Absolutely Small: How Quantum Theory Explains Our Everyday World - Michael D. Fayer (2010)

Chapter 13. What Holds Atoms Together: Diatomic Molecules

THE HYDROGEN MOLECULE is a diatomic molecule, that is, a molecule composed of only two atoms. Our investigation of hydrogen revealed how atoms can combine their atomic orbitals to form molecular orbitals. We need to extend our discussion of molecular orbitals to learn how more complicated molecules are built up from atoms. First, we will consider other diatomic molecules, for example, N2, O2, F2, and HF. N2, O2, and F2 (nitrogen, oxygen, and fluorine) are called homonuclear diatomics because the two atoms are the same. HF (hydrogen fluoride) is a heteronuclear diatomic because the two atoms are different. Examination of homonuclear diatomics will let us go beyond the hydrogen molecule, which is a special case. The nature of molecular orbitals for heteronuclear diatomics is an important step toward the study of polyatomic molecules that compose most of the molecular substances that surround us, from alcohol to fats.

The hydrogen molecule is the only neutral molecule that only uses electrons in its 1s orbitals to form a chemical bond. The electrons that an atom employs in bonding are called the valence electrons. In the molecules N2, O2, F2, and HF, 2s and 2p orbitals are involved in bonding. The 2s and 2p electrons are the valence electrons. N, O, and F are in the second row of the Periodic Table. Atoms in the third row of the Periodic Table, such as P, S, and Cl (phosphorus, sulfur, and chlorine), will have 3s and 3p valence electrons involved in bonding. Atoms in the third and higher rows of the Periodic Table can also employ d electrons in forming chemical bonds. Here we will focus on the second row elements. The second row elements are of great importance, and the ideas that will be developed are sufficiently general to cover the nature of chemical bonding of heavier elements.

SIGMA (σ) AND PI (π) BONDS

As shown in Figure 12.2, when two hydrogen atoms form the H2 molecule, the two hydrogen 1s orbitals combine to form the bonding molecular orbital. There is electron density along the line that connects the nuclei. A σ (sigma) bonding or antibonding molecular orbital has electron density along the line connecting the nuclei. In H2, we say that a σ bond is formed usingaσbonding MO. s orbitals always form σ bonds. There is no way to bring together two s orbitals and not have electron density along the line connecting the nuclei. This is not true for p orbitals.

Because of their shapes, there are two ways for a pair of p orbitals to come together, as is illustrated in Figure 13.1. The drawings in Figure 13.1 show the p orbitals very schematically. These are actually probability amplitude waves that have a diffuse probability distribution of finding the electron some distance from the nucleus. Here the outline gives a representation of the general shape of a p orbital. Better illustrations are shown in Figure 10.7. Recall that a p orbital has a nodal plane between the two lobes. A nodal plane is a plane in which the probability of finding the electron is zero. For a pz orbital, the nodal plane is the xy plane (see Figure 10.7). The probability of finding an electron in some region of space is frequently called the electron density. A high density means there is a high probability of finding the electron.

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FIGURE 13.1. A pair of p orbitals brought close to each other. Upper portion: the orbitals are brought together end to end. There is electron density along the line connecting the nuclei. Bottom portion: the orbitals are brought together side to side. There is no electron density along the line connecting the nuclei.

The upper portion of Figure 13.1 shows two p orbitals brought close to each other end to end. The lobes are pointing at each other. The nuclei are represented by the dots. The dashed line is the line connecting the nuclei. There is clearly electron density along the line connecting the two nuclei. The lower portion of the figure shows the two p orbitals brought close to each other side to side. The nodal plane is perpendicular to the plane of the page. The nuclei lie in the nodal plane. There is no electron density along the line connecting the nuclei. The lobes of the p orbitals have signs. One lobe is positive and the other negative. In Figure 13.1, the positive lobes were brought together next to each other in both the top and bottom portions of the figure.

