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

Chapter 12. The Hydrogen Molecule and the Covalent Bond

ONE OF THE GREAT TRIUMPHS of quantum mechanics is the theoretical explanation of the covalent bond. Two types of interactions hold atoms together, covalent bonds and ionic bonds. Ionic bonds are the type that occurs in a sodium chloride (NaCl) crystal. We know from Chapter 11 and our discussion of the Periodic Table that this salt crystal is composed of sodium cations, Na+1, and chloride anions, Cl-1. The ions in the crystal are held together by electrostatic interactions. Opposite charges attract. There are some complications because like charges repel, but it is possible to show that the attractions of the oppositely charged ions overcome the repulsions of the like charged ions. Such electrostatic interactions can be explained quite well with classical mechanics, although quantum theory is still needed to explain many properties in detail.

In contrast to ionic solids that are held together by electrostatic interactions, classical mechanics cannot explain the covalent bond. We saw in Chapter 11 that a hydrogen atom will tend to form one covalent bond with another atom to share one electron. This sharing brings the H atom to the helium closed shell configuration. But what is a covalent bond? Why do H atoms share electrons to form the H2 molecule, but helium atoms do not share electrons to form the He2 molecule? We will first investigate the nature of the covalent bond for the simplest molecule, H2, and then expand the discussion of covalent bonding for more complicated molecules in subsequent chapters. By the end of this chapter it will be clear why H2 exists and He2 does not exist.


Two hydrogen atoms, call them a and b, do not interact with each other when they are very far apart. When they are separated by a large distance, the electron on hydrogen atom a will only feel the Coulomb attraction to the proton in hydrogen atom a. The electron on hydrogen atom b will only interact with the proton on hydrogen atom b. We know how to describe these separated hydrogen atoms. Let’s say both are in their lowest energy state, the 1s state. The electron is describe by the 1s wavefunction, which is an atomic orbital. It describes the probability amplitude of finding the electron in a given region of space. The square of the wavefunction gives the probability of finding the electron. The 1s state of the hydrogen atom was discussed in some detail in Chapter 10 (see Figures 10.2- 10.4).


Now consider what happens when we start bringing the two hydrogen atoms closer and closer together. When they get close together but not too close they start to feel each other. This distance will be made quantitative below. The electron on hydrogen atom a starts to have attraction for the proton of hydrogen atom b, and it has repulsion for the electron on hydrogen atom b. In the same manner, the electron on hydrogen atom b is attracted to the proton of hydrogen atom a and is repelled by the electron of hydrogen atom a. In addition, the positively charged protons of hydrogen atom a and b repel each other because like charges repel.

It is possible to solve the Schrödinger Equation for this problem. It can’t be done exactly, but it can be done very accurately. What does the solution of the Schrödinger Equation give you? It gives the energies of the system, and it gives the wavefunctions. When we solved the particle in a box problem, we obtained the wavefunction for a single particle in a hypothetical one-dimensional box with infinite walls. When we solve the Schrödinger Equation for the hydrogen atom or other atoms, we get the energy levels and the atomic wavefunctions, the atomic orbitals. When we solve a molecular problem, we get the quantized energies of the molecular energy levels and the molecular wavefunctions. The molecular wavefunctions are usually called molecular orbitals. So for atoms we get atomic orbitals that describe the electron’s probability distribution about the atomic nucleus. This is a probability amplitude wave. The molecular orbital describes the probability distribution for the electrons in the molecule relative to the atomic nuclei of the atoms that make up the molecule. For the hydrogen molecule there are two electrons and two atomic nuclei, the two protons.


A very useful way to understand bonding of hydrogen atoms as they come together to form the hydrogen molecule uses a concept called the Born-Oppenheimer Approximation. As discussed in Chapter 5, Born won the Nobel Prize in 1954 for the Born probability interpretation of the wavefunction. Oppenheimer made major contributions to the field of physics. He is probably best known as the physicist who led the Manhattan Project during World War II that developed and tested the first atomic bomb. The Born-Oppenheimer Approximation works by placing the two hydrogen atom nuclei (the two protons) a fixed distance apart. Start with a distance that you suspect to be so far apart that the hydrogen atoms don’t feel each other. A quantum mechanical calculation of the energy is performed. If the atoms are far apart, then the energy will be just twice the energy of the 1s state of the hydrogen atom because you have two hydrogen atoms. Then you make the distance a little closer and do the calculation again. Then you make the distance even closer and do the calculation yet again. When the distance between the nuclei in the calculation becomes small enough, the atoms feel each other. If a chemical bond is going to form, that is, if the two hydrogen atoms are going to combine into a hydrogen molecule, then the energy must decrease. To form a bond, the energy of the molecule must be lower than the energy of the atoms when they are separated far apart.

