The Lewis structures, the VSEPR theory and the valence binding method form a powerful combination to describe covalent bonding and molecular structures. The results are satisfactory for most of our purposes. However, sometimes, chemists need a greater understanding of the structures and properties of what these methods provide.

   For example, none of these methods provides an explanation of the electronic spectra of the molecules, of the oxygen paramagnetism or of the stability of the H₂⁺ species. To answer these questions we need to describe the chemical bond by a different method. 

   This method, called molecular orbital theory, begins with a simple description of the molecules, but quickly becomes complex in the details. Theory assigns the electrons of a molecule to a series of orbitals that belong to the complete molecule, which are the so-called molecular orbitals. 

   In the same way as atomic orbitals, molecular orbitals are mathematical functions, and they can be related to the probability of finding electrons in certain regions of a molecule. As also happens with atomic orbitals, a molecular orbital can only contain two electrons, and these electrons must have opposite spins.

   But what happens to the atomic orbitals when the two H atoms come together to form a chemical bond? When the atoms approach, the wave functions of each atom combine and interfere in the way:

• Constructive: corresponds to the addition of the two mathematical functions (the positive sign means the waves in phase)
  

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• Destructive: corresponds to the subtraction of two mathematical functions (the negative sign means that the waves are not in phase).

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The constructive interference of the two wave functions leads to a greater probability of finding the electron between the nuclei. The increase in electronic charge density between the cores makes them attract more to each other, forming a chemical bond. The electronic probability or electronic charge density in the orbital is (1s_A+1s_B)², the square of the new function (1s_A+1s_B), where 1s_A and 1s_B are the two 1s orbitals of the two atoms of H.

   The result of this constructive interference is a bonding molecular orbital because it produces a high electron charge density between the nuclei. A high electronic charge density between the atomic nuclei reduces the repulsions between the positively charged nuclei and causes a strong bond. This molecular bonding orbital, designated by σ1s, it has one less energy than the 1s atomic orbitals.

   The molecular orbital formed by the subtraction of the two 1s orbitals leads to a reduced electronic probability between the nuclei. This produces a molecular anti-bonding orbital because it produces a very low electron density between the nuclei. The electronic probability or electronic charge density in the orbital σ*1s is (1s_A-1s_B)², the square of the new function (1s_A-1s_B), where 1s_A and 1s_B are the two 1s orbitals of the two H atoms.

   With a low charge density between the atomic nuclei, the nuclei are not shielded from each other, producing strong repulsions and the bond weakens, hence the term antibonding. This antibonding molecular orbital, designated by σ*1s, has an energy greater than that of the 1s atomic orbitals.

Diatomic molecules of the elements of the first period

When we want to assign electrons to molecular orbitals, some ideas must be taken into account, such as: 

• The number of molecular orbitals (OM) that are formed is equal to the number of atomic orbitals that are combined.

• When two atomic orbitals combine, two molecular orbitals are formed, one of them is a bonding OM with less energy than the atomic orbitals. The other is an anti-bonding OM with a greater energy.

• In the configuration of the ground state, the electrons are placed in the lowest available OMs.

• The maximum number of electrons in a given OM is two (Pauli exclusion principle)

• In the configuration of the ground state, the electrons are placed in the OM of identical energy individually before being matched (Hund’s rule)

If a molecular species is stable, it has more electrons in entangle orbitals than in antibonding orbitals. The link order is half the difference between the number of bonding and antibonding electrons, that is: 

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With these ideas we can describe some molecular species of the elements of the first period, such as:

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Molecular orbitals of the elements of the second period

For the diatomic molecules and the ions of H and He, only 1s orbitals had to be combined. In the second period the situation is more interesting because 2s and 2p orbitals are available. As a result, eight molecular orbitals are formed. 

   Molecular orbitals formed by combination of 2s atomic orbitals are similar to those obtained from 1s atomic orbitals, except that they have higher energy. However, when 2p atomic orbitals are combined, the situation is different.

The 2p atomic orbitals can be combined in two possible ways:

• The addition of two 2p orbitals along the internuclear axis to form the molecular orbital σ2p. This orbital increases the electronic density between the nuclei, forming a chemical bond.

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• The addition of two 2p orbitals with opposite signs forms an antibonding orbital σ*2p. This orbital has a nodal plane perpendicular to the internuclear axis, like all antibonding orbitals. 

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• The addition of 2p orbitals perpendicular to the internuclear axis forms a molecular orbital π2p.This orbital increases the electron density between the nuclei, contributing to a multiple chemical bond.

