Excitation trajectories

Figure 3.7. Excitation trajectories as a function of resonance offset for (a) a 90° pulse and (b) a 180° pulse. The offset moves from zero (on-resonance) to - -yB Hz in steps of 0.2yB] (as in Fig. 3.6). The 90° pulse has a degree of offset-compensation as judged by its ability to generate transverse magnetisation over a wide frequency bandwidth. In contrast the 180° pulse performs rather poorly away from resonance, leaving the vector far from the target South Pole and with a considerable transverse component.

Figure 9.13. Simulated excitation trajectories as a function of resonance offset for (a) the Gaussian, (b) the half-Gaussian and (c) the Gaussian-270.

Figure 9.17. Simulated zig-zag excitation trajectories of nuclear spin vectors at different resonance offsets for the DANTE sequence. 30 pulses of 3° each were separated by delays of 0.33 ms, producing an effective 10 ms selective pulse.

The control of such structures has been extensively studied. The main approaches are methods of non exciting trajectories [3], adaptative control [4], control using a knowledge model. In this last case, we can find singular perturbation methods [5], LQR methods based on an approximated linearized model [6] or non linear decoupling methods [7]. 

Figure Al.6.27. Equipotential contour plots of (a) the excited- and (b), (c) ground-state potential energy surfaces. (Here a hamionic excited state is used because that is the way the first calculations were perfomied.) (a) The classical trajectory that originates from rest on the ground-state surface makes a vertical transition to the excited state, and subsequently undergoes Lissajous motion, which is shown superimposed, (b) Assuming a vertical transition down at time (position and momentum conserved) the trajectory continues to evolve on the ground-state surface and exits from chaimel 1. (c) If the transition down is at time 2 the classical trajectory exits from chaimel 2 (reprinted from [52]).

Figure Al.6.28. Magnitude of the excited-state wavefimction for a pulse sequence of two Gaussians with time delay of 610 a.u. = 15 fs. (a) (= 200 a.u., (b) ( = 400 a.u., (c) (= 600 a.u. Note the close correspondence with the results obtained for the classical trajectory (figure Al. 6.27(a) and (b)). Magnitude of the ground-state wavefimction for the same pulse sequence, at (a) (= 0, (b) (= 800 a.u., (c) (= 1000 a.u. Note the close correspondence with the classical trajectory of figure Al.6.27(c)). Although some of the amplitude remains in the bound region, that which does exit does so exclusively from chaimel 1 (reprinted from [52]).

A DIET process involves tliree steps (1) an initial electronic excitation, (2) an electronic rearrangement to fonn a repulsive state and (3) emission of a particle from the surface. The first step can be a direct excitation to an antibondmg state, but more frequently it is simply the removal of a bound electron. In the second step, the surface electronic structure rearranges itself to fonn a repulsive state. This rearrangement could be, for example, the decay of a valence band electron to fill a hole created in step (1). The repulsive state must have a sufficiently long lifetime that the products can desorb from the surface before the state decays. Finally, during the emission step, the particle can interact with the surface in ways that perturb its trajectory. 

The first classical trajectory study of iinimoleciilar decomposition and intramolecular motion for realistic anhannonic molecular Hamiltonians was perfonned by Bunker [12,13], Both intrinsic RRKM and non-RRKM dynamics was observed in these studies. Since this pioneering work, there have been numerous additional studies [9,k7,30,M,M, ai d from which two distinct types of intramolecular motion, chaotic and quasiperiodic [14], have been identified. Both are depicted in figure A3,12,7. Chaotic vibrational motion is not regular as predicted by tire nonnal-mode model and, instead, there is energy transfer between the modes. If all the modes of the molecule participate in the chaotic motion and energy flow is sufficiently rapid, an initial microcanonical ensemble is maintained as the molecule dissociates and RRKM behaviour is observed [9], For non-random excitation initial apparent non-RRKM behaviour is observed, but at longer times a microcanonical ensemble of states is fonned and the probability of decomposition becomes that of RRKM theory. 

For some systems qiiasiperiodic (or nearly qiiasiperiodic) motion exists above the unimoleciilar tlireshold, and intrinsic non-RRKM lifetime distributions result. This type of behaviour has been found for Hamiltonians with low uninioleciilar tliresholds, widely separated frequencies and/or disparate masses [12,, ]. Thus, classical trajectory simulations perfomied for realistic Hamiltonians predict that, for some molecules, the uninioleciilar rate constant may be strongly sensitive to the modes excited in the molecule, in agreement with the Slater theory. This property is called mode specificity and is discussed in the next section. 

Apparent non-RRKM behaviour occurs when the molecule is excited non-randomly and there is an initial non-RRKM decomposition before IVR fomis a microcanonical ensemble (see section A3.12.2). Reaction patliways, which have non-competitive RRKM rates, may be promoted in this way. Classical trajectory simulations were used in early studies of apparent non-RRKM dynamics [113.114]. 