Figure 12.1 is a plot of the energy of two hydrogen atoms as they are brought closer and closer together. As mentioned, when the two H atoms are very far apart, they don’t interact with each other. Each is just a hydrogen atom, and each has the energy of the hydrogen 1s atomic orbital. We are going to take this as the zero of energy. The hydrogen atom itself has a negative energy as described in Chapter 10. That energy represents the binding of the electron to the proton (nucleus). Here we are interested in the change in energy when the two H atoms interact. We want to understand the energy associated with chemical bond, so the zero of energy is the energy when there is no chemical bond. In Figure 12.1, the zero of energy is indicated in the diagram by the dashed line. This is the energy when the atoms are completely separated. The horizontal axis is the separation, r, of the two H atoms. As the H atoms are brought together the energy begins to decrease, then decreases more rapidly. The energy reaches a minimum at a separation, r0 (see Figure 12.1). As the atoms are brought even closer together the energy increases very rapidly, that is, when the atoms are too close, they repel each other. Because the energy goes down when the atoms are brought together, a chemical bond is formed between the two atoms.


FIGURE 12.1. A plot of the energy of two hydrogen atoms as they are brought close together. When the H atoms are very far apart, the energy of the system is the sum of the 1s orbital energies of two H atoms. This is taken to be the zero of energy, the dashed line. As the atoms come together, the energy decreases to a minimum. If they are brought even closer together, the energy increases rapidly.

The Bond Length Is the Distance That Gives the Lowest Energy

At the distance r0, the energy is the minimum. r0 is the separation of the H atoms that is the most stable (lowest energy). This distance is called the bond length. It is the separation between the two protons in a stable hydrogen molecule. The difference between the bottom of the “potential energy well” and the zero of energy is the dissociation energy. The dissociation energy is the amount of energy that would have to be put into a hydrogen molecule to break the chemical bond, which would produce two hydrogen atoms. The potential energy well for the hydrogen molecule is equivalent to a hole in the ground that a ball rolls into. The top of the hole is the zero of energy. The ball falls to the bottom of the hole to minimize the gravitational potential. Gravity pulls the ball down. To lift the ball out of the gravitational well requires energy, as the gravitational pull on the ball must be overcome. The deeper the hole, the more energy it will take to lift the ball out of the hole. With molecules, the deeper the potential energy well, the more energy it will take to get out of the well, that is, to break the chemical bond.

The distance scale along the r axis of Figure 12.1 is not shown. But it is interesting to discuss two distances. At what distance do the hydrogen atoms first really begin to feel each other? Figure 10.3 shows that the 1s orbital probability amplitude wave for a hydrogen atom becomes very small at a distance from the nucleus of approxi mately 3 Å (3 × 10-10 m). So one might expect that when two hydrogen atoms are a little closer than 6 Å they would start to interact. In Figure 12.1, the point where the potential energy curve (solid curve) just stops touching the zero of energy line (dashed line) is approximately 6 Å. So the atoms begin to feel each other when the atomic wavefunctions just begin to overlap significantly. The point r0 is the location of the minimum of the potential energy curve. It is the length of the bond. Experiments and calculations have deter mined this distance to be 0.74 Å. If the atoms are further apart or closer together than this distance, the energy is higher. The potential energy curve shown in Figure 12.1 is from an actual quantum mechanical calculation. It is a relatively low-level calculation that can be done completely with pencil and paper; no computers are necessary. This approximate calculation gives r0 = 0.80 Å, so it is a little off. If you want to see the monumental amount of math that goes into even this relatively simple calculation, see Michael D. Fayer, Elements of Quantum Mechanics, Chapter 17 (New York: Oxford University Press, 2001). Much more complicated quantum theoretical calculations of the H2 molecule can produce all of the properties of the hydrogen molecule with accuracy better than can be obtained through experimental measurements. Such accurate calculations are possible because the hydrogen molecule is so simple. For large molecules, experiments still beat calculations.


