The General LCAO Method

Most chemists are well acquainted with LCAO-MO theory. The numbers of atomic orbitals, even in large molecules, however, are miniscule compared to a nonmolecular solid, where the entire crystal can be considered one giant molecule. In a crystal there are in the order of 10 atomic orbitals, which is, for all practical purposes, an infinite number. The principle difference between applying the LCAO approach to solids, versus molecules, is the number of orbitals involved. Fortunately, periodic boundary conditions allow us to smdy solids by evaluating the bonding between atoms associated with a single lattice point. Thus, the lattice point is to the solid-state scientist, what the molecule is to the chemist. 

As a prelude to our development of the LCAO treatment of solids, it will be beneficial to briefly review the LCAO-MO method. The cyclic tt systems from organic chemistry are familiar, relatively simple, and, more importantly, resemble Bloch functions of periodic solids. Thus, they will be used as the introductory examples. 

The independent-electron approximation was discussed in the previous chapter. The molecular wave functions, ifi, are solutions of the Hartree-Fock equation, where the Fock operator operates on tfi, but the exact form of the operator is determined by the wave-function itself. This kind of problem is solved by an iterative procedure, where convergence is taken to occur at the step in which the wave function and energy do not differ appreciably from the prior step. The effective independent-electron Hamiltonian (the Fock operator) is denoted here simply as H. The wave functions are expressed as linear combinations of atomic functions, x- 

The latter two integrals can be represented as a square matrix, for which each matrix element corresponds to a particular combination for the values of /r and v. It is noted that because of the Hermitian properties of H, and S/j,v = Equation 5.2 

The variational method by Walter Ritz (1878-1909) indicates that  

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