The Extended Hiickel Method

The extended Hiickel method (EHM) developed by Hoffmann [44] is the simplest noniterative semiempirical method. Formally, the EHM equations coincide with the equations of Roothaan [Eqs. (2.4) and (2.10)], however, only the overlap integrals are calculated in them exactly, while the matrix elements are replaced by empirical parameters  

Nowadays the EHM is regarded as mainly a qualitative method not claiming to satisfy the demands listed in Sect. 1 of this chapter. Its chief advantage is a quite fair reproduction of the relative energies and the form of MO s, particularly, in the case of not very strongly polarized molecules. In view of its extreme simplicity, it may be useful in calculations on the systems with a practically unlimited size of the basis set. When DZ-type basis sets are used, this method is sufficiently effective for qualitative analysis of structures and reactions of organometallic compounds [45-47]. 

The extended Hiickel (EH) method is much like the simple Hiickel method in many of its assumptions and limitations. However, it is of more general applicability since it takes account of all valence electrons, o and n, and it is of more recent vintage because it can only be carried out on a practical basis with the aid of a computer. The basic methods of extended Hiickel calculations have been proposed at several times by various people. We will describe the method of Hoffmann [1], which, because of its systematic development and application, is the EHMO method in common use. 

The method is described most easily by reference to an example. We will use methane (CH4) for this purpose. 

The first choice we must make is the molecular geometry to be used. For methane, we will take the H-C-H angles to be tetrahedral and C-H bond distances of 1.1 A. We can try altering these dimensions later. 

Cartesian coordinates for the five atoms are listed in Table 10-1, and the orientation of the nuclei in Cartesian space is indicated in Fig. 10-1. (Even though the eigenvalues and MOs one finally obtains are independent of how CH4 is oriented in Cartesian space, it is generally a good idea to choose an orientation that causes some Cartesian and synunetry axes to coincide. The resulting expressions for MOs in terms of AOs are generally much simpler to sketch and interpret.) 

Next we must select the basis set of functions with which to express the MOs. The extended Hiickel method uses the normalized valence AOs for this purpose. For CH4, this means a Is AO on each hydrogen and a 2s, 2px, 2py, and 2pz AO on carbon. The inner-shell Is AO on carbon is not included. The AOs are represented by Slater-type orbitals (STOs). Except for the Is AOs on hydrogen, the exponential parameters of the STOs are determined from Slater s mles (Chapter 5). Various values for the hydrogen 1 s AO exponent have been suggested. These have ranged from the 1.0 given by Slater s 

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The Extended Hiickel method, for example, does not explicitly consider the effects of electron-electron repulsions but incorporates repulsions into a single-electron potential. This simplifies the solution of the Schrodinger equation and allows HyperChem to compute the potential energy as the sum of the energies for each electron. 

The Extended Hiickel method also allows the inclusion of d orbitals for third row elements (specifically. Si, P, S and Cl) in the basis set. Since there are more atomic orbitals, choosing this option results in a longer calculation. The major reason to include d orbitals is to improve the description of the molecular system. 

Note You cannot use the Extended Hiickel method or any one of the SCFmethods with the Cl option being turned on for geometry optimizations, molecular dynamics simulations or vibrational calculations, in the current version of HyperChem. 

Note You can not use the Extended Hiickel method, nor any of the other SCFmethods with the Cl option turned on, for geometry optimization or molecular dynamics simulations. 

The Extended Hiickel method neglects all electron-electron interactions. More accurate calculations are possible with HyperChem by using methods that neglect some, but not all, of the electron-electron interactions. These methods are called Neglect of Differential Overlap or NDO methods. In some parts of the calculation they neglect the effects of any overlap density between atomic orbitals. This reduces the number of electron-electron interaction integrals to calculate, which would otherwise be too time-consuming for all but the smallest molecules. 

Molecular orbital calculations for the parent vinyl cation, Cj H3, were first reported by Hoffmann (161), who used the extended Hiickel method, and more recently by Yonezawa and co-workers (162), who used a semiempirical SCF procedure. Both treated the problem of classical, 172 (R = H), versus bridged structures, 173, but the methods suffered from their inability to account satisfactorily for bond-length changes, and neither discussed the question of linear, 172a, versus bent, 172b, structures. 

