The Hiickel Method

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. 

The 7i-electron wave functions in the Hiickel method are given by... 

In Section 7.3.1.3 the 71-electron MOs of benzene were obtained by the Hiickel method using only the Ip AOs on the six carbon atoms. The ground configuration is... 

Although the Hiickel method has now been supplanted by more complete treatments for theoretical analysis of organic reactions, the pictures of the n orbitals of both linear and cyclic conjugated polyene systems that it provides are correct as to symmetry and the relative energy of the orbitals. In many reactions where the n system is the primary site of reactivity, these orbitals correctly describe the behavior of the systems. For that reason, the reader should develop a familiarity with the qualitative description of the n orbitals of typical linear polyenes and conjugated cyclic hydrocarbons. These orbitals will be the basis for further discussion in Chapters 9 and 11. 

The Hiickel method is essentially only used for educational purposes or for veiy qualitative orbital considerations. It has the ability to produce qualitatively correct MOs involving a computational effort which is within reach of doing by hand. 

For planar unsaturated and aromatic molecules, many MO calculations have been made by treating the a and n electrons separately. It is assumed that the o orbitals can be treated as localized bonds and the calculations involve only the tt electrons. The first such calculations were made by Hiickel such calculations are often called Hiickel molecular orbital (HMO) calculations Because electron-electron repulsions are either neglected or averaged out in the HMO method, another approach, the self-consistent field (SCF), or Hartree-Fock (HF), method, was devised. Although these methods give many useful results for planar unsaturated and aromatic molecules, they are often unsuccessful for other molecules it would obviously be better if all electrons, both a and it, could be included in the calculations. The development of modem computers has now made this possible. Many such calculations have been made" using a number of methods, among them an extension of the Hiickel method (EHMO) and the application of the SCF method to all valence electrons. ... 

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. 

Qualitatively, similar relationships are ascertained in heteroaromatic systems where the same conclusion is derived by a numerical calculation. In more elaborate calculations than the Hiickel method, such as the Pariser-Parr-Pople approximation 21>23>, similar distinct parallelisms are recognized 59> (Table 4.1). Essentially the same circumstances exist also... 

The useful part of the calculation in the Hiickel method is with regard to the 7r bond,... 

Despite its simplicity, the Hiickel method enables other useful properties to be deduced. For example, the wave function for the bonding molecular orbital is... 

Although the Hiickel method is most often applied to organic molecules, the H3+ case discussed shows that it can also be applied to some inorganic species. Suppose we consider the pyrrole molecule,... 

As should be evident, part of the problem in dealing with structures that contain atoms other than carbon is what values to use for a and /3. The values that have been suggested are based on correlating calculated properties with other known data. Because the Hiickel method is not a quantitative scheme for calculating properties of molecules, we will not address the issue of correcting the values of a and /3 further. 

Use the Hiickel method to determine whether H3 should have a linear or a ring structure. Calculate the electron density at each atom and the bond orders for the more stable structure. 

The triaryl radicals of germanium exhibit lower g values than their trialkyl counterparts. In part, this arises from increased delocalization of the unpaired spin density onto the aryl rings (and the Ar3Ge radicals do show hyperfme coupling to the ring protons). For example, spin densities for the radicals PhmMe3 mGe, calculated by the Hiickel method (Table 2), reveal that there is a linear correlation between the g value of the radical and... 

The Hiickel method attempts to solve these equations without explicitly writing down an expression for the effective Hamiltonian. It firstly assumes that the integrals Ha along the diagonal... 

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. 

The molecular orbital structure of 1,2,3-benzotriazine has been calculated using the Hiickel method, and the energy levels, charge densities, and wave functions were obtained.Theoretically the charge densities thus obtained should be of some value in predicting the positions of electrophilic and nucleophilic attack, but in the absence of information on reagents and reaction conditions, such predictions cannot be made with any degree of accuracy. 

Some relevant references concerning the application of the Hiickel method are Hoffmann, 1963 Cotton, 1971 lung Canadell, 1997. 

These properties of the d-shell chromophore (group) prove the necessity of the localized description of d-electrons of transition metal atom in TMCs with explicit account for effects of electron correlations in it. Incidentally, during the time of QC development (more than three quarters of century) there was a period when two directions based on two different approximate descriptions of electronic structure of molecular systems coexisted. This reproduced division of chemistry itself to organic and inorganic and took into account specificity of the molecules related to these classical fields. The organic QC was then limited by the Hiickel method, the elementary version of the HFR MO LCAO method. The description of inorganic compounds — mainly TMCs,— within the QC of that time was based on the crystal field... 

Fig. 4. HOMO coefficients of three betaines calculated by the Hiickel method.

Considering the proliferation of MO calculations in recent years, it is remarkable how few calculations have been reported for these heterocycles. Frontier orbitals of several betaines have been calculated using the Hiickel method and electroselectivity correctly predicted on the basis of the orbital symmetry.Similar results have been obtained for pyridinium-3-olates (427) using the Pariser- Parr Pople (PPP) method. The CNDO... 

The EH method (developed by Wolfsberg and Helmholz and by Hoffmann) is an extension of the Hiickel method in which the pi-electron approximation is not made, but all valence electrons are treated. The method is thus applicable to nonplanar, as well as planar, molecules. The valence-electron Hamiltonian is taken as the sum of one-electron Hamiltonians //va, = 2(/ eff(0, where Hcft(i) is not explicitly defined. The valence-electron wave function is the antisymmetrized product of spin-... 

The Hiickel method is the simplest of the quantitative MO techniques. It has the following characteristics ... 

As an example of the Hiickel method we will examine the allyl system. There are three basis orbitals, numbered as shown in 1. Atoms 1 and 2 tire bonded to each... 

Hiickel theory [1], the oldest quantum-mechanical approach to calculating the properties of organic molecules, has been outshined by the more rigorous semi-empirical and ab initio techniques for at least two decades. Even if the ab initio total energies of conjugated systems were found [2, 3] to parallel the Hiickel tt-electron energies, the Hiickel method is certainly too crude to be of substantial value for quantitative considerations. 

It is of course possible to calculate dE indirectly by difference from the delocalization energies of the reactants and of the products or transition state. Indeed most of the reported calculations have made use of this procedure, the ir energies usually being estimated by using the Hiickel method. The trouble with this approach is that the assumptions underlying the Hiickel method break down for compounds other than AH s, while in the case of AH s the PMO method is not only much simpler but also more... 

A typical failure of the Hiickel method in the case of non-alternant systems is its prediction that all the hydrocarbons XVII-XXIII should be aromatic of these only azulene (XXII) shows aromatic properties. 

A similar failure in the case of heterocyclic compounds is shown by the basicity of even heteroaromatic bases such as pyridine or quinoline. These all have similar basic strengths, as the simple PMO method predicts (Section IX). However the charge densities on the nitrogen atoms calculated51 by the Hiickel method are not the same (Table IV) application of Eq. (101) would then imply that the bases should differ considerably in strength. 

---