Figure 12.1 shows that a chemical bond will be formed between two hydrogen atoms to yield the H2 molecule, but it doesn’t show why. As mentioned in Chapter 11, a covalent bond involves the sharing of electrons by atoms. When a molecule is formed, the atomic orbitals combine to form molecular orbitals. For the hydrogen molecule, we start with two hydrogen atoms, Ha and Hb. Each has a single electron in an atomic 1s orbital. We will call these orbitals 1sa and 1sb. These two atomic orbitals are represented in the top portion of Figure 12.2 as circles. This is a simple schematic of the delocalized electron probability amplitude wave shown in Figures 10.2 through 10.4. The lower portion of the figure shows what happens when the two atoms are brought together so they are separated by the bond length, r0 (see Figure 12.1). Wavefunctions have signs. In this case, both signs are positive. The probability amplitude waves add together to form a molecular orbital. We discussed waves in some detail in Chapters 3 and 5. In Chapter 3, we saw that waves could be combined to have constructive or destructive interference. In Chapter 5, the interference of photons was explained in terms of the Born interpretation of the wavefunction as a probability amplitude wave. Here, the two atomic orbital electron probability amplitude waves combine constructively to form a molecular orbital. The molecular orbital is a probability amplitude wave. The absolute value squared of the wave gives the probability of finding the electrons in some region of space.


FIGURE 12.2. The upper portion is a schematic of two hydrogen atom 1s orbitals. These are actually delocalized electron probability amplitude waves, represented here simply as circles. The lower proportion shows what happens when the two atoms are brought together to form the H2molecule. The two atomic orbitals combine to make a molecular orbital.

Figure 12.3 shows a one-dimensional plot of the probability distribution (square of the wavefunction) of the two atomic orbitals 1sa and 1sb, and the square of the sum of the atomic orbitals. The atomic orbitals are centered on the H atom nuclei, which are separated by the bond length, r0. The protons (nuclei) are positively charged and repel each other. However, the molecular orbital concentrates the negatively charged electron density between the nuclei and holds them together. The important feature of the molecular orbital is that the electrons no longer belong to one atom or the other. The molecular orbital describes a delocalized probability distribution for the two electrons. Both electrons are free to roam over the entire molecule. The two electrons, which belonged to different atoms when the atoms were far apart, belong to the entire molecule. They are shared by the atoms, which are no longer independent.


FIGURE 12.3. A one-dimensional plot of the square of the two 1s orbitals that belonged to H atoms a and b (solid curves), and the square of the sum of the atomic orbitals, which is the square of the molecular orbital. The electron density is concentrated between the two nuclei.


Something very important has been left out of the discussion so far. The Pauli Principle (Chapter 11) states that at most two electrons can be in any orbital. This is true of atomic orbitals or molecular orbitals. To form the hydrogen molecule, we began with two hydrogen atoms. Each hydrogen atom has a 1s orbital. According to the Pauli Principle, it would be possible to put four electrons in these two orbitals. The two hydrogen atoms only have two electrons, but it would not violate the Pauli Principle to add another electron to each of the atomic 1s orbitals. We added together the two 1s orbitals (constructive interference) to form one molecular orbital. The Pauli Principle tells us that we can put at most two electrons in this molecular orbital. We started with two atomic orbitals that could hold four electrons, but now we have one molecular orbital that can only hold two electrons. Something is missing. You never lose or gain orbitals or places for electrons when forming molecules. If you start with two atomic orbitals, then two molecular orbitals will be formed, which can hold four electrons.

In Figures 12.2 and 12.3, the two 1s hydrogen atomic orbitals were added with the same sign to produce a molecular orbital that concentrates the electron density between the two atomic nuclei. The 1s orbitals are probability amplitude waves and can also be added with opposite sign. When the atomic orbitals are added with opposite sign, they destructively interfere. The addition with opposite sign is shown in Figure 12.4. Because the signs of the two atomic orbitals are opposite, there must be a place where the positive wave exactly cancels the negative wave. This is a node, as we discussed before for the particle in the box wavefunctions and atomic orbitals. As can be seen in the schematic in Figure 12.4, the destructive interference between the atomic orbitals and the resulting node pushes the electron density out from between the atomic nuclei. The negatively charged electrons are no longer screening the positively charged nuclei, which repel. In contrast to the curve shown in Figure 12.1, as the two hydrogen atoms are brought together from far apart, the energy goes up rather than down.


FIGURE 12.4. The left side is a schematic of two hydrogen atom 1s orbitals that are being added together. Note that the probability amplitude waves have opposite signs. The two atomic orbitals combine to make a molecular orbital. Because of the opposite signs, there is destructive interference, in contrast to Figure 12.2.

The molecular orbital that arises from constructive interference of the 1s atomic orbitals (see Figures 12.2 and 12.3), which is responsible for the H2 chemical bond, is called a bonding molecular orbital, or bonding MO. The molecular orbital that arises from destructive interference between the atomic orbitals is called an antibonding MO because it does not give rise to a bond, and in fact increases the energy as the atoms are brought together.

