Hamiltonian transfer integral terms

In the electronic Hamiltonian t +i, is the transfer integral, i.e. the re-electron wavefunction overlap between nearest neighbour sites in the polymer chain, and is equivalent to the parameter /3 in Equation (4.20), and c 1+,s. and clhS are creation and annihilation operators that create an electron of spin s ( 1/2) on the carbon atom at site n-f 1 and destroy an electron of spin, s at the carbon atom on site n, i.e. in effect transfer an electron between adjacent carbon atoms in the polymer chain. The elastic term is just the energy of a spring of force constant k extended by an amount ( +1— u ), where the u are the displacements along the chain axis of the carbon atoms from their positions in the equal bond length structure, as indicated in Fig. 9.8(b). The extent of the overlap of 7i-electron wavefunction will depend on the separation of nearest neighbour carbon atoms and is approximated by ... 

We have employed the PPP model with independent bond distortions for a fixed electron-phonon coupling constant. The distortions of the bond lead to change in the corresponding transfer integrals. The bond distortions are introduced such that the total bond-order of the chain remains a constant. In the absence of this constraint, the purely tt electron Hamiltonian would lead to a collapse of the chain to a point since the total energy of the system tends to decrease with decrease in chain length equivalently, the tt electron Hamiltonian does not have an in-built repulsive term which keeps the atoms apartsince this term comes from the a framework and the internuclear potentials. Imposition of the constraint of constant total bond-order serves the purpose of the a framework and the inter-nuclear potentials. 

The minimalist model which mimics the essential features derived by the computational studies on realistic systems is made by a one dimensional stack of planar conjugated molecules (Fig. 11), with one molecular orbital per molecule (for example the HOMO in case of hole transport). Each molecule j, associated with the orbital j), has mass m and can be displaced transversally by a length uj from its equilibrium position around which it oscillates with frequency to. The transfer integral T between the consecutive orbitals j) and y - -1) is modulated linearly by the term aa uj+i - uj) with a the electron-phonon coupling constant. The semiclassical Hamiltonian for this system reads... 

Other methods of calculating the O N separation dependent proton transfer rates, such as a Fermi Golden Rule approach (Siebrand et al. 1984), can also be employed. In this approach, two harmonic potential wells (e.g., O-H N and, O H-N) are considered to be coupled by an intermolecular term in the Hamiltonian. Inclusion of the van der Waals modes into this approximation involves integration of the coupling term over the proton and van der Waals mode wavefunctions for all initial and final states populated at a given temperature of the system. Such a procedure requires the reaction exothermicity and a functional form for the variation of the coupling as a function of well separation. In the present study, we employ the barrier penetration approach this approach is calculationally straightforward and leads to a clear qualitative physical picture of the proton transfer process. 

In these relations the operator B (Bn) describes the creation (annihilation) of a molecular excitation at lattice site n. We assume below that n 1,2,. ..,Ar, where N is the number of molecules in the chain and we consider one electronically excited molecular state. Then E-p is the on-site energy of a Frenkel exciton and Mnni is the hopping integral for molecular excitation transfer from molecule n to molecule n. In the summation in HF the terms with n = n are omitted. The Hamiltonian HF describes the Frenkel excitons in the Heitler-London approximation. 

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