Coupling external fields

Conversion of magnetic hyperfine coupling constants A into internal fields for Fe When the nuclear spin is decoupled from the electronic spin by application of an external field, the hyperfine coupling, according to (4.76), is defined as... 

In this review we discuss the theoretical frame which may serve as a basis for a DFT formulation of solvent effects for atoms and molecules embedded in polar liquid environments. The emphasis is focused on the calculation of solvation energies in the context of the RF model, including the derivation of an effective energy functional for the atomic and molecular systems coupled to an electrostatic external field. 

External fields have been widely used to align block copolymer microdomains. This approach relies on the ability to couple an external applied bias field to some molecular or supermolecular feature, and thus achieve directional control over the microdomains. External fields, such as an electric field [61-63], mechanical flow field [53-55], and temperature gradient [60], have been utilized to control the long-range orientation of block copolymers in the bulk state. 

The NMR spectrum of 1, 1, 2, trichloroethane shares a doublet for the methylene protons due to one adjacent non-equivalent CH protons. The spins of two protons adjacent to the one considered couple in three different manners relative to the external field, reinforcing not affecting and opposing the external field. 

External fields are introduced in the relativistic free-particle operator hy the minimal substitutions (17). One should at this point carefully note that the principle of minimal electromagnetic coupling requires the specification of particle charge. This becomes particularly important for the Dirac equation which describes not only the electron, but also its antiparticle, the positron. We are interested in electrons and therefore choose q = — 1 in atomic units which gives the Hamiltonian... 

Much information of interest for atomic and molecular systems involves properties other than energy, usually observed via the energy shifts generated by coupling to some external field. The desired property is then the derivative of the energy with respect to the external field, which may be obtained by two different approaches. The finite-field method solves the Schrodinger equation in the presence of the external field, yielding... 

The concept of quantum ferroelectricity was first proposed by Schneider and coworkers [1,2] and Opperman and Thomas [3]. Shortly thereafter, quantum paraelectricity was confirmed by researchers in Switzerland [4], The real part of the dielectric susceptibihty of KTO and STO, which are known as incipient ferroelectric compounds, increases when temperature decreases and becomes saturated at low temperature. Both of these materials are known to have ferroelectric soft modes. However, the ferroelectric phase transition is impeded due to the lattice s zero point vibration. These materials are therefore called quantum paraelectrics, or quantum ferroelectrics if quantum paraelectrics are turned into ferroelectrics by an external field or elemental substitution. It is well known that commercially available single crystal contains many defects, which can include a dipolar center in the crystal. These dipolar entities can play a certain role in STO. The polar nanoregion (PNR originally called the polar microregion) may originate from the coupling of the dipolar entities with the lattice [5-7]. When STO is uniaxially pressed, it turns into ferroelectrics [7]. 

Figure 3.39. Holographic setup for photorefractive molecular glasses. The sample is tilted toward the grating, allowing an applied external field to support the motion of the mobile charges. The phase shift of the refractive index grating can be determined by measuring the transmitted writing beam intensities (two-beam coupling).

Identify magnetically coupled iron and magnetic state by response to applied external field... 

The symbol V(q,Q) stands for a kinematic operator containing spin-orbit terms, electron-phonon couplings and, eventually, a coupling to external fields. The molecular Hamiltonian is given by ... 

Dipolar (or direct) coupling between nuclear dipoles is the second mechanism of spin-spin interaction (see Section 4.1.1.2. for scalar coupling). Acting through space209-212 it is dependent on the distance r between the dipoles as well as the angle between their vectors and the direction of the external field B0. 

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