SIGMA MOLECULAR ORBITALS

When the orbitals are brought sufficiently close together, they can form bonding and antibonding molecular orbits. First we will look at molecular orbitals formed from s and p atomic orbitals that give σ bonding and antibonding MOs. A σ MO has electron density along the line joining the nuclei. As discussed above, s orbitals can only form σ orbitals because of their spherical shape. p orbitals can also form σ MOs. Figure 13.2 shows σ bonding and antibonding MOs formed from both s and p orbitals. The upper part of the figure depicts the two possible ways that a pair of s orbitals can be combined. The s orbitals are waves and can have either a + or - sign associated with them. In the top portion, both s orbitals have + signs. When they combine, the s orbital waves constructively interfere to produce a σ bonding MO. In the second line of the figure, one s orbital is + and the other is -. When they combine, they destructively interfere to form an antibonding MO. The bonding MO concentrates the electron density between the nuclei, while the antibonding MO pushes the electron density to the outside, reducing the negative electron density between the nuclei. The positively charged nuclei repel each other more strongly, making this configuration antibonding.

The bottom portion of Figure 13.2 shows the results of combining two p orbitals to form σ molecular orbitals. The σ p bonding MO is generated by overlapping the + lobe of one p orbital with the + lobe of the other p orbital. There is constructive interference between the + lobes, resulting in high electron density between the atomic nuclei. There are two nodal planes that are perpendicular to the page. These are the two nodal planes that came from the two atomic p orbitals. In contrast, in the bottom line of the figure, the positive lobe of one p orbital is overlapped with the negative lobe of the other p orbital. The result is destructive interference, producing the antibonding MO. The electron density is pushed to the outside, and it is greatly reduced between the two nuclei. In addition to the two nodal planes that come from the atomic orbitals, there is a third nodal plane that arises because there is complete destructive interference between the positive and negative lobes of the two atomic p orbitals. In these bonding and antibonding MOs formed from atomic p orbitals, there is electron density along the line connecting the nuclei. Therefore, these are σ MOs.

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FIGURE 13.2. Upper portion: a pair of s orbitals are overlapped in two ways to give σ bonding (constructive interference) and σ antibonding (destructive interference) molecular orbitals. Lower portion: a pair of p orbitals are overlapped in two ways to give σ bonding (constructive interference) and σ antibonding (destructive interference) molecular orbitals. In all cases, there is electron density along the line through the nuclei.

PI MOLECULAR ORBITALS

s orbitals can only form σ MOs, but p orbitals can form σ MOs and another type of molecular orbital called π (the Greek letter pi) MOs. When p atomic orbitals come together end to end, they form σ MOs. When they come together side to side, they form π MOs, as shown in Figure 13.3. The upper portion of the figure shows two p orbitals forming a bonding molecular orbital. The positive lobe of one atomic orbital overlaps the positive lobe of the other, and the negative lobes overlap. There is constructive interference for both the positive and negative lobes. As can be seen in the figure, there is a great deal of electron density in the area between the two nuclei. However, there is no electron density directly along the line connecting the nuclei. There is a nodal plane that is perpendicular to the plane of the page because each atomic orbital has a nodal plane perpendicular to the page. The nodal plane contains the nuclei. In spite of the nodal plane, all of that electron density immediately above and below the line connecting the nuclei reduces the repulsion of the positive nuclear changes. The energy is lower than the separated atoms resulting in a π bonding MO.

The lower portion of Figure 13.3 shows the π antibonding MO. The two p atomic orbitals come together side to side, but the positive lobe of one orbital overlaps the negative lobe of the other orbital, and vice versa. The result is destructive interference between the lobes giving rise to the π antibonding MO. The antibonding MO has much less electron density between the nuclei. The result is that the energy is higher than the separated atoms, so this configuration of atomic orbitals produces an antibonding MO.

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FIGURE 13.3. Upper potion: a pair of p orbitals are overlapped side to side to give a π bonding molecular orbital (constructive interference). There is no electron density along the line connecting the nuclei. Lower portion: a pair of p orbitals are overlapped side to side to give a π antibonding molecular orbital (destructive interference). Note the signs of the lobes of the p atomic orbitals. The antibonding MO has a node between the nuclei.

BONDING IN DIATOMIC MOLECULES: THE FLUORINE MOLECULE

We are now ready to discuss bonding for diatomics with atoms other than hydrogen. Let’s start with the diatomic, F2, the fluorine molecule. We will use the same approach as we did for H2, but now there are more orbitals and more electrons involved. We imagine bringing two F atoms together and stopping at the point where the energy is lowest. This is the separation of the two F atoms when they are bonded (assuming they form a bond), as in Figure 12.5. We can draw an energy level diagram, as in Figure 12.6. We need to define the axis along which the two atoms approach each other because we have pz, px, and py orbitals. It matters whether we bring p orbitals together end to end or side to side. When the two atoms (labeled a and b) approach along the z axis (see Figure 13.4), the pz orbitals will come together end to end, while the px and py orbitals will come together side to side. Therefore, the pz atomic orbitals will form σ MOs and the px and py atomic orbitals will form π MOs.

