Bruce’s Notes – Molecular Orbitals

In CHEM 151, you learned about a theory of chemical bonding called valence bond theory.  In this theory, chemical bonds (which contained valence electrons being shared between two bonding atoms) are formed when orbitals from each of the bonding atoms containing each atom’s valence electrons overlap.  This theory, along with the concepts of hybridization, Lewis dot structures, and VSEPR, explain many aspects of covalent molecules quite well.

Molecular Orbital Theory uses a different philosophy.  The orbitals you learned about in CHEM 151 are called atomic orbitals – as each orbital belongs to the atom. You may recall that an (atomic) orbital is a mathematical equation that is a solution to the Schrodinger Equation that gives a probability density of the location of an electron in an atom.  In molecular orbital theory, once atoms are bonded into molecules the orbitals from each atom are no longer valid — instead the molecule has orbitals belonging to the entire molecule.  As we will see, molecular orbital theory does explains some aspects of bonding well that valence bond theory does not.

Objective 1:Explain the essential features of the molecular orbital theory.

Valence Bond Theory

This is generally the first theory of bonding students learn about in general chemistry, and was the theory you learned in CHEM 151. According to this theory, covalent bonds result when atoms share outer (valence) electrons.  The way atoms share these electrons is the orbitals containing these valence electrons overlap in space. An orbital gives the probability density of location of an electron in an atom.  You may recall that each orbital holds two electrons.

Another facet of valence bond theory you may remember is hybridization.  Hybrid orbitals are each a mixture of atomic orbitals from a single atom.  Many atomic orbitals involved in bonding are hybrid orbitals, and they are oriented to each other in a way that explains the geometries such as tetrahedral, trigonal bipyramidal, etc found in molecules.  Hybrid orbitals are still atomic orbitals, though, and remain localized on that atom.

Limitations of Valence Bond Theory

Valence bond theory assumes all bonds are localized bonds in a molecule: for example, in a CH4 molecule, each pair of electrons is confined to the region between the carbon nucleus and the hydrogen nucleus.  Also, experimental observations show that some substances (O2 for example) are paramagnetic while others (N2 for example) are diamagetic.  Valence bond theory has no explanation for this, but molecular orbital theory does.  Valence bond theory also not deal well with odd electron molecules such as NO, but, as we will see, molecular orbital theory does.

Molecular Orbital (MO) Theory

Molecular orbitals (MOs) are linear mathematical combinations of atomic orbitals (AOs) from different atoms.  The number of AOs combined is equal to the number of MOs resulting from the combination.

MOs are associated with the entire molecule, not a single atom.  However, like an atomic orbital, each MO can hold two electrons of opposite spin.  

Figure 8.29 in OpenStax shows the combining of two AOs (the s orbitals on the left) to make two MOs (on the right)

A diagram is shown that depicts a vertical upward-facing arrow that lies to the left of all the other portions of the diagram and is labeled, “E.” To the immediate right of the midpoint of the arrow are two circles each labeled with a positive sign, the letter S, and the phrase, “Atomic orbitals.” These are followed by a right-facing horizontal arrow that points to the same two circles labeled with plus signs, but they are now touching and are labeled, “Combine atomic orbitals.” Two right-facing arrows lead to the last portion of the diagram, one facing upward and one facing downward. The upper arrow is labeled, “Subtract,” and points to two oblong ovals labeled with plus signs, and the phrase, “Antibonding orbitals sigma subscript s superscript asterisk.” The lower arrow is labeled, “Add,” and points to an elongated oval with two plus signs that is labeled, “Bonding orbital sigma subscript s.” The heading over the last section of the diagram are the words, “Molecular orbitals."

Objective 2:Explain the relationship between bonding and antibonding molecular orbitals, both in terms of energy compared to the corresponding atomic orbitals and where electron density is concentrated relative to the nuclei.

There are two general types of molecular orbitals: bonding molecular orbitals and antibonding molecular orbitals.  If you look at the figure above, you can see there is one of each.