The extended Hiickel method has been used in a discussion of properties and reactivity of radicals and biradicals (75). We have found it possible to correlate the basicity constants, pKbh. of radical anions with extended Hiickel data (76). 

The definition of "concepts" must be accompanied by explicit recipes for computing them is actual cases. There is no more space in theoretical chemistty for "driving forces", "effects, etc. not accompanied by specific rules for their quantification. The impact of a new "concept will be greater if the rules of quantifications are not restricted to ad hoc methods, but related to methods of general use in molecular quantum mechanics. A concept based exclusively on some specific features of a given method, e g. the extended Hiickel method, is less robust than a concurrent concept which may be quantified also using other levels of the theory. 

Figure 1. Density of states for various Ag clusters computed for 4d-, 5 s-, and 5p-orbitals within the extended Hiickel method. (Reprinted from Ref [32], 1981, with permission from Elsevier.)...

Of the many discussions of the intrinsic properties of the Extended Hiickel method, that of Blyholder and Coulson (7a) can be specially recommended. The validity of the above mentioned cancellation of repulsions seems to be substantiated (7b). 

The electronic structures of furan, thiophene, and selenophene, their protonated complexes, and their anions have been calculated by the extended Hiickel method.6 The results of these calculations have been used to determine the influence of the heteroatom on the degree of aromaticity and electron density. 

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... 

In the usual formulation of the extended Hiickel method, the elements of the hamiltonian matrix are computed according to a simple set of arithmetic rules, and do not depend on the molecular orbitals. In this way, there is no need for the iterations required by more sophisticated methods, and in practice the results may be obtained nowadays in a question of seconds for any reasonably sized complex. 

Figure 1. a donation interaction between metal and hydride (left), and n backdonation interaction metal and carbonyl (right) as computed by the extended Hiickel method. 

The utility of the extended Hiickel method for qualitative analysis must nevertheless not hide its limitations when quantitative results are desired. Although it can be of some utility in predicting bond and dihedral angles, it is unappropriate for the prediction of bond distances or bond energies. As a result, it cannot be applied to any reaction where bonds are made or broken,... 

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. 

The high-resolution C spectrum has also been measured, and the following values obtained chemical shift (in acetone) 62.4 ppm upheld from CSg (130.9 ppm downfield from tetramethylsilane) J (geminal) CH, 205 Hz J (vicinal) CH, 13.4 Hz. The chemical shift value is in accord with an empirical equation based on the number and position of the nitrogen atoms in several five-membered heterocyclics, and also reflects the 7T-electron density of the system as calculated by the extended Hiickel method - and by the simple MO method. spectra of v-triazole and of its 1- and 2-methyl derivative have also been obtained. ... 

Fig. 6. Orbital correlation diagram for the DHP-cis-stilbene conrotatory path. R is the C(4a) — C(4b) separation. The dotted line indicates the ground state occupancy limit. The molecular orbitals were computed by the Extended Hiickel method )...

MI21104, 75JSP(54)167>. Correlation between molecular orbital energies calculated using the MIEH method and photoelectron spectra of naphthyridines showed more recognizable lone pair orbitals than did the extended Hiickel method (74MI21100). 

In Table 27 the simple Hiickel indices, Nr (Dewar number), Fr (free valence number) and Lr (localization energy) are given, together with the Mulliken overlap population pr, calculated by the extended Hiickel method and using the geometry of hexahelicene as determined by X-ray analysis141) to account for the non-planarity. 

As mentioned in Section III,A, benzo[c]furan (4) participates in thermal [jt4 -I- TCgJ-cycloadditions. With tropone and substituted tropones, compounds of type 206 (exo R = H, Cl, OMe) have been obtained, and 6,6-dimethylfulvene yielded 207 (endo). The extended Hiickel method and... 

Woodward and Hoffmann refer to this geometry of approach as the nonlineary to distinguish it from the alternative, more symmetrical linear approach (5), in which the sp orbital enters suprafacially. Although Hoffmann has reported calculations by the extended Hiickel method that are in agreement with the mode of addition 4 predicted by the pericyclic theory,6 there is little experimental information. 

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