LCAO from atomic orbitals to bands

Here the crystallographic indices in the subscripts refer to the hybrid molecular orbital directions. Because the Schrbdinger Equation governing electron motion is linear, any combination of wave functions that solve it will also be a solution. In other words, choosing the hybrid orbitals or the atomic orbitals as a starting point for the calculation must yield identical results. The most flexible and general approaeh is not to be restricted to specific hybrid orbitals but rather to consider all possible orbital-by-orbital interactions of the fundamental atomic states. These states apply to a given atom in any environment. Thus, their use is valid for any material in which the atom occurs. As an example of a specific interaction, one can ask how does the Px orbital on one atom interact with the orbital on another atom. 

It is through the phase factors that a given electron momentum is defined. One might have expected this as, from the discussion of Chapter 2, band structures represent the interference of electron waves with the periodic potential of the lattice. For the wave functions in Equation 5.10, the corresponding phase factors are  

An examination of the terms for the g values will show that these simply represent the interference behavior of the electron waves with given wave vectors k interacting with atoms at positions defined by the real-space vectors d and at the origin. The g values include the free-electron-like behavior of Chapter 2. The calculation of the g factors becomes more complex when second-nearest neighbors and beyond are included, but the method is the same. The energy of an electron with wave vector k is the determinant of a matrix representing the energies of all possible orbital pairs. For example, for a zincblende semiconductor with no d-orbitals the LCAO matrix is [c.f Ref 4]  

Additional rows and columns should be added to the matrix for more complex compounds with more than two atoms and the formula for the g s becomes much more complex. The matrix also expands when shallow-lying d-orbitals must be taken into accoimt. Simpler structures such as the diamond lattice have a smaller interaction matrix because there is no distinction between cation and anion sites. The matrix may also be modified by effects such as spin-orbit splitting. (Spin-orbit splitting is one of the corrections necessary to an accurate band calculation. It results from the interaction of the electron spin magnetic moment with the dot product of its velocity and the local electric field due to the positive atomic cores of the lattice.) Likewise, greater accuracy can be obtained if additional terms are included in the g values to accoimt for second and higher neighbors. 

We will consider some of the imphcations of the LCAO method next based on the above equations and ignoring detailed corrections. The simplest trend to understand is the behavior at T, where k=0. In this case Equation 5.13 yields go=4 and gi, g2, and g3=0. This makes the interaction matrix exceedingly simple and results in the following four energies [1]  

---