The Hiickel Hamiltonian

As explained in Section 2.4, cE creates an electron with spin a in the spin-orbital, Xn(r, O )- We define the Dirac ket state as 

a and are equivalent both have the effect of transferring 

As described in Section 3.3.1, for cyclic undimerized chains this Hamiltonian is diagonalized by the Bloch states, 

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A basic tenet of the Hiickel jr-electron theory is that the orbital energies add to give the TT-electron energy. Yet 1 demonstrated earlier that HF orbital energies do not add in this way, so the Hiickel Hamiltonian cannot be strictly identified with the HF Hamiltonian. 

The accepted wisdom is that the Hiickel Hamiltonian matrix should be identified with the matrix (h where G is the electron repulsion matrix of Chapter 6. The basis for this belief is that that the matrix (h has eigenvalues that do sum correctly to the electronic energy. 

Problem 11-14. Verify that these molecular orbitals for benzene are eigenfunctions of the Hiickel Hamiltonian, with the given eigenvalues. 

The standard treatment employs equal Coulomb parameters a for all six carbon atoms, and equal resonance parameters /3 for all six CC bonds. In effect, a is the origin and /3 the unit in a scaled representation of the Hiickel hamiltonian matrix as... 

Most modern Hiickel programs will accept the molecular structure as the input. In older programs, the input requires the kind of atoms present in the molecule (characterized by their Coulomb integrals a) and the way in which they are connected (described by the resonance integrals. ). These are fed into the computer in the form of a secular determinant. Remember that the Coulomb and resonance integrals cannot be calculated (the mathematical expression of the Hiickel Hamiltonian being unknown) and must be treated as empirical parameters. 

In this case the energy of the -subsystem described by the Hiickel Hamiltonian eq. (2.16) can be presented as... 

Here the 7r-system is treated with a very simple, but still quantum mechanical method e.g. by the Hiickel Hamiltonian and MO LCAO approximation (which in the particular case of the Hiickel Hamiltonian gives the exact answer). No explicit interaction, i.e. junction, between the subsystems was assumed at that time however, the effects of the geometry of the classically moving nuclei were very naturally reproduced by a linear dependence of the one-electron hopping matrix elements of the bond length ... 

Coulomb interactions dominate the electronic structure of molecules. The total spin S2 and Sz are nearly conserved for light atoms. We will consider spin-independent interactions in models with one orbital per site. In the context of tt electrons, the operators a+a and OpCT create and annihilate, respectively, an electron with spin a in orbital p. The Hiickel Hamiltonian is... 

The Hiickel Hamiltonian matrix from (1) is of order N and its solution yield... 

It must be emphasized, however, that this coincidence is due to the peculiar nature of the Hiickel Hamiltonian, with the short-range forces being dominant in the effective potential. 

All of these results show that the value of the band gap converges rapidly to zero with the growth of the skeleton toward the two-dimensional direction. Kertesz and Hoffmann (1983) have examined the relationship of the decrease in the band gap in the framework of the Hiickel Hamiltonian. By the same token, polyperylene, shown in Fig. 15e, is a graphitized version of poly(p-phenylene), which will be discussed in the next section, and has a smaller band gap, calculated to be 3.17 eV, than the 10.67-eV band gap calculated for poly(p-phenylene) (Tanaka et al., 1984c). It would be of interest to note that all of these polymers have the HO and LU bands of it nature, the major coefficients of which consist in the carbon atoms on the perimeter of the polymer chain. Therefore, it is predicted that these polymers will be electrically conductive per se or... 

We suppose that there is only one delocalised electron per site (an appropriate assumption for an alkali cluster). The Hiickel Hamiltonian for the cluster can then be written as ... 

Consider the Hiickel Hamiltonian-matrix, HI, for an alternant hydrocarbon, constructed on the basis of the simple Huckel-approximations (equations (6-2)-(6-5)) as an example, the matrix, HI, for butadiene is shown in equation (2-54). If the (m) starred atoms are labelled from 1, 2,..., rn, and the unstarred ones from m + 1, m + 2,. .., n, then by an exactly similar argument to that used when discussing the corresponding secular-determinant for an alternant hydrocarbon in 6.3, the matrix HI may be partitioned as in equation (D7). 

The Hiickel Hamiltonian-matrix for an n-annulene (Fig. 5-1) is the matrix in the determinant found in equation (5-1), with the quantities ( — x), along the diagonal, replaced by zero (i.e. a). As pointed out in Appendix A, this matrix is isomorphic with the adjacency matrix, A(C ), of the correspondingly-labelled molecular graph furthermore, the eigenvalues of the matrix, A(C ), are the annulene energy-levels we require (expressed in units of fi and... 

We write the Hiickel Hamiltonian for a polymer composed of a periodic sequence of monomers as,... 

Using this, the Hiickel Hamiltonian takes the form ... 

Hereafter, as we try to reach analytic conclusions, the k electrons of a conjugated hydrocarbon are described by the Hiickel Hamiltonian. The on-site energies are assumed to be equal and define the zero of energy ... 

From these symmetry-adapted SOMOs and (p, which are the canonical (symmetry-adapted) SOMOs of the Hiickel Hamiltonian, again obtained without diagonalization, one may define the localized SOMOs... 

It is interesting at this stage to compare these MOs to the HOMO and the LUMO of the whole molecule. Actually they are somewhat different. The action of the Hiickel Hamiltonian on I [Pg.377]

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