Wolf R J and Hase W L 1980 Quasiperiodic trajectories for a multidimensional anharmonic classical Hamiltonian excited above the unimolecular threshold J. Chem. Phys. 73 3779-90... 

Lenzer T, Luther K, Troe J, Gilbert R G and Urn K F 1995 Trajectory simulations of collisional energy transfer in highly excited benzene and hexafluorobenzene J. Chem. Phys. 103 626-41... 

Grigoleit U, Lenzer T and Luther K 2000 Temperature dependence of collisional energy transfer in highly excited aromatics studied by classical trajectory calculations Z. Phys. Chem., A/F214 1065-85... 

Collision-induced dissociation mass spectrum of tire proton-bound dimer of isopropanol [(CH2)2CHOH]2H. The mJz 121 ions were first isolated in the trap, followed by resonant excitation of their trajectories to produce CID. Fragment ions include water loss mJz 103), loss of isopropanol mJz 61) and loss of 42 anui mJz 79). (b) Ion-molecule reactions in an ion trap. In this example the mJz 103 ion was first isolated and then resonantly excited in the trap. Endothennic reaction with water inside the trap produces the proton-bound cluster at mJz 121, while CID produces the fragment with mJz 61. 

In the other types of mass spectrometer discussed in this chapter, ions are detected by having them hit a detector such as an electron multiplier. In early ICR instruments, the same approach was taken, but FT-ICR uses a very different teclmique. If an RF potential is applied to the excitation plates of the trapping cell (figure B 1.7.18(b)) equal to the cyclotron frequency of a particular ion m/z ratio, resonant excitation of the ion trajectories takes place (without changing the cyclotron frequency). The result is ion trajectories of higher... 

Figure Cl.5.8. Spectral jumping of a single molecule of terrylene in polyethylene at 1.5 K. The upper trace displays fluorescence excitation spectra of tire same single molecule taken over two different 20 s time intervals, showing tire same molecule absorbing at two distinctly different frequencies. The lower panel plots tire peak frequency in tire fluorescence excitation spectmm as a function of time over a 40 min trajectory. The molecule undergoes discrete jumps among four (briefly five) different resonant frequencies during tliis time period. Arrows represent scans during which tire molecule had jumped entirely outside tire 10 GHz scan window. Adapted from...

Figure C3.3.8. A typical trajectory for a soft collision between a hot pyrazine molecule and a CO2 bath molecule in which the CO 2 becomes vibrationally excited.

Figure C3.6.7(a) shows tire u= 0 and i )= 0 nullclines of tliis system along witli trajectories corresponding to sub-and super-tlireshold excitations. The trajectory arising from tire sub-tlireshold perturbation quickly relaxes back to tire stable fixed point. Three stages can be identified in tire trajectory resulting from tire super-tlireshold perturbation an excited stage where tire phase point quickly evolves far from tire fixed point, a refractory stage where tire system relaxes back to tire stable state and is not susceptible to additional perturbation and tire resting state where tire system again resides at tire stable fixed point.

Figure C3.6.7 Cubic (jir = 0) and linear (r = 0) nullclines for tire FitzHugh-Nagumo equation, (a) The excitable domain showing trajectories resulting from sub- and super-tlireshold excitations, (b) The oscillatory domain showing limit cycle orbits small inner limit cycle close to Hopf point large outer limit cycle far from Hopf point.

By using this approach, it is possible to calculate vibrational state-selected cross-sections from minimal END trajectories obtained with a classical description of the nuclei. We have studied vibrationally excited H2(v) molecules produced in collisions with 30-eV protons [42,43]. The relevant experiments were performed by Toennies et al. [46] with comparisons to theoretical studies using the trajectory surface hopping model [11,47] fTSHM). This system has also stimulated a quantum mechanical study [48] using diatomics-in-molecule (DIM) surfaces [49] and invoicing the infinite-onler sudden approximation (lOSA). 

The generalized Prony analysis of END trajectories for this system yield total and state resolved differential cross-sections. In Figure 5, we show the results. The theoretical analysis, which has no problem distinguishing between the symmetric and asymmetric str etch, shows that the asymmetric mode is only excited to a minor extent. The corresponding state resolved cross-section is about two orders of magnitude less than that of the symmetric stretch. 

To return to the simple picture of vertical excitation, the question remains as to how a wavepacket can be simulated using classical trajectories A classical ensemble can be specified by its distribution in phase space, Pd(p,Q), which gives the probability of finding the system of particles with momentum p and position q. In conUast, it is strictly impossible to assign simultaneously a position and momentum to a quantum particle. 

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