The HUckel MO Method

The most celebrated semiempirical Tr-electron theory is the Huckel molecular-orbital 

Hamiltonians, a separation of variables is possible (Section 6.2). We have 

Since is not specified, there is no point in trying to solve (17.6) directly. Instead, the variational method is used. 

The next assumption in the HMO method is to approximate the tt MOs as LCAOs. In a minimal-basis-set calculation of a planar conjugated hydrocarbon, the only AOs of tt symmetry are the carbon Ipir orbitals, where by Ipir we mean the real 2p AOs that are perpendicular to the molecular plane. We thus write 

The key assumptions in the Hiickel theory involve the integrals in (17.10). The integral 7/ / is assumed to have the same value for every carbon atom in the molecule. (For benzene the six carbons are equivalent, and this is no assnmption. For 1,3-butadiene, CH2CHCHCH2, one would expect for an end carbon and a middle carbon to differ slightly.) Moreover, is assumed to be the same for carbon atoms in different planar hydrocarbons. The integral is assumed to have the same value for any two carbon atoms bonded to each other and to vanish for two nonbonded carbons. The integral Srr is 

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A thiepin is formally isoelectronic with the 8ic-electron 1,3,5,7-cyclooctatetraene and 1,3,5-cycloheptatrienide ion and, if planar, may actually be antiaromatic. Recently, the question of the antiaromaticity of thiepin has been the subject of interest for both synthetic and theoretical chemists. The apparent instability of the thiepin ring system is in good agreement with theoretical calculations. Dewar and Trinajstic 68) have reported that the thiepin is considered to be weakly antiaromatic (RE = — 1.45 kcal mol-1) based on PPP SCF MO calculations. On the other hand, Hess Jr. and Schaad 69) have found it to be substantially antiaromatic (RE = —0.232 J) by using the Huckel MO method. This result was also supported by a graph-theoretical treatment by Aihara 70). 

If calculated corrections for the anisotropic contribution 8 (C,) are introduced into the experimentally measured values of 5" (Cj), then the corrected values of the chemical shifts reveal their common linear dependence on the values of jt-electron deficiency calculated by the Huckel MO method... 

For qualitative purposes the circle itself isn t even necessary We could locate the Huckel MOs by simply working with the polygons themselves The circle is needed only when Frost s method is used quantitatively In those cases the radius of the circle has a prescribed value allowing each MO to be assigned a specific energy... 

The photocyclodehydrogenation of thienyl ethylenes is well-defined when both thiophene rings are bound via a C(2) atom to the ethylenic bond as in (70). In other cases, however, more cyclization products are possible. To predict the photocyclization mode for heterohelicenes the F s rule fails in many cases, because correction factors for the hetero atoms in the Huckel MO calculation have to be introduced and the systems are not well comparable with carbocyclic diaryl ethylenes. A better reaction parameter in these cases is the Mulliken overlap population (nrs)51), introduced by Muszkat52) for these cases. The overlap populations of the atoms r and s in ground and excited state (nIiS and n s), are calculated using the extended Huckel method. Cyclizations should not occur when nr>s and An s (= nr>s — n s) have negative values. (This method can also be used for diaryl olefins, but in these cases calculation of F s is more simple.). 

Using the HMO approximation, the ir-electron wave function is expressed as a linear combination of atomic orbitals (for the case in which the plane of the molecule coincides with the y-z plane of the coordinate axis), in much the same way as described previously for the generalized MO method. Minimizing the total -jT-electron energy of the molecule with respect to the coefficients (the variation method) leads to a series of equations from which the coefficients can be extracted by way of a secular determinant. The mathematical operations involved in solving the equations are not difficult. We will not describe them in detail, but will instead concentrate on the interpretation of the results of the calculations. For many systems, the Huckel MO energies and atomic coefficients have been tabulated." ... 

Hearndon has considered the question of resonance and bond order from a theoretical viewpoint and the electronic structures of nine (unknown) heterobenzocyclo-butadienes have been discussed using the Huckel MO standard method. ... 

Semiempirical calculations for j or AEf were carried out by the SCF-MO method with integral approximations [23], by various versions of the CNDO [24 to 27] and INDO [1 to 4, 28] methods, by the MNDO method [29], by the extended Huckel and Pariser-Parr-Pople methods [30], and by a simple [31] MO method [32]. 

Wade and collaborators more recently continued to use extended Huckel MO methods to investigate stability patterns in deltahedral borane chemistry. Calculations were carried out for published experimental geometries and also for hypothetical structures in which all boron atoms lie on a spherical surface and all deltahedral faces are equilateral triangles with B—B = 1.70 A and B—H = 1.19 A. These calculations were found to reproduce at least semiquantitatively the patterns of chemical stability for the observed B H 2 (6 < n < 12) ions and... 

More sophisticated procedures involve taking the start MO coefficients from a semi-empirical calculation, such as Extended HUckel Theory (EHT) or Intermediate Neglect of Differential Overlap (INDO) (Sections 3.12 and 3.9). The EHT method has the advantage that it is readily parameterized for all elements, and it can provide start orbitals for systems involving elements from essentially the whole periodic table. An INDO calculation normally provides better start orbitals, but at a price. The INDO... 

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... 

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