Figure 13.5 shows the energy level diagram for two F atoms brought together along the z axis. In the diagram, the energy levels for the atomic orbitals for the two atoms (a and b) are shown on the right and left sides of the diagram. The corresponding bonding (b) and antibonding (*) MOs are shown in the center. σ MOs formed from s atomic orbitals have a subscript s. σ MOs formed from pz atomic orbitals have a subscript z, and π MOs formed from px or py atomic orbitals have a subscript x or y. The bonding MO is always lower in energy than the atomic orbitals that formed it, and the antibonding MO is always higher in energy. The three p atomic orbitals have the same energy. When quantum states have the same energy, they are said to be degenerate. In the diagram, the three p atomic orbitals are shown as three closely spaced lines even though they are degenerate. As shown, only the matching atomic orbitals of the same energy combine to make MOs. Quantum theory gives this result. States of identical energy can readily combine to make superposition states. In this case, atomic orbitals of identical energy on two different atoms can combine to make molecular orbitals. In general, only atomic states with similar energy can combine to make MOs. This will be important when we discuss heteronuclear diatomics below. For the homonuclear diatomics, the atomic orbitals are identical in energy. In the diagram, the three p orbitals on each atom, a total of six atomic orbitals, combine to form six molecular orbitals. The pz atomic orbitals produce the σ bonding and antibonding MOs, which are different in energy from the bonding and antibonding πx and πy MOs formed from the px and py atomic orbitals. However, the πx and πy bonding MOs have the same energy, and the πx and πyantibonding MOs have the same energy. The degenerate pairs of π MOs are shown as two closely spaced lines.

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FIGURE 13.4. Two atoms are brought together along the z axis. Pzorbitals will approach end to end; pxand pyorbitals will approach side to side.

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FIGURE 13.5. The energy level diagram for two fluorine atoms, a and b, brought together to form molecular orbitals. The atomic orbital energies are on the right and left sides. The bonding (b) and antibonding (*) MO energy levels are in the center. There are σ and π MOs. The three atomic p orbitals have the same energies. These are shown as three closely spaced lines. The spacings between the levels are not to scale.

Fluorine has nine electrons. So a fluorine atom will have two electrons in the 1s orbital, two electrons in the 2s orbital, and five electrons in the three 2p orbitals. Two F atoms have a total of 18 electrons. We now need to place the 18 electrons in the proper molecular orbitals in a manner equivalent to the way we placed the electrons in the atomic orbitals when we were building up the Periodic Table in Chapter 11 and the hydrogen molecule in Chapter 12. As before, we need to follow the three rules for filling the MOs. First is the Pauli Principle, which states that at most two electrons can be in an orbital and the two must have opposite spins. Opposite spins are represented by one up arrow and one down arrow. Second, electrons are placed in the lowest energy level first, consistent with the Pauli Principle. Third is Hund’s Rule, which states that electrons will not pair their spins if possible. In F2, Hund’s Rule will not change the results of placing the electrons in the proper orbitals. When we consider oxygen, O2, it will be important.

Figure 13.6 is the MO energy level diagram for F2 with the electrons placed in the proper orbitals. The atomic orbital energy levels shown in Figure 13.5 are not included. Only the MO energy levels are shown. The first two electrons go into the σ bonding MO formed from 1s orbitals. The next two electrons go in the σ antibonding MO formed from 1s orbitals. The electrons in the bonding MO are lower in energy than the atomic orbitals of the separated atoms, but the electrons in the antibonding MO are the same amount higher in energy. Therefore, these four electrons do not contribute to bonding in F2. The next four electrons go in the σ bonding and antibonding MOs formed from the 2s atomic orbitals. Again, these do not contribute to bonding because there are two electrons in the bonding MO and two electrons in the antibonding MO.

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FIGURE 13.6. The molecular orbital energy level diagram for the F2, fluorine diatomic molecule. The atomic orbital energies are not shown. Two fluorine atoms have 18 electrons. These have been placed in the orbitals following the rules discussed for atomic orbitals in Chapter 11. There is one more filled bonding MO than antibonding MO. F2has a single bond.

