Section 15-7 An Example: Polyacetylene with Uniform Bond Lengths

For noncyclic, one-dimensional systems, then, eigenfunctions have the form ψk = exp(ikx)Uk(x), −π/a < k ≤ π/a

(15-16)

Just as was true for cyclic cases, we can approximate this Bloch functional form with a Bloch sum:

n−1

ψk = exp(iksa)Uk(sa), −π/a < k ≤ π/a

(15-17)

s=0

In chemistry, it is Bloch sums that are normally used.

Two- or three-dimensional periodic systems can be treated similarly, with separate k vectors for each of the independent translational directions.

15-6 A Retrospective Pause

We have reviewed some familiar systems and presented Bloch functions and sums for one-dimensional cyclic and linear periodic systems. The practical lesson to this point is that one can generate useful one-electron wavefunctions for some such systems simply by choosing a basis set of AOs that is identical from one unit cell to the next and then modulating that set with exponential (or sine and cosine) functions. This is most easily done by using the exponential or trigonometric functions as “coefﬁcient waves” from which coefﬁcient values are plucked at points where a unit cell basis set is centered (i.e., by taking Bloch sums). In a case like the π electrons of benzene, each unit cell is a carbon atom and the basis set is a 2pπ AO at each atom. The coefﬁcients determined by exp(ij φ) with j = 0, ±1, ±2, 3 serve to deﬁne the wavefunctions completely. Once these are known, the energy values follow. In cases where several AOs are present in each unit cell the situation is complicated because the mix of these AOs may change as j (or k) changes. This added complexity is easily handled by computer programs working within the paradigm of the variational method. What is important for us to understand as chemists is what the computer is doing in such cases, why it is doing it, and what it means for the chemical and physical properties of the system.

15-7 An Example: Polyacetylene with Uniform Bond Lengths

Consider the molecule produced by addition polymerization of acetylene. The standard chemical representation for the all trans version of the molecule is shown in Fig. 15-7a.

The resonance diagram makes it plausible that this polymer, like benzene, will have equal C–C bond lengths, intermediate in value between the lengths of single and double bonds. For the moment let us assume this to be true and examine the MOs for the π orbitals of this system. To keep matters simple, we begin by treating the system at the simple H¨uckel level. This means that the hydrogen atoms can be ignored. Also, since only nearest-neighbor π interactions are accounted for, we can pretend that the carbon framework is linear (Fig. 15-7b).

Chapter 15 Molecular Orbital Theory of Periodic Systems

Figure 15-7 (a) All-trans polyacetylene in its two “resonance structures.” (b) The linear model appropriate for a nearest-neighbor, π -only calculation.

First we must choose our unit cell. The simplest choice is to make it a single carbon atom with the characteristic translational distance being a C–C bond length, shown as a in Fig. 15-7b.

Next we must choose the periodic basis set Uk(x) for the π MOs. The normal choice for this (and also the choice dictated by the simple H¨uckel method) is a 2pπ AO on each carbon. Furthermore, it is normal to assume that this basis does not change as we go from one MO to another, so we can dispense with the subscript k on U .

(Hence, no variational calculation will be necessary in ﬁnding the wavefunctions for this system.)

Equation (15-17) tells us how to produce wavefunctions. We simply take exp(ikx) times our basis set, with k taking on all values between −π/a and π/a, and pick off values at discrete points corresponding to carbon atom positions. Since it is more convenient to work with the real forms of solutions, we in effect choose a pair of k values (e.g., −π/4a and π/4a) so that we can generate a pair of trigonometric coefﬁcient waves [cos(πx/4a) and sin(πx/4a)]. The π MOs for regular polyacetylene produced from Bloch sums are shown in Fig. 15-8 for selected values of k. These can be used to illustrate some important points:

1. The k = 0 situation merely reproduces the basis set.

2. As |k| increases, the coefﬁcient wave goes to shorter wavelength and more nodes.

3. Only one MO results from each of the ±π/a extremes of the k range—these are nondegenerate solutions.

4. The pair of MOs drawn for ±π/4a are degenerate, as are those for ±π/2a. There fore, they can be mixed, either to generate complex functions or else simply to shift the phase. For example, the pair at k =±π/2a can be added or subtracted to produce the equally valid alternative pair shown in Fig. 15-9. In this way a phase shift of any degree could be created. There is no requirement that the MOs be “lined up” in a simple way with the atoms, as we have done in Figs. 15-8 and 15-9. (It is, however, usually more convenient.)

Section 15-7 An Example: Polyacetylene with Uniform Bond Lengths

Figure 15-8

Coefﬁcient waves times a periodic basis set of one 2pπ AO on each carbon. (a) k = 0, so the coefﬁcient wave is a constant and the periodic basis is unmodulated. (b)–(d) Plus and minus k-value exponentials are mixed to give trigonometric coefﬁcient waves. In (d), one of these waves has nodes at every carbon so no Bloch sum function exists for this case.

