Coupling Hamiltonians

Straightforward stepwise integration of the coupled Hamiltonian and LvN differential equations would be computationally inaccurate, because the fast electronic oscillations would demand very small time-steps, while the slow... 

The simplest example of angular momentum coupling is provided by the scaled spin-rotation coupling Hamiltonian... 

Within this reduced base, the coupling Hamiltonian // ) L X describes the energy exchange between the states 0, [2]) and 1, [0]). This kind of exchange coupling is, as a matter of fact, the one used in the adiabatic frame. Indeed, the... 

As demonstrated by Griffin, Levitt, and coworkers in the late 1980s [21, 93], it is also possible to recouple homonuclear dipolar couplings through interference between isotropic chemical shifts and the rotor revolution. This phenomenon, called rotational resonance, occurs when the spinning frequency is adjusted to a submultiple of the isotropic chemical shift difference, i.e., ncor = ct> so o) °. To understand this experiment, the dipolar coupling Hamiltonian in (10) is transformed... 

The so-called HORROR experiment by Nielsen and coworkers [26] introduced continuous rf irradiation recoupling to homonuclear spin-pairs and initiated the later very widely used concept of /-encoded recoupling. Using a irreducible spherical approach as described above, the HORROR experiment (Fig. 2d) is readily described as starting out with the dipolar coupling Hamiltonian in (10) and x-phase rf irradiation in the form Hrf = ncor(Ix +SX), also here without initial constraint on n. The dipolar coupling Hamiltonian transforms into tilted frame (rotation n/2 around Iy + Sy)... 

Substituting Eq. (14) into Eq. (12), and neglecting any correlation between the density matrix and the spin-lattice coupling Hamiltonian, one obtains to first order... 

The dipolar-coupling Hamiltonian (TTy) describes the through-space coupling between two nuclear spins l and Ij. The dipolar coupling has an rk- dependence, and is key to the determination of internuclear distances in both solid-state and solution-state NMR. The high-field truncated form of the dipolar Hamiltonian is given by... 

The main effect of taking spin-orbit interaction into account will be an admixture of singlet character to triplet states and triplet character to singlet states. The spin-orbit coupling Hamiltonian can to a good approximation be described by an effective one-electron operator Hso ... 

Figure 5. Schematic representation of a two-ion resonance-coupled system. Energy gaps Ea and Eb are equal. The coupling Hamiltonian // represents the electrostatic interaction of the electron of ion A with the electron of ion B.

In many respects, particularly since the coupling between the ions is so we k, it is convenient to consider the system as fluctuating from state 20 to state xu- The probability of such a transfer of energy is proportional to the square of the matrix element of the coupling Hamiltonian, namely,... 

Each of the resonances which are separated by the chemical shifts may be further split into several lines by spin-spin interactions. These can be described by an isotropic coupling Hamiltonian of the type... 

The effective-mode transformation described here is closely related to earlier works which led to the construction of so-called interaction modes [75, 76] or cluster modes [77, 78] in Jahn-Teller systems. The approach of Refs. [54,55,72] generalizes these earlier analyses to the generic form - independent of particular symmetries - of the linear vibronic coupling Hamiltonian Eq. (8). 

Here we should stress the important property of the self-energy (24), it is determined only by the coupling Hamiltonians and the retarded GFs of the isolated leads G R = [E — HR) 1 (i = L, R)... 

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