Now the p electrons come into play. There are a total of 10, five from each F atom. The first two go into the σ bonding MO formed from the pz atomic orbitals. Then four electrons go into the πx and πy bonding MOs. Four electrons can go into these π molecular orbitals because there are two MOs, and each can take two electrons according to the Pauli Principle. The last four electrons go into the πx and πy antibonding MOs. The four electrons in the π antibonding MOs cancel the bonding effect of the four electrons in the π bonding MOs. Therefore, the π electrons do not contribute to bonding. However, nothing cancels the two electrons in the σ bonding MO formed from the pzatomic orbitals because there are no electrons in the corresponding antibonding MO. The net result is that one pair of bonding electrons is not canceled out, so F2 has a bond order of 1 like H2. We say that F2 has a single bond, and it is a σ bond. The single covalent bond comes from having two electrons in a bonding MO. Molecular orbitals are probability amplitude waves that span the entire molecule. The atomic nuclei share these electrons.

THE NEON MOLECULE DOESN’T EXIST

We can use the same MO energy level diagram shown in Figure 13.6 to consider the hypothetical diatomic neon molecule, Ne2. Neon is immediately to the right of fluorine on the Periodic Table. Figure 13.7 shows the results of placing the 20 electrons from two neon atoms into the MO energy level diagram. The first 18 electrons go in the same as in F2. However, there are two more electrons, and they must go in the 116 antibonding MO. Therefore, for every pair of electrons in a bonding MO there is a pair of electrons in an antibonding MO. The result is no bonding. The molecule Ne2 does not exist. Other noble gas homonuclear diatomics do not exist. The example of Ne2 indicates why. A noble gas atom has a closed shell. Two noble gas atoms have just the right number of electrons to fill all of the bonding and antibonding MOs. Therefore, no net bonds are formed.

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FIGURE 13.7. The MO energy level diagram for the hypothetical molecule, Ne2. Two neon atoms have 20 electrons. There are the same number of bonding and antibonding electrons, so there is no bond. Ne2does not exist.

THE OXYGEN MOLECULE: HUND’S RULE MATTERS

One atom to the left of fluorine on the Periodic Table is oxygen. O2 is an important example that introduces a couple of new ideas. Figure 13.8 shows the MO energy level diagram filled with O2’s 16 electrons, eight from each oxygen atom. The bonding and antibonding MOs arising from the 1s and 2s orbitals are filled. These do not contribute to bonding. There are two electrons in the 118 bonding MO and none in the corresponding antibonding MO. In addition, there are four electrons in the two π bonding MOs but only two electrons in the π antibonding MOs. The result is one σ bond and one π bond. Oxygen has a bond order of 2. It has a double bond. As will be discussed further below, a double bond is stronger and shorter than a single bond.

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FIGURE 13.8. The MO energy level diagram for oxygen, O2. There is one extra pair of σ bonding electrons and one extra pair of π bonding electrons. O2has a double bond. Note the unpaired electrons in the π antibonding MOs.

O2 is the first example where Hund’s Rule comes into play and is important. Note that in filling the energy levels with electrons, the last two electrons have their spins unpaired. It is possible to have unpaired spins without violating the Pauli Principle because there are two distinct π antibonding MOs. 120 comes from the side-to-side overlap of the two px atomic orbitals (see Figure 13.3), and 121 , comes from the side-to-side overlap of the two py atomic orbitals. Hund’s Rule says that the electrons will go into orbitals unpaired if the result doesn’t violate the Pauli Principle and doesn’t require using a much higher energy orbital. The two π antibonding MOs have identical energy, so Hund’s Rule comes into play.

An electron has a magnetic moment. In some sense it acts like a tiny bar magnet. It has a north pole and a south pole. The term spin for the electron quantum number comes from classical mechanics. In classical mechanics, a charge distribution that is spinning has a magnetic moment. An electron is a probability amplitude wave. It has a delocalized charge distribution. It has a magnetic moment, but it is not literally spinning. That is a classical idea. Dirac, who gave us the concept of absolute size (see Chapter 2), also showed why an electron has a magnetic moment by combining quantum theory and Einstein’s Theory of Relativity. Although an electron is not actually spinning, the name stuck, and the magnetic moment of the electron is important.

