Interaction Hamiltonians

In the following, the individual spin interactions are summarized together with some of their properties. 

The largest interaction is the Zeeman interaction. It essentially defines the nuclear polarization (cf. eqn (2.2.3)), 

Here the coupling tensor between the spin vector operator I and the applied magnetic field Bq is given by the unit matrix 1. The prefactor is listed in Table 3.1.1. The energy 

The second laigest interaction is the quadrupole interaction. It is expressed by the operator 

Nuclei with spin quantum number / 5 exhibit an electric quadrupole moment, which couples to the electric field gradient established by the electrons surrounding the nucleus. The quadrupole interaction, therefore, is a valuable sensor of the electronic structure. Following Li, possesses the smallest quadrupole moment (cf. Table 2.2.1). 

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In order to evaluate equation B1.2.6, we will consider the electric field to be in the z-direction, and express the interaction Hamiltonian as... 

In diatomic VER, the frequency Q is often much greater than so VER requires a high-order multiphonon process (see example C3.5.6.1). Because polyatomic molecules have several vibrations ranging from higher to lower frequencies, only lower-order phonon processes are ordinarily needed [34]- The usual practice is to expand the interaction Hamiltonian > in equation (03.5.2) in powers of nonnal coordinates [34, 631,... 

The non-bonded interaction energy, the van-der-Waals and electrostatic part of the interaction Hamiltonian are best determined by parametrizing a molecular liquid that contains the same chemical groups as the polymers against the experimentally measured thermodynamical and dynamical data, e.g., enthalpy of vaporization, diffusion coefficient, or viscosity. The parameters can then be transferred to polymers, as was done in our case, for instance in polystyrene (from benzene) [19] or poly (vinyl alcohol) (from ethanol) [20,21]. 

In other words that a negaton initially in a state of momentum p, energy Vp2 + m2 helicity s, would remain forever in that state (since it does not interact with anything). Let us, however, compute the left-hand side of Eq. (11-123) with the -matrix given in terms of the interaction hamiltonian (11-121). To lowest order the diagrams indicated in Fig. 11-6 contribute and give rise to the following contribution to the matrix element of S between one-particle states... 

Similar considerations lead to the transformation properties of the one-photon states and of the photon in -operators which create photons of definite momentum and helicity. We shall, however, omit them here. Suffice it to remark that the above transformation properties imply that the interaction hamiltonian density Jf mAz) = transforms like a scalar under restricted inhomogeneous Lorentz transformation... 

The interaction hamiltonian density jin A x) transforms like a scalar ... 

The basic physics governing the measurement of the EDM in all types of electrically neutral systems is almost the same as discussed in this section. If the system under consideration has a magnetic moment p and is exposed to a magnetic field B, then the interaction Hamiltonian can be written... 

If the system under consideration also possesses an electric dipole moment d and is exposed to an electric field then the interaction Hamiltonian can be written... 

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 ... 

If the interaction Hamiltonian in the Coulomb term is expanded in a series about the separation vector, the first term of the expansion is a dipole-dipole interaction, the second a dipole-quadrupole interaction, etc.<4> Again reverting to a classical analog (dipole oscillators), the energy of interaction between the two dipoles is inversely proportional to the third power of the... 

The interaction Hamiltonian with the specified short-range interactions between the monomers is given by the following expression [124,125] ... 

The interaction between both subsystems is cast into a form where the physical charge density of the surrounding medium rm(i X) = appears explicitly, and the interaction Hamiltonian describes now the coupling of the solute... 

Empirically corrected DFT theories almost invariably go back to second-order perturbation theory with expansion of the interaction Hamiltonian in inverse powers of the intermolecular distance, leading to R 6, R x, and R 10 corrections to the energy in an isotropic treatment (odd powers appear if anisotropy is taken into account [86]). 

The interaction Hamiltonian is decomposed into a sum of terms characterized by their commutation relations with Sz ... 

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... 

If the transition is of an electric dipole nature, the interaction Hamiltonian can be written as // = p E, where p is the electric dipole moment and E is the electric field of the radiation. The electric dipole moment is given by p =, where r is the... 

Provided that a transition is forbidden by an electric dipole process, it is still possible to observe absorption or emission bands induced by a magnetic dipole transition. In this case, the transition proceeds because of the interaction of the center with the magnetic field of the incident radiation. The interaction Hamiltonian is now written as // = Um B, where is the magnetic dipole moment and B is the magnetic field of the radiation. 

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