What is the molecular orbital theory?

The Molecular Orbital Theory (TOM) is a mathematical model developed to explain the physicochemical properties of molecules, such as the absorption and emission of radiation, electrical conductivity, as well as the electronic nature of their bonds. This considers, unlike the valence bond theory (VTE), that the electrons are delocalized throughout the molecule, without being probabilistically located in the bonds.

Therefore, the molecular orbital theory applies the same quantum principles that dictate wave functions for atomic orbitals, thus describing the energetic state of electrons in atoms; only now, it addresses the so-called molecular orbitals, originated from the linear combinations of the atomic orbitals of the atoms that are bonded.

The paramagnetism of liquid oxygen, and consequently its attraction to magnets, is explained thanks to the molecular orbital theory. Source: Pieter Kuiper via Wikipedia.

The theory of the molecular orbital works therefore, working with the molecular orbitals, their electronic and spatial distributions, as well as the diagrams of their respective energies, which allow us to observe the energy gaps that separate them. Above all, it explains the paramagnetism of certain substances, such as molecular oxygen in a liquid state (see above).

2 , described by Lewis structures and the valence bond theory, has all its electrons paired, so in theory it should be diamagnetic. However, in practice it has been shown that it is actually paramagnetic, that is, it has unpaired electrons; specifically a pair of unpaired electrons.

Another of the most important triumphs of the molecular orbital theory is that it helps to predict the possible existence of diatomic molecules, as well as their relative stabilities. Thus, the molecular orbital theory is correct in predicting molecules such as B 2 and Li 2 , even though they can only be detected in the vapor phase at high temperatures. Likewise, this theory rejects the existence of the hypothetical He 2 or Ne 2 molecules .

Types of bonds and molecular orbitals

As with the valence bond theory, in the molecular orbital theory several types of bonds are considered depending on the directionality of the overlap of the atomic orbitals that participate in the chemical bond . Among the most important and present in molecular nature, we have the sigma and pi bonds.


Formation of σ1s and σ1s * bonds during the overlap of two 1s atomic orbitals. Source: Gabriel Bolívar.

The sigma bonds, σ, are established when the atomic orbitals overlap frontally, relative to an imaginary internuclear axis connecting the two approaching nuclei.

Consider, for example, the σ bond that forms when two 1s atomic orbitals, say a hydrogen atom, come close to each other (image above). Since the 1s orbitals are symmetrical, their overlap will always be frontal; therefore, they will always generate σ 1s and σ 1s * molecular orbitals .

In contrast, in the σ 1s * orbital we have a nodal plane between the two nuclei. This means that the probability of finding an electron in that space is equal to zero. Note also that in the σ 1s * orbital the electrons describe positions around each of the two nuclei; not around the molecule as a whole.


Sigma and pi bonds derived from the overlap of p orbitals. Source: V8rik at en.wikipedia, CC BY-SA 3.0 <>, via Wikimedia Commons

Now consider pi or pi molecular orbitals and bonds. These occur when the orbital overlap occurs in a direction perpendicular to the internuclear axis, which is arbitrarily fixed on any of the axes of the Cartesian plane. Assuming we are talking about a 2p x orbital , the 2p y orbital will be free to establish a pi bond (see vertical blue and white lobes).

When two 2p and orbitals overlap, they give rise to two molecular orbitals: π 2py and π 2py *; the former has the highest electron density above and below the internuclear axis, while the latter resembles a four-petal flower, where the probability of finding the electrons is much lower.

On the other hand, the 2p orbitals, say 2p x , can also overlap frontally to originate a σ bond and two molecular orbitals σ 2px and σ 2px *. As in the cases of σ 1s and σ 1s *, the σ 2px orbital shows a higher electron density between the two nuclei; which is the opposite in σ 2px *, where the electrons are oriented to the outer sides.


The binding molecular orbitals are those that contribute to the stability of the molecule. That is, they must have less energy compared to the atomic orbitals of the individual atoms before they bond and form the molecule. In these orbitals the electrons are delocalized through all the dimensions of the molecule as a whole.

In OM diagrams, these will always be located below the atomic orbitals that were combined to form them. This will be seen in more detail in the next sections.

Anti-bonding agents

Anti-bonding molecular orbitals, on the other hand, are those that destabilize the molecule. They are symbolized with an asterisk (*), and their energies are higher than that of the atomic orbitals that originated them. In these orbitals the electrons are delocalized quite irregularly, as if the molecule were electronically fragmented.

In OM diagrams, these will always be located above the atomic orbitals that were combined to form them.