Bonding MOs

Since atomic orbitals are wave functions, the waves can interact constructively or destructively. In a bonding molecular orbital, the wave functions combine constructively.  In a bonding MO, most of the electron density is between adjacent nuclei (see the diagram above).  Therefore, bonding MOs are lower in energy that the combined atomic AOs.  As a result, they energetically favors bond formation.  An electron in a bonding orbital stabilizes a molecule.  There are both sigma (σ) and pi  (π ) MOs (more on that later).

Antibonding MOs

In an antibonding molecular orbital, the wave functions combine destructively.  In these MOs, most of the electron density is not between the nuclei (see the diagram above).  In fact, there is a node (a region of zero probability of finding the electrons) between the nuclei.  Therefore, bonding MOs are higher in energy than the combined atomic AOs.  As a result, they do not favor bond formation.  An electron in a bonding orbital destabilizes a molecule. Antibonding MOs are signified by an asterisk *, so you will see σ* and  π*  antibonding MOs.

Objective 3:Given a molecular orbital diagram for the diatomic molecule or ion built from elements of the first or second period, predict the placement of electrons, the bond order and the number of unpaired electrons.  Also, determine whether the molecule is paramagnetic or diamagnetic.

Molecular orbital diagrams

The combination of atomic orbitals and the relative energies of the molecular orbitals are shown by an energy-level (or molecular orbital) diagram.  We can think of an MO diagram as a “comparison” diagram – where the atomic orbitals of the individual atoms in the molecule are shown on the same diagram as the molecular orbitals resulting from their combination, allowing for a comparison of energy.

Figure 8.35 in Open Stax shows a MO diagram for the H2 molecule:

A diagram is shown that has an upward-facing vertical arrow running along the left side labeled “E.” At the bottom center of the diagram is a horizontal line labeled, “sigma subscript 1 s,” that has two vertical half arrows drawn on it, one facing up and one facing down. This line is connected to the right and left by upward-facing, dotted lines to two more horizontal lines, each labeled, “1 s,” and each with one vertical half-arrow facing up drawn on it. These two lines are connected by upward-facing dotted lines to another line in the center of the diagram, but farther up from the first, and labeled, “sigma subscript 1 s superscript asterisk.” The left and right sides of the diagram have headers that read, ”Atomic orbitals,” while the center header reads, “Molecular orbitals.” The bottom left and right are labeled “H” while the center is labeled “H subscript 2.”

Each hydrogen atom in the molecule has one valence electron (1s1).  These are shown on the atomic orbitals (1 for each atom) on the left and right sides of the diagram. The resulting two MOs are shown in the center (note the σ1s bonding MO is lower in energy than the AOs and the σ1s* antibonding MO is higher in energy.  The two electrons in the H2 molecule will fill the MOs following the Aufbau principle lowest energy first and Hund’s rules (if you have equal energy orbitals, one electron goes in each first with the same spin before they are filled).  Therefore, the two electrons are in the bonding MO.

Calculating Bond Order from MO diagrams

Once the MO diagram is filled, we can calculate the bond order as follows:

\text{Bond order }\!\!~\!\!etext{ =}\frac{\text{ }\!\!\#\!\!\text{ electrons in bonding MO -- }\!\!\#\!\!\text{ electrons in antibonding MO }\!\!~\!\!\text{ }}{\text{2}}

A bond order of 1 corresponds to a single bond, 2 a double bond, etc. A bond order 0 means there will be no bonding and that the molecule does not exist.

For H2, the bond order will be:

\text{Bond order }\!\!~\!\!\text{ =}\frac{\text{ }\!\!\#\!\!\text{ electrons in bonding MO -- }\!\!\#\!\!\text{ electrons in antibonding MO }\!\!~\!\!\text{ }}{\text{2}}=\frac{2-0}{2}=1

Next, let’s look at the bond order of He2.  He2 has 4 electrons (2 from each He atom)

Figure 8.36 in Open Stax shows a MO diagram for the He2 molecule:

A diagram is shown that has an upward-facing vertical arrow running along the left side labeled, “E.” At the bottom center of the diagram is a horizontal line labeled, “sigma subscript 1 s,” that has two vertical half arrows drawn on it, one facing up and one facing down. This line is connected to the right and left by upward-facing, dotted lines to two more horizontal lines, each labeled, “1 s,” and each with one vertical half-arrow facing up and one facing down drawn on it. These two lines are connected by upward-facing dotted lines to another line in the center of the diagram, but farther up from the first, and labeled, “sigma subscript 1 s superscript asterisk.” This line has one upward-facing and one downward-facing vertical arrow drawn on it. The left and right sides of the diagram have headers that read, “Atomic orbitals,” while the center header reads, “Molecular orbitals.” The bottom left and right are labeled, “H e,” while the center is labeled, “H e subscript 2.”

