Trajectory propagation methods

ions in FAIMS experience directed drift, anisotropic diffusion, and Coulomb repulsion. The drift proceeds along E that is orthogonal to electrodes (3.2.3). The diffusion and Coulomb force transpose ions in aU directions, but the separation is due to ions hitting electrodes and, at fixed residence time fres, the motion parallel to them 

The displacement of y-th out of k ions due to electric fields is given by a modified equation (Equation 1.11)  

FIGURE 4.1 Evaluation of the mean Coulomb force in FAIMS simulation, exemplified for the initial packet of five ions. (From Shvartsburg, A.A., Tang, K., Smith, R.D., J. Am. Soc. Mass Spectrom., 15, 1487, 2004.) The values marked near each ion are kdifA- 

The displacement due to diffusion (A d) is randomly selected (for each ion separately) with the probability distribution given by Equation 1.21 for ID longitudinal diffusion (2.2.4 and 2.4.3)  

Same procedure is adopted as the statistical diffusion simulation in SIMION 7.0. The result is added to Axe by Equation 4.1 and ions with x lying inside an electrode are identified. As those ions have hit an electrode, they are removed from simulation and CTq in Equation 4.2 is adjusted.  

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The center of the wavepacket thus evolves along the trajectory defined by classical mechanics. This is in fact a general result for wavepackets in a hannonic potential, and follows from the Ehrenfest theorem [147] [see Eqs. (154,155) in Appendix C]. The equations of motion are straightforward to integrate, with the exception of the width matrix, Eq. (44). This equation is numerically unstable, and has been found to cause problems in practical applications using Morse potentials [148]. As a result, Heller inboduced the P-Z method as an alternative propagation method [24]. In this, the matrix A, is rewritten as a product of matrices... 

Nevertheless, this simple propagation method provides an intriguing picture of the evolution of the quantum mechanical wavepacket, at least for short times. It readily demonstrates that for short times the center of the wavepacket follows essentially a classical trajectory ( Ehrenfest s theorem, Cohen-Tannoudji, Diu, and Laloe 1977 ch.III). Figure 4.2 depicts an example the evolution of the two-dimensional wavepacket follows very closely the classical trajectory that starts initially with zero momenta at the Franck-Condon point. 

The acceleration would be similar for SEDI, making it practical for medium-size ions. The expense of TC method is not dominated by rotation operations. Hence their cutting will produce only modest gains and optimization efforts would have to aim at the trajectory propagation. 

We compare the symplectic methods developed so far in Fig. 3.2, for t e [0,100] using planar phase space and the Lennard-Jones oscillator (the one degree-of-freedom problem with Hamiltonian H(q,p) = p j2 -I- simulation time increases. In each case, we observe an oscillation in the fluctuation of the total energy. Additionally we compute trajectories propagated at different stepsizes, and plot the maximum deviation... 

It is possible to parametarize the time-dependent Schrddinger equation in such a fashion that the equations of motion for the parameters appear as classical equations of motion, however, with a potential that is in principle more general than that used in ordinary Newtonian mechanics. However, it is important that the method is still exact and general even if the trajectories aie propagated by using the ordinary classical mechanical equations of motion. 

Quantum chemical methods, exemplified by CASSCF and other MCSCF methods, have now evolved to an extent where it is possible to routinely treat accurately the excited electronic states of molecules containing a number of atoms. Mixed nuclear dynamics, such as swarm of trajectory based surface hopping or Ehrenfest dynamics, or the Gaussian wavepacket based multiple spawning method, use an approximate representation of the nuclear wavepacket based on classical trajectories. They are thus able to use the infoiination from quantum chemistry calculations required for the propagation of the nuclei in the form of forces. These methods seem able to reproduce, at least qualitatively, the dynamics of non-adiabatic systems. Test calculations have now been run using duect dynamics, and these show that even a small number of trajectories is able to produce useful mechanistic infomiation about the photochemistry of a system. In some cases it is even possible to extract some quantitative information. 

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