Interaction Hamiltonian electronic

Hamiltonians equivalent to (1) have been used by many authors for the consideration of a wide variety of problems which relate to the interaction of electrons or excitons with the locaJ environment in solids [22-25]. The model with a Hamiltonian containing the terms describing the interaction between excitons or electrons also allows for the use of NDCPA. For example, the Hamiltonian (1) in which the electron-electron interaction terms axe taken into account becomes equivalent to the Hamiltonians (for instance, of Holstein type) of some theories of superconductivity [26-28]. 

In addition, the numerator will be nonzero only for double substitutions. Single substitutions are known to make this expression zero by Brillouin s theorem. Triple and higher substitutions also result in zero value since the Hamiltonian contains only one and two-electron terms (physically, this means that all interactions between electrons occur pairwise). 

The continuum model with the Hamiltonian equal to the sum of Eq. (3.10) and Eq. (3.12), describing the interaction of electrons close to the Fermi surface with the optical phonons, is called the Takayama-Lin-Liu-Maki (TLM) model [5, 6], The Hamiltonian of the continuum model retains the important symmetries of the discrete Hamiltonian Eq. (3.2). In particular, the spectrum of the single-particle states of the TLM model is a symmetric function of energy. 

Orientational disorder and packing irregularities in terms of a modified Anderson-Hubbard Hamiltonian [63,64] will lead to a distribution of the on-site Coulomb interaction as well as of the interaction of electrons on different (at least neighboring) sites as it was explicitly pointed out by Cuevas et al. [65]. Compared to the Coulomb-gap model of Efros and Sklovskii [66], they took into account three different states of charge of the mesoscopic particles, i.e. neutral, positively and negatively charged. The VRH behavior, which dominates the electrical properties at low temperatures, can conclusively be explained with this model. 

The relativistic treatment of electron EDM begins by replacing the nonrelativistic Hamiltonian H and the interaction Hamiltonian Hi by their relativistic counterparts... 

Consequently, we introduce the second approximation which is to use an approximate electrostatic potential in Eq.(4-21) to determine inter-fragment electronic interaction energies. Thus, the electronic integrals in Eq. (4-21) are expressed as a multipole expansion on molecule J, whose formalisms are not detailed here. If we only use the monopole term, i.e., partial atomic charges, the interaction Hamiltonian is simply given as follows ... 

Electronic Hamiltonian, conical intersections, spin-orbit interaction, 559 Electronic states ... 

Unfortunately, the Schrodinger equation for multi-electron atoms and, for that matter, all molecules cannot be solved exactly and does not lead to an analogous expression to Equation 4.5 for the quantised energy levels. Even for simple atoms such as sodium the number of interactions between the particles increases rapidly. Sodium contains 11 electrons and so the correct quantum mechanical description of the atom has to include 11 nucleus-electron interactions, 55 electron-electron repulsion interactions and the correct description of the kinetic energy of the nucleus and the electrons - a further 12 terms in the Hamiltonian. The analysis of many-electron atomic spectra is complicated and beyond the scope of this book, but it was one such analysis performed by Sir Norman Lockyer that led to the discovery of helium on the Sun before it was discovered on the Earth. 

Any Hamiltonian for an impurity in a semiconductor must include terms that describe the interactions between the nuclei, the interactions of electrons with the nuclei, and the electron-electron interactions. The latter are the hardest part of the problem. Typically, a one-particle description is used. 

The most obvious defect of the Thomas-Fermi model is the neglect of interaction between electrons, but even in the most advanced modern methods this interaction still presents the most difficult problem. The most useful practical procedure to calculate the electronic structure of complex atoms is by means of the Hartree-Fock procedure, which is not by solution of the atomic wave equation, but by iterative numerical procedures, based on the hydrogen model. In this method the exact Hamiltonian is replaced by... 

