First neighbor approximation

A Tc-electron model is necessarily oversimplified because, among other reasons, it lies upon the so-called a — n separation. This means that the n electrons are described by an effective model Hamiltonian as discussed in the previous sections while the effect of the a electrons (a core) is taken into account only by modifying the matrix elements of the effective Hamiltonian for the n electrons. Another drastic simplification is the first-neighbor approximation discussed also in the previous sections. These approximations may considerably limit the power of the... 

Alternant hydrocarbons exhibit certain symmetry properties which are reflected by the special arrangement of their orbital energy levels and the specific form of their molecular orbitals. These special properties are often referred to as the pair-theorem or, using the terminology of physics, it is called the particle-hole symmetry. It holds under the first-neighbor approximation in the 7c-electron system. In the name of this symmetry, the term particle refers simply to the electrons of the molecule. We say that the occupied levels are filled with particles. 

The alternant nature of the system, together with the first-neighbor approximation, have been extensively utilized in the above demonstration it is obvious that these are necessary conditions for the fulfillment of the particle-hole symmetry. The fact that the system is a hydrocarbon was used only in the sense that all a parameters have a common value thus they could be eliminated from the Hamiltonian by an appropriate shift of the energy origin. The presence of a heteroatom with a different a destroys the particle-hole symmetry. However, the system used not be a hydrocarbon in the strict sense, silicon or other atoms would also qualify provided they form a rc-electron system in a planar arrangement. 

In spite of the formal similarities, the parametrization of the PPP and the CNDO Hamiltonians are quite different. In particular, the CNDO model is not restricted to the first-neighbor approximation. For this reason the particle-hole symmetry does not apply for... 

The simplest form of the quasi-chemical theory that is of interest in the present connection is the first-order approximation, which considers the distribution of nearest-neighbor pairs as affected by nonzero values of the linear combination,... 

It will be observed that entropies of dilution (as indicated by i) are highly variable from one system to another. This is contrary to the theory developed from consideration of lattice arrangements, according to which pi should be approximately 1/2 and nearly independent of the system. For polystyrene in methyl ethyl ketone, the entropy of dilution is nearly zero i.e., this solvent is a poor one not because of an adverse energy of interaction but because of the low entropy. First neighbor interactions apparently contribute to the entropy as well as to the energy, a point which was emphasized in Chapter XII. It will be noted also that cyclic solvents almost without exception are more favorable from the standpoint of the entropy than acyclic ones. 

For POM, a matrix algorithm for the statistical mechanical treatment of an unperturbed -A-B-A-B- polymer chain with energy correlation between first-neighboring skeletal rotations is described. The results of the unperturbed dimensions are in satisfactory agreement with experimental data. In addition, if the same energy data are used, the results are rather close to those obtained by the RIS scheme usually adopted. The RIS scheme is shown to be also adequate for the calculation of the average intramolecular conformational energy, if the torsional oscillation about skeletal bonds is taken into account in the harmonic approximation. 

H -O distance of 1.85 A, and the nonbonded O -O distance of 2.85 A, with approximately fourfold first-neighbor coordination. The H- -O and O - -O distances are 0.1 A longer than the ice values and 0.1 A shorter than the gas-phase water dimer values. This is qualitatively consistent with the concept that the liquid lies between the gas and the solid. 

The various absorption lines have been assigned with the help of MS-Xa calculations. Because the main influence on the shape of the XANES spectra is from the atoms in the first neighbor shell, only constituent clusters have been treated in the simulations, e.g., P033 for P406 and P043- for P4O10. In terms of this approximation, some details of the spectra (e.g., intensity and line widths of the absorptions) could be simulated sufficiently, whereas the predicted term values calculated for transitions near the absorption edge lie too close to each... 

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. 

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... 

Fig. 3. The derivative of the Pt NMR absorption signal in a slightly alloyed platinum sample. (In this experiment the field is swept slowly while the sample is continuously irradiated at fixed frequency. Lock-in detection records the derivative.) Most of the Pt nuclei resonate at a field/frequency ratio of approximately 1.14 G/kUz, oft scale to the right (dashed curve). Those Pt atoms that are first neighbors to an alloying impurity resonate at 1.127 G/kUz, and second and third neighbors give separate resonances as well. The results show that the susceptibility is site dependent. [Reproduced with permission from Weisman and Knight (77). Copyright 1968 American Physical Society.]...

Since nearest neighbor metal ions in a crystal lattice are usually separated by at least a few Angstroms (very seldom less than 5 A), the interaction between an electron on each metal ion is represented quite well to a first-order approximation by a simple classical dipolar interaction between electrons located at a point on each metal atom. The splitting AH produced by this spin-spin interaction is given by... 

We have applied this method first to formaldehyde and to l-oxy-3-aza-butadiene (see Figure 1). Both calculations have shown, in the MP2 and T2T2 approximation, that the method provides about 90% of the correlation energy, if one takes into account only excitations to the first neighbors as compared with the case where the orbitals are not localized and one takes into account all excitations.8,13 Since in the case of the larger nucleotide bases we could not perform a full T2T2 calculation, we could use the above described... 

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