Section 15-5
Section 15-5 General Form of One-Electron Orbitals in Periodic Potentials
for benzene MOs, are also shown (at the positions labeled 1–6), and it is obvious that the same set of coefficients is produced by each function. Increasing j has caused the coefficient wave to oscillate with a shorter wavelength, but the extra oscillations do not register because they occur between the special points of interest where we are sampling the function (i.e., at the points where the carbon nuclei reside). In other words, the extra oscillations in our benzene coefficient functions are just “empty wiggling” and have no effect on the MO, so we cut off the j range where the meaningful wiggling starts to become empty wiggling. The reason that the particle in a ring does not show this behavior is that the exponential (or sine, cosine) functions in that system actually are
the wavefunctions, while in benzene these functions are sampled only at discrete points where AOs are located, and the resulting coefficients are then used to produce MOs.
Notice that the MOs for benzene as given by Eq. (15-4) are produced from an equation having the following form: There is an exponential term that, for each MO, supplies the coefficients for various carbons in the molecule, and there is a basis set of functions located on the various carbons. This basis set has the same “periodicity” as does the molecule. (In the case of benzene, we have so far taken this to be six identical 2pπ AOs.) Wavefunctions for all periodic systems have this same form—an exponential
(or equivalent trigonometric) expression times a periodic basis. This is the content of Bloch’s theorem, which we prove in the next section.
EXAMPLE 15-1 What coefficients are generated for a benzene MO by a coefficient wave of the type shown in Fig. 15-6 with j = −3? j = 0?
SOLUTION The arguments of the cosine function of Eq. (15-6) will be the negative of those of the j = 3 case. Because the cosine is symmetric about an argument of zero, we obtain the same curve for j = −3 as for j = +3, hence the same MO coefficients. This demonstrates that the MO generated by j − 6 is the same as that generated by j . For j = 0, the cosine arguments all vanish, the cosines all equal +1, and the curve of Fig 15-6 becomes a straight line at a coefficient value of
√
+1/ 6. This produces the lowest-energy MO of benzene.
15-5 General Form of One-Electron Orbitals in PeriodicPotentials—Bloch’s Theorem
We will establish in this section the general mathematical nature of eigenfunctions for periodic hamiltonians. This is the content of Bloch’s theorem. The actual proof will be carried through for cyclic “periodic” systems, and those results will then be extended to noncyclic periodic systems. The rationale for this is that a nearly infinite linear periodic structure can be treated as cyclic without introducing error. That is, a sufficiently long extended chain of atoms can be assumed to have the same wavefunctions and energies as the same chain joined end to end to form a cycle. For short chains and rings, this is not the case; the lowest-energy MO for hexatriene is not as low in energy as that for benzene, nor is it uniform over the whole chain (as it is in benzene). But for long enough chains, the difference becomes negligible.
Bloch’s theorem, as it applies to periodic cycles, states that eigenfunctions have the form
j (φ) = exp(ij φ)Uj (φ)
(15-7)
Chapter 15 Molecular Orbital Theory of Periodic Systems
√
where i = −1, j is an integer and Uj (φ) has the periodicity of the cycle. If the cycle belongs to the Cn point group, then U (φ) = U (φ + 2π/n). Functions of the form of Eq. (15-7) are called Bloch functions. It was stated in Section 13-8 that every wavefunction or orbital is a member of a basis for an irreducible representation for the point group of the system, so it follows that the wavefunctions we seek to characterize have to be bases for irreducible representations of the Cn point group. These groups are discussed in Section 13-11, where it is shown that the functions = exp(iφ) and ∗ = exp(−iφ) are bases for irreducible representations for these groups, which is to say that they are eigenfunctions for the rotation operator, no matter what the value of the integer n in Cn. We consider the more general functions exp(±ij φ) with j an integer (so that the function obeys the cyclic continuity condition) and ask what happens when
