Extended Huckel approximation

Figure 29 Band structure and Fermi surface of (TMTTF)2X calculated within the extended Huckel approximation, (taken from ref. 10)...

With the fundamental assumption of the Extended-Huckel approximation that the interaction between the nuclei is the same as that between the electron clouds localized on different atoms, Eq.(2.30) reduces to ... 

As we will see a little later, Eq.(2.52b) has the same form as the total energy in the Extended-Huckel approximation, when 5 is assumed to be small. However, one... 

We will now discuss general features of the solutions of the secular equations to be solved within the Extended-Huckel approximation. 

The expression for the chemical bond energy Eq.(2.229c), which is based on surface state energies and the phase shift, is an alternative to Eq.(2.42), which relates the bond energy to changes in bond-order and Mulliken population. Within the Extended-Huckel approximation the expression becomes ... 

The intermolecular interaction is very sensitive to the anisotropy of the conduction molecular orbitals (HOMO, LUMO,. ..) and the compactness of the intermolecular contacts. So that it may be appropriate to use the anisotropy of the intermolecular overlap integral (5) of the conduction molecular orbitals as a semi-quantitative measure of the dimensionality of the system. If we adopt the simple extended Huckel approximation (transfer integral (t)(xS), we can visualize the dimensionality of the molecular conductor as a form of simple band structure (or the shape of Fermi surface (FS)). 

As mentioned above, HMO theory is not used much any more except to illustrate the principles involved in MO theory. However, a variation of HMO theory, extended Huckel theory (EHT), was introduced by Roald Hof nann in 1963 [10]. EHT is a one-electron theory just Hke HMO theory. It is, however, three-dimensional. The AOs used now correspond to a minimal basis set (the minimum number of AOs necessary to accommodate the electrons of the neutral atom and retain spherical symmetry) for the valence shell of the element. This means, for instance, for carbon a 2s-, and three 2p-orbitals (2p, 2p, 2p ). Because EHT deals with three-dimensional structures, we need better approximations for the Huckel matrix than... 

Extended Huckel provides the approximate shape and energy ordering of molecular orbitals. It also yields the approximate form of an electron density map. This is the only requirement for many qualitative applications of quantum mechanics calculations, such as Frontier Orbital estimates of chemical reactivity (see Frontier Molecular Orbitals on page 141). 

Other approximate, more empirical methods are the extended Huckel 31> and hybrid-based Hiickel 32. 3> approaches. In these methods the electron repulsion is not taken into account explicitly. These are extensions of the early Huckel molecular orbitals 4> which have successfully been used in the n electron system of planar molecules. On account of the simplest feature of calculation, the Hiickel method has made possible the first quantum mechanical interpretation of the classical electronic theory of organic chemistry and has given a reasonable explanation for the chemical reactivity of sizable conjugated molecules. 

Polydiacetylene crystals. The enhancement of x because of one-dimensional electron delocalization is strikingly corroborated in the polydiacetylene crystals. Their structure is that of a super alternated chain with four atoms per unit cell and the Huckel approximation yields four bands for the ir-electrons, two valence and two conduction bands. When depicted in the extended Jones zone, each pair can be viewed as arising by a discontinuity at the middle of the Brillouin zone of the polyene chain. The dominant contribution to X(2n 1) comes from the critical point at the edge of the extended Jones zone (initially at the center of the reduced B.Z.). The complete expressions are derived in (4,22) and calculated for different polydiacetylenes. We reproduce the values of x 2 for TCDU and PTS in table IV. The calculated values are in good agreement... 

Prior to considering semiempirical methods designed on the basis of HF theory, it is instructive to revisit one-electron effective Hamiltonian methods like the Huckel model described in Section 4.4. Such models tend to involve the most drastic approximations, but as a result their rationale is tied closely to experimental concepts and they tend to be inmitive. One such model that continues to see extensive use today is the so-called extended Huckel theory (EHT). Recall that the key step in finding the MOs for an effective Hamiltonian is the formation of the secular determinant for the secular equation... 

Very small differences in bond strength between different silica polymorphs were found. Since the Extended Huckel Method is too approximate to calculate reliably the small diffferences in energy between low-density material, containing micropores, and high-density material without micropores, work was initiated to study the same problem but now with two rigorous techniques that are currently considered to be state of the art. 

A wide range of theoretical methods has been applied to the study of the structure of small metal clusters. The extremes are represented on the one hand by semi-empirical molecular orbital (Extended Huckel) (8 ) and valence bond methods (Diatomics-In-Molecules) ( ) and on the other hand by rigorous initio calculations with large basis sets and extensive configuration interaction (Cl) (10). A number of approaches lying between these two extremes have been employed Including the X-a method (11), approximate molecular orbital methods such as CNDO (12) and PRDDO (13) and Hartree-Fock initio molecular orbital theory with moderate Cl. 

Electronic band structures were calculated for several polymeric chains structurally analogous to polyacetylene (-CH-CH) and carbyne (-CbC). Ihe present calculations use the Extended Huckel molecular orbital theory within the tight binding approximation, and values of the calculated band gaps E and band widths BW were used to assess the potential applic ilitf of these materials as electrical semiconductors. Substitution of F or Cl atoms for H atoms in polyacetylene tended to decrease both the E and BW values (relative to that for polyacetylene). Rotation about rhe backbone bonds in the chains away from the planar conformations led to sharp increases in E and decreases in BW. Substitution of -SiH or -Si(CH,) groups for H in polyacetylene invaribly led to an increase in E and a decrease in BW, as was generally the case for insertion of Y ... 

Recent evidence favors Dg symmetry and a pentagonal bipyramidal structure for the heptafluoride. Claassen et al. (2) review the earlier debate about structure of lower symmetry. They provide convincing evidence of D symmetry from the first observation of Raman spectra of the vapor state and re-examination of the infrared spectra. Their data—including five fundamentals in Raman (two polarized), five fundamentals in infrared, no coincidences between Raman and infrared, and one fundamental inactive—are consistent only with symmetry. This is confirmed by Falconer et al. (4) who interpret their electric-deflection experiments as indicating a symmetry-forbidden dipole moment. Extended Huckel-molecular-orbital calculations (5) also predict Dg symmetry. The adopted structural parameters are from our approximate analysis of the electron-diffraction data of Thompson and Bartell (6). The authors gave a radial distribution curve and suggested only a gross (unrefined) structure because of the probable presence of... 

Q.18.10 What is the primary difference between the Extended Huckel and Huckel semi-empirical methods Why would somebody use a simpler approximation instead of one that is more accmate ... 

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