Molecular Orbital Theory (MOT) is a modern and complete approach to chemical bonding. In its simplest form, it is based on a small number of premises, listed below:

1. Covalency is important, at least to some extent, in all bonding interactions.

2. Molecules are considered to consist of a framework of nuclei and a series of 1-electron orbitals characteristic of the nuclear framework. These orbitals are called molecular orbitals, or MOs. Electrons are fed into these MOs in accord with the rules of Pauli and Hund, just as electrons are placed in the atomic orbitals of atoms. The electrons in MOs are considered to belong to the molecule as a whole, rather than to one particular nucleus. In MOT, there is no a priori assumption about the localization of electron pairs. In this sense, the theory is very different from Lewis theory, in which such localization is assumed from the start.

3. To obtain mathematical forms for the MOs, the Schrodinger equation is solved by assuming that the MOs can be approximated as linear combinations of atomic orbitals (LCAO) of the constituent atoms. The AOs are considered to mix, or overlap, to form the MOs.

A linear combination is simply a weighted sum of a series of AOs:

This approach to molecular orbital formation is called the LCAO-MO approximation.

There are 2 aspects of the overlap of atomic orbitals that must be addressed. These are the symmetry of overlap and the sign of overlap. We address symmetry first.

A. Symmetry of Overlap. There are 3 ways in which 2 orbitals, each on a separate atom, can overlap. The first overlap symmetry is sigma overlap, in which the two orbitals approach each other and overlap along the line joining the nuclei of the atoms (the internuclear axis). The second is called pi overlap, and involves 2 lobes of each AO. Overlap occurs above and below the plane containing the internuclear axis, but not along it. In pi overlap, the plane containing the internuclear axis and perpendicular to the plane of the overlapping orbitals is a nodal plane. Finally, delta overlap involves 4 lobes of each overlapping AO, and produces two nodes each of which contains the internuclear axis. Delta overlap is important for compounds of the transition metals, which frequently use d orbitals in bonding. It will not concern us in this course. The 3 overlap symmetries are presented in Figure MO-1. It should be clear that two s orbitals can overlap in sigma fashion only; two p orbitals can overlap in either sigma or pi symmetry depending on their relative orientations; and two d orbitals can overlap in sigma, pi, or delta fashion, again depending on relative orientation. As a general rule, sigma overlap is more extensive and therefore stronger than pi overlap, which in turn is stronger than delta overlap. We will make use of these relative overlap strengths later in construction of MO energy level diagrams.

B. Sign of Overlap. The sign of overlap can be positive (+), negative (-), or zero (i.e., no overlap). Overlap is positive when the overlapping lobes of the two AOs have the same sign. This type of overlap is analogous to constructive interference in wave interaction, and leads to a buildup of probability density between nuclei. Due to this buildup, the MO has increased stability relative to the non-overlapping AOs. The MO formed by positive overlap of two AOs is called a bonding MO, because electrons in it will be localized between nuclei and will tend to hold the nuclei together. Positive overlap of 2 s orbitals is illustrated in Figure MO-2. Overlap is considered to be negative when the overlapping lobes have opposite sign. This type of overlap leads to destructive wave interference and results in a depletion of probability density between nuclei and a decrease in stability relative to the non-interacting AOs. An MO formed by negative overlap is called an antibonding MO because electrons in it will avoid the region between the nuclei, leading to nuclear repulsion and the opposite of what we would normally consider to be a good bonding situation. Cancellation of probability density between nuclei leads to a node, as shown for the antibonding overlap of two s orbitals in Figure MO-2. Finally, zero overlap results when there are equal amounts of ++ and +- overlap between 2 orbitals. In this situation, the AOs do not interact and are said to be nonbonding toward one another. A nonbonding interaction between an s orbital on one atom and a p orbital on another is shown in Figure MO-2. Two orbitals which have zero overlap with each other are said to be orthogonal.