q
such functions are subjected to a rotation. Let the rotation operator be symbolized Cn for a rotation of q times 2π/n. The n-fold cyclic system is invariant to this rotation if q is an integer. It is not difficult to show that (see Section 13-11)
q
Cnf (φ) = f (φ − 2πq/n)
(15-8)
(For example, clockwise rotation by 60◦ brings to each point in the circle the value that used to be 60◦ in the counterclockwise direction.) Then
q
Cn exp(ij φ) = exp[ij (φ − 2πq/n)] = exp(−2πiqj/n) exp(ijφ)
(15-9)
Equation (15-9) shows that exp(ij φ) is an eigenfunction for the rotation operator, so exp(ij φ) is a basis for an irreducible representation, with j any integer. We can build additonal flexibility into this eigenfunction if we write an exponential function in a more complicated way: exp[i(j + nN)φ] with N also an integer. We know that this must still be a basis function because (j + nN) must still be an integer. If we operate
q
with Cn, we find that (Problem 15-6)
q
Cn exp [i(j + nN)φ] = exp(−2πiqj/n) exp [i(j + nN)φ]
(15-10)
This shows that our newest, most general, exponential has the same eigenvalue for
q
Cn no matter what integer value we choose for N. [Observe that the eigenvalue in Eq. (15-10) depends on q, j , and n, but not N .] Therefore, since we have an unlimited number of choices for the integer N , we have an unlimited number of degenerate
q
eigenfunctions for Cn for each choice of j . We can mix together such degenerate eigenfunctions in any way we please and still have an eigenfunction. Let us take the linear combination
∞
AN exp[i(j +nN)φ]
(15-11)
N =−∞
This can be factored into the form
∞
exp(ij φ) AN exp(inNφ)
(15-12)
N =−∞
The sum is a function of φ which we can call U (φ). It has the same periodicity in φ as the cycle because n is equal to the number of identical cells in the cycle and N is an integer. It is actually the general Fourier series expansion of a periodic function.
For N = 0, the contribution is constant (A0) at all φ. For N = ±1, the functions repeat n times around the cycle and clearly have the periodicity of the system. For N = ±2 the functions repeat 2n times, so they obviously still have the periodicity of the system.
Since all the individual members of the sum have the periodicity of the system, so does the sum itself. The specific nature of the resultant function U (φ) depends on the choice of coefficients AN .
We have shown that exp(ij φ)U (φ)
(15-13)
with U (φ) n-fold periodic is a general mathematical form for bases for irreducible representations for the Cn group. But wavefunctions, ψ(φ), for a periodic potential must also be bases for such irreducible representations. Therefore, wavefunctions must be expressible in this form: ψj (φ) = exp(ij φ)U (φ), j = 0, ±1, ±2, . . .
(15-14)
We are not claiming that all functions of the form Eq. (15-13) are wavefunctions for the n-fold periodic potential. Rather, we are claiming that all such wavefunctions belong to the class of functions represented by Eq. (15-13). Only certain choices of U (φ) will serve to make Eq. (15-13) become a wavefunction for the system, and the choice of an appropriate function U (φ) is not necessarily the same at different values of j . To make this dependence on j explicit, we include it as a subscript on U : ψj (φ) = exp(ij φ)Uj (φ), j = 0, ±1, ±2, . . .
(15-15)
with Uj (φ) n-fold periodic. This completes the proof of Bloch’s theorem for periodic cycles.
Before extending this result to noncyclic systems, a number of comments and clar ifications should be made. First, the function Uj (φ) may still seem to be something of a mystery. Think of it this way. For benzene π MOs we know we need to create a basis set having two properties: It should be like a 2pπ AO near any carbon atom, and it should be the same at each carbon atom. Bloch’s theorem does not comment on whether U should look like a 2pπ AO at a carbon atom, but it does require that it be the same at each carbon atom, that it be the same at the midpoints between adjacent carbons, that it be the same at a point one bohr above each carbon atom, etc. In short, U must be symmetric for the six-fold rotation operation. It is up to us to figure out what that six-fold symmetric basis set should look like in detail, and to construct U appropriately.
