Let's talk about air, or more precisely, about electronic structure of its two most abundant components: dinitrogen, N2, and dioxygen, O2 (nitrogen and oxygen, respectively, for short).  Each of these molecules is built from two identical atoms, and each of these atoms has an s and three p atomic orbitals available for bonding (we are concerned only with the valence orbitals and electrons, and neglect 1s atomic orbitals with their electrons).

Although nitrogen's and oxygen's orbitals have slightly different energies and sizes, for simplicity we can disregard this aspect for now, and concentrate just on their relative energies and shapes. When the two atoms of these diatomic molecules approach each other to within the bonding distance, the atomic orbitals on the atoms start to overlap making new molecular orbitals (MO).  These new orbitals are formed by "mixing" of atomic orbitals (forming linear combinations of them, or in simplistic terms, merging parts of them). The best such "mixing" is obtained between atomic orbitals that have the same energy and symmetry, or in simpler terms, the orbitals  "mix" more effectively (stronger), if they have matching energies and shapes. It is best illustrated on the actual example. Before we start "mixing" remember the rules: for each bonding combination we also produce an antibonding combination, and the total number of orbitals must be preserved. For N2 and O2 molecules it means that we have to produce 8 MOs, half of them bonding.

OK, it's mixing time!  At the bottom (lowest energy) are two s atomic orbitals. Mixing them in "additive" manner generates a σ type MO, and the combination in the "subtractive" manner generates the antibonding σ* orbital. The first of these is the lowest energy MO, and the second one is just above it.  These interactions are shown schematically below; on the left the interaction lines are drawn and on the right the shapes of the resulting MO are presented (broken and solid lines represent the opposite signs of the wavefunctions, and the black dots show positions of nuclei).  Above the σ orbitals, we have three possible combinations of p orbitals. Since the p orbitals on one atom are perpendicular to each other, their combinations may take place only pairwise with matching symmetries.  One pair (pz in our example) may overlap head-on, forming  σ and σ* types of molecular orbitals. The remaining two pairs of p orbitals (px's and  py's) may overlap only sideways, forming two sets of π and two sets of  π* molecular orbitals. πx and πy have the same energy (we call them degenerate) and shapes, but are perpendicular to each other.  The same is true about the πx* and πy* pair.

  MO-8: the highest energy antibonding combination of two pz orbitals, forming a σ* type MO
MO-6 and MO-7: the antibonding interaction of two px and two py orbitals, forming two π* type MOs
MO-4 and MO-5: the bonding interaction of two px and two py orbitals, forming two π type MOs
MO-3: the bonding interaction of two pz atomic orbitals, forming a σ type MO
MO-2: the antibonding combination of two s atomic orbitals, forming a σ* type MO
MO-1: the lowest energy bonding interaction of two s atomic orbitals, forming a σ type MO

This way we have constructed a generic MO scheme for any diatomic molecule built of atoms  from the second row with p-electrons, such as N2, O2, F2, or carbon monoxide for example. Interestingly, there are occasionally some complications, such as an additional "mixing" of orbitals of similar energy and symmetry. Such "complications" are present, for example, in N2 molecule (see below), or in CO molecule.

In the "lighter" second-row atoms, the 2s and 2p atomic orbitals are close in energy (because of smaller nuclear charge). What it means is that  MO-1 and MO-3 that have the same symmetry (shown in blue below) are now close enough to interact ("mix") further. As the result, MO-1 has its energy lowered, and MO-3 has its energy raised, to the point that it is higher than the energy of π MOs (MO-4 and MO-5). Similarly, the interaction of  MO-2 and MO-8 (both have the same symmetry, shown in read below) leads to lowering of the energy of MO-2 and increase in energy of MO-8. From the MO point of view MO-1, MO-2, MO-3, and MO-8 are now "mixtures" of  s and p atomic orbitals.  It sounds like "hybridization", doesn't it? We should not be mixing these concepts here, but yes, they are equivalent.

In the "heavier" second-row atoms, the energy separation between the s and 2p atomic orbitals is greater (larger nuclear charges involved), and thus the "extra" mixing is weaker and no change in energy levels is observed.  The "bottom line" is that we have two generic molecular orbital energy ordering schemes to consider. One with "weak" s-p mixing, shown on the left, applies to "heavy-element" diatomics (O2, F2), and one with "strong" mixing, shown on the right, governs the the light-element diatomics (B2, C2, N2). Hetero-diatomic molecules are a little bit more complicated: CO for example follows the scheme on the right.

