Look at the computer-generated model of the nicotine molecule shown at the upper right corner of this window. The colors of the sculpted folds represent different electron densities— probabilities of finding an electron at any given point. The sculpting itself is created by the positive cores that remain after the “atoms” in this molecule have been stripped of their valence electrons, which are now able to range over the entire molecule, occupying orbitals whose spatial extent and properties depend entirely on the charges and geometric distribution of the collection of positive centers.

Welcome to the world of molecular orbitals !

This is a big departure from the simple Lewis and VSEPR models that were based on the one-center orbitals of individual atoms. The more sophisticated hybridization model recognized that these orbitals will be modified by their interaction with other atoms. But all of these valence-bond models, as they are generally called, are very limited in their applicability and predictive power, because they fail to recognize that distribution of the pooled valence electrons is governed by the totality of positive centers.

The molecular orbital model is by far the most productive of the various models of chemical bonding, and serves as the basis for most quantiative calculations, including those that lead to many of the computer-generated images that you have seen elsewhere in these units. In its full development, molecular orbital theory involves a lot of complicated mathematics, but the fundamental ideas behind it are quite easily understood, and this is all we will try to accomplish in this lesson.

Chemical bonding occurs when the net attractive forces between an electron and two nuclei exceeds the electrostatic repulsion between the two nuclei. For this to happen, the electron must be in a region of space which we call the binding region. Conversely, if the electron is off to one side, in an anti-binding region, it actually adds to the repulsion between the two nuclei and helps push them away.

The easiest way of visualizing a molecular orbital is to start by picturing two isolated atoms and the electron orbitals that each would have separately. These are just the orbitals of the separate atoms, by themselves, which we already understand. We will then try to predict the manner in which these atomic orbitals interact as we gradually move the two atoms closer together. Finally, we will reach some point where the internuclear distance corresponds to that of the molecule we are studying. The corresponding orbitals will then be the molecular orbitals of our new molecule.

The simplest molecule: hydrogen molecule ion

To see how this works, we will consider the simplest possible molecule, H₂⁺. This is the hydrogen molecule ion, which consists of two nuclei of charge +1, and a single electron shared between them.

Figure 8.67 Overlap of Atomic Orbitals

As two H nuclei move toward each other, the 1s atomic orbitals of the isolated atoms gradually merge into a new molecular orbital in which the greatest electron density falls between the two nuclei. Since this is just the location in which electrons can exert the most attractive force on the two nuclei simultaneously, this arrangement constitutes a bonding molecular orbital. Regarding it as a three- dimensional region of space, we see that it is symmetrical about the line of centers between the nuclei; in accord with our usual nomenclature, we refer to this as a σ (sigma) orbital.

Bonding and antibonding molecular orbitals

Orbitals are conserved

There is one minor difficulty: we started with two orbitals (the 1s atomic orbitals), and ended up with only one orbital. Now according to the rules of quantum mechanics, orbitals cannot simply appear and disappear at our convenience. For one thing, this would raise the question of at just what internuclear distance do we suddenly change from having two orbitals, to having only one? It turns out that when orbitals interact, they are free to change their forms, but there must always be the same number. This is just another way of saying that there must always be the same number of possible allowed sets of electron quantum numbers.

In-phase and out-of-phase wave combinations

How can we find the missing orbital? To answer this question, we must go back to the wave-like character of orbitals that we developed in our earlier treatment of the hydrogen atom. You are probably aware that wave phenomena such as sound waves, light waves, or even ocean waves can combine or interact with one another in two ways: they can either reinforce each other, resulting in a stronger wave, or they can interfere with and partially destroy each other. A roughly similar thing occurs when the “matter waves” corresponding to the two separate hydrogen 1s orbitals interact; both in-phase and out-of-phase combinations are possible, and both occur. The in-phase, reinforcing interaction yields the bonding orbital that we just considered. The other, corresponding to out-of-phase combination of the two orbitals, gives rise to a molecular orbital that has its greatest electron probability in what is clearly the antibonding region of space. This second orbital is therefore called an antibonding orbital.

