Extended Hiickel approximation

In the Extended Hiickel approximation, the charges in the unselected part are treated like classical point charges. The correction of these classical charges to the diagonal elements of the Hamiltonian matrix may be written as ... 

Using the extended Hiickel approximation, we obtain the corresponding eigenvalues E (k) and coefficients C (k) from the eigenvalue equation " ... 

The calculation of the exact band structure from first principles, however, is rather complex and requires considerable simplifications. The usual and very successful method to calculate the band structure of organic charge transfer salts is a tight-binding method, called extended Hiickel approximation. In this approximation, one starts from the molecular orbitals (MO) which are approximated by linear combinations of the constituent atomic orbitals. Each MO can be occupied by two electrons with antiparallel spins. These valence electrons are assumed to be spread over the whole molecule. Usually, only the highest occupied molecular orbital (HOMO) and the lowest unoccupied molecular orbital (LUMO) are relevant and are, therefore, considered in most band-structure calculations [41]. 

It is shown that the LCAO molecular Hartree-Fock equations for a closed-shell configuration can be reduced to a form identical with that of the Hoffmann extended Hiickel approximation if (i) we accept the Mulliken approximation for overlap charge distributions and (ii) we assume a uniform charge distribution in calculating two-electron integrals over molecular orbitals. Numerical comparisons indicate that this approximation leads to results which, while unsuitable for high accuracy calculations, should be reasonably satisfactory for molecules that cannot at present be handled with facility by standard LCAO molecular Hartree-Fock methods. 

Fig. 4-6. Orbital energies (eV) as a function of angle for C1F3 as calculated in the extended Hiickel approximation. [Adapted from Gavin.5]...

The Hiickel and extended Hiickel approximations are one-electron approximations. They are quite useful but they do not take us beyond Hartree-Fock. To do so we need either to use the weak version of the CNDO approximation, as given in Eq. (19) above or to make a further simplification, the strong CNDO approximation, in which we allow only electrons (of opposite spins) which are on the same site to repel. In other words, in the two-center Coulombic repulsion integrals [ijlij], we allow only i = j and... 

The other assumption associated with the hybridization model is that the nd, (n + l)s and (n -I- l)p valence orbitals have the same valence state ionization energies. Molecular orbital calculations using the extended Hiickel approximation have indicated that the changes in the valence state ionization energies do not cause changes in the non-bonding wavefunctions. 

We will discuss in some detail how and to what extent these effects are described in the simple extended Hiickel approximation. 

Note In the extended Hiickel approximation, the transition energy is merely the energy difference between the orbitals involved. In other words, in this approximation, the states arising from the same configuration have the same energy. Since the approximation is a crude one, it is not expected that the calculated and observed transition energies match perfectly. 

The modulated structure was solved more recently by Lam et al. [101] and the results show that the structure modulation is essentially associated with the Co-S apical coordination, denoting possible fluctuations in the Co oxidation state. This picture is also consistent with the band structure calculation performed in the framework of the extended Hiickel approximation, indicating as most favourable a partial charge transfer of slightly less than one electron per perylene unit, as... 

The simplest approximation to the Schrodinger equation is an independent-electron approximation, such as the Hiickel method for Jt-electron systems, developed by E. Hiickel. Later, others, principally Roald Hoffmann of Cornell University, extended the Hiickel approximations to arbitrary systems having both n and a electrons—the Extended Hiickel Theory (EHT) approximation. This chapter describes some of the basics of molecular orbital theory with a view to later explaining the specifics of HyperChem EHT calculations. 

HyperChem currently supports one first-principle method ab initio theory), one independent-electron method (extended Hiickel theory), and eight semi-empirical SCFmethods (CNDO, INDO, MINDO/3, MNDO, AMI, PM3, ZINDO/1, and ZINDO/S). This section gives sufficient details on each method to serve as an introduction to approximate molecular orbital calculations. For further details, the original papers on each method should be consulted, as well as other research literature. References appear in the following sections. 

The off-diagonal elements of Extended Hiickel theory, (fi v) represent the effects of bonding between the atoms and are assumed to be proportional to the overlap, Sj y. An approximation for differential overlap referred to as the Mulliken approximation... 

One of the simplest approaches to comprehensive molecular orbital calculations is the extended Hiickel method. This method was developed by Roald Hoffman in the 1960s, and it was applied to hydrocarbon molecules. From the discussion presented in Chapters 2 and 3, we know that one of the first things that has to be done is to choose the atomic wave functions that will be used in the calculations. One of the most widely used types of wave functions is that known as the Slater wave functions (see Section 2.4). In the extended Hiickel method, the molecular wave functions are approximated as... 

All this suggests a further simplification, which has proved to be eminently successful in many cases. It is known that independent electron treatments, such as the Hiickel (HMO) treatment2 or the extended Hiickel treatment (EHT)172, which do not take the electron-electron interaction explicitly into account, yield—by and large—orbitals derived from sophisticated SCF calculations. In particular, the HMO and ETH molecular orbitals reflect faithfully the symmetry and nodal properties of their counterparts obtained from SCF treatments. 

Only for a special class of compound with appropriate planar symmetry is it possible to distinguish between (a) electrons, associated with atomic cores and (7r) electrons delocalized over the molecular surface. The Hiickel approximation is allowed for this limited class only. Since a — 7r separation is nowhere perfect and always somewhat artificial, there is the temptation to extend the Hiickel method also to situations where more pronounced a — ix interaction is expected. It is immediately obvious that a different partitioning would be required for such an extension. The standard HMO partitioning that operates on symmetry grounds, treats only the 7r-electrons quantum mechanically and all a-electrons as part of the classical molecular frame. The alternative is an arbitrary distinction between valence electrons and atomic cores. Schemes have been devised [98, 99] to handle situations where the molecular valence shell consists of either a + n or only a electrons. In either case, the partitioning introduces extra complications. The mathematics of the situation [100] dictates that any abstraction produce disjoint sectors, of which no more than one may be non-classical. In view if the BO approximation already invoked, only the valence sector could be quantum mechanical9. In this case the classical remainder is a set of atomic cores in some unspecified excited state, called the valence state. One complication that arises is that wave functions of the valence electrons depend parametrically on the valence state. 

At the opposite extreme from the ab initio SCF methods is the Wolfberg-Helmholtz approximation which Hoffmann 6> has applied extensively to organic problems under the term extended Hiickel method . While this has the advantage of requiring very little computation time, the results are so unreliable that the method is essentially useless for the calculation of potential surfaces. Not only are the errors in heats of atomization comparable with those given by ab initio SCF but they are not even the same for isomers. A good example is provided by cyclopropanone (1) which is predicted 7> to be less stable than the isomeric zwitterion 2, a result at variance with the available evidence ) concerning the... 

The continued success of the extended Hiickel method in transition metal chemistry, where it was the method of choice until the mid 1980 s is surely related to the problems of other semiempirical methods in this area of chemistry. While methods like MOP AC [21] or AMI [22] have been extremely productive in the field of organic chemistry, they have found little success in transition metal chemistry. These methods are based in equation 2, similar to 1, but with the very significant difference that the Fock matrix F is computed from the molecular orbitals, in an iterative way, though through an approximate formula. 

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