴1s (blue), resulting from addition of the wave functions of the 1s orbitals. The other is an antibonding orbital, ␴★ 1s (red), at higher energy resulting from subtraction of the waves that describe the combining 1s orbitals. In all ␴-type MOs, the electron density is symmetrical about an imaginary line connecting the two nuclei. The terms “subtraction of waves,” “out of phase,” and “destructive interference in the region between the nuclei” all refer to the formation of an antibonding MO. Nuclei are represented by dots. σ1s (bonding) Atomic orbital 1s CHAPTER 9: Molecular Orbitals in Chemical Bonding ase ct) a of-ph Out- ap (subtr l over and 2px Figure 9-3 Production of ␴2p and x ␴★ 2px molecular orbitals by overlap of 2px orbitals on two atoms. How we name the axes is arbitrary. We designate the internuclear axis as the x direction. This would involve rotating Figures 9-2, 9-3, and 9-4 by 90° so that the internuclear axes are perpendicular to the plane of the pages. 2px ★ σ2p (antibonding) x In-ph overl ase ap (a dd) Energy 356 σ2p (bonding) x Atomic orbitals (head-on overlap) Molecular orbitals For any two sets of p orbitals on two different atoms, corresponding orbitals such as px orbitals can overlap head-on. This gives ␴p and ␴★ p orbitals, as shown in Figure 9-3 for the head-on overlap of 2px orbitals on the two atoms. If the remaining p orbitals overlap (py with py and pz with pz ), they must do so sideways, or side-on, forming pi (␲) molecular orbitals. Depending on whether all p orbitals overlap, there can be as many as two ␲p and two ␲★ p orbitals. Figure 9-4 illustrates the overlap of two corresponding 2p orbitals on two atoms to form ␲2p and ␲★ 2p molecular orbitals. There is a nodal plane along the internuclear axis for all pi molecular orbitals. If one views a sigma molecular orbital along the internuclear axis, it appears to be symmetrical around the axis like a pure s atomic orbital. A similar cross-sectional view of a pi molecular orbital looks like a pure p atomic orbital, with a node along the internuclear axis. If we had chosen the z axis as the axis of head-on overlap of the 2p orbitals in Figure 9-3, side-on overlap of the 2px –2px and 2py –2py orbitals would form the ␲-type molecular orbitals. and ␲★ 2p Figure 9-4 The ␲2p and molecular orbitals from overlap of one pair of 2p atomic orbitals (for instance, 2py orbitals). There can be an identical pair of molecular orbitals at right angles to these, formed by another pair of p orbitals on the same two atoms (in this case, 2pz orbitals). se t) pha rac of- (subt t Ou rlap ove 2py or 2pz 2py or 2pz Inp ove hase rlap (ad d) ★ ★ π2p or π2p (antibonding) y z π2p or π2p (bonding) y Atomic orbitals (side-on overlap) z Molecular orbitals Energy The number of molecular orbitals (MOs) formed is equal to the number of atomic orbitals that are combined. When two atomic orbitals are combined, one of the resulting MOs is at a lower energy than the original atomic orbitals; this is a bonding orbital. The other MO is at a higher energy than the original atomic orbitals; this is an antibonding orbital. 9-2 Molecular Orbital Energy Level Diagrams 357 9-2 MOLECULAR ORBITAL ENERGY LEVEL DIAGRAMS Figure 9-5 shows molecular orbital energy level diagrams for homonuclear diatomic molecules of elements in the first and second periods. Each diagram is an extension of the right-hand diagram in Figure 9-2, to which we have added the molecular orbitals formed from 2s and 2p atomic orbitals. For the diatomic species shown in Figure 9-5a, the two ␲2p orbitals are lower in energy than the ␴2p orbital. Molecular orbital calculations indicate, however, that for O2, F2, and hypothetical Ne2 molecules, the ␴2p orbital is lower in energy than the ␲2p orbitals (see Figure 9-5b). Atomic orbitals Molecular orbitals Atomic orbitals Atomic orbitals Molecular orbitals 夹 σ2p 夹 夹 π2p π2p y z 夹 夹 π2p π2p y z π2p y 2p π2p π2p y π2p σ2s夹 夹 σ2s 2s 2s 2s σ2s σ2s 夹 σ1s σ1s夹 1s 1s 1s σ1s H2 through N2 z σ2p z 2s Atomic orbitals 2p 2p σ2p Energy (not drawn to scale) Spectroscopic data support these orders. 