Huckel matrix

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... 

Let us look at the Mathcad output for a Huckel matrix in more detail. We select the matrix for ethylene to preserve simplicity. 

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 Huckel methods perform the parameterization on the Fock matrix elements (eqs. (3.50) and (3.51)), and not at the integral level as do NDDO/INDO/CNDO. This means that Huckel methods are non-iterative, they only require a single diagonalization of the Fock (Huckel) matrix. The Extended Huckel Theory (EHT) or Method (EHM), developed primarily by Hoffmann again only considers the valence electrons. It makes use of Koopmans theorem (eq. (3.46)) and assigns the diagonal elements in the F... 

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. 

Admitting sp hybridization, for E, Civ, and Csv molecular symmetries, the s and z functions always belong to the same symmetry (Table 2.9), so that the row basis vector is now 1x3 giving a 3 x 3 Huckel matrix, whose diagonalization is not so easy. 

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