Hamiltonian Hiickel-Hubbard

We have outlined above the procedure for the construction of orthonormal molecular-orbital and atomic-orbital Gel fand states and for the conversion of the latter to the non-orthogonal valence bond states. We require, in addition, a freeon Hamiltonian to compute the spectra of the several polyene systems. For this we employ the freeon, reduced Hiickel-Hubbard Hamiltonian which has the following form ... 

The calculation of the ethylene spectrum from the Hiickel-Hubbard Hamiltonian is a freshman exercise. However a good approximation to the spectrum is obtained by linear interpolation between the MO and AO limits. See Fig. 3.1... 

The application of second-order perturbation theory to the Hiickel-Hubbard Hamiltonian for z close to one (Section 4.2) yields the freeon, antiferromagnetic, Heisenberg exchange Hamiltonian,... 

In this Section the ferromagnetic and the antiferromagnetic Heisenberg exchange Hamiltonians are derived from the Hiickel-Hubbard Hamiltonian in which the value of the coupling constant, z takes on values from - 1 to + 1. The M = N = 2, linear, extended Hiickel-Hubbard spectrum is plotted in Fig. 4.1. (Compare with Fig. 3.1)... 

The freeon theory of ferromagnetism is based on the negative-U, Hiickel-Hubbard Hamiltonian (NUHH). However the conventional positive-U, Hiickel-Hubbard (PUHH) Hamiltonian employs a positive U which is a measure of the repulsion of a bare pair of electrons on a single site. From whence comes NUHH There are three positions to take with respect to NUHH ... 

Tight-binding approximation (TBA) Hiickel-Hubbard Hamiltonian Hopping integral t Peierls distortion Magnetic material Nonmagnetic material... 

The term model Hamiltonian is more general. From this point of view the effective Hamiltonians and pseudo-Hamiltonians could also be considered as model Hamiltonians . However, for clarity, we suggest that the term model Hamiltonian should be used when some simplified form of an approximate Hamiltonian has been guessed from a preliminary physical analysis. In contrast with the effective Hamiltonians and the pseudo-Hamiltonians that can be obtained by means of well defined mathematical procedures, the model Hamiltonians are generally parametrized from experiment. They would involve the semi-empirical Hamiltonians of quantum chemistry and solid-state physics (Hiickel, Hubbard, Pariser-Parr-Pople (PPP),...). 

MO models with electron repulsion Hiickel-Hubbard Hamiltonian... 

The vibrational consequences of 7r-electron fluctuations discussed in Sections II and III drew on both molecular spectroscopy and solid-state physics. The analysis of NLO and EA spectra in Section IV combined PPP models for molecules with quantum cell models of alternating chains. We proposed at the outset to relate the conjugated polymers in Fig. 6.2 to alternating Fliickel or PPP chains and have so far discussed vibrational and optical implications of 7r-electron models rather than the Hamiltonian, Eq. (7), or its mathematical properties. The analysis holds for any H(8) with appropriate vibrational or optical susceptibilities. Equation (7) is sufficiently general to encompass Hiickel, Hubbard, extended Hubbard, PPP, and other models with suitable choices of U and Vp,. This generality is an extremely useful feature of solid-state models. 

This is a very popular form of the Hamiltonian that in the solid state physics literature is known as the Hubbard model. It has all the convenience of the Hiickel approximation plus the most important part of the electron correlation known as the Coulomb blockade electrons (of opposite spins) which are on the same site i repel one another with an energy [ i ii]. If all the atoms are the seune and each contributes one valence orbital, then (20) has the even simpler form of the simple Hubbard Hamiltonian ... 

Flocke et al (FSK) view the Hubbard model in this problem as essentially the Hiickel model, but supplemented by the penalty term (Hubbard U) associated with double occupancy of an atomic orbital. In its simple form all C - C bonds are characterized by a common hopping parameter, t say, and all C atoms have the common electron-electron repulsion energy U already mentioned above. As PKS stress, the Hubbard Hamiltonian constitutes the simplest gauge-invariant Hamiltonian taking into account electron correlation. The extent of electron correlation is readily varied through the ratio U/t a continuous variation from the (uncorrelated) Hiickel approximation U = 0,t 7 0) to a fully correlated limit t = O U 0) being available. 

The one-electron part of the Hubbard Hamiltonian is mostly simplified in the same manner as in the Hiickel model, cf Eqs. (10.5 — 10.7). Having a homogenous system, the on-site repulsion parameters can also be taken equal. This uniform value is often denoted by U, and within this condition the Hubbard Hamiltonian takes the form ... 

While the solution of the Hiickel model is mathematically very simple (it consists of the diagonalization of the matrix of the one-electron integrals h ), the diagonalization of the Hubbard Hamiltonian is not always straightforward (Whaley Falicov 1987). One-determinantal wave functions can be calculated analogously to the Hartree-Fock theory by an iterative procedure this corresponds to averaging the two-electron term. Such wave functions are only approximations to the exact solution of the Hubbard Hamiltonian whose exact eigenfunctions can be expressed only as linear combinations of determinants. 

Hiickel and PPP theory emphasize common aspects of 7T electrons in molecules whose geometry is specified in advance. The backbones of the representative conjugated polymers in Fig. 6.2, for example, immediately fix the Hiickel or PPP Hamiltonians. Effects due to substituents R or sulfur heteroatoms in polythiophene (PT) are approximated by site energies or neglected as perturbations of TT-TT tran.sitions. Such a priori information about Ha and V(/ ) comes from molecules and contrasts sharply with the adjustable parameters of extended Hubbard models. The a- conjugation encountered in polysi-... 

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