Figure 12.5 is a schematic of the energy curves for the bonding and antibonding MOs of the H2 molecule. As discussed in connection with Figure 12.1, as the two hydrogen atoms are brought together, the energy goes down and reaches a minimum before increasing again. This is the bonding MO curve. In contrast, the curve for the antibonding MO shows that the energy increases as the two atoms get close enough together to feel each other. The energy continues to increase as the atoms get progressively closer together. There is no reduction in energy. If the electrons are in the antibonding MO, the two atoms will not form a bond because the energy of the system is always higher than having separated atoms.


We started with two atomic orbitals, 1sa and 1sb, that were associated with two hydrogen atoms, Ha and Hb. These two atomic orbitals give rise to two molecular orbitals, the bonding and antibonding MOs. The rules we used for filling the atomic orbitals with electrons apply here as well. The Pauli Principle says no more than two electrons can go into an MO and they must have opposite spins (spins paired, one up arrow and one down arrow). Electrons go into the lowest energy levels first so long as this doesn’t violate the Pauli Principle. Hund’s Rule states that electrons will be unpaired if that doesn’t violate the first two rules. We will not need Hund’s Rule until the next chapter when we talk about larger molecules. Now we are in a position to see why the hydrogen molecule H2 exists, but the helium molecule He2 does not exist.


FIGURE 12.5. A schematic plot of the energy curves of the bonding and antibonding molecular orbitals for two hydrogen atoms as they are brought closer together. In contrast to the bonding MO, the energy of the antibonding MO increases as the atomic separation is decreased.


At the atomic separation that corresponds to the bond length, that is, the separation you find in the actual molecule, the bonding MO is always lower in energy than the separated atoms and the antibonding MO is always higher in energy. This is a rigorous result from quantum mechanics. It is a good approximation to say that the energy decrease of the bonding MO is equal to the energy increase of the antibonding MO.

A simple diagram that is used to reflect the atomic orbitals coming together to form molecular orbitals is shown in Figure 12.6. We will use this type of diagram in subsequent chapters. The two 1s atomic orbitals, one for each H atom, are depicted on the left and right sides of the figure. The lines through them are the zero of energy for the molecular orbitals. That is, the lines are the energy of the atoms when they are so far apart that they do not feel each other. In the center are the energy levels of the bonding and antibonding Os. These are called σb (b for bonding) and σ* (* for antibonding), and σ is the Greek letter sigma. σ designates a certain type of bond, called a σ bond, that will be discussed in Chapter 13. The dashed lines connecting the atomic orbitals to the MOs are used to indicate that both atomic orbitals combine to make both MOs when the atoms form the molecule.


FIGURE 12.6. An energy level diagram that represents the combination of the two atomic 1s orbitals to form the bonding and antibonding MOs at the bond length separation r0, which is the distance of the energy minimum of the bonding MO. The bonding MO is lower in energy and the antibonding MO is higher in energy than the atomic orbitals by the same amount. The bonding MO is called σb, and the antibonding MO is called σ*.

Figure 12.6 shows the MO energy level diagram for the two energy states that are involved in forming a hydrogen molecule. We have not “put in” the two electrons yet. This diagram is analogous to the many electron energy level diagram, Figure 11.1. We have the energy levels, but now we need to put in the electrons to see what happens. There are two electrons, one from each of the hydrogen atoms. We know that electrons are placed in the lowest possible energy level so long as the number of electrons does not violate the Pauli Principle, that is, at most two electrons can be in any given orbital with spins paired. This applies to MOs as well as atomic orbitals. Figure 12.7 shows the MO energy level diagram with the two electrons (arrows). The two electrons go into the lowest energy level, σb, with spins paired. When the atoms are well separated, the electrons have the energy represented by the lines though the 1s atomic orbitals. The bonding MO has substantially lower energy. It is this reduction in energy that holds the molecule together. The two electrons are in a molecular orbital. Neither is associated with a particular atom. A covalent bond involves sharing of the electrons by the atoms.