C. The Basic 2-Orbital MO Diagram. Suppose that we have two atoms, A and B, each with one valence orbital. Suppose further that these orbitals have proper symmetry to overlap in sigma fashion. There are then two MOs which can be constructed using the LCAO-MO approach. The first MO can be obtained by adding the two AOs together, so that constructive interference occurs between atoms when lobes of the same sign overlap. This linear combination gives a bonding MO, y. The second MO can be obtained by subtraction of one AO from the other, so that destructive interference occurs between the atoms due to overlap of lobes of different sign. This linear combination of AOs gives an antibonding MO, y*. These so-called ++ (bonding) and +- (antibonding) linear combinations are pictured in Figure MO-3. MO theory emphasizes the relative energies of the MOs that result from the overlap of AOs on neighboring atoms. This information is summed up in an energy level diagram. The basic diagram for the overlap of 2 valence orbitals, one on atom A and one on atom B, is shown in Figure MO-3. Several features of this diagram are relevant to all MO energy level diagrams:

Now we apply these basic ideas to obtain MO diagrams for some simple molecules. In the examples which follow, we will make frequent use of the main idea behind our generic 2-orbital diagram: that whenever two orbitals on two neighboring atoms have suitable symmetry for overlap, they will combine together in 2 linear combinations, viz. their sum and their difference, to give bonding and antibonding MOs. Our approach throughout the following section will be qualitative--i.e., we will not attempt to actually solve the Schrodinger equation or to obtain any quantitative energies for the MOs. However, keep in mind that it is relatively easy these days to perform quantitative MO calculations using Molecular Modelling software (e.g., HyperChem), so that we could easily get the energies if we wanted them.

Section MO-2. Examples of the Molecular Orbital Approach

A. The Hydrogen Molecule. The hydrogen molecule provides a special case of the generic 2-orbital situation. Each H atom has 1 valence orbital (1s), and these have the same energy. Thus the MO diagram is as shown in Figure MO-4. The two electrons of the H₂ molecule are fed into the lowest energy MO according to the Pauli Principle. We end up with 2 electrons in the bonding MO and none in the antibonding MO. This situation is equivalent to having a single 2-electron bond. We define a quantity called the bond order (BO), or number of bonds, as one-half the difference between the number of bonding electrons and the number of antibonding electrons:

For H₂ the bond order is (2-0)/2 = 1. The electron pair is shared equally between the two H atoms because the two 1s orbitals contribute equally to the bonding MO. The bond is completely non-polar, and is thus a pure covalent bond. We can conclude that the bonding electron pair spends half of its time on each H atom. Finally, we note in the figure that y and y* are labelled by their symmetry properties as s and s*.

B. The HCl Molecule. Our first step in developing the MO picture of HCl is to set the molecule up in a convenient coordinate system. This is important for HCl because we recognize in advance that Cl will use p orbitals in bonding. These have directional character, of course, so it will be necessary for us to specify which p orbital(s) are involved. The usual convention is to orient the molecule so that the molecular axis coincides with the z axis of a Cartesian coordinate system, as shown in Figure MO-5. The valence AOs of the system are listed below. Our problem is to decide which orbitals of Cl can overlap with the hydrogen 1s orbital.

It is easy to tell by inspection that the 3s and 3p_z orbitals of Cl can overlap with 1s of the hydrogen atom. The relative orientations of the overlapping orbitals is indicated in Figure MO-5. It should be clear from the figure that the 3p_x,y orbitals of Cl have zero overlap (are orthogonal to) 1s of H. We conclude, therefore, that they will remain non-bonding in the HCl molecule. We label them ppⁿ . The superscript indicates that the orbitals are non-bonding. The orbitals are labelled as having p symmetry because of their sideways orientation to the H atom.

Having recognized which orbitals of the two atoms are allowed to interact, we can proceed to form MOs via overlap. We begin by taking an INCORRECT APPROACH, which, though wrong, is very tempting and consequently common. By explicitly doing it incorrectly, we raise consciousness about the potential errors in constructing MOs, so that the error is less likely to occur in future. It is very tempting to conclude, based on overlap considerations above, that 1s(H) overlaps 3s(Cl) to give a bonding and an antibonding MO, and that 1s(H) then overlaps 3p_z(Cl) to give a second bonding/antibonding pair. However, this is wrong, because 4 MOs are generated from only 3 AOs. Creation and/or destruction of orbitals is not allowed in Quantum Mechanics. Conservation of orbitals is the inviolable rule. So we conclude that we are allowed to generate only 3 MOs here, because we input only 3 AOs. A second, more physical reason for concluding that the 4 MO approach is incorrect is that we know from experience that there is only 1 bond in HCl, not the 2 bonds expected from the formation of 2 bonding MOs!