Second, U depends on j . What does this mean? Suppose we compare the lowest and highest-energy π MOs for benzene. The lowest is totally bonding and the highest is totally antibonding. Ordinarily, we use the identical set of 2pπ AOs as basis functions for both MOs, but this is not required. Perhaps the pπ AOs for the bonding MO would be more appropriately chosen to be slightly larger, so they overlapped better, and maybe those for the antibonding MO should be slightly smaller, to reduce antibonding interactions. These objectives could be achieved, for example, by making U a mixture of 2pπ and 3pπ AOs on each center and letting the nature of the mixture depend on j .
This makes for a more involved variational calculation, so it is normally not included in Chapter 15 Molecular Orbital Theory of Periodic Systems cases like this one where such effects are expected to be small. (3pπ character should not mix in to a significant extent because the 3pπ AO is considerably higher in energy than the 2pπ AO.) However, there are many cases where a unit cell contains several AOs in the same symmetry class and of similar energy (e.g., 2s, 2pσ ), and it is then quite important to allow the mix of these to vary with j . A given band, for example, could start out being mainly 2s at low energies and end up being mainly 2pσ at the high-energy end. This means that the appropriate description of the function U must be redetermined for each choice of wavenumber j .
Third, careful comparison of Eqs. (15-15) and (15-4) shows that they are not exactly the same. Equation (15-15) instructs us to find a periodic function Uj (φ) and multiply it by exp(ij φ) at every point in φ. Think of the sine or cosine related to the exponential and imagine what this means as we multiply it times a 2pπ on some carbon. Say the cosine is increasing in value as it sweeps clockwise past the carbon nucleus at 2:00 on a clock face. This produces a product of cosine and 2pπ that is unbalanced—smaller toward 1:00 than toward 3:00, because the cosine wave modulates Uj (φ) everywhere. But Eq. (15-4) is different. It instructs us to take the value of the cosine at 2:00 and simply multiply the 2pπ AO on that atom by that number. The 2pπ AO is not caused to become unbalanced. Only its size in the MO is determined by the cosine. Equation (15-4) is called a Bloch sum. Such sums are approximations to Bloch functions, but any errors inherent in this form are likely to be quite small if the basis functions and unit cell are sensibly chosen. (Using Bloch sums is similar in spirit to the familiar procedure of approximating a molecular wavefunction as a linear combination of basis functions.)
Extending Bloch’s theorem to linear periodic structures requires identifying an appropriate substitute for the coordinate φ and reconsidering the appropriate values for j (which we will call k in noncyclic systems). We require that these substitutions yield eigenfunctions and eigenvalues for the linear system that are the same as what results when it is treated as a cycle. Let us suppose that we have some very large number, n, of unit cells in the cycle, with a repeat distance of a. This means that the cycle has a Cn axis and also a circumference of na, so the linear coordinate (call it x) ranges from 0 to na as the angular coordinate φ ranges from 0 to 2π . When we move s steps around the cyclic polymer, the exponential’s argument changes by a factor of 2π ij s/n, with j an integer. The same number of steps should give us the same effect on the exponential for the linear polymer. If we choose our exponential’s argument to be of the form ikx, then s steps creates the factor iksa, so we require that iksa = 2πij s/n. This gives k = 2πj/na, with j an integer. We see that, for extremely large values of n, k becomes an effectively continuous variable (points separated by 2π/na). Also, we see that k is uniformly distributed (j = 0, ±1, ±2, etc.), which affects our DOS calculations, as was mentioned in Section 15-2.