When two electron spins are paired, the north pole of one little magnet is matched with the south pole of the other. The magnetic property of one electron cancels the magnetic property of the other. However, in O2, two of the electrons are not paired. Their spins point in the same direction. The result is that the O2 molecule is referred to as being paramagnetic. It will respond to a magnet. O2 is a gas at room temperature, but if you make it very cold, below -183° C (—297° F) it will become a liquid. Water above 100° C is a gas, but if you cool it below 100° C, it becomes a liquid. O2 is the same, but you need to make it much colder. It is possible to put liquid O2 in a test tube hanging from a string. If you bring a magnet up to the test tube, you can actually pull the test tube around with the magnet. The electron spins (little bar magnets) of O2 are lined up by the magnetic field of the external macroscopic magnet. These lined-up little bar magnets add together to make the liquid O2 magnetic, and the test tube of O2 is attracted to the external magnet. The correct prediction that O2 is paramagnetic from the analysis we have done using the MO energy level diagram is remarkable. The magnetic moment of O2 is strictly a quantum effect, and our prediction that O2 is paramagnetic comes from the application of Hund’s Rule. By following some rules, we drew some lines to represent the energy levels. Then following more rules, we put some arrows up and down on the energy level lines (we put in the electrons). Using these lines and arrows we can say that the oxygen molecule is magnetic while the fluorine and nitrogen molecules are not.

THE NITROGEN MOLECULE

Figure 13.9 shows the filled MO energy level diagram for nitrogen, N2. The nitrogen atom is immediately to the left of oxygen on the Periodic Table. Note that there is a switch in the ordering of the bonding MOs derived from the p electrons. Detailed quantum mechanical calculations give the ordering and the actual energies of the MO energy levels. For nitrogen, the ordering is different than for O2 and F2. The nitrogen atom has seven electrons, so N2 has a total of 14 electrons. As with F2 and O2, the 1s and 2s electrons do not play a role in bonding because they fill both the bonding and antibonding MOs. Filling these MOs uses up eight of the 14 electrons. The other six electrons fill the three bonding MOs, one σ MO and two π MOs. There are no electrons in the π antibonding MOs or the σ antibonding MO formed from the pz orbitals. Therefore, N2 has a bond order of 3, a triple bond. A triple bond will be stronger and shorter than a double or single bond. Note that there are no unpaired electrons in N2. N2 is not paramagnetic. At low temperature, below—196° C (—320° F), N2 is a liquid. However, you cannot move a test tube of liquid N2 with a magnet because it does not have unpaired spins.

SINGLE, DOUBLE, AND TRIPLE BONDS

In discussing bonding in Chapter 11 based on an atom’s position in the Periodic Table, we used the idea that an atom would form covalent bonds to share electrons so that it could reach the noble gas configuration. For the second row elements we have been discussing here, nitrogen, oxygen, and fluorine, the noble gas is neon. We said that F, which is one electron from the Ne configuration, would share one electron with another atom. O, which is two electrons from the Ne configuration, would share two electrons, and N, which is three electrons from Ne, would share three electrons. Here we saw that F2 has a single bond, O2 has a double bond, and N2 has a triple bond. The type of bond between atoms, that is single, double, or triple, can be indicated as F—F, O = O, and N≡N. It is through the bonds that atoms share electrons. A covalent bond is an electron pair sharing bond. A double bond shares two pairs of electrons. A triple bond shares three pairs of electrons. When a bonding MO is exactly canceled by its corresponding antibonding MO, the electrons are not really shared by the atoms. They are in molecular orbitals, but the bonding MO produces constructive interference of the probability amplitude waves and the antibonding MO generates destructive interference. These cancel each other. The electrons in this situation are referred to as lone pairs. These are pairs of electrons that do not contribute to bonding. It is the F2 single bond, a shared pair of electrons, that provides the extra electron each F atom needs to give it the Ne configuration. In O2, the double bond (sharing of two pairs of electrons) provides two extra electrons for each O atom to give them the Ne configuration. In N2, the triple bond (sharing of three pairs of electrons) provides three extra electrons for each nitrogen atom to give them the Ne configuration.

In the series F2, O2, and N2, we have a single bond, double bond, and triple bond. The electron sharing gives each atom the Ne configuration. The next element to the left of nitrogen in the Periodic Table is carbon. One might assume that carbon would make a quadruple bond to form C2 and get to the Ne configurations. C2 doesn’t exist as a stable molecule. You can see why by looking at Figure 13.9, the MO diagram for N2, and removing the two electrons with the highest energy, the 122 bonding MO. This would be the C2 electron configuration. However, it would have a double bond from the four electrons in the two π bonding MOs, not a quadruple bond. These two bonds would mean that the carbons in C2 would only have gained two extra electrons through sharing, not the four extra electrons each carbon needs to obtain the Ne configuration. Carbon needs to make four bonds to obtain the Ne configuration by forming molecules such as CH4. It can’t make four bonds by forming the C2 molecule, and C2 doesn’t exist.