In order to approach the problem of MO formation correctly, we must ignore one of the two recognized overlap possibilities (i.e., 1s(H) with 3s(Cl) and with 3p_z(Cl)). To decide which overlap we may safely ignore, we list out what we know about the relative atomic orbital energies:

We now state a general principle that follows from a very large number of quantitative molecular orbital calculations. We make no attempt to prove it, but simply present it on faith. It will prove to be very useful in developing qualitative MO diagrams from here on out. The principle may be stated as follows: the extent of interaction of 2 AOs is inversely proportional to the energy difference (DE) between them. We thus expect less interaction between 3s(Cl) and 1s(H) than between 3p_z(Cl) and 1s(H). We assume that 3s(Cl) remains nonbonding, and call it ssⁿ. 1s(H) and 3p_z(Cl) then overlap to give bonding and antibonding MOs of the following form:

The final result is that we get 3 MOs -- ssⁿ, y y* -- from 3 AOs -- 3s,3p_z(Cl) and 1s(H). The MO energy level diagram is shown in Figure MO-6. We note that the non-bonding orbitals have the same energy in the molecule as in the originating atom. This will generally be true. Again, as for H₂, the MOs are labelled by their overlap symmetry, and non-bonding orbitals are labelled with a superscript n. The ps bonding orbital is polarized toward Cl because it is closer in energy to 3p_z(Cl) than to 1s(H). Similarly, ps* is polarized toward H. This is consistent with the idea that HCl is polar. The highest occupied MO, or HOMO, is pⁿ; the lowest unoccupied MO, or LUMO, is ps^*. The HOMO and the LUMO are very important in understanding electronic transitions, Lewis donor ability (because the electron pair which is donated is usually the HOMO pair) and Lewis acceptor ability (because the electron pair is usually accepted into the most stable available empty orbital of the acid). The bond order in HCl is (2-0)/2 = 1. Bond order considerations show that promotion of an electron from HOMO to LUMO in HCl weakens the H-Cl bond.

C. The BeH₂ molecule. In the gas phase, beryllium hydride consists of linear monomeric molecules, as expected from VSEPR theory. It appears to be somewhat more complex than H₂ and HCl because it contains 3 atoms rather than 2. However, if we consider it to be assembled from a beryllium atom and a hydrogen molecule, rather than 2 hydrogen atoms, it turns out to be easy to obtain the MO diagram. So here's the approach: we will assemble BeH₂ by inserting a Be atom into the H₂ molecule. We will then look for overlap between the valence AOs of Be and the valence MOs of H₂ (s and s*) in order to get the MOs of BeH₂.

Again we begin by orienting the molecule in a convenient coordinate system. As for HCl, we put the molecular axis along z, as in Figure MO-7. The valence orbitals of the system are as follows:

At least at first, overlaps may be more difficult to visualize when 3 atoms are involved, because now we are dealing with single orbitals that have lobes on more than one atom; specifically, s and s* of H₂ each have 2 lobes, one on one H atom and the second on the other H atom. In looking for successful overlaps of Be orbitals with the MOs of H₂, we look for similar distribution of + and - lobes in space. Overlap can be successful (i.e., nonzero) only if each + lobe of a Be orbital finds a + lobe of the H₂ MO and at the same time each - lobe of the Be orbital finds a - lobe of the H₂ MO to interact with. Based on this criterion, it should now be quite easy to visualize that the beryllium 2s orbital, 2s(Be), overlaps s(H₂) to give bonding and antibonding MOs. These overlaps and the resulting MOs are shown in Figure MO-7. Note that there are no nodes in the resulting bonding MO, called ss, and 2 nodes, one between the Be and each H, in the antibonding MO ss*. It is similarly easy to see that 2p_z(Be) overlaps s*(H₂) to give a second set of bonding and antibonding MOs, as shown in the figure. These are called ps and ps*. Finally, 2p_x,y(Be) have zero overlap with the MOs of H₂ and remain non-bonding in the beryllium hydride molecule. This pair of degenerate p-symmetry orbitals is labelled pⁿ. Counting up the MOs produced, we find 6, just as we should because we began with 6 AOs (4 from Be and 1 from each H; or 4 from Be and 2 from H₂). The energies of the 6 MOs are as shown in the energy level diagram in Figure MO-8. There are several notable features of this diagram:

The approach of "insertion" of an atom into H₂ can be used for any binary dihydride, linear or bent. To illustrate the bent case, we look next at water.