We can also ask about the range of k, which should correspond to the 0–2π range of φ. We expect “empty wiggling” to occur when the sin or cos equivalent of exp(ikx) has a wavelength shorter than 2a. The largest nonredundant value of k should come when cos(kx) goes from cos(0) to cos(2π ) as x goes from 0 to 2a. Therefore kx = k(2a) = 2π, so k has a range of 2π/a. Since the convention is to center k about zero, k runs from −π/a to π/a. (Note that this range becomes infinite for the free particle in a constant potential, for which a is zero.) The range of k over which all unique wavefunctions for the periodic system are produced once and only once (−π/a
for benzene MOs, are also shown (at the positions labeled 1–6), and it is obvious that the same set of coefficients is produced by each function. Increasing j has caused the coefficient wave to oscillate with a shorter wavelength, but the extra oscillations do not register because they occur between the special points of interest where we are sampling the function (i.e., at the points where the carbon nuclei reside). In other words, the extra oscillations in our benzene coefficient functions are just “empty wiggling” and have no effect on the MO, so we cut off the j range where the meaningful wiggling starts to become empty wiggling. The reason that the particle in a ring does not show this behavior is that the exponential (or sine, cosine) functions in that system actually are
the wavefunctions, while in benzene these functions are sampled only at discrete points where AOs are located, and the resulting coefficients are then used to produce MOs.
Notice that the MOs for benzene as given by Eq. (15-4) are produced from an equation having the following form: There is an exponential term that, for each MO, supplies the coefficients for various carbons in the molecule, and there is a basis set of functions located on the various carbons. This basis set has the same “periodicity” as does the molecule. (In the case of benzene, we have so far taken this to be six identical 2pπ AOs.) Wavefunctions for all periodic systems have this same form—an exponential
(or equivalent trigonometric) expression times a periodic basis. This is the content of Bloch’s theorem, which we prove in the next section.
EXAMPLE 15-1 What coefficients are generated for a benzene MO by a coefficient wave of the type shown in Fig. 15-6 with j = −3? j = 0?
SOLUTION The arguments of the cosine function of Eq. (15-6) will be the negative of those of the j = 3 case. Because the cosine is symmetric about an argument of zero, we obtain the same curve for j = −3 as for j = +3, hence the same MO coefficients. This demonstrates that the MO generated by j − 6 is the same as that generated by j . For j = 0, the cosine arguments all vanish, the cosines all equal +1, and the curve of Fig 15-6 becomes a straight line at a coefficient value of
√
+1/ 6. This produces the lowest-energy MO of benzene.
15-5 General Form of One-Electron Orbitals in PeriodicPotentials—Bloch’s Theorem
We will establish in this section the general mathematical nature of eigenfunctions for periodic hamiltonians. This is the content of Bloch’s theorem. The actual proof will be carried through for cyclic “periodic” systems, and those results will then be extended to noncyclic periodic systems. The rationale for this is that a nearly infinite linear periodic structure can be treated as cyclic without introducing error. That is, a sufficiently long extended chain of atoms can be assumed to have the same wavefunctions and energies as the same chain joined end to end to form a cycle. For short chains and rings, this is not the case; the lowest-energy MO for hexatriene is not as low in energy as that for benzene, nor is it uniform over the whole chain (as it is in benzene). But for long enough chains, the difference becomes negligible.
Bloch’s theorem, as it applies to periodic cycles, states that eigenfunctions have the form
j (φ) = exp(ij φ)Uj (φ)
(15-7)
Chapter 15 Molecular Orbital Theory of Periodic Systems
√
where i = −1, j is an integer and Uj (φ) has the periodicity of the cycle. If the cycle belongs to the Cn point group, then U (φ) = U (φ + 2π/n). Functions of the form of Eq. (15-7) are called Bloch functions. It was stated in Section 13-8 that every wavefunction or orbital is a member of a basis for an irreducible representation for the point group of the system, so it follows that the wavefunctions we seek to characterize have to be bases for irreducible representations of the Cn point group. These groups are discussed in Section 13-11, where it is shown that the functions = exp(iφ) and ∗ = exp(−iφ) are bases for irreducible representations for these groups, which is to say that they are eigenfunctions for the rotation operator, no matter what the value of the integer n in Cn. We consider the more general functions exp(±ij φ) with j an integer (so that the function obeys the cyclic continuity condition) and ask what happens when
