ABSTRACT
Contents 6.1 Introduction . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 129 6.2 Delocalized orbital approach . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 133
6.2.1 Methods . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 133 6.2.2 Applications . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 135
6.3 Localized orbital approach . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 143 6.3.1 Methods . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 143 6.3.2 Applications . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 146
6.4 Concluding remarks . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 152 Acknowledgments . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 153 References . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 153
6.1 Introduction There are two computational approaches to the electronic structures of periodic and nonperiodic solids-the delocalized and localized orbital approaches [98]. The delocalized orbital approach or the so-called crystalline orbital (CO) theory [3-5,18,24,63,72,107,123] is strongly based on the periodic boundary conditions. It views an extended system of one-dimensional periodicity as a ring of K identical repeat units. A symmetry-adapted orbital such as a canonical Hartree-Fock (HF) orbital in this approach is delocalized over the entire ring and has the Bloch form [14]:
ψpk(r) = K −1/2 ∑
∑ m
Cµpk exp (imka ) χµ (r − ma) , (6.1)
where Cµpk is a CO coefficient, a is the lattice vector that outlines the unit cell, and χµ (r − ma) is the µth atomic orbital (AO) centered in the mth unit cell. Each orbital is characterized by k, which is the linear momentum (in atomic
units) of an electron in this orbital, and can take one of K distinct values:
ka π
= 2m K
, ∀m = 1, 2, . . . , K . (6.2)
In the delocalized orbital approach, K is a crucial parameter with dual meaning. It is the number of k-vector sampling points in the first Brillouin zone (BZ) according to Equation (6.2). It is also the nominal size of the system, which should thus be commensurate with the number of the nearest neighbor unit cells included in the lattice sums of particle-particle interactions.
