Bloch sums

The structure of the p band may be obtained by writing as a linear combination of the three p Bloch sums corresponding to the atomic px, py, and p2 orbitals. That is,... 

Using the same type of LCAO expansion as for molecules (see eq. (2.8)), the crystal orbitals are expanded as Bloch sums of the basis function centred at site fi in cell / ... 

Figure 3.9. (a) The diamond structure viewed as two interlocking FCC sublattices displaced by I a along 1 1 1). In this projection along the [0 01] direction, only the top face of each cube is shown, (b) The unit cell, (c) Some possible sign combinations of the basis atomic orbitals used to construct LCAO COs from two Bloch sums, (d) A qualitative CO energy-level diagram for the center of the Brillouin zone, F = k(0, 0, 0). 

Crystal orbitals are built by combining different Bloch orbitals (which we will henceforth refer to as Bloch sums), which themselves are linear combinations of the atomic orbitals. There is one Bloch sum for every type of valence atomic orbital contributed by each atom in the basis. Thus, the two-carbon atom basis in diamond will produce eight Bloch sums - one for each of the s- and p-atomic orbitals. From these eight Bloch sums, eight COs are obtained, four bonding and four antibonding. For example, a Bloch sum of s atomic orbitals at every site on one of the interlocking FCC sublattices in the diamond structure can combine in a symmetric or antisymmetric fashion with the Bloch sum of s atomic orbitals at every site of the other FCC sublattice. 

Alternatively, a Bloch sum of s atomic orbitals could combine with a Bloch sum of p atomic orbitals. The symmetric (bonding) combinations of the basis atomic orbitals for the latter case are illustrated for one CC4 subunit in Figure 3.9c. The actual COs are delocalized over aU the atoms with the space group symmetry of the diamond lattice. A LCAO-CO construction from Bloch sums is thus completely analogous to a... 

The one-electron wave function in an extended solid can be represented with different basis sets. Discussed here are only two types, representing opposite extremes the plane-wave basis set (free-electron and nearly-free-electron models) and the Bloch sum of atomic orbitals basis set (LCAO method). A periodic solid may be considered constmcted by the coalescence of these isolated atoms into extended Bloch-wave functions. On the other hand, within the free-electron framework, in the limit of an infinitesimal periodic potential (V = 0), a Bloch-wave function becomes a simple... 

One defines a Bloch sum (or BO) for each atomic orbital in the chemical point group (or lattice point), and COs are then formed by taking linear combinations of the Bloch sums. 

First, take the simplest possible case, a monatomic solid with a primitive Bravais lattice containing one atomic orbital per lattice point. The COs are then equivalent to simple Bloch sums. The wavelength. A, of a Bloch sum is given by the following relation ... 

It can be seen how the phase of a Bloch sum changes in a periodic lattice by considering a simple one-dimensional lattice of (tr-bonded) p atomic orbitals, with a repeat distance d. Figure 4.4 shows such a chain and the sign combinations of the atomic... 

For lattices with more than one atom per lattice point, combinations of Bloch sums have to be considered. In general, the LCAO approach requires that the result be the same number of MOs (COs in sohds) as the number of atomic orbitals (Bloch sums in sohds) with which was started. Thus, expressing the electron-wave functions in acrystaUine sohd as linear combinations of atomic orbitals (Bloch sums) is really the same approach used in the 1930s by Hund, Mulliken, Htickel, and others to construct MOs for discrete molecules (the LCAO-MO theory). 

Consider a J-electron system, such as a transition metal compound. The valence d atomic orbitals do not range far from the nucleus, so COs comprised of Bloch sums of d orbitals and, say, O 2p orbitals, tend to be narrow. As the interatomic distance increases, the bandwidth of the CO decreases because of poorer overlap between the d and p Bloch SUMS. In general, when the interatomic distance is greater than a critical value, the bandwidth is so small that the electron transfer energy becomes prohibitively large. Thus, the condition for metallic behavior is not met insulating behavior is observed. 

For each t)q)e of atomic orbital in the basis set, which is the chemical point group, or lattice point, one defines a Bloch sum (also known as Bloch orbital or Bloch function). A Bloch sum is simply a linear combination of aU the atomic orbitals of that type, under the action of the infinite translation group. These Bloch sums are of the exact... 

In this equation, N is equal to the number of unit cells in the crystal. Note how the function in Eq. 5.27 is the same as that of Eq. 5.19 for cyclic tt molecules, if a new index is defined ask = liij/Na. Bloch sums are simply symmetry-adapted linear combinations of atomic orbitals. However, whereas the exponential term in Eq. 5.19 is the character of the yth irreducible representation of the cychc group to which the molecule belongs, in Eq. 5.27 the exponential term is related to the character of the Mi irreducible representation of the cychc group of infinite order (Albright, 1985). This, in turn, may be replaced with the infinite linear translation group because of the periodic boundary conditions. It turns out that SALCs for any system with translational symmetry are con-stmcted in this same manner. Thus, as with cychc tt systems, there should never be a need to use the projection operators referred to earher to generate a Bloch sum. 

Now, how Bloch sums combine to form COs must be considered, for example, hke the cr- or 7r-combinations between the Bloch sums of metal d orbitals and oxygen p orbitals in a transition metal oxide. Bloch sums are used as the basis for such COs ... 

The vector notation used to this point is concise, but it will be instructive to resolve the vectors into their components. Each atom is located at a position pai + qhj + rck, where a, b, c are lattice parameters and i, j, k are unit vectors along the x, y, z axes. A Bloch sum may thus be written as ... 

Thus, the energy of a Bloch sum of s atomic orbitals in the SC lattice (with one atom per lattice point) is ... 

The inclusion of second (and often third terms) is particularly important for certain stracmre types. For example, in the BCC and CsCl lattices the second-nearest neighbors are only 14 percent more distant than the first-nearest (Harrison, 1989). Accounting for the six second-nearest neighbor interactions in the FCC and BCC lattices, the energies of a Bloch sum of s atomic states are given by E qs. 5.42 and 5.43, respectively ... 

Since there are two atoms per primitive cell, or lattice point, and consideration is still on just s atomic orbitals, two separate Bloch sums are required. These combine to form two COs with energies given by the sum and difference in energy between the nearest neighbor and second-nearest neighbor interactions. The nearest neighbor... 

As in the case of diamond, two different Bloch sums ofatomic orbitals are needed, one for each distinct atomic site ... 

At r, the and p Bloch sum interactions are not symmetry allowed. Nor is the hybridization between the d and py Bloch sums. The d orbital at the origin does, however, have a tt antibonding interaction with the axial p orbitals (not shown) directly above and below it. In other words, the d orbital at the origin only interacts with two of the six first-nearest neighbor oxygen atoms. 

Figure 5.13. A schematic illustrating the superposition of the and p Bloch sums at T viewed down [001]. The only symmetry-allowed d -p interactions at T are those between the dxz orbitals and the axial oxygen p orbitals directly above and below them (not shown). Thus, the dispersion at T is driven primarily by d-d interactions.

Figure 5.15. A schematic illustration of the superposition of the d and Bloch sums at N viewed down [001]. The net d-d interaction is ir bonding while the p-d interactions are anti bonding.

Why are projection operators not requited to generate Bloch sums for crystaUine... 

A wave number must again be associated with every state for each wave number we construct eight different Bloch sums, four for the bond orbital types,... 

In contrast, in CsCl wc sufficiently simplified the matrix elements that we were able to use single Bloch sums, of the kind in Eq. (3-19), as eigenstates we cannot do that here. 

Consider first a matrix element from Eq. (3-22), where both Bloch sums... 

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