q
such functions are subjected to a rotation. Let the rotation operator be symbolized Cn for a rotation of q times 2π/n. The n-fold cyclic system is invariant to this rotation if q is an integer. It is not difficult to show that (see Section 13-11)
q
Cnf (φ) = f (φ − 2πq/n)
(15-8)
(For example, clockwise rotation by 60◦ brings to each point in the circle the value that used to be 60◦ in the counterclockwise direction.) Then
q
Cn exp(ij φ) = exp[ij (φ − 2πq/n)] = exp(−2πiqj/n) exp(ijφ)
(15-9)
Equation (15-9) shows that exp(ij φ) is an eigenfunction for the rotation operator, so exp(ij φ) is a basis for an irreducible representation, with j any integer. We can build additonal flexibility into this eigenfunction if we write an exponential function in a more complicated way: exp[i(j + nN)φ] with N also an integer. We know that this must still be a basis function because (j + nN) must still be an integer. If we operate
q
with Cn, we find that (Problem 15-6)
q
Cn exp [i(j + nN)φ] = exp(−2πiqj/n) exp [i(j + nN)φ]
(15-10)
This shows that our newest, most general, exponential has the same eigenvalue for
q
Cn no matter what integer value we choose for N. [Observe that the eigenvalue in Eq. (15-10) depends on q, j , and n, but not N .] Therefore, since we have an unlimited number of choices for the integer N , we have an unlimited number of degenerate
q
eigenfunctions for Cn for each choice of j . We can mix together such degenerate eigenfunctions in any way we please and still have an eigenfunction. Let us take the linear combination
∞
AN exp[i(j +nN)φ]
(15-11)
N =−∞
This can be factored into the form
∞
exp(ij φ) AN exp(inNφ)
(15-12)
N =−∞
The sum is a function of φ which we can call U (φ). It has the same periodicity in φ as the cycle because n is equal to the number of identical cells in the cycle and N is an integer. It is actually the general Fourier series expansion of a periodic function.
For N = 0, the contribution is constant (A0) at all φ. For N = ±1, the functions repeat n times around the cycle and clearly have the periodicity of the system. For N = ±2 the functions repeat 2n times, so they obviously still have the periodicity of the system.
Since all the individual members of the sum have the periodicity of the system, so does the sum itself. The specific nature of the resultant function U (φ) depends on the choice of coefficients AN .
We have shown that exp(ij φ)U (φ)
(15-13)
with U (φ) n-fold periodic is a general mathematical form for bases for irreducible representations for the Cn group. But wavefunctions, ψ(φ), for a periodic potential must also be bases for such irreducible representations. Therefore, wavefunctions must be expressible in this form: ψj (φ) = exp(ij φ)U (φ), j = 0, ±1, ±2, . . .
(15-14)
We are not claiming that all functions of the form Eq. (15-13) are wavefunctions for the n-fold periodic potential. Rather, we are claiming that all such wavefunctions belong to the class of functions represented by Eq. (15-13). Only certain choices of U (φ) will serve to make Eq. (15-13) become a wavefunction for the system, and the choice of an appropriate function U (φ) is not necessarily the same at different values of j . To make this dependence on j explicit, we include it as a subscript on U : ψj (φ) = exp(ij φ)Uj (φ), j = 0, ±1, ±2, . . .
(15-15)
with Uj (φ) n-fold periodic. This completes the proof of Bloch’s theorem for periodic cycles.
Before extending this result to noncyclic systems, a number of comments and clar ifications should be made. First, the function Uj (φ) may still seem to be something of a mystery. Think of it this way. For benzene π MOs we know we need to create a basis set having two properties: It should be like a 2pπ AO near any carbon atom, and it should be the same at each carbon atom. Bloch’s theorem does not comment on whether U should look like a 2pπ AO at a carbon atom, but it does require that it be the same at each carbon atom, that it be the same at the midpoints between adjacent carbons, that it be the same at a point one bohr above each carbon atom, etc. In short, U must be symmetric for the six-fold rotation operation. It is up to us to figure out what that six-fold symmetric basis set should look like in detail, and to construct U appropriately.