Well, what's next?  Now, we have to fill the molecular orbitals with all available valence electrons, starting with the lowest energy orbitals and moving up in such a way as to place each electron in the lowest-energy available orbital.  Each orbital may accommodate no more than two electrons (with opposite spins).


Let's start with N2 which has 10 valence electrons (5 from each nitrogen atom), filling the five lowest-in-energy orbitals in the scheme of the strong s-p mixing (on the right right above). Two degenerate π-type MOs (MO-4 and MO-5, from the picture at the top of the page) correspond to the two π bonds,  and the s-type MO (MO-3) serves as the HOMO. Two degenerate π* orbitals (MO-6 and MO-7 from the picture at the top) act as the LUMO's.  Formally counting the bonding (4 pairs) and antibonding (1 pair) electrons leaves us with the conclusion that dinitrogen has a triple bond, in agreement with the Lewis structure.

Indeed, the N-N bond lengths is only 1.098 Å (for comparison a typical N-N single bond is ca. 1.47 Å and an average N-N double bond is ca. 1.24 Å. This is a sign of a very strong bond (BDE = 226 kcal/mol), great stability and very low reactivity. That's why nitrogen is an "inert" gas.


It is oxygen's turn. Here we have 12 valence electrons (6 from each oxygen atom) filling the unperturbed orbital diagram (see above). Filling the lowest 5 MOs (MO-1 to MO-5) is a piece of cake. We have already done that. But we are left with two electrons, and we have two equal energy (degenerate) orbitals (MO-6 and MO-7) that we can use.  It turns out that the lowest energy arrangement is such that each electron ends up in a different orbital, and both electrons have the same spin (as shown). This behavior is in agreement with Hunds' rule (which is based in part on minimizing the electron-electron repulsion).  Thus, the antibonding MO-6 and MO-7 serve as the HOMOs (sometimes also called SOMOs for "Singly Occupied MOs") and the highest energy orbital (MO-8) is the LUMO. But the most important conclusion is that the oxygen molecule has two unpaired electrons!


Now formally counting electrons in bonding orbitals (4 pairs) and antibonding orbitals (1 pair and two single electrons) gives us the bond order of two, as represented in Lewis structure A.

This Lewis structure is not, however, exactly correct (that's a polite way to say it is wrong ).  It neglects to inform us of the two unpaired electrons, and it suggests the existence of a "normal" π bond.  Oxygen molecule has two unpaired electrons, and its π-system according to the MO picture is better described as two "half π-bonds".   Accordingly, a better Lewis structure would be that of B, with the unpaired electrons shown in red for emphasis.  The O-O bond length in O2 is 1.208 Å close to the expected value for a double bond (a typical O-O single bond is ca. 1.48 Å) and its BDE is 119 kcal/mol. The two unpaired electrons make O2 a magnetically active substance. It is attracted into a magnetic field (it is paramagnetic). Substances without unpaired electrons are weakly repelled from the magnetic field (they are diamagnetic).

The most important consequence of the electronic structure of O2 is its reactivity. This molecule is a diradical, its two electrons need pairing.   Dioxygen participates in numerous radical chain reactions, the combustion being a violent manifestation of its reactivity. It also easily accepts electrons (that's where the word "oxidation" comes from), and is, thus, responsible for providing energy for many living organisms (us included).  The other side of the coin is that oxygen-derived radicals may be responsible for cellular damage implicated in aging. Yes..., breathing may be hazardous to your health.

And the final word about molecules with unpaired electrons. The unpaired electrons behave as little magnets. In the magnetic field their spins may align themselves with the magnetic field (the lowest energy state) or against the magnetic field (the highest energy state).  A molecule with two unpaired electrons can exist in three states (both spins up aligned with the magnetic field, one up and one down, or both spins down against the magnetic field). We call such a molecule a triplet.   A molecule with one unpaired electron can exist in two states (spin up, or spin down) and it is called a doublet. All (mono)radicals are doublets.  A molecule with all paired electrons is called a singlet.  The O2 molecule in the air around us is a triplet.

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