Invariant trajectory

The motion of the normalized magnetization vector n Xt) of a spin i with offset Vj under the action of a basis sequence is called an invariant trajectory if the vector returns after to its initial orientation, that is,... 

If the basis sequence creates a nonvanishing effective field this is only possible if n X0) is oriented parallel (or antiparallel) to the direction of Bf During the basis sequence, the time evolution of the components n jKt), n y t), and n j t) of the invariant trajectory can be expressed with the help of the coefficients a it) defined in Eq. (90) as... 

Effective autorelaxation rates of the invariant trajectory of an uncoupled spin i during a basis sequence can be defined as... 

Although the invariant trajectory approach was derived for uncoupled spins i and j, it also reflects qualitatively the cross-relaxation and autorelaxation behavior in coupled spin systems (Griesinger and Ernst, 1988 Bax, 1988a). 

Based on Eq. (139), a relationship between A and the contribution of of longitudinal cross-relaxation to the effective cross-relaxation rate [see Eq. (131)] can be derived. For two spins with the same offset v, = Vj, the invariant trajectories are identical [n j Kt) = and the... 

As discussed in Section IV [Eq. (131)], the effective cross-relaxation rate between two invariant trajectories is given by... 

Method D. Compensating delays are introduced during the composite pulse R, whenever magnetization on an invariant trajectory is oriented along the z axis, that is, after (or during) one or several square pulses (Griesinger et al., 1988 Kerssebaum, 1990 Cavanagh and Ranee, 1992). 

Figure Al.2.7. Trajectory of two coupled stretches, obtained by integrating Hamilton s equations for motion on a PES for the two modes. The system has stable anhamionic synmretric and antisyimnetric stretch modes, like those illustrated in figrne Al.2.6. In this trajectory, semiclassically there is one quantum of energy in each mode, so the trajectory corresponds to a combination state with quantum numbers nj = [1, 1]. The woven pattern shows that the trajectory is regular rather than chaotic, corresponding to motion in phase space on an invariant torus.

As already mentioned, the motion of a chaotic flow is sensitive to initial conditions [H] points which initially he close together on the attractor follow paths that separate exponentially fast. This behaviour is shown in figure C3.6.3 for the WR chaotic attractor at /c 2=0.072. The instantaneous rate of separation depends on the position on the attractor. However, a chaotic orbit visits any region of the attractor in a recurrent way so that an infinite time average of this exponential separation taken along any trajectory in the attractor is an invariant quantity that characterizes the attractor. If y(t) is a trajectory for the rate law fc3.6.2] then we can linearize the motion in the neighbourhood of y to get... 

As a consequence of this observation, the essential dynamics of the molecular process could as well be modelled by probabilities describing mean durations of stay within different conformations of the system. This idea is not new, cf. [10]. Even the phrase essential dynamics has already been coined in [2] it has been chosen for the reformulation of molecular motion in terms of its almost invariant degrees of freedom. But unlike the former approaches, which aim in the same direction, we herein advocate a different line of method we suggest to directly attack the computation of the conformations and their stability time spans, which means some global approach clearly differing from any kind of statistical analysis based on long term trajectories. 

Recent mathematical work suggests that—especially for nonlinear phenomena—certain geometric properties can be as important as accuracy and (linear) stability. It has long been known that the flows of Hamiltonian systems posess invariants and symmetries which describe the behavior of groups of nearby trajectories. Consider, for example, a two-dimensional Hamiltonian system such as the planar pendulum H = — cos(g)) or the... 

A more vexing issue is that one of the e in (3.37) equals zero. To see this, note that the function Xins(t) [where Xjns is the instanton solution to (3.34)] can be readily shown to satisfy (3.37) with o = 0. Since the instanton trajectory is closed, it can be considered to start arbitrarily from one of its points. It is this zero mode which is responsible for the time-shift invariance of the instanton solution. Therefore, the non-Gaussian integration over Cq is expected to be the integration over... 