Once we have the MOs, also sometimes called “crystal orbitals” (COs) for inﬁ nite systems, we can consider their energies. In general, these would come from computations involving nuclear-electronic interactions, kinetic energies, etc. But the simple H¨uckel method takes the much easier approach of relating energy to bond order.

Chapter 15 Molecular Orbital Theory of Periodic Systems Figure 15-9 An alternative to the Bloch sums shown in Fig. 15-8c. These have the same coefﬁcient wavelength but the wave is displaced by a/2.

(See Section 8-9.) We can see at once that the MO at k = 0 is totally bonding. By analogy with the totally bonding MO of benzene, its energy is α + 2β. The pair of MOs due to k = ±π/2a are nonbonding. This is obvious from Fig. 15-8, but is also not hard to see in Fig. 15-9, which shows the MOs as having equal numbers of bonding and antibonding interactions. These MOs, then, have E = α. Finally, the totally antibonding MO at k = π/a has an energy of α − 2β. (Note that the pairing theorem holds. Compare the coefﬁcients of this MO with those at k = 0.)

EXAMPLE 15-2 What is the energy of an MO in Fig. 15-8 having k = ±π/4a?

SOLUTION The ﬁgure shows that these are triads of atoms having bonding interactions within themselves. Interactions between triads are nonbonding, since only nearest-neighbor interactions are taken into account in the HMO method. So E lies between α and α + 2β. Within a triad, the

√

√

coefﬁcient ratios (if we choose the cosine set) go as cos(−π/4), cos(0), cos(π/4) or 2/2, 1, 2/2.

√

√

√

Normalizing for a triad gives 1/2, 1/ 2, 1/2. Eq. (8-58) gives E = α + 2β(1/2 2 + 1/2 2) = α + 1.414β. This is the same as the result we obtained for the allyl radical in Section 8-6, which makes sense since, because they are nonbonding to their neighbors in this CO, they behave like isolated allyl systems.

The H¨uckel π MO energies for regular polyacetylene can be plotted versus k. The result appears in Fig. 15-10 along with a plot of the density of states (DOS). Each carbon atom brings one π electron to the polymer, and these ﬁll the lower-energy half of the MOs, so the highest occupied MO (HOMO) for the polymer is at E = α, |k| = π/2a.

Because the CO energies are identical for |k| and − |k|, we can observe all the unique energies by plotting over the range 0 ≤ k ≤ π/a. The k-range that produces all unique energies (rather than all the independent wavefunctions) is called the reduced ﬁrst
Brillouin zone (RFBZ).

The term Fermi energy is often used in reference to electrons having the highest energy in the ground state of the system. Here, if we ignore the effects of thermal energy, the Fermi energy is the HOMO energy. When there is a gap between the HOMO and LUMO energies, there is some disagreement in use of this term. Physicists, for good theoretical reasons, place the Fermi level midway between HOMO and LUMO energies; chemists often continue to equate it to the HOMO energy.

Section 15-7 An Example: Polyacetylene with Uniform Bond Lengths α – 2β

Fermi

energy

E

α

E

E

α + 2β

0

π/a

0

– 0 +

k

DOS

COBO (a)

(b)

(c)

Figure 15-10 (a) Energy versus k for a chain of π AOs as calculated by the simple H¨uckel method with Bloch sums. (b) Sketch of the variation of density of states versus energy. (c) Sketch of the crystal orbital bond order (COBO) function. Solid line is bond orders between neighbors.

Dashed line is bond orders between second-nearest neighbors (reduced by an arbitrary factor of ﬁve to reﬂect smaller extent of overlap between π AOs on second-nearest-neighbor atoms). (Adapted from Hoffmann et al. [1].)

It is apparent that the energy span of the π band depends on the magnitude of β and that this in turn should depend on a, the distance between the carbon atom unit cells.

The simple H¨uckel method in fact assumes β to be roughly proportional to AO overlap. (The extended H¨uckel method makes the assumption explicit.) This means that the π band of polyacetylene should become “wider” as the distance between carbons decreases. Standard terminology is to refer to the span between lowest and highest energies in a band as the band width, even though normal graphical representations show this as a vertical distance.)

There is yet another graphical representation of the relationships within these MOs that is useful to chemists. This is a plot of the amount of Mulliken overlap population between AOs as a function of E. This quantity, called crystal orbital overlap population (COOP), allows one to see at a glance how the bonding interactions change in a band. The simple H¨uckel analog of COOP is the crystal orbital bond order (COBO).