Meanwhile, the non-binding orbitals, as well as their electrons, are those that do not contribute or harm the stability of the molecule. Their energies are very similar to those of atomic orbitals.

Link order

Bond order comes to be in molecular orbital theory, just as bond number is in valence bond theory: a measure of the bond strength in a molecule. Thus, a bond order equal to 1 corresponds to a simple bond (-). And a bond order (oe) equal to 2, corresponds to a double bond (=). So on.

This order is determined from the electron counts in the OM diagrams for a particular molecule. For this, the following formula must be applied:

oe = (Number of bonding electrons – Number of bonding electrons) / 2

The bonding electrons help stabilize the molecule, while the anti-bonding electrons destabilize it. Therefore, the more antibonding electrons there are, the lower the oe, and the molecule will tend to be very unstable. When the oe is equal to 0, it means that the molecule does not exist (or at least in theory).

OM diagrams

In the OM diagrams the energies of the molecular orbitals are represented, and their electronic filling is also visualized, which obeys the Aufbauf and Hund rules, as well as the Pauling exclusion principle. Consider for example the following two diagrams:

The horizontal lines on the sides, colored black, represent the energies of the atomic orbitals. Instead, the central horizontal lines, colored purple, are the energies of the molecular orbitals. Note how the bonding and anti-bonding MOs (*) are distributed in relation to the atomic orbitals of the individual atoms.

Also note that in the diagram on the right, the OM σ 2p change places with π 2p . This occurs with the molecules of O 2 , F 2 , and the hypothetical Ne 2 .


In the following examples we will proceed to fill the OMs with electrons, calculate the bond order, and make predictions about the bond strength or stability of the molecule in question.


OM diagram for the dihydrogen molecule. Source: Gabriel Bolívar.

Starting with the simplest molecule, that of dihydrogen, H 2 , two H atoms are linked by combining their two 1s atomic orbitals to form two molecular orbitals σ 1s and σ 1s *. The two electrons then proceed to fill the molecular orbitals.

The σ 1s orbital is filled first , because it is the one with the lowest energy (Aufbauf’s Rule). Then the second electron must finish filling the σ 1s orbital before going up to the σ 1s * orbital (Hund’s Rule). And finally, this second electron orients its spin in the opposite direction to the first electron (Pauling’s exclusion principle). Thus, the two electrons are located in the bonding orbital σ 1s .

What about the binding order? In the bonding molecular orbital σ 1s we have 2 electrons; while in the antibonding molecular orbital σ 1s * we have none. Therefore, the calculation would be as:

oe = (2-0) / 2

= 1

Being oe equal to 1, it means that the bond that joins the two hydrogen atoms is simple: HH. Generally, if this value is equal to or greater than 1, the molecule is said to exist and is stable.

I have 2

OM diagram for the hypothetical dihelium molecule. Source: Gabriel Bolívar.

Now suppose we have the He 2 molecule . Above we see that its OM diagram is very similar to that of H 2 , with 2 additional electrons moving to the σ 1s * orbital . Since there are a total of 4 electrons adding the two He atoms, then there must also be 4 electrons in all the resulting molecular orbitals.

Determining the oe for the He 2 molecule we will have:

oe = (2-2) / 2

= 0

This means that there is no possible bond between the two helium atoms. In fact, to date the existence of this molecule has not been identified, which is consistent with the predictions of the molecular orbital theory.


OM diagram for the B2 molecule. Source: Gabriel Bolívar.

Note that the electrons of the inner shells are also taken into account in the formation of molecular orbitals; not only those of Valencia.

Thus, in the example of the diboron molecule, B 2 , each boron atom contributes 5 electrons in total, 3 of which are of valence; these are those of its 2s and 2p orbitals. The two electrons of the 2p orbitals are positioned in different π 2p molecular orbitals with parallel spins (Hund’s rule).

We then proceed to calculate the binding order:

oe = (6-4) / 2

= 1

Therefore, the molecule is expected to have a single BB bond. B 2 is a molecule that only exists in the vapor phase at very high temperatures, because boron under normal conditions adopts more complex network structures and designs.

2 and C 2-

Let’s first consider the OM diagram for the C 2 molecule :

OM diagram for the C2 molecule. Source: Gabriel Bolívar.

Now, the two new electrons contributed by the carbon atoms are positioned again in the molecular π 2p orbitals but with opposite spins (Pauling’s exclusion principle).