D. The Water Molecule. This system can be treated by partially inserting an oxygen atom into H₂ so as to form a "V". We place the resulting molecule in a coordinate system such that the z axis bisects the H-O-H angle and the plane of the molecule is in the xz plane. This is shown in Figure MO-9. The valence orbitals of the system are also shown in the figure. We proceed to look for overlaps of the orbitals of oxygen with the MOs of H₂. It should be clear that both 2s and 2p_z of oxygen are capable of overlap with s(H₂). This gives us the same 3-orbital situation (2 orbitals on one species overlapping 1 on the second species) that we encountered in HCl. This time, though, we won't make the common mistake. We know that we must get 3 MOs from the interaction of the 3 valence orbitals. We proceed by finding the 2 which are most likely to overlap and treat the third as nonbonding. Since c(O) > c(H), it follows that E(2s) < E(2p) < E(s). Therefore 2p and s are most likely to overlap. We assume 2s(O) to be non-bonding and label it ssⁿ. We then do a simple 2-orbital overlap of 2p_z and s to produce the MOs p_zs (bonding) and p_zs* (antibonding) shown in the figure. Note the two nodes in the antibonding orbital between the central oxygen atom and the two hydrogen atoms, consistent with the antibonding nature of this orbital. Proceeding with our search for overlaps, we see that 2p_x(O) overlaps s*(H₂) to give a bonding and an antibonding pair labelled p_xs and p_xs*. 2p_y(O) has zero overlap with s and s* of H₂, so it remains nonbonding in water and is labelled appropriately as p_yⁿ. We obtain, as we should, 6 MOs from 6 AOs. These are shown on the energy level diagram in Figure MO-10.

E. Homonuclear and Heteronuclear Diatomics. These are a bit more complex than the molecules that we have considered up to this point, because both atoms contribute a full set of s and p valence orbitals to the bonding. We begin by considering homonuclear species, in two stages, then proceed to the heteronuclear case.

1) Homonuclear diatomics, A₂, first approximation diagram. We locate the molecule along the z axis of a coordinate system. Each atom contributes valence orbitals ns and np_x,y,z for MO formation. To obtain the so-called first approximation diagram, we consider simple pair overlaps of like orbitals on the two atoms. Each pair overlaps to give y and y*, just as in H₂. This gives the energy level diagram in Figure MO-11. Note that in drawing the diagram, we have made use of the earlier-mentioned fact that s overlap is generally stronger and more extensive than p overlap. Consequently, the pp-pp* separation is less than that between ps and ps*. This diagram adequately accounts for the electronic structures of the elements late in a period, where the energy separation between the s and p valence orbitals is large. However, for the molecule, B₂, it predicts the configuration (ss)²(ss*)²(ps)². All electrons are expected to be paired up, with the consequence that the molecule is predicted to be diamagnetic. Similarly, C₂ is predicted to have 2 unpaired electrons, hence to be paramagnetic. The experimental facts, however, are that diboron is paramagnetic and dicarbon diamagnetic, just opposite to prediction. The implication of these observations is that for some reason the energy of the ps MO is higher than that of pp. How do we account for this?

2) Allowance for s-p Overlap. Our simple pairwise overlaps ignore the fact that the ns orbital of atom A₁ can overlap the np_z orbital of atom A₂, and vice versa. We can allow for this overlap by mixing -- adding and subtracting -- the MOs ss and ps. This will put sp overlap into both MOs. The effect of overlapping two MOs is just the same as overlapping two AOs: 2 new MOs are created, one higher in energy and one lower than the original pair. Mixing of ss and ps is shown in Figure MO-12. The major effect of the mixing is that the ps orbital (now p_ss) is raised above pp in energy. We obtain the new energy level diagram shown in Figure MO-13. You should convince yourself that in terms of this new diagram, B₂ is expected to be paramagnetic and C₂ diamagnetic, in agreement with experiment.

The major initial success of MO Theory, which led to its rapid acceptance by chemists, was its ability to rationalize both the double bond character and the paramagnetism (2 unpaired electrons) of O₂. The electron configuration for O₂, using either Figures MO-11 or MO-13, is

By Hund's Rule, the 2 electrons in pp* should remain unpaired, which accounts for the paramagnetism; and a simple bond order calculation gives a result of (6-2)/2 = 2. Contrast this quite natural result with the predictions of Lewis Theory, according to which dioxygen may be either doubly bonded and diamagnetic, or singly bonded and paramagnetic, but cannot be both doubly bonded and paramagnetic!

