Hamiltonian hopping term

In the next step, to analyze the resonant processes associated with charge exchange between He+ and He , we consider a spin-less Newns-Anderson Hamiltonian [18] where the level and the hopping terms Tis /a that have been neglected to calculate ElHe" "] and [He , are intioduced. This Hamiltonian reads... 

Hubbard initiated a treatment where some electron interaction is included in a localized description with the aim to deal with magnetic features. His model has become a popular and widely used vehicle for the study of electron correlation. It is based on the reduction of the electron repulsion terms to intraatomic ones and to use only the form with the largest pair parameter, (Hint)Ao gAO gAO, for each atom A. Interatomic coupling is represented by a one-particle operator which moves particles from one atom to another, so-called hopping terms. The Hamiltonian is then... 

FVom the above discussion, it follows that the construction of parameterized hamiltonians for molecular orbital calculations may lead to certain operator relationships being violated in a limited basis. It is also clear that some of these operator relations can be restored at the expense of introducing various approximations in the evaluation of integrals. Atomic parameters may be derived from consideration of the separated atoms limit, while interatomic parameters are commonly associated with overlap integrals and possibly other functions of the interatomic distance. For instance, it is often assumed that when r is a spin orbital on atom A and s is one on atom B, a suitable form for the hopping term is... 

The first two terms in equation (1.6) are the kinetic and the elastic energy of tiie system, the third is the electron hopping term, and the fourth corresponds to the correlation energy. If C/=0, equation (1.6) reduces to the Su-Schrieffer-Heeger Hamiltonian with a Peierls gap of... 

Consider the state where the nominal O 2p electrons at each site are six, that is, the 2p orbitals are fully occupied and the 3d orbitals are partially occupied by an integer number of electrons per site. The electron energy gap is considered based on the above Hamiltonian model, without the electron hopping term. The energy gap is defined by the energy required to remove an electron from a site and add this electron at another site, which is far away from the initial site in the system ... 

Figure 7.5. The three terms in the effective second order hamiltonian obtained from the Hubbard model in the limit t < U. (a.) Thehopping term, which allows spin-up or spin-down electrons to move by one site when the neighboring site is unoccupied, (b) The exchange term, which allows electrons to exchange spins or remain in the same configuration, at the cost of virtual occupation of a site by two electrons of opposite spin (intermediate configuration), (c) The pair hopping term, which allows a pair of electrons with opposite spins to move by one site, either with or without spin exchange, at the cost of virtual occupation of a site by two electrons of opposite spin (intermediate configuration). Adapted from Ref. [87].

In the Hamiltonian Eq. (3.39) the first term is the harmonic lattice energy given by Eq. (3.12). It depends only on A iU, i.e., the part of the order parameter that describes the lattice distortions. On the other hand, the electron Hamiltonian Hcl depends on A(.v), which includes the changes of the hopping amplitudes due to both the lattice distortion and the disorder. The free electron part of Hel is given by Eq. (3.10), to which we also add a term Hc 1-1-1 that describes the Coulomb interne-... 

At this point, we have reached the stage where we can describe the adatom-substrate system in terms of the ANG Hamiltonian (Muscat and Newns 1978, Grimley 1983). We consider the case of anionic chemisorption ( 1.2.2), where a j-spin electron in the substrate level e, below the Fermi level (FL) eF, hops over into the affinity level (A) of the adatom, whose j-spin electron resides in the lower ionization level (I), as in Fig. 4.1. Thus, the intra-atomic electron Coulomb repulsion energy on the adatom (a) is... 

The interest here is in the energy levels of molecular systems. It is well known that an understanding of these energy levels requires quantum mechanics. The use of quantum mechanics requires knowledge of the Hamiltonian operator Hop which, in Cartesian coordinates, is easily derived from the classical Hamiltonian. Throughout this chapter quantum mechanical operators will be denoted by subscript op . If the classical Hamiltonian function H is written in terms of Cartesian momenta and of interparticle distances appropriate for the system, then the rule for transforming H to Hop is quite straightforward. Just replace each Cartesian momentum component... 

The Schrodinger equation for nuclear motion contains a Hamiltonian operator Hop,nuc consisting of the nuclear kinetic energy and a potential energy term which is Eeiec(S) of Equation 2.7. Thus... 

Here indexes a = 1,2 numerate spinor components. Because we ignore direct hopping of d-electrons on the lattice, in the Hamiltonian (10) a quadratic term with the A -operators is absent, and A -operators enter linearly via the hybridization term. However in the second order perturbation with respect to hybridization such a term should appear it describes the induced hopping on the lattice. 

The summation over n is eliminated by the projection operator P and its complementary projection operator Q = I — P. The operator P projects to the IV-electron states with Nr electrons in the R-system. By acting on the states of eq. (1.227) it cuts off all the terms with electron distributions different from the required one. The general technique with the projection operator P in the previous section leads to the effective Hamiltonian with the intersubsystem electron hopping Wr projected out ... 

For one hole in the half-filled band, the exact energy spectrum of the chain with free ends formed by L unit cells is spin-degenerate, similar to the spectrum of the uniform Hubbard chain with U = oo. In the case of periodic boundary conditions an electron hopping between the first and the last unit cells of the chain leads to the additional term to the Hamiltonian (9). For one hole in the half-filled band this term has the following form ... 

Note, in general case the effective Hamiltonian contains also the Coulomb interaction between doped holes and the interaction of quasiparticles via the phonon field. We dropped these terms here, because they do not contribute directly to the spin susceptibility. The hopping matrix element is Uj = tfj e rE /hcot [ + < SiSj >] where r . is the bare hopping inte-... 

Hopping and hybridization terms introduce strong coupling between the QE, stripon and spinon fields, which is expressed by a coupling Hamiltonian of the form (for p-type cuprates) ... 

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