Equation (4.24) indicates that the quantum number of the transverse x-vibration is an adiabatic invariant of the trajectory. At T=0 becomes the instantaneous zero-point spread of the transverse vibration (2co,) in agreement with the uncertainty principle. 

As stated by inequality (2.81) (see also section 4.2 and fig. 30), when the tunneling mass grows, the tunneling regime tends to be adiabatic, and the extremal trajectory approaches the MEP. The transition can be thought of as a one-dimensional tunneling in the vibrationally adiabatic barrier (1.10), and an estimate of and can be obtained on substitution of the parameters of this barrier in the one-dimensional formulae (2.6) and (2.7). The rate constant falls into the interval available for measurements if, as the mass m is increased, the barrier parameters are decreased so that the quantity d(Vom/mn) remains approximately invariant. 

Although, in general, there may be many distinct invariant measures, we can single out one particular equilibrium measure by demanding that the spatial average over the distribution for (almost) all initial points xq be equal to the temporal average over the trajectory, [xq, x, X2,. ... ... 

Invariance Properties.—Before delving into the mathematical formulation of the invariance properties of quantum electrodynamics, let us briefly state what is meant by an invariance principle in general. As we shall be primarily concerned with the formulation of invariance principles in the Heisenberg picture, it is useful to introduce the concept of the complete description of a physical system. By this is meant at the classical level a specification of the trajectories of all particles together with a full description of all fields at all points of space for all time. The equations of motion then allow one to determine whether the system could, in fact, have evolved in the way... 

The point q = p = 0 (or P = Q = 0) is a fixed point of the dynamics in the reactive mode. In the full-dimensional dynamics, it corresponds to all trajectories in which only the motion in the bath modes is excited. These trajectories are characterized by the property that they remain confined to the neighborhood of the saddle point for all time. They correspond to a bound state in the continuum, and thus to the transition state in the sense of Ref. 20. Because it is described by the two independent conditions q = 0 and p = 0, the set of all initial conditions that give rise to trajectories in the transition state forms a manifold of dimension 2/V — 2 in the full 2/V-dimensional phase space. It is called the central manifold of the saddle point. The central manifold is subdivided into level sets of the Hamiltonian in Eq. (5), each of which has dimension 2N — 1. These energy shells are normally hyperbolic invariant manifolds (NHIM) of the dynamical system [88]. Following Ref. 34, we use the term NHIM to refer to these objects. In the special case of the two-dimensional system, every NHIM has dimension one. It reduces to a periodic orbit and reproduces the well-known PODS [20-22]. 

A typical trajectory has nonzero values of both P and Q. It is part of neither the NHIM itself nor the NHIM s stable or unstable manifolds. As illustrated in Fig. la, these typical trajectories fall into four distinct classes. Some trajectories cross the barrier from the reactant side q < 0 to the product side q > 0 (reactive) or from the product side to the reactant side (backward reactive). Other trajectories approach the barrier from either the reactant or the product side but do not cross it. They return on the side from which they approached (nonreactive trajectories). The boundaries or separatrices between regions of reactive and nonreactive trajectories in phase space are formed by the stable and unstable manifolds of the NHIM. Thus once these manifolds are known, one can predict the fate of a trajectory that approaches the barrier with certainty, without having to follow the trajectory until it leaves the barrier region again. This predictive value of the invariant manifolds constitutes the power of the geometric approach to TST, and when we are discussing driven systems, we mainly strive to construct time-dependent analogues of these manifolds. 

So far, the discussion of the dynamics and the associated phase-space geometry has been restricted to the linearized Hamiltonian in eq. (5). However, in practice the linearization will rarely be sufficiently accurate to describe the reaction dynamics. We must then generalize the discussion to arbitrary nonlinear Hamiltonians in the vicinity of the saddle point. Fortunately, general theorems of invariant manifold theory [88] ensure that the qualitative features of the dynamics are the same as in the linear approximation for every energy not too high above the energy of the saddle point, there will be a NHIM with its associated stable and unstable manifolds that act as separatrices between reactive and nonreactive trajectories in precisely the manner that was described for the harmonic approximation. 

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