Figure 8.68 formation of Bonding and antibonding MOs through electron cloud

When the two 1s wave functions combine out-of-phase, the regions of high electron probability do not merge. In fact, the orbitals act as if they actually repel each other. Notice particularly that there is a region of space exactly equidistant between the nuclei at which the probability of finding the electron is zero. This region is called a nodal surface, and is characteristic of antibonding orbitals. It should be clear that any electrons that find themselves in an antibonding orbital cannot possibly contribute to bond formation; in fact, they will actively oppose it.

We see, then, that whenever two orbitals, originally on separate atoms, begin to interact as we push the two nuclei toward each other, these two atomic orbitals will gradually merge into a pair of molecular orbitals, one of which will have bonding character, while the other will be antibonding. In a more advanced treatment, it would be fairly easy to show that this result follows quite naturally from the wave-like nature of the combining orbitals.

What is the difference between these two kinds of orbitals, as far as their potential energies are concerned? More precisely, which kind of orbital would enable an electron to be at a lower potential energy? Clearly, the potential energy decreases as the electron moves into a region that enables it to “see” the maximum amount of positive charge. In a simple diatomic molecule, this will be in the internuclear region— where the electron can be simultaneously close to two nuclei. The bonding orbital will therefore have the lower potential energy.

A molecule is assumed to be a new entity consisting of positively charged nuclei and electrons.

The electrons in the molecule are contained in the molecular orbitals, which is the simplest form of the model are constructed from the atomic orbitals of the constituent atoms.

The model correctly predicts relative bond strength, magnetism and bond polarity.

It correctly portrays electrons as being delocalized in polyatomic molecule. The main disadvantage of the model is that it is difficult to apply qualitatively to polyatomic molecules.

We will discuss more in detail about this theory with an example of H₂ moelcule

When two H nuclei lie near each other as in H₂, their AOs overlap. The two AOS of combining the AOs are as follows:

Adding he wave functions together: This combination forms a bonding MO, which has a region of high electron density between the nuclei. Additive overlap is analogous to light waves reinforcing each other, making the resulting amplitude

Subtracting the wave functions from each other: This combination forms an antibonding MO, which has a region of zero electron density ( a node) between the nuclei.

Figure 8.69 fFormation of Bonding and antibonding MOs

Ref: commons.wikimedia.org/

According to MO theory, a bonding MO is lower in energy than the atomic orbitals from which it is constructed. Electrons in this type of MO are lower in energy in the molecule than in the separated atoms and thus favor molecule formation.

An antibonding MO is higher in energy than the atomic orbitals from which it is constructed. Electrons in this type of MO are higher in energy in the molecule than in the separated atoms and thus do not favor molecule formation.

σ MOs have their electron probability centered on a line passing through the nuclei.

П Mos have their electron probability concentrated above and below the line connecting the atoms.

Bond order=  # of bonding MOs-# of Antibonding MOs
                                                            2

Bond order is measure of bond strength.

Watch the following video:

Understanding Molecular Orbital Theory

Filling Molecular Orbitals with Electrons: Electrons fill MOs just as they fill AOs.

MOs are filled in order of increasing energy ( Aufbau Principle)

An MO has a maximum capacity of two electrons with opposite spins (exclusion principle)

Orbitals of equal energy are half-filled, with spins parallel, before any of them is completely filled ( Hind’s Rule).

Understanding molecular orbital diagrams

Homonuclear Diatomic Molecules of Period 1 elements

The bonding MO in H₂ is spread mostly between the nuclei, with the nuclei attracted to the intervening electrons, An electron in this MO can delocalize its charge over a much larger volume than it is possible. Because the electron density-electron repulsions are reduced, the bonding MO is lower in energy than the isolated AOs. Therefore when electrons occupy this orbital, the molecule is more stable than the separate atoms. In contrast, the antibonding MO has a node between the nuclei

Molecular Orbital Theory II: MO’s of the H2 Molecule

Figure 8.70 MO Diagram of H₂

Ref: commons.wikimedia.org/

The diagram below is for He₂ molecule. Bond order indicates that He₂ doesn’t exist.

