Hiickel determinant

Solution of the ag block of the Hiickel determinant for Cso leads to two linear combinations of ag irreducible symmetry. The first of these is the linear combination resulting from the projection of the lag polynomial of Table 3.20 on the 80 vertices, so that each coefficient is of equal value, while the second linear combination is identical to the one found by projecting the level 6 polynomial of the table onto the 80 vertices and rendering the result orthogonal to the first. 

The operator of the perturbed system is written as H = H° + h, where H° is the Hiickel operator of the parent molecule and h the perturbation operator. We could, of course, set up a new Hiickel determinant for the perturbed system, but having the eigenvalues s p and the associated eigenfunctions with the operator H of the parent system at hand, we already have a good starting point for the new system. Therefore, the wavefunctions are set up as linear combinations of the functions i//j° (Equation 4.40). 

Since the Hiickel determinant contains only information about the number of unsaturated carbons and how they are connected together, it is sometimes referred to as a topological determinant. (Topology refers to properties that are due to the connectedness of a figure, but are unaffected by twisting, bending, etc.)... 

The Hiickel determinant summarizes the connectedness of the unsaturated system, and is independent of cis-trans isomerism or bond length variation. 

This treatment is much simpler than butadiene, because only two carbon atoms are involved. It should be easy to apply Hiickels approximations to get the following 2X2 Hiickel determinant ... 

The only real dilference between equation 15.26 and the earlier Hiickel determinants is the presence of (3 in the upper right and lower left corners. This is because the molecule is cyclic and the first carbon atom is adjacent to the sixth carbon atom. 

A second example for the pi-orbitals of formaldehyde will suffice to illustrate the hetero-pi method. Once again we only have two orbitals, 2p C and a 2p O. Note this method ignores the nonbonded orbitals on oxygen and the C-H sigma bonds as well as the two inner shell Is orbitals we are only treating the pi-orbitals. This time the Hiickel determinant becomes... 

In this section, the systematic search for conical intersections based on the Longuet-Higgins phase-change rule is described. For conciseness sake, we limit the present discussion to Hiickel-type systems only, unless specifically noted otherwise. The first step in the antilysis is the determination of the LH loops containing a conical intersection for the reaction of interest. 

For this class of thiazoles most of the chemical and physicochemical studies are centered around the protomeric equilibrium and its consequences. The position of this equilibrium may be determined by spectroscopic and titrimetric methods, as seen in each section. A simple HMO (Hiickel Molecular Orbitals) treatment of 2-substituted compounds however, may, exemplify general trends. This treatment considers only protomeric forms 1 and 2 evidence for the presence of form 3 has never been found. The formation energy reported in Table 1 is the energy difference in f3 units. 

The second step determines the LCAO coefficients by standard methods for matrix diagonalization. In an Extended Hiickel calculation, this results in molecular orbital coefficients and orbital energies. Ab initio and NDO calculations repeat these two steps iteratively because, in addition to the integrals over atomic orbitals, the elements of the energy matrix depend upon the coefficients of the occupied orbitals. HyperChem ends the iterations when the coefficients or the computed energy no longer change the solution is then self-consistent. The method is known as Self-Consistent Field (SCF) calculation. 

The Hiickel MO method is based on the LCAO method for diatomic molecules discussed in Section 7.2.1. Extension of the LCAO method to polyatomic molecules gives a secular determinant of the general type... 

In the case of benzene, Hiickel treatment of the six 2p orbitals on the carbon atoms and perpendicular to the plane of the ring leads to the secular determinant... 

In an early investigation (66T539) the two highest occupied and the two lowest unoccupied orbitals were calculated on the basis of an extended Hiickel theory to determine the electron transition responsible for the long wavelength UV absorption. An Ai- Bi, [Pg.197]

The u) parameter determines the weight of the charge on the diagonal elements. Since Ga is calculated from the results (MO coefficients, eq. (3.90)), but enters the Hiickel matrix which produces the results (by diagonalization), such schemes become iterative. Methods where the matrix elements are modified by the calculated charge are often called charge iteration or self-consistent (Hiickel) methods. 

One important stracture in molecules are polar bonds and, as a result, polar molecules. The polarity of molecules had been first formulated by the Dutch physicist Peter Debye (1884-1966) in 1912, as he tried to build a microphysical model to explain dielectricity (the behaviour of an electric field in a substance). Later, he related the polarity of molecules to the interaction between molecules and ions. Together with Erich Hiickel he succeeded in formulating a complete theory about the behaviour of electrolytes (Hofimann, 2006). The discovery of the dipole moment caused high efforts in the research on physical chemistry. On the one hand, methods for determining the dipole momerrt were developed. On the other hand, the correlation between the shape of the molectrle and its dipole moment was investigated (Estermanrr, 1929 Errera Sherrill, 1929). 

Conjugation of the 7t-electrons of the carbon-carbon double bond with the LUMO sulfur 3d-orbitals would be expected to stabilize the Hiickel 4n -I- 2 (n = 0) array of n-electrons in the thiirene dioxide system. No wonder, therefore, that the successful synthesis of the first member in this series (e.g. 19b) has initiated and stimulated several studies , the main objective of which was to determine whether or not thiirene dioxides should be considered to be aromatic (or pseudo-aromatic ) and/or to what extent conjugation effects, which require some sort of n-d bonding in the conjugatively unsaturated sulfones, are operative within these systems. The fact that the sulfur-oxygen bond lengths in thiirene dioxides were found to be similar to those of other 802-containing compounds, does not corroborate a Hiickel-type jr-delocalization... 

In the above two equations, the former value is valid for basic SI units and the latter value for / in moles per cubic decimetre and a in nanometres. The parameter a represents one of the difficulties connected with the Debye-Hiickel approach as its direct determination is not possible and is, in most cases, found as an adjustable parameter for the best fit of experimental data in the Eq. (1.3.29). For common ions the values of effective ion radii vary from 0.3 to 0.5. Analogous to the limiting law, the mean activity coefficient can be expressed by the equation... 

Extended Hiickel molecular-orbital (EHMO) calculations using structural parameters from the X-ray determination of Cp(CO)3Mo 3In (Fig. 29) and ideal symmetry Cih were carried out for this compound in order to investigate the extent of the indium-molybdenum ir-bonding.122 The HOMO of the compound is the 3e molecular orbital, which is In—Mo [Pg.54]

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. 

By using the determinant form of the electronic wave functions, it is readily shown that a phase-inverting reaction is one in which an even number of electron pairs are exchanged, while in a phase-preserving reaction, an odd number of electron pairs are exchanged. This holds for Hiickel-type reactions, and is demonstrated in Appendix A. For a definition of Hiickel and Mobius-type reactions, see Section III. 

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