Figure 15-10c shows the COBO for the linear polyacetylene π system as a function of k. The lower-energy COs have a high degree of net bond order, but this falls off to become zero at E = α and then negative at higher energies. This is just what we know must be happening. The H¨uckel MO energies are, after all, directly proportional to bond order, so low energy must go with large positive net bond order, etc. Also, the pairing theorem holds, so the nearest-neighbor COBO curve must behave antisymmetrically through E = α. Although this particular COBO curve is not especially subtle, it is a good ﬁrst example because we understand so well what it is telling us.

A useful feature of COOP or COBO curves is that one has complete freedom as to which AOs (or groups of AOs) one can look at in this way. For example, the dashed line in Fig. 15-10c shows the overlap populations between π AOs on second-nearest
neighbors in linear polyacetylene. (Even though all overlaps are formally assumed to be zero in the H¨uckel method, once we have the MOs we are at liberty to calculate the overlaps that actually exist between the AOs in these MOs.) We see that the 1,3 interactions start out with positive overlap at low E, decrease to a negative value at E = α, and then rise to a positive value at high energy. This behavior is less obvious than the 1,2 COOP curve, though examination of the MOs at k = 0, π/2a, and π/a Chapter 15 Molecular Orbital Theory of Periodic Systems in Fig. 15-8 makes sense of it. (COOP curves are often shown without an explicit numerical overlap scale because they are usually used for qualitative comparisons.)

EXAMPLE 15-3 Compare Fig. 15-8(c) to Fig. 15-9 with regard to second-nearest neighbor COBO values.

SOLUTION The values should be equal since these are equivalent orbitals. It is easy to see that the value should be negative in Fig. 15-8(c) because half of the next-nearest-neighbor coefﬁcient products are negative, and the other half are zero. In Fig. 15-9, they are all negative. Because the

√

normalization constant in 15-8(c) is larger by a factor of 2, the sum is the same in each case.

These are nonbonding orbitals, so their energies equal α. Fig. 15-8(c) shows that the corresponding second-nearest-neighbor COBO is indeed negative.

The π system of polyacetylene is a convenient ﬁrst example. It provides elementary examples of the unfamiliar (to chemists) band, DOS, and COOP plots. Also, the MOs are trivial to generate because they are identical to the Bloch sums of the basis functions.

We have already seen similar behavior in the context of diatomic molecules, where the analog to a Bloch sum is a symmetry orbital (SO). (See Chapter 7.) In the case of H2, we saw that use of a minimal basis set ( 1sA and 1sB) gives two symmetry orbitals (1sA ± 1sB) and that these are identical to the MOs for H2. However, when we go to larger basis sets (1s, 2s, 2px, 2py, 2pz on each center) we ﬁnd that symmetry orbitals may no longer be the same as MOs—some MOs are mixtures of symmetry orbitals of like symmetry (e.g., the 2sσg SO mixes with the 2pσg SO). We will now explore a case where the analogous mixing of Bloch sums occurs for a polymer.

Consider what happens to our treatment of the linear chain of carbon atoms when we use an all-valence basis set (2s, 2px,y,z) on each carbon. Since our chain lies along the x-coordinate, the symmetries of the 2s and 2px AOs is σ , while that for 2py and 2pz is π. We can use the extended H¨uckel (EH) method to deal with this basis. At very large internuclear distances, interactions between AOs on different atoms vanish, so our energy diagram becomes the simple two-level picture for isolated atoms (Fig. 15-11a).

As the atoms move closer together, bands of ﬁnite width begin to develop (Fig. 15-11b).

At ﬁrst, while the bands are still fairly narrow compared to the distance between E2s and E2p, we can describe them as almost pure s and pure p in character. That is, the MOs are essentially identical to Bloch sums of the basis set at this point. The 2s band rises in energy from a totally bonding set of 2s AOs to a totally antibonding set of 2s AOs.

The 2py and 2pz bands (degenerate) behave similarly. The 2px band looks peculiar in that it drops in energy as k increases. Sketching out the Bloch sums for 2px at k = 0 and π/a (Fig. 15-12) shows why this happens. The 2px AOs in their original basis set arrangement are antibonding. The k = 0 Bloch sum keeps them that way. At the other extreme, k = π/a and the Bloch sum reverses the sign of every second coefﬁcient, making all of the interactions bonding. This is just the opposite of what happens for the other bands. We say that the 2s, 2px, and 2py bands “run up” as |k| increases and that the 2px band “runs down.” The simple rule is that bands run up when unit cell basis functions are symmetric across the unit cell (so they have phase agreement on the two ends) and run down for antisymmetric basis functions. Note that the 2px band is wider than the 2py, 2pz band because the 2px orbitals overlap more strongly due to their orientation. Note also how the DOS plot (Fig. 15-11c) reﬂects the degeneracy of

Section 15-7 An Example: Polyacetylene with Uniform Bond Lengths

Figure 15-11 (a) Valence AO energies for an EHMO calculation of a linear chain of carbon atoms. (b) EHMO band diagram calculated at an interatomic distance of 220 pm. (c) DOS diagram and (d) COOP curve between adjacent carbons, for the same calculation. (Adapted from Hoffmann et al. [1].)

