Tight-binding Hamiltonian

Fig. 5. Band gap as a function of nanotube radius calculated using empirical tight-binding Hamiltonian. Solid line gives estimate using Taylor expansion of graphene sheet results in eqn. (7).

For the description of the random Hamiltonian we employ TB-LMTO formalism in the most tight binding representation . The Hamiltonian for the binary random alloy takes the form ... 

We write a tight-binding (TB) Hamiltonian which, in general, is non-orthogonal,... 

Let us now discuss the calculation of Eband more detail. The tight-binding hamiltonian is written as ... 

Rose and Benjamin (see also Halley and Hautman ) utilized molecular dynamic simulations to compute the free energy function for an electron transfer reaction, Fe (aq) + e Fe (aq) at an electrodesolution interface. In this treatment, Fe (aq) in water is considered to be fixed next to a metal electrode. In this tight-binding approximation, the electron transfer is viewed as a transition between two states, Y yand Pf. In Pj, the electron is at the Fermi level of the metal and the water is in equilibrium with the Fe ion. In Pf, the electron is localized on the ion, and the water is in equilibrium with the Fe" ions. The initial state Hamiltonian H, is expressed as... 

The simplest model of a solid is a linear chain of N atoms, with one end of the chain corresponding to the surface. If an atomic orbital r> (r = 1,..., N) is associated with the rth atom, then, in the tight-binding approximation, the matrix elements of the Hamiltonian for the solid, can... 

In order to understand the concept, it is customary to present a sort of paradox arising from the solution of the Hamiltonian (11) in the tight-binding approximation (Eq. (12)). 

The Hamiltonian of valence electrons (39), in the so-called orthogonal representation (or in the most localized representation, neglecting orbital overlap) can be mapped on a tight-binding form Hamiltonian... 

Within the Landauer approach, the computation of the quantum conductance is thus traced back to the knowledge of the electronic structure—i.e., the Hamiltonian—of the target system molecule+leads . The best-developed implementations of the Landauer framework employ tight-binding Hamilto-... 

Using the Dirac notations a) = ipa( ) and assuming that ipa( ) are or-thonormal functions (a (3) = 5ap we can write the single-particle matrix (tight-binding ) Hamiltonian in the Hilbert space formed by 4>a ( )... 

Finally, in the second quantized form the tight-binding Hamiltonian is... 

Compare the tunneling Hamiltonian (54) and the tight-binding Hamiltonian (2), divided into left and right parts... 

The Hamiltonians (54) and (55) are essentially the same, only the first one is written in the eigenstate basis Ar), g), while the second in the tight-binding basis a), (3) of the left lead and <5), y) of the right lead. Now we want to transform the TB Hamiltonian (55) into the eigenstate representation. 

Canonical transformations from the tight-binding (atomic orbitals) representation to the eigenstate (molecular orbitals) representation play an important role, and we consider it in detail. Assume, that we find two unitary matrices SR and SR, such that the Hamiltonians of the left part Hi and of the right part Hi can be diagonalized by the canonical transformations... 

The tight-binding band structure calculations were based upon the effective one-electron Hamiltonian of the extended Huckel method. [5] The off-diagonal matrix elements of the Hamiltonian were calculated acording to the modified Wolfsberg-Helmholtz formula. All valence electrons were explicitly taken into account in the calculations and the basis set consisted of double- Slater-type orbitals for C, O and S and a single- Slater-type orbitals for H. The exponents, contraction coefficients and atomic parameters were taken from previous work [6],... 

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