Time-averaged Hamiltonian

Nuclear spin relaxation is caused by fluctuating interactions involving nuclear spins. We write the corresponding Hamiltonians (which act as perturbations to the static or time-averaged Hamiltonian, detemiming the energy level structure) in tenns of a scalar contraction of spherical tensors ... 

For all practical applications, exact effective Hamiltonians can be derived numerically. The advantage of this approach is exactness, flexibility, and convenience. A disadvantage of this approach is that it provides only limited insight into the way a given multiple-pulse sequence is able to create a desired effective Hamiltonian. A more intuitive approach is provided if the effective Hamiltonian can be approximated by a time-averaged Hamiltonian (see Section IV.C). 

Hamiltonian ft is well approximated by the time-averaged Hamiltonian in the toggling frame ... 

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. 

In cases where the Hamiltonians (typically due to phase or amplitude switching in the rf fields) are discontinuously time-dependent, the average Hamiltonian may conveniently be set up using the semi-continuous Baker-Campbell-Hausdorff (scBCH) expansion [56] as... 

If i and j are different nuclei (for instance 13C and H) the hetero-nuclear dipolar interaction, which is often strong, can be removed by dipolar decoupling, which consists of irradiation nucleus j (say H) at its resonance frequency while observing nucleus i (say 13C). The time-averaged value of the Hamiltonian is then zero. 

When K is in equilibrium, 0 must be constant in time, and this will be die case if il is a function of llie (time-indepeiidem) Hamiltonian H of K. Ensemble averages are now assumed to coincide with temporal averages. When, in particular. K is in diathermic equilibrium with its surroundings one can show that

[Pg.1607]

Let us consider a system in equilibrium, described in the absence of external perturbations by a time-independent Hamiltonian Ho. We will be concerned with equilibrium average values which we will denote as (...), where the symbol (...) stands for Trp0... with p0 = e H"/ Vre the canonical density operator. Since we intend to discuss linear response functions and symmetrized equilibrium correlation functions genetically denoted as Xba(, 0 and CBA t,t ), we shall assume that the observables of interest A and B do not commute with Ho (were it the case, the response function %BA(t, t ) would indeed be zero). This hypothesis implies in particular that A and B are centered A) =0,... 

The type of correlated method that has enjoyed the most widespread application to H-bonded systems is many-body perturbation theory, also commonly referred to as Mpller-Plesset (MP) perturbation theory This approach considers the true Hamiltonian as a sum of its Hartree-Fock part plus an operator corresponding to electron correlation. In other words, the unperturbed Hamiltonian consists of the interaction of the electrons with the nuclei, plus their kinetic energy, to which is added the Hartree-Fock potential the interaction of each electron with the time-averaged field generated by the others. The perturbation thus becomes the difference between the correct interelectronic repulsion operator, with its instantaneous correlation between electrons, and the latter Hartree-Fock potential. In this formalism, the Hartree-Fock energy is equed to the sum of the zeroth and first-order perturbation energy corrections. 

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