夹 σ2p 2p (a) “Homonuclear” means consisting only of atoms of the same element. “Diatomic” means consisting of two atoms. 1s σ1s (b) Figure 9-5 Energy level diagrams for first- and second-period homonuclear diatomic molecules and ions (not drawn to scale). The solid lines represent the relative energies of the indicated atomic and molecular orbitals. (a) The diagram for H2, He2, Li2, Be2, B2, C2, and N2 molecules and their ions. (b) The diagram for O2, F2 and Ne2 molecules and their ions. O2 through Ne2 358 CHAPTER 9: Molecular Orbitals in Chemical Bonding Diagrams such as these are used to describe the bonding in a molecule in MO terms. Electrons occupy MOs according to the same rules developed for atomic orbitals; they follow the Aufbau Principle, the Pauli Exclusion Principle, and Hund’s Rule. (See Section 5-17.) To obtain the molecular orbital description of the bonding in a molecule or ion, follow these steps: See the Saunders Interactive General Chemistry CD-ROM, Screen 10.10, Molecular Electron Configurations. 1. Draw (or select) the appropriate molecular orbital energy level diagram. 2. Determine the total number of electrons in the molecule. Note that in applying MO theory, we will account for all electrons. This includes both the inner-shell electrons and the valence electrons. 3. Add these electrons to the energy level diagram, putting each electron into the lowest energy level available. a. A maximum of two electrons can occupy any given molecular orbital, and then only if they have opposite spin (Pauli Exclusion Principle). b. Electrons must occupy all the orbitals of the same energy singly before pairing begins. These unpaired electrons must have parallel spins (Hund’s Rule). 9-3 BOND ORDER AND BOND STABILITY Now we need a way to judge the stability of a molecule once its energy level diagram has been filled with the appropriate number of electrons. This criterion is the bond order (bo): Electrons in bonding orbitals are often called bonding electrons, and electrons in antibonding orbitals are called antibonding electrons. (number of bonding electrons) ⫺ (number of antibonding electrons) Bond order ⫽ ᎏᎏᎏᎏᎏᎏᎏᎏ 2 Usually the bond order corresponds to the number of bonds described by the valence bond theory. Fractional bond orders exist in species that contain an odd number of electrons, such as the nitrogen oxide molecule, NO (15 electrons) and the superoxide ion, O2⫺ (17 electrons). A bond order equal to zero means that the molecule has equal numbers of electrons in bonding MOs (more stable than in separate atoms) and in antibonding MOs (less stable than in separate atoms). Such a molecule would be no more stable than separate atoms, so it would not exist. A bond order greater than zero means that more electrons occupy bonding MOs (stabilizing) than antibonding MOs (destabilizing). Such a molecule would be more stable than the separate atoms, and we predict that its existence is possible. But such a molecule could be quite reactive. The greater the bond order of a diatomic molecule or ion, the more stable we predict it to be. Likewise, for a bond between two given atoms, the greater the bond order, the shorter is the bond length and the greater is the bond energy. The bond energy is the amount of energy necessary to break a mole of bonds (Section 15-9); therefore, bond energy is a measure of bond strength. 9-4 Homonuclear Diatomic Molecules 359 Problem-Solving Tip: Working with MO Theory MO theory is often the best model to predict the bond order, bond stability, or magnetic properties of a molecule or ion. The procedure is as follows: 1. Draw (or select) the appropriate MO energy level diagram. 2. Count the total number of electrons in the molecule or ion. 3. Follow the Pauli Exclusion Principle and Hund’s Rule to add the electrons to the MO diagram. bonding e’s ⫺ antibonding e’s 4. Calculate the bond order: Bond order ⫽ ᎏᎏᎏᎏ . 