Second, U depends on j . What does this mean? Suppose we compare the lowest and highest-energy π MOs for benzene. The lowest is totally bonding and the highest is totally antibonding. Ordinarily, we use the identical set of 2pπ AOs as basis functions for both MOs, but this is not required. Perhaps the pπ AOs for the bonding MO would be more appropriately chosen to be slightly larger, so they overlapped better, and maybe those for the antibonding MO should be slightly smaller, to reduce antibonding interactions. These objectives could be achieved, for example, by making U a mixture of 2pπ and 3pπ AOs on each center and letting the nature of the mixture depend on j .
This makes for a more involved variational calculation, so it is normally not included in Chapter 15 Molecular Orbital Theory of Periodic Systems cases like this one where such effects are expected to be small. (3pπ character should not mix in to a significant extent because the 3pπ AO is considerably higher in energy than the 2pπ AO.) However, there are many cases where a unit cell contains several AOs in the same symmetry class and of similar energy (e.g., 2s, 2pσ ), and it is then quite important to allow the mix of these to vary with j . A given band, for example, could start out being mainly 2s at low energies and end up being mainly 2pσ at the high-energy end. This means that the appropriate description of the function U must be redetermined for each choice of wavenumber j .
Third, careful comparison of Eqs. (15-15) and (15-4) shows that they are not exactly the same. Equation (15-15) instructs us to find a periodic function Uj (φ) and multiply it by exp(ij φ) at every point in φ. Think of the sine or cosine related to the exponential and imagine what this means as we multiply it times a 2pπ on some carbon. Say the cosine is increasing in value as it sweeps clockwise past the carbon nucleus at 2:00 on a clock face. This produces a product of cosine and 2pπ that is unbalanced—smaller toward 1:00 than toward 3:00, because the cosine wave modulates Uj (φ) everywhere. But Eq. (15-4) is different. It instructs us to take the value of the cosine at 2:00 and simply multiply the 2pπ AO on that atom by that number. The 2pπ AO is not caused to become unbalanced. Only its size in the MO is determined by the cosine. Equation (15-4) is called a Bloch sum. Such sums are approximations to Bloch functions, but any errors inherent in this form are likely to be quite small if the basis functions and unit cell are sensibly chosen. (Using Bloch sums is similar in spirit to the familiar procedure of approximating a molecular wavefunction as a linear combination of basis functions.)
Extending Bloch’s theorem to linear periodic structures requires identifying an appropriate substitute for the coordinate φ and reconsidering the appropriate values for j (which we will call k in noncyclic systems). We require that these substitutions yield eigenfunctions and eigenvalues for the linear system that are the same as what results when it is treated as a cycle. Let us suppose that we have some very large number, n, of unit cells in the cycle, with a repeat distance of a. This means that the cycle has a Cn axis and also a circumference of na, so the linear coordinate (call it x) ranges from 0 to na as the angular coordinate φ ranges from 0 to 2π . When we move s steps around the cyclic polymer, the exponential’s argument changes by a factor of 2π ij s/n, with j an integer. The same number of steps should give us the same effect on the exponential for the linear polymer. If we choose our exponential’s argument to be of the form ikx, then s steps creates the factor iksa, so we require that iksa = 2πij s/n. This gives k = 2πj/na, with j an integer. We see that, for extremely large values of n, k becomes an effectively continuous variable (points separated by 2π/na). Also, we see that k is uniformly distributed (j = 0, ±1, ±2, etc.), which affects our DOS calculations, as was mentioned in Section 15-2.
We can also ask about the range of k, which should correspond to the 0–2π range of φ. We expect “empty wiggling” to occur when the sin or cos equivalent of exp(ikx) has a wavelength shorter than 2a. The largest nonredundant value of k should come when cos(kx) goes from cos(0) to cos(2π ) as x goes from 0 to 2a. Therefore kx = k(2a) = 2π, so k has a range of 2π/a. Since the convention is to center k about zero, k runs from −π/a to π/a. (Note that this range becomes infinite for the free particle in a constant potential, for which a is zero.) The range of k over which all unique wavefunctions for the periodic system are produced once and only once (−π/a