The HUckel Matrix

The Hilckel matrix encountered in the Hiickel theory of conjugated systems (Hiickel, 1931,1932a, 1932b) may be considered to be an augmented vertex-adjacency matrix (Trinajstic, 1992). In the Hiickel theory, the aim is to solve the following secular equation set up for a given conjugated systan (Streitwieser, 1961)  

the Hiickel matrix X is a kind of augmented vertex-adjacency matrix  

The Hiickel theory was widely used in the early days of quantum chemistry, before the age of computers, and in spite of its simplicity, it is still in use e.g., papers appear continually in Physics Reviews B using the Hiickel theory in a solid-state 

FIGURE 2.20 Graph Gg representing the carbon skeleton of 1,2-divinylcyclobutadiene. 

FIGURE 2.21 A classical example of a pair of isospectral graphs. 

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Then the Huckel matrix for the conjugated i -system is constructed. The a-values of the Huckel matrix of each atom i of the conjugated system are adjusted to the <7-chaige distribution by Eq. (13). 

As mentioned above, HMO theory is not used much any more except to illustrate the principles involved in MO theory. However, a variation of HMO theory, extended Huckel theory (EHT), was introduced by Roald Hof nann in 1963 [10]. EHT is a one-electron theory just Hke HMO theory. It is, however, three-dimensional. The AOs used now correspond to a minimal basis set (the minimum number of AOs necessary to accommodate the electrons of the neutral atom and retain spherical symmetry) for the valence shell of the element. This means, for instance, for carbon a 2s-, and three 2p-orbitals (2p, 2p, 2p ). Because EHT deals with three-dimensional structures, we need better approximations for the Huckel matrix than... 

The tipper triangular part of the Huckel matrix for ethylene, Fq, (6-48), exclusive of the diagonal elements consists of only one element. It can be entered into Program QMOBAS by making the dimension of the matrix 2 and changing the data statement to enter 1 in the 1.2 position... 

In the Mathcad calculation of eigenvalues and eigenvectors of the Huckel matrix for ethylene (j g), the eigenvalues are given in the order upper followed by lower. The matr ix E for this order is... 

The form of the SCF matrix is the same as the Huckel matrix hence, we substitute... 

The w parameter determines the weight of the charge on the diagonal elements. Since Qa is calculated from the results (MO coefficients, eq. (3.90)), but enters the Huckel 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 (Huckel) methods. 

The equivalence of equations (25) and (28) is obtained by setting 17 = exp(iZ). In order to determine the limit of the ir energy we observe that the Huckel matrix of order 2N + 1 is obtained by adding one row and one column to that of order 2N furthermore, being tridiagonal, aU eigenvalues are simple. By the interlace theorem [16], the eigenvalues of the matrix of odd orders 2A 1 separate those of the matrix of order 2N. One has explicitly ... 

Exercise 5.16 Set up the Huckel matrix for benzene and find its eigenvalues. Remember that if the carbon atoms are labeled clockwise from 1 to 6, then atoms 1 and 6 are nearest neighbors. Show that the six eigenvalues are identical to those given by Eq. (5.131). Find the total energy and compare it with the result given by Eq. (5.132). 

We now turn to the localized description of polyenes. As the zeroth-order approximation, we assume that the polyene consists of n noninteracting double bonds or ethylenic units. To obtain the energy corresponding to this description we note that the Huckel matrix for ethylene is... 

Do this first by using the general expression and then by setting up the Huckel matrix, diagonalizing it and adding up the occupied orbital energies. Note that when Pi= p = P e recover our old result, 6a + p. 

The cyclopropenyl system is perturbed so that the Huckel matrix element H22 = a- -cP, where c is positive. 

If we allow for sp mixing onto oxygen, the Huckel matrix for the (s z h ) basis of Ai symmetry becomes ... 

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