2 冢 冣 5. Use the bond order to evaluate stability. 6. Look for the presence of unpaired electrons to determine if a species is paramagnetic. See the Saunders Interactive General Chemistry CD-ROM, Screen 10.11, Homonuclear Diatomic Molecules. 9-4 HOMONUCLEAR DIATOMIC MOLECULES The electron distributions for the homonuclear diatomic molecules of the first and second periods are shown in Table 9-1 together with their bond orders, bond lengths, and bond energies. The Hydrogen Molecule, H2 The overlap of the 1s orbitals of two hydrogen atoms produces ␴1s and ␴★ 1s molecular orbitals. The two electrons of the molecule occupy the lower energy ␴1s orbital (Figure 9-6a). Because the two electrons in an H2 molecule are in a bonding orbital, the bond order is one. We conclude that the H2 molecule would be stable, and we know it is. The energy associated with two electrons in the H2 molecule is lower than that associated with the same two electrons in separate 1s atomic orbitals. The lower the energy of a system, the more stable it is. As the energy of a system decreases, its stability increases. ls ␴ls ls Energy Energy ␴ls ls ␴ls H (a) H2 2⫺0 H2 bond order ⫽ ᎏ ⫽ 1 2 ls ␴ls H He (b) Figure 9-6 Molecular orbital diagrams for (a) H2 and (b) He2. He2 He 360 CHAPTER 9: Molecular Orbitals in Chemical Bonding The Helium Molecule (Hypothetical), He2 2⫺2 He2 bond order ⫽ ᎏ ⫽ 0 2 6⫺4 B2 bond order ⫽ ᎏ ⫽ 1 2 The energy level diagram for He2 is similar to that for H2 except that it has two more electrons. These occupy the antibonding ␴★ 1s orbital (see Figures 9-5a and 9-6b and Table 9-1), giving He2 a bond order of zero. That is, the two electrons in the bonding orbital of He2 would be more stable than in the separate atoms. But the two electrons in the antibonding orbital would be less stable than in the separate atoms. These effects cancel, so the molecule would be no more stable than the separate atoms. The bond order is zero, and the molecule would not exist. In fact, He2 is unknown. The Boron Molecule, B2 The boron atom has the configuration 1s22s22p1. Here p electrons participate in the bonding. Figure 9-5a and Table 9-1 show that the ␲py and ␲pz molecular orbitals are lower in energy than the ␴2p for B2. Thus, the electron configuration is ␴★2 1s ␴1s2 Orbitals of equal energy are called degenerate orbitals. Hund’s Rule for filling degenerate orbitals was discussed in Section 5-17. ␴★2 2s ␴2s2 ␲2p 1 ␲2p 1 y z The unpaired electrons are consistent with the observed paramagnetism of B2. Here we illustrate Hund’s Rule in molecular orbital theory. The ␲2py and ␲2p orbitals are equal in z energy and contain a total of two electrons. Accordingly, one electron occupies each orbital. The bond order is one. Experiments verify that B2 molecules exist in the vapor state. The Nitrogen Molecule, N2 10 ⫺ 4 N2 bond order ⫽ ᎏ ⫽ 3 2 In the valence bond representation, N2 is shown as SNmNS, with a triple bond. Experimental thermodynamic data show that the N2 molecule is stable, is diamagnetic, and has a very high bond energy, 946 kJ/mol. This is consistent with molecular orbital theory. Each nitrogen atom has seven electrons, so the diamagnetic N2 molecule has 14 electrons. ␴1s2 ␴★2 1s ␴2s2 ␴★2 2s ␲2py2 ␲2p 2 z ␴2p2 Six more electrons occur in bonding orbitals than in antibonding orbitals, so the bond order is three. We see (Table 9-1) that N2 has a very short bond length, only 1.09 Å, the shortest of any diatomic species except H2. The Oxygen Molecule, O2 Among the homonuclear diatomic molecules, only N2 and the very small H2 have shorter bond lengths than O2, 1.21 Å. Recall that VB theory predicts that O2 is diamagnetic. Experiments show, however, that it is paramagnetic, with two unpaired electrons. MO theory predicts a structure consistent with this observation. For O2, the ␴2p orbital is lower in energy than the ␲2py and ␲2p orbitals. Each oxygen atom has eight electrons, so the z O2 molecule has 16 electrons. ␴1s2 10 ⫺ 6 O2 bond or
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