Figure 15-12 2px and 2py basis sets at k = 0, π/a.

the 2py, 2pz (or 2pπ ) band as a large area. Because the data leading to Fig. 15-11 are produced by the extended H¨uckel (EH) method, we see the characteristic magniﬁcation of antibonding effects (Chapter 10), leading to asymmetries in DOS and COOP curves.

The variational method underlying EH calculations on this system evaluates interac tions between various Bloch sums at each k value. Because the Bloch sums (BSs) for 2s and 2px AOs have the same symmetry (σ ), a nonzero hamiltonian matrix element may exist between them, permitting mixing. However, the higher energy of the 2px BS compared to the 2s BS discourages much mixing as long as the overlap between them is small. At large distances, the overlap is indeed small, so we see the relatively

Chapter 15 Molecular Orbital Theory of Periodic Systems “pure” bands of Fig. 15-11. Since the 2pπ band disagrees in symmetry with the σ BSs, it remains unmixed with them regardless of C–C distance.

EXAMPLE 15-4 How should the areas compare in the DOS of 2s and 2p regions in Fig. 15-11?

SOLUTION To the extent that these regions remain “pure,” we should have three times more area in the p regions because, with three times as many basis functions, there are three times as many states. (The integral over DOS gives the number of states.)

Recalculating the band diagram for all-valence linear carbon at a smaller C–C dis tance gives the band diagram in Fig. 15-13a. Several things have happened: The lowest band has dropped signiﬁcantly in energy at k = 0 and risen slightly at k = π/a, so it is wider. The π band has gotten slightly wider too. The highest band has shot up to very high energies and has undergone a noticeable change of shape: It has a large hump at intermediate k values that did not show at large C–C separation. What is responsible for these changes?

Even without BS mixing we should expect band widths to increase as a result of increased overlap between AOs on adjacent carbons. Hence, at ﬁrst glance the lowestenergy band could be simply a more spread out 2s band. This cannot be right, though, because the EH method on a pure 2s band would give an antibonding-caused energy rise at k = π/a that is larger than the bonding-caused energy drop at k = 0. Instead, we are seeing a smaller rise. Something must be happening to either enhance the drop at k = 0 or counteract some of the rise at k = π/a. The DOS plots with 2s and 2pσ portions shaded (Fig. 15-13b, c) indicate what is happening. We see that the lowest

Figure 15-13 (a) EHMO band diagram for a linear chain of carbon atoms separated by 140 pm.

(b) DOS plot, with 2s AO contributions shaded. (c) DOS plot with 2pσ AO contributions shaded.

(From Hoffmann et al. [1].)

Section 15-7 An Example: Polyacetylene with Uniform Bond Lengths

Figure 15-14 Band diagram for a linear chain of carbon atoms showing some COs and intended correlations. Positive and negative phases are indicated by shading or lack of shading. (From Hoffmann et al. [1].)

band is still primarily 2s for small k, but becomes mainly 2pσ at larger k. The highest band shows opposite behavior. It is mostly 2pσ at smaller k and 2s at larger k. Mixing is occurring between Bloch sums.

Another way to see these features is to sketch the MOs that the computer program reports for various k values. These are shown for k = 0, π/a in Fig. 15-14. It is obvious that the lowest band is all 2s at k = 0 and all 2pσ at k = π/a, whereas the highest band is exactly the reverse. The bands have switched character. The pure 2s band we were discussing, that should have gone to very high energy, is represented by a dashed line in Fig. 15-14, and the pure 2pσ band by another, and these dashed lines do just what we argued they should.

The variational program is ﬁguring out, at each value of k, how to get the minimum energy σ band, minimum-energy π band, etc. and is not seeking to maintain 2s or 2pσ purity in the bands. Evidently, for the lowest band, it ﬁnds pure 2s to be lowest in energy at k = 0 and pure 2pσ to be lowest at k = π/a. At intermediate k values it ﬁnds a mixture to be best.

The dashed lines in Fig. 15-14 are the bands for Bloch sums of pure 2s and 2pσ AOs.

They differ from the bands for variational COs, where the Bloch sums are mixed. Even though they are not real bands, these dashed lines are nevertheless useful for guessing in advance of calculation what a band diagram will look like. The dashed lines are sometimes referred to as intended correlations because they show where the band that starts out as 2s, for instance, “intends” to be at k = π/a. These two dashed lines are forced to cross at some value of k, but we have seen (Section 7-6) that such crossings