Figure 8.71 MO Diagram of He₂

Ref: commons.wikimedia.org/

Figure 8.71 MO Diagram of H₂ ⁺

This scheme of bonding and antibonding orbitals is usually depicted by a molecular orbital diagram such as the one shown here for the dihydrogen ion H₂⁺. Atomic valence electrons (shown in boxes on the left and right) fill the lower-energy molecular orbitals before the higher ones, just as is the case for atomic orbitals. Thus, the single electron in this simplest of all molecules goes into the bonding orbital, leaving the antibonding orbital empty. Since any orbital can hold a maximum of two electrons, the bonding orbital in H₂⁺ is only half-full. This single electron is nevertheless enough to lower the potential energy of one mole of hydrogen nuclei pairs by 270 kJ— quite enough to make them stick together and behave like a distinct molecular species. Although H₂⁺ is stable in this energetic sense, it happens to be an extremely reactive molecule— so much so that it even reacts with itself, so these ions are not commonly encountered in everyday chemistry.

Bonding in dihydrogen

If one electron in the bonding orbital is conducive to bond formation, might two electrons be even better? We can arrange this by combining two hydrogen atoms– two nuclei, and two electrons. Both electrons will enter the bonding orbital, as depicted in the Figure. We recall that one electron lowered the potential energy of the two nuclei by 270 kJ/mole, so we might expect two electrons to produce twice this much stabilization, or 540 kJ/mole.

Figure 8.72  Energy correlation in MO Diagram of H₂ ⁺

Bond order is defined as the difference between the number of electron pairs occupying bonding and nonbonding orbitals in the molecule. A bond order of unity corresponds to a conventional “single bond”.

Figure 8.73 MO Diagram of H₂

Experimentally, one finds that it takes only 452 kJ to break apart a mole of hydrogen molecules. The reason the potential energy was not lowered by the full amount is that the presence of two electrons in the same orbital gives rise to a repulsion that acts against the stabilization. This is exactly the same effect we saw in comparing the ionization energies of the hydrogen and helium atoms

Figure 8.74 MO Diagram of He₂ ⁺

***MO’s in dihelium

With two electrons we are still ahead, so let’s try for three. The dihelium positive ion is a three-electron molecule. We can think of it as containing two helium nuclei and three electrons. This molecule is stable, but not as stable as dihydrogen; the energy required to break He₂+ is 301 kJ/mole. The reason for this should be obvious; two electrons were accommodated in the bonding orbital, but the third electron must go into the next higher slot— which turns out to be the sigma antibonding orbital. The presence of an electron in this orbital, as we have seen, gives rise to a repulsive component which acts against, and partially cancels out, the attractive effect of the filled bonding orbital.

Taking our building-up process one step further, we can look at the possibilities of combining to helium atoms to form dihelium. You should now be able to predict that He₂ cannot be a stable molecule; the reason, of course, is that we now have four electrons— two in the bonding orbital, and two in the antibonding orbital. The one orbital almost exactly cancels out the effect of the other. Experimentally, the bond energy of dihelium is only .084 kJ/mol; this is not enough to hold the two atoms together in the presence of random thermal motion at ordinary temperatures, so dihelium dissociates as quickly as it is formed, and is therefore not a distinct chemical species.

MO’s in diatomic molecules containing second-row atoms

The four simplest molecules we have examined so far involve molecular orbitals that derived from two 1s atomic orbitals. If we wish to extend our model to larger atoms, we will have to contend with higher atomic orbitals as well. One greatly simplifying principle here is that only the valence-shell orbitals need to be considered. Inner atomic orbitals such as 1s are deep within the atom and well-shielded from the electric field of a neighboring nucleus, so that these orbitals largely retain their atomic character when bonds are formed.

MO’s in dilithium

Figure 8.75 MO Diagram of Li₂

For example, when lithium, whose configuration is 1s²2s¹, bonds with itself to form Li₂, we can forget about the 1s atomic orbitals and consider only the σ bonding and antibonding orbitals. Since there are not enough electrons to populate the antibonding orbital, the attractive forces win out and we have a stable molecule. The bond energy of dilithium is 110 kJ/mole; notice that this value is less than half of the 270 kJ bond energy in dihydrogen, which also has two electrons in a bonding orbital. The reason, of course, is that the 2s orbital of Li is much farther from its nucleus than is the 1s orbital of H, and this is equally true for the corresponding molecular orbitals. It is a general rule, then, that the larger the parent atom, the less stable will be the corresponding diatomic molecule.