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FIGURE 13.9. The MO energy level diagram for nitrogen, N2. There is one extra pair of σ bonding electrons and two extra pairs of p bonding electrons. N2has a triple bond.

F2 has a single bond, O2 has a double bond, and N2 has a triple bond. Table 13.1 illustrates how the bond order strongly affects the properties of the bond. As the bond order increases, the bond length gets shorter and the bond energy increases. The bond energy is the energy that needs to be put into the molecule to break the bond. Breaking the bond means separating the atoms to a distance so far apart that they no longer feel each other. As will be discussed in the next chapter, carbon can make single, double, and triple bonds to another carbon atom while at the same time forming bonds to other atoms, such has hydrogen. Before we can discuss molecules that are larger than diatomics, we need to go beyond homonuclear diatomic molecules and examine heteronuclear diatomics to see how molecular orbitals are formed from nonidentical atoms.

HETERONUCLEAR DIATOMICS

In homonuclear diatomics, the MOs are formed from atomic orbitals with identical energies. In a heteronuclear diatomic, for example hydrogen fluoride (HF), the two atoms are different. Because the atoms are different, the atomic orbitals’ energies of one atom will not match those of the other. For HF, the hydrogen atom has a single electron in the 1s orbital. F has nine electrons in the 1s, 2s, and 2p orbitals. Both F2 and H2 have single bonds. Looking at Figure 13.6, the single bond in F2 isaσbond due to the bonding MO, 124 . This bonding MO was formed from two 2pz atomic orbitals, one on each F atom. H2 has a single σ bond due to the bonding MO formed from two 1s orbitals (see Figure 12.7). To make HF, the question is, which orbital on F will combine with the 1s orbital on H to give MOs involved in bonding? Quantum theory calculations show that states (atomic orbitals) that are close in energy can combine to produce electron sharing MOs. Atomic orbitals with significantly different energies form MOs that are essentially equal to the individual atomic orbitals, and they do not contribute to bonding.

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TABLE 13.1. The Effect of Bond Order on Bond Properties.

The hydrogen 1s orbital has an energy of—2.2 × 10-18 J. (Recall that the negative sign means that the electron is bound.) The energy of the fluorine 1s orbital (measured in the F2 molecule) is —1.1 × 10-16 J. So the 1s orbital of F is about 50 times lower in energy than the 1s orbital of H. This is a tremendous difference in energy, and the H 1s will not form a MO with the F 1s. In contrast, an F 2p orbital has an energy of—2.8 × 10-18 J. This energy is about 25% below the H 1s energy, which is close in energy. So an F 2p and the H 1s orbital energies are similar enough that they will form true MOs.

There are three F 2p orbitals, 2pz, 2px, and 2py. To decide which of these will interact with the H 1s, we need to define how we bring the atoms together. Let’s say we bring the H atom and the F atom toward each other along the z axis as shown at the top of Figure 13.10. The two circles reflect the correct relative sizes of the H and F atoms. Fluorine’s 2py orbital has its lobes perpendicular to the z axis, as shown in the middle portion of the figure. (The orbitals are not drawn to scale.) When the 2py overlaps the hydrogen 1s orbital, the positive 2py lobe will constructively interfere with the 1s orbital, but the negative lobe will destructively interfere. The net result is that there is no net constructive or destructive interference. The same is true for the 2px orbital. The 2py and 2px orbitals will not form bonding or antibonding MOs in the HF molecule. The lower portion of the figure shows the 2pz orbital’s positive lobe overlapping with the 1s orbital, which is also taken to be positive. This overlap will result in constructive interference of the probability amplitude waves, and can produce a bonding MO. Since there is electron density along the line connecting the nuclei, the bond will be a σ bond. If the F negative 2pz lobe overlaps with the H positive 1s lobe, there will be destructive interference that gives rise to an antibonding MO.

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FIGURE 13.10. Top: H and F atoms brought together along the z axis. Circles show the relative size of the atoms. Middle: overlap of the H 1s and F 2pyorbitals. There is equal constructive (+) and destructive (-) interference in the overlap region. No MO formation. Bottom: overlap of the H 1s and the F 2pzorbitals. There is constructive interference in the overlap region.

