Hamiltonian interaction term

Here, Uo is the potential part of Ho, while Uml pertains to Hmt. All Hamiltonian interaction terms [both kinetic and potential, except (loo(2> (Eq. 3)] were expressed, in terms of raising and lowering operators, in explicit totally symmetric form, thus allowing an analytical calculation of the required matrix elements (i Hmi 1 ) for the Hamiltonian matrix. 

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]. 

All discussions of transport processes currently available in the literature are based on perturbation theory methods applied to kinetic pictures of micro-scattering processes within the macrosystem of interest. These methods do involve time-dependent hamiltonians in the sense that the interaction operates only during collisions, while the wave functions are known only before and after the collision. However these interactions are purely internal, and their time-dependence is essentially implicit the over-all hamiltonian of the entire system, such as the interaction term in Eq. (8-159) is not time-dependent, and such micro-scattering processes cannot lead to irreversible changes of thermodynamic (ensemble average) properties. 

To describe atoms with several electrons, one has to consider the interaction between the electrons, adding to the Hamiltonian a term of the form Ei< . Despite this complication it is common to use an approximate wave function which is a product of hydrogen-like atomic orbitals. This is done by taking the orbitals in order of increasing energy and assigning no more than two electrons per orbital. 

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] ... 

When we include the Zeeman interaction term, gpBB-S, in the spin Hamiltonian a complication arises. We have been accustomed to evaluating the dot product by simply taking the direction of the magnetic field to define the z-axis (the axis of quantization). When we have a strong dipolar interaction, the... 

We now notice that we could write a Hamiltonian operator that would give the same matrix elements we have here, but as a first-order result. Including the electron Zeeman interaction term, we have the resulting spin Hamiltonian ... 

In Equation 7.33 we have written out both the g-value and the zero-field coefficient of the basic S2 interaction term in the form of diagonal 3x3 matrices in which all off-diagonal elements are equal to zero. The diagonal elements were indexed with subscripts x, y, z, corresponding to the Cartesian axes of the molecular axes system. But how do we define a molecular axis system in a (bio)coordination complex that lacks symmetry The answer is that if we would have made a wrong choice, then the matrices would not be diagonal with zeros elsewhere. In other words, if the spin Hamiltonian would have been written out for a different axes system, then, for example, the g-matrix would not have three, but rather six, independent elements ... 

In HMO theory the assumed separability is partially justified by the symmetry of the Hamiltonian. In the present instance there is no such symmetry and neglect of interaction terms is tantamount to assuming a classical Hamiltonian. 

In order to take into account these intra-atomic terms, and in a way similar to the Stoner s model, Hubbard ), see also adds to the Hamiltonian (11) an exchange interaction term ... 

In contrast to the one-electron terms, the reduction of the 4x4 Dirac-Breit Hamiltonian to the 2x2 Breit-Pauli Hamiltonian is very tedious for the two-electron terms as each interaction term has to be transformed according to the Foldy-Wouthuysen protocol. As the derivation can be found for example in Refs. (56-58) and in detail in Ref. (21), we only present here the transformed terms and discuss their dimension. The two-electron Breit-Pauli operator gBP (i, j) reads... 

The nuclear spins give rise to additional terms in the Breit-Pauli Hamiltonian due to the interaction of the electrons with the magnetic moment of the nuclei and the electrostatic interaction with the electric quadrupole interaction of the nuclei. The magnetic interaction term of the spins with the nuclei is of the same type as the spin-spin interaction and following Abragam and Pryce (61) can be written as... 

The vibration-rotation interaction term makes the Hamiltonian for nuclear motion of a polyatomic molecule difficult to deal with. Frequently, this term is small compared to the other terms. We shall make the initial approximation of omitting Tvib rot. The rotational kinetic energy TTOt involves the moments of inertia of the molecule, which in turn depend on the instantaneous nuclear configuration. However, the vibrational motions are much faster than the rotational motions, so that we can make the approximation of calculating the moments of inertia averaged over the vibrational motions. 

The Hamiltonian of the system consists of the sum of internal Hamiltonians and an interaction term... 

R m can be expressed in terms of the probabilities of all possible transitions involving different spin states. From the time dependent perturbation theory, if Ho is the static Hamiltonian (first two terms in Eq. (3.13)) with eigenvalues E = ha> and H, is the time dependent perturbing Hamiltonian (third term in Eq. (3.13)), which describes an interaction with energy EdO and whose explicit form will depend on the kind of interaction, the probability to have a transition from the m to the n state is proportional to... 

Since there is a non-Abelian nature to this theory, we return to the nonrelativistic equation that describes the interaction of a fermion with the electromagnetic field. The Pauli Hamiltonian is modified with the addition of a interaction term [9]... 

For the 2 2g and 5T2g-terms of the Oh-reference the Griffith theory could be appropriate. In the case of the Cl-interacting terms 3Tig or 4 Tig, the Figgis isotropic Hamiltonian can be applied. These theories offer the magnetic susceptibility formulae in closed forms. However, these approaches... 

Two - particle energy correction correction to electron - electron correlation energy due to the phonon field. This non-adiabatic term represents full attractive contribution, and can be compared to the reduced form of Frohlich effective Hamiltonian which maximizes attractive contribution of electron - electron interaction and that can be either attractive or repulsive (interaction term of the BCS theory). For superconducting state transition at the non-adiabatic conditions, the two-particle correction is unimportant - see [2],... 

Given the lattice Hamiltonian Eq. (5), which casts the interactions in terms of site-specific and site-site interaction terms, a complementary diabatic representation can be constructed which diagonalizes the Hamiltonian excluding the electron-phonon interaction, Hq = He + f7ph. This leads to the form... 

Now we see clear the problem while the new dot Hamiltonian (154) is very simple and exactly solvable, the new tunneling Hamiltonian (162) is complicated. Moreover, instead of one linear electron-vibron interaction term, the exponent in (162) produces all powers of vibronic operators. Actually, we simply remove the complexity from one place to the other. This approach works well, if the tunneling can be considered as a perturbation, we consider it in the next section. In the general case the problem is quite difficult, but in the single-particle approximation it can be solved exactly [98-101]. 

The second strategy we mention in this rapid survey replaces the QM description of the solvent-solvent and solute-solvent with a semiclassical description. There is a large variety of semiclassical descriptions for the interactions involving solvent molecules, but we limit ourselves to recall the (1,6,12) site formulation, the most diffuse. The interaction is composed of three terms defined in the formula by the inverse power of the corresponding interaction term (1 stays for coulombic interaction, 6 for dispersion and 12 for repulsion). Interactions are allowed for sites belonging to different molecules and are all of two-body character (in other words all the three- and many-body interactions appearing in the cluster expansion of the Hss and HMS terms of the Hamiltonian (1.1)... 

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