3) Heteronuclear diatomics, A-B. This is similar to the homonuclear case, except that the electronegativities of the bonded atoms are different. We assume that c_B > c_A, and again begin with simple pair overlaps of corresponding AOs on the two atoms. We obtain the diagram in Figure .MO-14. We note that the bonding orbitals are now biased toward the more electronegative atom B, whereas the antibonding orbitals are biased toward A. We also see that ss* is now a fairly high-energy orbital. As an example of the application of this diagram, we consider the very important CO molecule. According to the energy level diagram, its 10 valence electrons should be configured as follows:

This configuration implies a bond order of (8-2)/2 = 3 (1 sigma and 2 pi bonds). The lone pairs are located in ss, which belongs primarily to O, and ss*, which belongs primarily to C. The C lone pair in ss* is the HOMO. Therefore when CO functions as a Lewis base, it will donate the C lone pair, not the O lone pair.

F. One-Dimensional (Linear Chain) and Two-Dimensional (Cyclic) Molecules. What we have seen so far plus the results of many quantitative MO calculations reveal some useful general principles about MO energy and nodes:

1) Chain molecules. For a chain molecule consisting of n atoms, each contributing a single valence orbital for bonding, the following general statements can be made:

We can easily draw the LCAO-MOs for several chain sizes. For convenience, we use s orbitals. MOs for 2-atom through 6-atom chains are pictured in Figure MO-15. Note in the figure that by convention, the leftmost AO is always taken with positive sign. The first node is always in the center. As we move to the next higher MO, this node splits and moves out symmetrically toward the ends of the chain. New nodes are always added 1 at a time to the center.

As an example of the application of these ideas, we consider the p system of ozone, O₃, for which the Lewis structure is given in Figure MO-16. The pairs of electrons represented with dots are involved in the p bonding. There are therefore 4 electrons involved in the p system of the molecule. The molecule is bent, due to the three electron groups surrounding the central oxygen, but still constitutes a linear 3-atom chain. The p bonds of ozone will be formed from the three unhybridized p orbitals which are perpendicular to the molecular plane. From these three orbitals, and based on the results in Figure MO-15, we expect to get 3 p MOs, one bonding, one nonbonding, and one antibonding. These are pictured in Figure MO-16. The 4 p electrons (the p bond pair and one lone pair) are placed in the two lowest energy MOs, y and yⁿ. The net O-O p bond order is (2-0)/(2)(2) = 1/2 (the second factor of 2 in the denominator spreads the p bond over both O-O pairs). This particular bonding situation, in which 4 electrons, 2 bonding and 2 nonbonding, are spread over 3 atoms is quite common. It is referred to as three center, 4 electron (3c-4e) bonding.

2) Cyclic Molecules. For cyclic systems of n atoms, each contributing 1 orbital, we can make the following general statements:

From these generalizations, we can easily construct LCAO-MOs for various ring sizes. Again, for simplicity of visualization, we use s orbitals. For a 3-membered ring, there should be a single low energy bonding MO with no nodes, and a pair of degenerate antibonding MOs each with 1 node. These are pictured in Figure MO-17. A 4-membered ring should have a single zero-node MO, a pair of 1-node nonbonding orbitals, and a single 2-node antibonding orbital. For pictorial representations of these MOs, and for MOs of higher ring sizes, see Figure MO-17.

As an example of the application of these ideas, we can use them to treat the p systems of unsaturated cyclic hydrocarbons. To test your understanding of the concepts, you might like to try to construct the p MOs for benzene. The inorganic chemist is interested in MOs of cyclic systems because they can be used as a starting point for MO diagrams of AB_n-type molecules. The general approach is to insert the A atom into a ring (or more generally, a cluster) of n B atoms, much as we did for BeH₂ and H₂O. Let's illustrate the approach by applying it to ammonia, arguably one of the most important known inorganic materials.