As discussed above, the fluorine 1s orbital energy is so much lower than the hydrogen 1s, that the fluorine 1s electrons do not participate in bonding. The outermost electrons in an atom, that is, the last shell that is filled, are the ones that contribute to bonding. These are the valence electrons. For elements in the second row of the Periodic Table, like fluorine, the 2s and 2p are the valence electrons. In making molecular orbital energy level diagrams, usually only the orbitals associated with the valence electrons are shown because these are the orbitals that can be involved in bonding. Figure 13.11 shows the MO energy level diagram for HF with the F 1s orbital and electrons left out. The energy level spacings are not to scale. As discussed in connection with Figure 13.10, the H 1s atomic orbital will combine with the F 2pz atomic orbital to form bonding (σb) and antibonding (σ*) MOs. This is indicated in the diagram by the dashed lines. This diagram is similar to the energy level diagram in Figure 13.5 except that now the atomic orbitals that form the MOs do not have identical energies.

Fluorine has nine electrons. Two are in the 1s orbital, which leaves seven. Hydrogen has one electron. So there are a total of eight valence electrons to place in the MO energy levels. The first two go into the level labeled 2s. The fluorine 2s orbital is much lower in energy than the hydrogen 1s, and these electrons do not participate in bonding. Therefore, the 2s molecular orbital is essentially the same as the fluorine 2s atomic orbital. The two electrons in the 2s orbital are a lone pair. The next two electrons go into the σb bonding MO. The final four electrons go into the 2px and 2py orbitals. Again, these are basically atomic orbitals of fluorine. They do not play a role in the bonding. These four electrons comprise two more lone pairs. While the lone pairs do not play a role in bonding, in polyatomic molecules they influence the shapes of molecules, which will be discussed in Chapter 14. The net result is that there are two electrons in the bonding MO and none in an antibonding MO. Therefore, HF has a single bond. Hydrogen and fluorine share a pair of electrons in the bonding MO. For H, the sharing provides the additional electron necessary to achieve the helium rare gas configuration. For F, the sharing provides the extra electron needed to obtain the neon rare gas configuration.

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FIGURE 13.11. Molecular orbital energy level diagram for HF. The atomic orbitals of the valence electrons are shown at the right and left. The F 2pzatomic orbital combines with the H 1s atomic orbital to give a bonding (σb) and antibonding (σ*) MO. σbis filled with one H electron and one F electron. σ* is unfilled. The net is one bond. The other F electrons do not participate in the bonding. They are lone pairs of electrons.

VISUAL MODELS OF MOLECULES

HF like F2, O2, and N2 are diatomics and therefore linear molecules. In the next chapter, we will talk about molecules with more complex shapes. The structure of molecules can be shown in a number of ways. HF can be written as H-F to indicate there is a single bond. In more complicated molecules, this type of representation can show which atoms are bonded to each other and the bond order. However, it cannot display the three-dimensional geometry or give a feel for what the molecule actually looks like. Now to say a molecule looks like something is fundamentally incorrect. HF has two nuclei surrounded by the probability amplitude waves that are the electrons. Nonetheless, there are representations that are useful in discussing the nature of molecules. Figure 13.12 displays two such representations of HF. The top portion is a ball-and-stick molecular model. It shows the connection between the atoms and their relative size. H is light in color and F is dark in color. The bond between the atoms is much too long. The bottom portion of the figure shows a space-filling model. Most of the electron density is inside of the overlapping spheres. The sizes and the internuclear separations are correct. The colors and the sharp line between the atoms are to aid the eye. There is no actual separation of the electrons between the atoms.

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FIGURE 13.12. Representations of the HF molecule. H: light; F: dark. Top: a ball-and-stick version that shows how the atoms are bonded and the relative sizes of the atoms. Bottom: a space-filling version that is more realistic.

The material presented in this chapter and in the next chapter is necessary to understand bonding in polyatomic molecules. In the next chapter, we need to extend the ideas presented here to molecules with more than two atoms. Polyatomic molecules have shapes, and to understand the shapes, we will introduce new ideas, hybrid atomic orbitals. The material developed in Chapters 13 and 14 will be used in subsequent chapters to examine a wide variety of problems such as what are trans fats and why are they different from other fats.