As indicated, we shall denote electrons and nuclei with Roman (/) and Greek (a) indices, respectively. In terms of kinetic-energy operators for electrons ( e) and nuclei (7k) and the Coulombic potential-energy interactions of electron-electron (Dee), nuclear-nuclear (UNN), and nuclear-electron (VW) type, we can write the supermolecule Hamiltonian as... 

The problem of T[p] is cleverly dealt with by mapping the interacting many-electron system on to a system of noninteracting electrons. For a determinantal wave function of a system of N noninteracting electrons, each electron occupying a normalized orbital >p, (r), the Hamiltonian is given by... 

Here Zg is the number of tt electrons provided by atom is essentially an ionization potential for an electron extracted from in the presence of the part of the framework associated with atom r alone (a somewhat hypothetical quantity), is a framework resonance integral, and is the coulomb interaction between electrons in orbitals < >, and <(>,. The essential parameters, in the semi-empirical form of the theory, are cug, and and from their definition these quantities are expected to be characteristic of atom r or bond r—s, not of the particular molecule in which they occur (for a discussion see McWeeny, 1964). In the SCF calculation, solution of (95) leads to MO s from which charges and bond orders are calculated using (97) these are used in setting up a revised Hamiltonian according to (98) and (99) and this is put back into (95) which is solved again to get new MO s, the process being continued until self-consistency is achieved. It is now clear that prediction of the variation of the self-consistent E with respect to the parameters is a matter of considerable difficulty. 

We know from Chapter 1 that the probability P,f of indncing an optical transition from a state i to a state / is proportional to (1 //1), where in the matrix element Ip, and P f denote the eigenfnnctions of the ground and excited states, respectively, and H is the interaction Hamiltonian between the incoming light and the system (i.e., the valence electrons of the center). In general, we can assnme that // is a sinnsoidal... 

In the vicinity of the atomic absorption edges, the participation of free and bound excited states in the scattering process can no longer be ignored. The first term in the interaction Hamiltonian of Eq. (1.11) leads, in second-order perturbation theory, to a resonance scattering contribution (in units of classical electron scattering) equal to (Gerward et al. 1979, Blume 1994)4... 

When the electric field gradient at the nucleus exerted by the electrons is nonzero, the nuclear levels will be split. The eigenvalues of the quadrupolar interaction Hamiltonian are given by... 

Symmetry tools are used to combine these M objects into M new objects each of which belongs to a specific symmetry of the point group. Because the hamiltonian (electronic in the m.o. case and vibration/rotation in the latter case) commutes with the symmetry operations of the point group, the matrix representation of H within the symmetry adapted basis will be "block diagonal". That is, objects of different symmetry will not interact only interactions among those of the same symmetry need be considered. 

The binding corrections to h q)erfine splitting as well as the main Fermi contribution are contained in the matrix element of the interaction Hamiltonian of the electron with the external vector potential created by the muon magnetic moment (A = V X /Lx/(47rr)). This matrix element should be calculated between the Dirac-Coulomb wave functions with the proper reduced mass dependence (these wave functions are discussed at the end of Sect. 1.3). Thus we see that the proper approach to calculation of these corrections is to start with the EDE (see discussion in Sect. 1.3), solve it with the convenient... 

The weak interaction contribution to hyperfine splitting is due to Z-boson exchange between the electron and muon in Fig. 6.7. Due to the large mass of the Z-boson this exchange is effectively described by the local four-fermion interaction Hamiltonian... 

These fields become important in magnetic resonance spectroscopies, where the interaction between electronic and nuclear spins is considered in phenomenological Spin Hamiltonians. Note... 

The direct dipole-dipole interaction between electron spins given in Eq. (14) can also contribute to D and E in the spin Hamiltonian. Various estimates of its contribution have shown it to be much smaller than the spin-orbit terms for transition-metal ions. For systems in which the crystal field is greatly distorted, this term can become large, however, and it is found to be the major source of D in the spin Hamiltonian of organic triplet-state molecules, where the spin-orbit terms are small as a result of the small size of the spin-orbit coupling parameter. 

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