3) The Ammonia molecule, NH₃. We can obtain MOs for this molecule by partially inserting a nitrogen atom into a (hypothetical) triangular H₃ molecule and looking for overlap of N orbitals with the MOs of H₃. The molecule is placed in a coordinate system with the z axis through N and the center point of the H₃ plane, and one N-H bond in the xz plane, as shown in Figure MO-18. The valence orbitals of nitrogen are, of course, 2s and 2p_x,y,z. The valence MOs of H₃ are shown in Figure MO-17 and are reproduced in Figure MO-18. It is clear that both 2s and 2p_z of N can overlap y₁ of H₃. But since c(N) > c(H), and therefore E(2s) < E(2p) < E(y₁), the main overlap will occur between 2p_z(N) and y₁(H₃). 2s(N) will be considered nonbonding and labelled ssⁿ. Simple 2-orbital overlap of 2p_z and yⁿ produces bonding and antibonding MOs labelled p_zs and p_zs*, as shown in the figure. Overlap of 2p_x(N) with y₂*(H₃) and of 2p_y with y₃*(H₃) are also shown. In all, 7 MOs are obtained from 7 AOs. The MO energy level diagram is presented in Figure MO-19. If we wish to refine it further, we can allow the neglected overlap of ssⁿ and p_zs to occur. This will have the effect of moving p_zs up above p_xys, consistent with the Lewis basicity of NH₃.

We close out this section by stressing the importance of thinking by analogy. MO diagrams which we have already developed can be used, at least qualitatively, for other isostructural species. For example, the cation BH₂⁺ is isoelectronic and isostructural with BeH₂ and can be discussed in terms of the MO diagram for the latter species. The amide ion, NH₂^-, is analogous to H₂O; and the imide ion, NH^2- , is analogous to HCl. This type of reasoning can save you much work. Thus, for example, if you were interested in analyzing the electronic absorption spectrum of phosphine, PH₃, you could do so in terms of the MO diagram for NH₃ already at hand, rather than develop a diagram from scratch.

Section MO-3. Band Theory of Solids.

One very important application of MO theory is in understanding the electronic structures of metals and of materials which behave as semiconductors. The latter materials are the basis for the modern electronics and computer industries. It turns out to be relatively simple to obtain at least a qualitative understanding of the energy levels in these materials, and therefore to understand a number of their physical and chemical properties.

A. Some definitions. One way in which we classify solid materials is by the magnitude and temperature dependence of their electrical conductivities. We recognize 3 classes, as in the following table:

An insulator is simply a semiconductor with a very low conductivity. We can rationalize these conductivity behaviors using MO theory.

B. MO Theory of Solids. Consider a linear chain of n identical atoms, each bringing in a valence s orbital for MO formation. If n = 2, 2 MOs are formed, one bonding and one antibonding. If n = 3, we obtain 3 MOs, bonding, nonbonding, and antibonding. If we run n right up to Avogadro's number, we expect to obtain N_o MOs, ranging from fully bonding (+++++...) to fully antibonding (+-+-+-...), with a whole bunch of other MOs between these extremes. This is shown in Figure MO-20. The energy spacing between lowest and highest MOs is determined primarily by the overlap between neighboring atoms, so will stay finite even though the number of atoms in the chain reaches toward the infinite! We thus have a huge number of MOs crammed into a finite energy interval. They will be so close together in energy that they will form, for all practical purposes, a continuous band of energy levels. For this reason the MO theory of solids is often called Band Theory. A very important result of this treatment is that each of the MOs in the band is delocalized over all of the atoms in the chain.

The same ideas apply to a 3-dimensional, close-packed aggregate of atoms. A band of MOs will be formed from each type of valence AO on the atoms. Thus we obtain an s band, a p band, a d band, and so on, as shown in Figure MO-20. All of the MOs are delocalized over all of the atoms in the aggregate, so electrons in them can be considered to be everywhere at once! The highest filled band of MOs is called the valence band; the lowest unfilled band is called the conduction band; and the energy separation between the top of the valence band and the bottom of the conduction band is called the band gap. A partially filled band is simultaneously the valence band and the conduction band, so in this case the band gap is essentially zero.

Now let's apply this picture to understand the electronic nature of the various classes of materials given above. A conductor (which is usually a metal) is a solid with a partially full band, as shown in Figure MO-21. An electron in the highest occupied MO is easily promoted to the next higher empty delocalized MO, where it is then free to roam over the whole solid lattice under the influence of an applied electric field; i.e., the solid conducts electricity due to this facile electron movement. The high reflectivity of metals is also due to the availability of a proliferation of empty MOs above the HOMO. Electrons in the filled MOs of the partially-filled band can absorb and then re-emit light of many wavelenths in making transitions to empty MOs in the band. This gives the metal surface a shiny reflective appearance. An example of a conductor is Na metal. It has an s band consisting of N MOs, where N is the number of Na atoms in the crystal. The band contains N electrons (one from each Na atom) arrayed in N/2 pairs. These N/2 pairs go in the N/2 bonding MOs, which leaves N/2 antibonding MOs empty but readily accessible. Thus Na exhibits the characteristic properties of a metal, and is a conductor.