Why doesn’t the helium molecule, He2, exist? Two separated He atoms each have 1s orbitals with two electrons in them. Therefore the MO diagram is the same as that shown in Figure 12.6. However, now we need to put four electrons into the MO energy levels. Figure 12.8 shows the MO diagram with the four electrons. The first electron goes into the bonding MO because it is the lowest energy state. The second electron also goes into the bonding MO with the opposite spin of the first. The Pauli Principle says that no two electrons can have all of their quantum numbers the same. The two electrons in the bonding MO have different spin quantum numbers, s = +1/2 and s = -1/2. There are only these two spin quantum numbers, so the third electron cannot go into the bonding MO. It must go into the next lowest energy level, which is the antibonding MO. The fourth electron can also go into the antibonding MO with the opposite spin. There are two electrons in the bonding MO and two electrons in the antibonding MO. The two electrons in the bonding MO lower the energy relative to the separated atoms, but the two electrons in the antibonding MO raise the energy just as much as the bonding electrons lowered the energy. The net result is that there is no decease in energy relative to the separated atoms. A molecule is held together because the bonded atoms have lower energy than the separated atoms. For helium atoms, there is no reduction in energy that would produce a stable configuration. Therefore, there is no bond. We will see this same type of behavior for the noble gas neon in the next chapter.


FIGURE 12.7. The MO diagram for the hydrogen molecule. The two electrons (arrows), one from each hydrogen atom, go into the lowest energy level with their spins paired. The energy is lower than the separated atoms. An electron pair sharing bond is formed.


FIGURE 12.8. The MO diagram for the hypothetical helium molecule. There are four electrons (arrows), two from each helium atom. Two go into the bonding MO. Because of the Pauli Principle, the other two go into the antibonding MO. There is no net reduction in energy and, therefore, no bond.

To see the predictive capabilities of the simple diagrams like those shown in Figures 12.7 and 12.8, consider four molecules, or possible molecules. They are the hydrogen molecule ion 093 , the hydrogen molecule H2, the helium molecule ion 094 , and the helium molecule He2095 is composed of two hydrogen nuclei (protons) and one electron. Like the atomic cation ion Na+, it is positively charged because it has one fewer electrons than it has protons. 096is the molecular ion composed of two helium nuclei with two protons in each nucleus and three electrons. So it has four positive charges (four protons) and three negatively charged electrons.

Figure 12.9 shows the MO energy level diagrams for the four molecules. The atomic energy levels have been omitted. 097 has only one electron, so it goes into the lowest energy level, the bonding MO. The energy is lower than the separated atoms, but by only approximately half as much as for the H2 molecule, which has two electrons in the bonding MO. H2 has a full covalent bond. We say it has a bond order of 1. 098 has a bond order of 1/2. 099 has three electrons. The first two can go into the bonding MO, but, because of the Pauli Principle, the third electron must go into the antibonding MO. Two electrons lower the energy relative to the separated atoms, but one electron raises the energy. There is a net lowering of the energy. The 100 molecular ion exists in nature, and it has a bond order of 1/2. As discussed, He2 has two bonding electrons and two antibonding electrons. It has no bond. The bond order is zero. The He2 molecule does not exist.


FIGURE 12.9. The MO energy level diagrams for four molecules, the hydrogen molecule ion 102, the hydrogen molecule H2, the helium molecule ion 103, and the helium molecule He2.

Table 12.1 gives some quantitative information about these four molecules. It gives the number of bonding electrons, the number of antibonding electrons, and the net, which is the number of bonding electrons minus the number of antibonding electrons. It also gives the bond order. The last two columns are of particular interest.

These are the results of experimental measurements on the molecules. First consider the bond length. The lengths are in Å (Ångstroms, 10-10 m). The 104 molecule has a bond order of 1/2 and a bond length of 1.06 Å. In contrast, H2, which has a full bond with bond order 1, has a bond length of 0.74 Å. The additional bonding electron in the H2 bonding MO holds the atoms together tighter and therefore, closer. 105 has a bond order of 1/2 and a bond length of 1.08 Å, which is only a little longer than 106 . Of course, He2 is not a molecule, so it does not have a bond length. The last column is the bond energy in units of 10-19 J. The relative strength of the bonds is interesting. The H2molecule with a bond order of 1 has a considerably stronger bond than the two molecular ions, which have bond orders of 1/2. These simple MO diagrams allow us to see if a bond will exist, and they give information on how strong the bond will be.


TBLE 12.1. Properties of the hydrogen molecule ion 108, the hydrogen molecule H2, the helium molecule ion 109, and the helium molecule He2.

In this chapter, we have used the ideas of molecular orbitals to look at the simplest molecules. The discussion only involved atoms that have 1s electrons. All other atoms and molecules involve more electrons and more orbitals. In the next chapter, the ideas introduced here will be used to examine diatomic molecules involving larger atoms, such as the oxygen molecule, O2, and the nitrogen molecule, N2. These two molecules are the major components of the air we breathe.