Determining your link order we will have:

oe = (8-4) / 2

= 2

Note that the π 2p molecular orbitals add 4 bonding electrons to the formula. As this order is equal to 2, it means that the C 2 molecule has a double bond, C = C. Again, the C 2 molecule , also called diatomic carbon, exists only in the vapor phase at high temperatures, and is one of the simplest allotropic forms of carbon.

And what about C 2- ? His OM diagram is as follows:

OM diagram for the C22- anion. Source: Gabriel Bolívar.

The two new electrons (red arrows) are positioned in the bonding molecular orbital σ 2p . This is so assuming that each carbon atom contributes one of the two negative charges (for an oxidation state of -1).

Calculating your link order we will have:

oe = (10-4) / 2

= 3

That is, the C 2- anion , also called the acetylide anion, has a triple bond, [C≡C] 2- . It is a relatively stable anion with a high binding force; however, depending on its counterpart ions, it can produce explosive compounds.

2 and N +

The famous nitrogen molecule, N 2 , can also be perfectly described using molecular orbital diagrams:

OM diagram for the N2 molecule. Source: Gabriel Bolívar.

Note that this diagram is exactly the same as for the C 2- anion . This means that N 2 and C 2- are isoelectronic. However, this fact does not imply that both species behave in the same way. Neutral N 2 is much more stable than negative C 2- , even though both have a bond order equal to 3, N≡N.

And what about the N + cation ? Let’s look at your OM diagram:

OM diagram for the N2 + cation. Source: Gabriel Bolívar.

Since the N + cation has one less electron, it removes the σ 2p molecular orbital . N 2 is diamagnetic, while N + is paramagnetic. And its binding order becomes:

oe = (9-4) / 2

= 2.5

A 2.5 bond would be represented by two lines and a point. However, such a thing does not make much sense in valence bond theory or in Lewis structures. Because this bond order is less than 3, the bond strength present in N + is less than that of N 2 , making it more unstable.

2 , O 2- and O 2+

Let us now look at another very important molecule for life: molecular or diatomic oxygen, O 2 . According to the valence bond theory and Lewis structures, it should be diamagnetic; but experimentally it has been shown that it is paramagnetic, so it has unpaired electrons somewhere.

Below we compare the OM diagrams for O 2 and the ions O 2- (oxide) and O 2+ (oxydication):

OM diagrams for molecular oxygen and two of its main ions. Source: Gabriel Bolívar.

The OM diagram for O 2 shows, unlike the previous ones, that the molecular orbitals σ 2p and π 2p change their energetic position. Likewise, we actually see that there are two unpaired electrons in the π 2p * orbitals , which explains the paramagnetic character of oxygen (mentioned at the beginning of the article).

On the other hand, we also have the OM diagram for the anion O 2- , infinitely diffused in the Cosmos (moons, planets, comets, asteroids, etc.). It has two extra electrons (red arrows), which finish filling the π 2p * orbitals , pairing all the electrons. Consequently, O 2- is diamagnetic.

Similarly, we consider the OM diagram for oxydication O 2+ . It has two fewer electrons than O 2 , leaving the π 2p * orbitals empty . All of its electrons are paired, and therefore it is diamagnetic.

The bond orders for O 2 , O 2- and O 2+ are, respectively: 2 (O = O), 1 [OO] 2- and 3 [O≡O] 2+ . Therefore, the O 2+ would have the strongest bond.

Advantages and disadvantages


Among the advantages of the molecular orbital theory we can mention the following:

-It allows to evaluate binding orders that ordinarily would not make much sense in the valence bond theory

-It correlates well with diamagnetism and paramagnetism of molecules

-The distance that separates the molecular orbitals in the diagrams is equal to ΔE, and serves to explain the electronic transitions product of the absorption of photons

-Applies not only to homonuclear diatomic molecules, but also to heteronuclear molecules, such as CO 2 and benzene

-It extends its model to other types of compounds, such as inorganic complexes, which is why it supports the theory of the field of ligands

-The consideration that electrons are delocalized throughout the molecule is convenient to explain many of the physicochemical properties


And among some of the disadvantages of the molecular orbital theory we have, finally:

-It is very abstract and requires a deep mathematical understanding to fully understand it

-The diagrams for molecules such as CO 2 , CO, H 2 O and others, can be too tedious to elaborate and explain

-It does not say anything about specific regions or bonds of a large molecule

-Neither does it provide any information regarding molecular geometry (trigonal plane, square plane, tetrahedral, etc.).

-It is not as graphic as the valence bond theory

The molecular orbital theory, in conclusion, is a theory that complements the valence bond theory to have a broader and more complete picture of the molecular spectrum.

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