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• The addition of two 2p orbitals with opposite signs forms an antibonding orbital π*2p. This orbital has a nodal plane perpendicular to the internuclear axis, like all antibonding orbitals. 

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The best overlap of the p orbitals is along a straight line, that is, frontally. This combination gives rise to molecular orbitals of σ type: σ2p and σ*2p. When forming the bonding and antibonding combinations along the internuclear axis, we must take into account the phase of the 2p orbitals.

   Only one pair of obitals can be combined frontally. The other two pairs must be combined in a parallel or lateral way to form the molecular orbitals of π type: π2p and π*2p.

 The link π is weaker than the link σ. The antibonding orbital π*2p is formed by subtraction of two p orbitals perpendicular to the internuclear axis. In addition to the nodal plane, which contains the nuclei, a node is formed between the nuclei and this is a characteristic of the anti-bonding character. There are four π type molecular orbitals (two bonding and two antibonding orbitals) because there are two pairs of 2p atomic orbitals located in parallel.

   It is necessary to take into account that both the 2s and 2p orbitals form molecular orbitals (σ2s and σ2p) whose electronic density is in the same region, between the nuclei. These two σ orbitals have a shape and energy so similar that they mix with each other to form σ modified orbitals. Each of these σ modified orbitals contains a fraction of the orbitals σ2s and σ2p. The σ2s modified orbital decreases in energy, and the σ2p modified orbital increases in energy, resulting in a different order of energy levels. The important question of this mixture is that the σ2p modified orbital has a higher energy than that of the orbitals π2p.

 For the molecular orbitals in O₂ and F₂, the situation is as expected because the energy difference between the 2s and 2p orbitals is large. Little mixing of the s and p orbitals occurs. For other diatomic molecules of elements of the second period, for example C₂ and N₂, the orbitals π2p have lower energy than σ2p the ones because the energy difference between the 2s and 2p orbitals is smaller and the 2s-2p orbital interactions affect the way they are combine atomic orbitals. This leads to the modified orbitals σ2s and σ2p. 

Diagram of molecular orbitals

The way to assign the electrons to the molecular orbitals of the diatomic molecules is the following:

• We start filling the orbitals σ1s and σ*1s.
• We add electrons in order of increasing in the available molecular orbitals of the second main layer. 

  In the same way that we could order the molecular orbitals of the valence shell of an atom, we can also order the molecular orbitals of the second layer of a diatomic molecule in order of increasing energy. We can assign the electrons to those orbitals by obtaining a molecular orbital diagram.

For example: If we assign the 8 valence electrons of the C₂ molecule, we obtain:

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The experimental data indicate that the C₂ molecule is diamagnetic, not paramagnetic, and this configuration is incorrect. Here we can see the importance of the energy level diagram. The assignment of the 8 electrons in the following molecular orbital diagram is consistent with the diamagnetism of C_2. 

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This modified energy level diagram is used for homonuclear diatomic molecules of elements with atomic numbers from three to seven.

Heteronuclear diatomic molecules

The ideas developed for homonuclear diatomic molecules can be extended, to give us an idea of ​​the linkage of heteronuclear diatomic species. But to illustrate the differences we will consider the construction of σ bonding and antibonding orbitals from the 2s orbitals of the C and O atoms in the CO bond.

   The link combination is:

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 and the antibonding combination is: 

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where the coefficients c₀ and c_c reflect the mixing percentage of the orbitals. In the homonuclear case, the two atoms are the same and the coefficients are equal because there is the same probability of finding the electrons in the orbital associated with any of the nuclei. 

  If the nuclei are different, a greater probability of finding the electrons in the orbital associated with the more electronegative element is expected. As a result, in our example, it is expected that c₀ is greater than c_c in the bonding orbital. Thus, the bonding molecular orbital has a greater contribution from the 2s orbital of oxygen than from the 2s orbital of carbon, which means that the σ2s orbitals are more like a 2s orbital of oxygen. In the antibonding orbital the situation is the reverse, with c*0 less than c*c, being the antibonding orbital more like the 2s carbon orbital.

For example: If we apply the aufbau principle to the molecular orbital diagram for O₂ and the aforementioned to carbon monoxide (CO), which has 10 valence electrons, we obtain the following configuration:

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Thus, the bond order of the CO is three,  according to the experimental data. The CO has a link energy greater than the NO, which has a link order of 2.5, as it was predicted by the molecular orbital diagram.

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Here you can see that the NO molecule is paramagnetic, a fact that coincides with the experimental data.

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