Hiickel matrix

We can now assign the four carbon p-orbitals, one to each carbon. For simplicity, we will label them with the subscript corresponding to the number of the carbon atom to which the AO belongs. We will use the symbol p to denote AOs and P for MOs. We can now write the Hiickel matrix as a square matrix involving the AOs as shown in Figure 7-20. 

The diagonal elements of the Hiickel matrix represent the energies of the contributing AOs, which in this case are all a. Each of the bonds (in this case tpi-Wi> arid 3- 4) is assigned the overlap energy and all other elements of... 

Figure 7-20. The Hiickel matrix (above) and the eigenvalues and eigenvectors for 1,3-butadiene.

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. 

Another solution is to use the master equation in its discrete from and to perform an exact mode analysis on the resulting Hiickel matrix arbitrarily truncated. In such a case the truncation is directly associated with the finite length of the chain which is taken into account in the calculation. In fact, this procedure, proposed by Jones and Stockmayer does not lead to a closed expression for the OACF, but to an infinite series of expressions corresponding to different truncations. This makes the comparison of the J S mddel with experiments rather lengthy. Since similar ideas can now be accounted for by closed expressions, we will not present here the detailed discussion of the JS model (for a more complete discussion, see Ref. 

In analogy with the bond order-weighted edge adjacency matrix, a resonance-weighted edge adjacency matrix E was also proposed, replacing the bond orders with parameters kc-x used in the Hiickel matrix and related to the resonance integral Pc-x of the bond between the heteroatom X and the carbon atom by the relationship ... 

If we then construct Hiickel MOs using the LCAO method, assuming for simplicity orthonormal3 basic AOs, we must solve the secular equation of Figure 2.6. After diagonalization of the Hiickel matrix, we get ... 

Admitting sp hybridization, for X, C2v, and C3v 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 Hiickel matrix, whose diagonalization is not so easy. 

As a second step, we admit full mixing of s and z AOs on fluorine within X symmetry. The Hiickel matrix will be ... 

Since the AOs of the hybridized basis are now properly directed along the z axis, we see that in the range 0 < w < 45° (namely, from pure p to equivalent digonal hybrids) /3a, decreases and /3, h increases from the value assumed for co = 0. The transformed Hiickel matrix ... 

If we allow for sp mixing onto oxygen, the Hiickel matrix for the (s z hz) basis of A symmetry becomes ... 

The elements of the Hiickel matrix are given in terms of just two negative unspecified parameters,31 the diagonal a and the off-diagonal /3 for the nearest neighbours, introduced in a topological way as ... 

Rule 4 enables us to determine the coefficients of NBMOs without diagonalizing the Hiickel matrix.287 Start by assigning an arbitrary value of a to one of the atoms with nonvanishing coefficients. Then assign multiples or fractions of a to the other atoms in the same set, using rule 3 that the coefficients on atoms that are attached to an atom with Cnbmo, = 0 must add up to zero. Finally, the NBMO must be normalized (Equation 4.25), whereby the value of a is defined. 

Equation (25) refers to an Hiickel matrix of order 2N where there are N off diagonal elements equal to jS+ and — 1 equal to /3. The eigenvalues fall in the same intervals as in the previous cases, i.e. 

The Hiickel matrix of order 2N is symmetrical and tridiagonal, and therefore has exactly 2N non-degenerate real eigenvalues. Since equation (25) is, a part from a constant factor, the secular determinant of the Hiickel matrix [6], it has exactly N non-zero distinct roots. 

The interaction elements in the Hiickel matrix are thus replaced by... 

The 19-dimensional Hiickel matrix thus will be resolved into five blocks, one of dimension 5, two identical blocks of dimension 4, and three identical blocks of dimension 2. In Table 4.9 we display the blocks for each irrep and the corresponding SALCs for one component. The corresponding secular equations are ... 

Here TV is the number of vertices in a graph (which corresponds to the number of atoms in a conjugated molecule), Xj are the roots of the characteristic polynomial of the aromatic system, and xf are the roots of the acyclic polynomial of the polyene-like reference system. In essence this corresponds to the procedure of the Hiickel method to solve for the eigenvalues Xj of the Hiickel matrix in units of Finally, gj is the orbital occupancy number. The method was applied to a large number of conjugated hydrocarbons - with results for TREPE that usually show a similar trend as the HSREPE values. Aiha-ra76.77 extended the concept to three-dimensional systems, in particular polyhedral boranes. However, soon afterward, controversial difficulties arose with this approach. ... 

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

Hosoya (2013) and Hosoya et al. (1994, 2001) observed that two nonisomorphic graphs may possess identical distance-spectra. We already mentioned isospectral graphs when presenting the Hiickel matrix (see Section 2.19). A pair of the two polyhedral graphs on eight vertices that possess the same distance-spectra are shown in Figure 4.1. 

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