Since the bonding MO is filled with the first two electrons, the second two go into the antibonding.  The bond order is:

\text{Bond order }\!\!~\!\!\text{ =}\frac{\text{ }\!\!\#\!\!\text{ electrons in bonding MO -- }\!\!\#\!\!\text{ electrons in antibonding MO }\!\!~\!\!\text{ }}{\text{2}}=\frac{2-2}{2}=0

The bond order of zero means there is no bond between the two He atoms – the He2 molecule does not exist.  We could also write a molecular orbital electron configuration for He2:(σ1s)2 (σ*1s)2 

MO Diagram Practice

MO Diagrams for Second Row Diatomic Molecules

When we get to the second row of the periodic table, things get more complicated as there are valence 2s and 2p orbitals.  As previously seen, though, the same rules still hold:

  • The # of MOs formed equals the # of AOs combined
  • AOs combine most effectively with other atomic orbitals of similar energy

This means in a diatomic molecule, the two 2s orbitals (one from each atom) will combine to make two MOs (one bonding and one antibonding). The six 2s orbitals (three from each atom) will combine to make six MOs (three bonding and three antibonding).

There is one other complication introduced by the 2p.  Combining of 2p AOs yields two types of MOs: sigma (σ) and pi  (π ).

Figure 8.30 in OpenStax shows the σ2p MOs (both bonding and antibonding).  The MOs are along the internuclear axis.

Two horizontal rows of diagrams are shown. The upper diagram shows two equally-sized peanut-shaped orbitals with a plus sign in between them connected to a merged orbital diagram by a right facing arrow. The merged diagram has a much larger oval at the center and much smaller ovular orbitals on the edge. It is labeled, “sigma subscript p x.” The lower diagram shows two equally-sized peanut-shaped orbitals with a plus sign in between them connected to a split orbital diagram by a right facing arrow. The split diagram has a much larger oval at the outer ends and much smaller ovular orbitals on the inner edges. It is labeled, “sigma subscript p x superscript asterisk”.

Figure 8.31 in OpenStax shows the π2p MOs (both bonding and antibonding).  The MOs are perpendicular the internuclear axis.

Two horizontal rows of diagrams are shown. The upper and lower diagrams both begin with two vertical peanut-shaped orbitals with a plus sign in between followed by a right-facing arrow. The upper diagram shows the same vertical peanut orbitals bending slightly away from one another and separated by a dotted line. It is labeled, “pi subscript p superscript asterisk.” The lower diagram shows the horizontal overlap of the two orbitals and is labeled, “pi subscript p.”

What we learned for atomic orbitals still holds: fill the orbitals from low to high energy, half-filling each degenerate (same energy) orbital first before pairing electrons in the same orbital.  

Figure 8.37 in OpenStax shows the valence shell MO diagrams for all of the second row elements:

A graph is shown in which the y-axis is labeled, “E,” and appears as a vertical, upward-facing arrow. Across the top, the graph reads, “L i subscript 2,” “B e subscript 2,” “B subscript 2,” “C subscript 2,” “N subscript 2,” “O subscript 2,” “F subscript 2,” and “Ne subscript 2.” Directly below each of these element terms is a single pink line, and all lines are connected to one another by a dashed line, to create an overall line that decreases in height as it moves from left to right across the graph. This line is labeled, “sigma subscript 2 p x superscript asterisk”. Directly below each of these lines is a set of two pink lines, and all lines are connected to one another by a dashed line, to create an overall line that decreases in height as it moves from left to right across the graph. It is consistently lower than the first line. This line is labeled, “pi subscript 2 p y superscript asterisk,” and, “pi subscript 2 p z superscript asterisk.” Directly below each of these double lines is a single pink line, and all lines are connected to one another by a dashed line, to create an overall line that decreases in height as it moves from left to right across the graph. It has a distinctive drop at the label, “O subscript 2.” This line is labeled, “sigma subscript 2 p x.” Directly below each of these lines is a set of two pink lines, and all lines are connected to one another by a dashed line to create an overall line that decreases very slightly in height as it moves from left to right across the graph. It is consistently lower than the third line until it reaches the point labeled, “O subscript 2.” This line is labeled, “pi subscript 2 p y,” and, “pi subscript 2 p z.” Directly below each of these lines is a single blue line, and all lines are connected to one another by a dashed line to create an overall line that decreases in height as it moves from left to right across the graph. This line is labeled, “sigma subscript 2 s superscript asterisk.” Finally, directly below each of these lines is a single blue line, and all lines are connected to one another by a dashed line to create an overall line that decreases in height as it moves from left to right across the graph. This line is labeled. “sigma subscript 2 s.”

Notice the order or the orbitals change between N2 and O2.  In either case, the orbitals will still fill in order of increasing energy.

Figure 8.40 (in example 8.6) shows the filled valence orbital diagram for O2.  See if you can come up with the same filling using the 12 valence electrons for O2 and the diagram for O2 in Figure 8.37.

A diagram is shown that has an upward-facing vertical arrow running along the left side labeled, “E.” At the bottom center of the diagram is a horizontal line labeled, “sigma subscript 2 s,” that has two vertical half arrows drawn on it, one facing up and one facing down. This line is connected to the right and left by upward-facing, dotted lines to two more horizontal lines, each labeled, “2 s,” and with two vertical half arrows drawn on them, one facing up and one facing down. These two lines are connected by upward-facing dotted lines to another line in the center of the diagram, but farther up from the first and labeled, “sigma subscript 2 s superscript asterisk.” This horizontal line has two vertical half-arrow drawn on it, one facing up and one facing down. Moving further up the center of the diagram is a horizontal line labeled, “sigma subscript 2 p subscript x,” which lies below two horizontal lines, lying side-by-side, and labeled “pi subscript 2 p subscript y,” and “pi subscript 2 p subscript z.” Both the bottom and top lines are connected to the right and left by upward-facing, dotted lines to three more horizontal lines, each labeled, “2 p,” on either side. These sets of lines each hold three upward-facing and one downward-facing half-arrow. They are connected by upward-facing dotted lines to another single line and then pair of double lines in the center of the diagram, but farther up from the lower lines. They are labeled, “sigma subscript 2 p subscript x superscript asterisk,” “pi subscript 2 p subscript y superscript asterisk,” and “pi subscript 2 p subscript z superscript asterisk,” respectively. The lower of these two central, horizontal lines each contain one upward-facing half-arrow. The left and right sides of the diagram have headers that read, ”Atomic orbitals,” while the center header reads, “Molecular orbitals.”

OpenStax Table 8.3 shows electron configurations and bond order for the second row diatomics.

Examining the diagram above, we can determine that O2 is paramagmetic.  You may remember from the study of coordination chemistry that paramagnetism is indicated if at least one d orbital is half filled with only 1 electron.  The same holds true for MOs.  Also, if all molecular orbitals are empty or fully filled (no unpaired spins) it is diamagnetic.

MO Diagrams for Second Row Diatomic Molecules- practice

Using Figure 8.37 in OpenStax, fill out the MO diagram for N2 and answer the following questions:

Heteronuclear diatomic molecules

The molecules we have looked at so far are homonuclear diatomic molecules — both atoms are the samre element.  MO theory also works well for heteronuclear diatomic molecules such as NO and OF and ions such as CN– or NO+. For example, CN– would have the same filled MO diagram as N2 as they each have 10 valence electrons.  You should be able to complete MO diagrams and determine bond order and magnetism for heteronuclear diatomic molecular and diatomic ions as well, including those with an odd number of valence electrons.

Heteronuclear diatomic molecules- practice

Using Figure 8.37 in OpenStax, fill out the MO diagram for CN (you can use the MO diagram for either C2 or N2) and answer the following questions:

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