Hamiltonian for electrons

There are cases in which the angular momentum operators themselves appear in the Hamiltonian. For electrons moving around a single nucleus, the total kinetic energy operator T has the form ... 

These ideas can be applied to electrochemical reactions, treating the electrode as one of the reacting partners. There is, however, an important difference electrodes are electronic conductors and do not posses discrete electronic levels but electronic bands. In particular, metal electrodes, to which we restrict our subsequent treatment, have a wide band of states near the Fermi level. Thus, a model Hamiltonian for electron transfer must contains terms for an electronic level on the reactant, a band of states on the metal, and interaction terms. It can be conveniently written in second quantized form, as was first proposed by one of the authors [Schmickler, 1986] ... 

The leading term in T nuc is usually the magnetic hyperfine coupling IAS which connects the electron spin S and the nuclear spin 1. It is parameterized by the hyperfine coupling tensor A. The /-dependent nuclear Zeeman interaction and the electric quadrupole interaction are included as 2nd and 3rd terms. Their detailed description for Fe is provided in Sects. 4.3 and 4.4. The total spin Hamiltonian for electronic and nuclear spin variables is then ... 

Holme, T. A., and Levine, R. D. (1988), An Algebraic Hamiltonian for Electronic Nuclear Degrees of Freedom Based on the Vector Model, Inti. J. Quant. Chem. 34, 457. 

In quantum mechanics the splitting of electronic terms is described by using the degenerate version of the perturbation theory. The Hamiltonian for electrons in the atom in the crystal environment acquires the form ... 

Here h i), z =l or 2, stands for the core Hamiltonian for electron i moving in the field of nuclei a and b separated by distance R, and g(l,2) is the repulsion potential between two electrons away from each other by distance rl2. 

We now show how the many-electron Hamiltonian developed in the previous chapter may be extended to include magnetic interactions which arise from the presence of nuclear spin magnetic moments. Equation (3.140) represents the Hamiltonian for electron i in the presence of other electrons we present it again here ... 

A general model Hamiltonian for electron transfer in an electrochemical environment must contain terms for the different components of the system, i.e., the reactant, the electrode and the solvent, and their corresponding interactions ... 

Finally we must examine the property of additivity of the exchange Hamiltonian. Let us consider, for instance, four interacting electrons and let us specifically examine the exchange coupling between electrons 1 and 2. The effect of electrons 3 and 4 is treated through the effective (mean) potentials SV n) and SV r2) seen by electrons 1 and 2 (Hartree-Fock approximation). Under these conditions the general Hamiltonian for electrons 1 and 2 may be written ... 

In the nrl (after removing the rest mass terms) the two diagonal blocks of the FW-transformed Dirac operator become the nonrelativistic Hamiltonian for electrons and positrons respectively. We shall see later (section... 

Although the transformed operator is not fully block diagonal, the block L li is, nevertheless, an effective Hamiltonian, with its (right) eigenfunctions in the model space of electronic states. Unlike with the FW transformation one only gets an effective Hamiltonian for electrons, there is an alternative similarity transformation for positronic states. 

While the next step for the FW transformation was a renormalization by means of the transformation Wb in order to achieve an overall unitary transformation (which can actually be done before or after the projection to positive energy states), we now apply a transformation that simultaneously block-diagonalizes the Hamiltonian for electrons, such that it isolates a block corresponding to the model space, reestablishes unitarity of this... 

It is an open question whether it is possible, starting from the QED Hamiltonian with electronic, positronic, and photonic degrees of freedom, to construct an effective Hamiltonian for electrons only. If this is possible, the effective Hamiltonian will certainly not be simple, it will e.g. also contain a three-electron interaction [86]. 

M. Berrondo, J. P. Daudey and O. Goscinski Effective One-Body Hamiltonian for Electrons in Atoms Chem. Phys. Letters 62, 34 (1979). 

Here, h r) is a one-particle Dirac Hamiltonian for electron in a field of the finite size nucleus and y is a potential of the interelectron interaction. In order to take into account the retarding effect and magnetic interaction in the lowest order on parameter a (a is the fine structure constant), one could write [23]... 

We allow the overall wave function to be a product of two wave functions, one for each individual electron, as shown in Equation (4.9). The effective Hamiltonian for electron I can then be calculated using Equation (4.10), where is the effective potential energy that electron I feels with respect to electron 2 and is given by Equation (4.1 I). 

We then substitute the resulting value for (p t ) into the effective potential energy equation for electron 2. This value is then used in the equation that corresponds to Equation (4.10) to determine the effective Hamiltonian for electron 2. Then the Schrodinger equation corresponding to Equation (4.12) for electron 2 is solved in order to determine a new value for (p r2)- The whole process is repeated in an iterative manner until the wave functions for 0(r ) and rj) no longer change with time. We call this a SCF and the two resulting wave functions obtained by this method are known as the Hartree-Fock orbitals. 

A Model Hamiltonian for Electron and Ion Transfer Reactions at Metal Electrodes... 

The quantum mechanical description of these two fermions is straightforward in a formal sense considering the one-particle Dirac Hamiltonians for electron e and proton p plus the interaction operator Vep,... 

T. Itoh. Derivation of Nonrelativistic Hamiltonian for Electrons from Quantum Electrodynamics. Rev. Mod. Phys., 37(1) (1965) 159-165. 

The tight-binding Hamiltonian for electrons in graphene, considering that electrons can hop to both nearest- and next-nearest-neighbor atoms, has the form (in units h = 1)... 

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