Nearest-neighbor approximation

Impurity-cluster size nearest-neighbor approximation... 

Full consideration of intrachain interactions (beyond the nearest-neighbor approximation) and interchain interactions is included in the self-consistent field and leads to an increase in the transition temperatures in Eqs. (2.2.24) and (2.2.25) by a factor46 of C 2) 1.645. 

In both types of lattices the C0 value obtained in the present approximation is a significant improvement over that associated with the F. G. approximation. On the other hand, as judged by the trend of the entries in Table IV, convergence to the correct C0 value is very slow, and considerable further refinement seems to be required to approach the exact quantities. Unfortunately, the next higher approximation becomes extremely cumbersome, since it can be shown that 30 additional variables are required to characterize the problem. It is likely that the next nearest neighbor approximation as given here represents an upper limit for practical calculations. 

The energy of the py (or p band is obtained by cyclic permutation. The behavior of Eq. 5.46, for two-dimensional square lattices of px and py atomic orbitals, using the first-nearest neighbor approximation, is shown later in Figure 5.5. 

The energy bands of tetrahedral solids have been studied in terms of LCAO s for many years the first study was that of Hall (1952), who used a Bond Orbital Approximation, keeping only nearest-neighbor interbond matrix elements in order to obtain analytic expressions for the bands over the entire Brillouin Zone. The recent study by Chadi and Cohen (1975), which did not use either of Hall s approximations, is the source of the interatomic matrix elements between. v and p orbitals, which appear in the Solid Stale Table. Pantelides and Harrison (1975) used the Bond Orbital Approximation but not the nearest-neighbor approximation and found that accurate valence bands could be obtained by adjusting a few matrix elements at the same time very clear interpretations of many features of the bands were achieved. The main features of the Pantiledes-Harrison interpretation will be presented here. 

In practice, the summations over n in these expressions are truncated to a reasonable value for which the integrals xix r) H xv r - na)) and Xix(r) Xv(r - na)) are negligible. This is the so-called nth nearest-neighbor approximation. 

When the overlap integrals are neglected within the first nearest-neighbor approximation, the energy e ka,kb,kc) of the BO ip(ka, h,kc) is given by... 

The spin variables can adopt the two values af= + 1. The nonvanishing terms of the spin Hamiltonian in the nearest-neighbor approximation and in absence of an external magnetic field are... 

Expanding these expressions to second order in 2J/Eq, we obtain the same results as those following from our expressions (3.84) and (3.78). In the nearest-neighbor approximation the HLA exciton energy is EHLA(k) = J cos k, and the correction is given by... 

To simplify the analysis here we confine ourselves to discussing the simplest case, when k = Jt2 = 0. In this case the quantity (12.24) is equal to the energy of interaction of the dipole pa in the plane n3 = const with the lattice dipoles p13 located in the plane to3 = const. It was shown long ago (26 27) that the electric field created by such a lattice of dipoles decreases exponentially at distances of the order of the lattice constant. Thus in (12.23) when summing over m3, one can confine oneself to the nearest-neighbor approximation. In this approximation the system of equations (12.23,12.24) can be written as follows... 

It is well known that for ideal linear molecular chains of finite length without mixing of Frenkel with CT states, only the usual space quantization of exciton states inside the band appears. Nearest neighbor approximation (this model is often used for analysis of spectra of J-aggregates, see, e.g. (42)) yields the energy of Frenkel exciton states ... 

This is the nearest-neighbor approximation. We encounter a similar problem inside the F q because we have somehow truncated the summations over h and 1. These problems will be discussed later in this chapter. 

In the periodic Anderson model state of the electrons of the crystal containing impurities in the 7i-electron approximation and the nearest neighbor approximation is described by the effective Hamiltonian, having the following standard form [5] ... 

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