Bonding of the Period 2 s-block Diatomic molecule

Both Li and Be occur s metal under normal conditions, but MO theory predict their stability as dilithium and diberyllium.

These molecules have both inner 1s and outer 2s orbitals. Is orbitals are core orbitals. We ignore inner or core 1s orbitals since only valence orbitals interact enough to form molecular orbitals. Like the MOS formed from 1s AOs, those formed from 2s AOs are also σ orbitals, cylindrically symmetrical around the internuclear axis. Bonding σ_2s and antibonding σ*_2s MOS form and valence electrons occupy bonding σ2s MO with opposite spins for lithium molecule and σ*2s MO for Be­₂ molecule. Bond oder in Li₂= 1/2(2-0)= 1 and bond order for Be₂= ½(2-2)=0. Therefore, ground state of Be₂ molecule doesn’t exist.

Molecular orbitals from the atomic p-orbital combination:

As we move from Be to B, atomic 2p orbitals become involved. We will first consider the shapes and energies of the MOs resulting from their combinations. P orbital can overlap in two ways, end to end combination will give a pair of σ MOs, σ2p and σ*2p.  Side to side combination will give a pair of п MOs. п _2p and п*_2p. Similar to MOs formed from s orbitals, bonding MOs from p-orbital combinations have their greatest electron density between the nuclei whereas antibonding MOs from p-orbital combinations have a node between the nuclei and most of their electron density outside the internuclear region.

• Ms formed from 2s orbitals are lower in energy than MOs formed from 2p orbitals because 2s AOs are lower in energy than 2p AOs.
• Bonding MOs have lower in energy than anti-bonding MOs. Therefore σ2p is lower in energy than σ*2p and п2p is lower in energy than п*2p.
• p-orbitals can interact more extensively end to end than side to side. Thus σ2p is lower in energy than п2p.  Similarly, destabilizing effect of  σ*2p MO is greater than that of the п*2p MO.

Energy order of MO derived from p orbital:

σ2p< п2p< п*2p< σ*2p

When all six p –orbitals combine to form MOs, two of the six combine end to end to produce sigma and sigma* MOs. Other four p-orbitals produce two pi-bonding MOs and two antibonding MOs.Two pi-bonding MOs are degenerate as well as two pi-antibonding MOs.

But since B, C and N atoms are larger and p orbitals are not completely filled up, atomic orbitals repel each other and energy of 2p orbitals become closer to the energy of the 2s orbital. As a result, some mixing occurs between the 2s orbital of one atom and the end-on 2p orbital of another atom. This orbital mixing lowers the energy of the σ2s and σ*2s MOs and raises the energy of the σ2p and σ*2p MOs. That creates dengerate п2p MOs lower in energy than σ2p.

Figure 8.76 Degenerate Pi orbitals

Ref: commons.wikimedia.org/

After Be , porbitals are involved in MO formation. Here is the MO diagram for B₂ molecule including 1 2p electron from each atom.  Possible energy levels look like the following diagram.

Figure 8.76 MO diagram for Li₂-N₂

Ref: commons.wikimedia.org/

Following the above energy scheme: B₂ molecule

Figure 8.77 MO diagram for B₂

Ref: commons.wikimedia.org/

Following diagram is for N₂ molecule

Figure 8.78 MO diagram for N₂

Ref: commons.wikimedia.org/

Bonding in the p-block homonuclear diatomic molecules

The B₂ molecule has six outer electrons to place in its MOs. Four of these fill σ_2s and σ*_2s MOs. The remaining two electrons occupy the two п_2p MOs, one in each orbital, in keeping with Hund’s rule. With four electrons in the bonding MOs and two electrons in the antibonding MOs, the bond order of B₂ is ½(4-2)=1, therefore B₂ is diamagnetic.

The two additional electrons present in C₂ will fill the п_2p. Since C₂ has two more bonding electrons thanB₂, it has bond order of 2, and bond is expected to be stronger, shorter. With all electrons paired, C₂ is diamagnetic.

In N₂, two additional electrons fill the σ2p MO. The resulting bond order is 3.  According to theory, bond energy is higher, bond length is shorter and N₂ is diamagnetic.