An insulator is a solid with a full band and a large band gap, as shown in Figure MO-21. The MOs in the conduction band are so high in energy that they are not thermally populated by the Boltzmann distribution, and there is no conductivity at ordinary temperatures. An example of an insulator is solid carbon in the diamond modification. Diamond consists of a covalently bonded network of carbon atoms (a fcc array of C atoms with more C atoms in half the tetrahedral holes), constructed from sp³ hybrid orbitals. N carbon atoms contribute 4N sp³ hybrids, which overlap strongly to give 2N bonding MOs and 2N antibonding MOs which are separated in energy by 5.47 eV from the bonding MOs. The 4N electrons exactly fill the band of bonding MOs. The antibonding band is not thermally accessible, so diamond does not conduct.

A semiconductor is a solid with a full band and a small band gap, as shown in Figure MO-21. There is a small thermal population of the conduction band at normal temperature, hence a small conductivity. For example, silicon has a diamond modification similar to that of carbon, but a band gap of only 1.12 eV, due to poorer overlap of the sp³ hybrids of the larger Si atoms. Since the antibonding band will be occupied to a small extent via the Boltzmann distribution, Si exhibits a small conductivity at room T.

The group 4A elements, which have a number of valence electrons equal to twice the number of MOs in the bonding band, are uniquely structured to show semiconductivity. The elements C through Sn all exhibit a diamondlike crystal form, but with a band gap which decreases in magnitude for the larger atoms as orbital overlap becomes weaker. The trend in band gap down family 14 is shown below:

Thus Si and Ge are semiconductors at room T, and Sn is a conductor. Pure compounds which are electronically analogous to the group 14 elements are also semiconductors. These include the compounds boron nitride, BN, and gallium arsenide, GaAs. Note that these compounds contain one element from group 13 and one element from group 15, in a 1:1 stoichiometric ratio. They thus have exactly the same number of valence electrons as a group 14 element, and will arrange these electrons in a group-14 type band structure. They are often called 3-5 compounds, to indicate that they consist of elements taken from groups 13 and 15. Similarly, 2-6 compounds such as ZnS and CdS (both of which have the zincblende structure, which is analogous to the diamond structure) function as semiconductors. Generally, band gaps vary with position in the periodic table, but tend to decrease with increasing MW of the semiconductor.

The temperature dependence of conductivity is readily understood within the framework of band theory. For a conductor, promotion of electrons is facile within a band at any T. However, As T increases, vibrational motions of the metal atoms in the lattice increases and interferes with the motion of the conducting electrons. The result is a decrease in conductivity as T increases. For a semiconductor, an increase in T causes an exponential increase in the population of the conduction band, because of the Boltzmann distribution. Therefore the conductivity of semiconductors increases dramatically with T. Because an insulator is actually a semiconductor with a large band gap, the conductivity of an insulator should also increase markedly if the temperature is made high enough.

1.Below are several descriptions of overlaps. For each description,

In all cases, consider atoms A and B to lie on the z axis of a Cartesian coordinate system. Consider + lobes of orbitals to point in the positive z direction.

3. Draw a diagram to illustrate each described overlap:

4. Use MO theory to determine the bond order for He₂⁺.

5. How will the bond strength in HCl be affected by promotion of an electron from HOMO to LUMO?

6. How will the B-H bond length and bond strength in BH₂⁺ (a gas-phase fragment) be affected by promotion of an electron from HOMO to LUMO?

7. Determine the electron configurations, bond orders, and numbers of unpaired electrons for the species O₂⁺, O₂, O₂^-, and O₂^2-.

8. Consider the hypothetical gas phase molecule BH₃, which is trigonal planar. Starting with the AO's of B and the MO's of triangular (cyclic) H₃, obtain the MO diagram for BH₃.

9. Consider the hypothetical square planar hydride MH₄. Starting with the s and p valence AOs of M and the MOs of square (cyclic) H₄, obtain the MO diagram for MH₄.