Approximate trajectories

Related to the previous method, a simulation scheme was recently derived from the Onsager-Machlup action that combines atomistic simulations with a reaction path approach ([Oleander and Elber 1996]). Here, time steps up to 100 times larger than in standard molecular dynamics simulations were used to produce approximate trajectories by the following equations of motion ... 

Figure 3.2 An actual phase-space trajectory (bold curve) and an approximate trajectory generated by repeated application of Eq. (3.17) (series of arrows representing individual time steps). Note that each propagation step has an identical At, but individual Ap values can be quite different. In the illustration, the approximate trajectory hews relatively closely to the actual one, but this will not be the case if too large a time step is used...

Figure 8.40 Approximate trajectory taken in ethanol-water-CTAB phase space during the EISA process. Point A corresponds to the initial composition of entrained solution, Point B is near the drying line, and Point C corresponds to the dried product. Reproduced with permission from [180]. Copyright (1999) Wiley-VCH...

The distribution of errors is a property of the exact trajectories. If we generate an approximate trajectory based on the finite difference formula, we should generate an error distribution that is consistent with what we know about the true solution. 

If the errors are zero, we obtain the most probable trajectory within the framework of the stochastic difference equation. This trajectory is not exact and is within cr from the exact trajectory [Eq. (13)]. What are the approximations made In Section III.C we argue that the approximate trajectory is a solution of the slow modes in the system where the high-frequency modes are filtered out. 

The last hmitrng expression is the Gauss action, Sc, for classical mechanics. It clearly has a minimum that satisfies the equations of motion (when the action is zero). The action is non-negative, which makes it easier to identify the trae minimum. The non-negativity is an important difference from the classical action formulation that we introduced at the beginning and makes the calculations with the Ssdet and Sg significantly more stable. The approximate trajectories that are produced by optimization of Ssdet are stable. An exponential solution of the type expjmf] (co positive) cannot be obtained in a reasonable formulation of a boundary value problem if the boundaries are not explosive . However, an explosive solution can be obtained with initial value formulation. 

At this point we may continue in one of two directions. We may use a single approximate trajectory at the neighborhood of the exact trajectory that is, the trajectory that was obtained by the minimization of the discrete action. Alternatively, we recognize that the exact trajectory deviates from the optimal trajectory by errors distributed normally (keep in mind that the error distribution is a property of the exact trajectory). We may sample errors (and plausible trajectories) from the appropriate distribution of coordinates in the neighborhood of the trajectory with filtered high-frequency modes. The sampling in the neighborhood of the optimized trajectory should be normalized (we approximate one trajectory) ... 

In turn, the change in energy, AH, on the approximate trajectories (6.76) is computed, yielding... 

The following steps are used to obtain an approximate trajectory using... 

From what we have just said, the structure in e(a)) evidently arises from the "bump" in <(f> (t)> at t = Tj, Apparently, (f>(t) has come back to the general vicinity of maximum overlap to be low compared to that at T=0. This is much clearer from the sketch in Fig. 3, which shows the potential surface governing the approximate trajectory of the wavepacket. 

It is important to recognize that the time-dependent behaviour of tire correlation fimction during the molecular transient time seen in figure A3.8.2 has an important origin [7, 8]. This behaviour is due to trajectories that recross the transition state and, hence, it can be proven [7] that the classical TST approximation to the rate constant is obtained from A3.8.2 in the t —> 0 limit ... 

The obvious defect of classical trajectories is that they do not describe quantum effects. The best known of these effects is tunnelling tln-ough barriers, but there are others, such as effects due to quantization of the reagents and products and there are a variety of interference effects as well. To circumvent this deficiency, one can sometimes use semiclassical approximations such as WKB theory. WKB theory is specifically for motion of a particle in one dimension, but the generalizations of this theory to motion in tliree dimensions are known and will be mentioned at the end of this section. More complete descriptions of WKB theory can be found in many standard texts [1, 2, 3, 4 and 5, 18]. 

Quasiclassical Trajectory Calculations on a H -I- D2 Reaction at 2.20 eV in. The Extended Bom-Oppenheimer Approximation... 

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). 

For larger systems, various approximate schemes have been developed, called mixed methods as they treat parts of the system using different levels of theory. Of interest to us here are quantuin-seiniclassical methods, which use full quantum mechanics to treat the electrons, but use approximations based on trajectories in a classical phase space to describe the nuclear motion. The prefix quantum may be dropped, and we will talk of seiniclassical methods. There are a number of different approaches, but here we shall concentrate on the few that are suitable for direct dynamics molecular simulations. An overview of other methods is given in the introduction of [21]. 

In the full quantum mechanical picture, the evolving wavepackets are delocalized functions, representing the probability of finding the nuclei at a particular point in space. This representation is unsuitable for direct dynamics as it is necessary to know the potential surface over a region of space at each point in time. Fortunately, there are approximate formulations based on trajectories in phase space, which will be discussed below. These local representations, so-called as only a portion of the FES is examined at each point in time, have a classical flavor. The delocalized and nonlocal nature of the full solution of the Schtddinger equation should, however, be kept in mind. 

The Hemian-Kluk method has been developed further [153-155], and used in a number of applications [156-159]. Despite the formal accuracy of the approach, it has difficulties, especially if chaotic regions of phase space are present. It also needs many trajectories to converge, and the initial integration is time consuming for large systems. Despite these problems, the frozen Gaussian approximation is the basis of the spawning method that has been applied to... 

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. 

For the case of intramolecular energy transfer from excited vibrational states, a mixed quantum-classical treatment was given by Gerber et al. already in 1982 [101]. These authors used a time-dependent self-consistent field (TDSCF) approximation. In the classical limit of TDSCF averages over wave functions are replaced by averages over bundles of trajectories, each obtained by SCF methods. 

Abstract. A stochastic path integral is used to obtain approximate long time trajectories with an almost arbitrary time step. A detailed description of the formalism is provided and an extension that enables the calculations of transition rates is discussed. 

The two sources of stochasticity are conceptually and computationally quite distinct. In (A) we do not know the exact equations of motion and we solve instead phenomenological equations. There is no systematic way in which we can approach the exact equations of motion. For example, rarely in the Langevin approach the friction and the random force are extracted from a microscopic model. This makes it necessary to use a rather arbitrary selection of parameters, such as the amplitude of the random force or the friction coefficient. On the other hand, the equations in (B) are based on atomic information and it is the solution that is approximate. For ejcample, to compute a trajectory we make the ad-hoc assumption of a Gaussian distribution of numerical errors. In the present article we also argue that because of practical reasons it is not possible to ignore the numerical errors, even in approach (A). 

To exemplify both aspects of the formalism and for illustration purposes, we divide the present manuscript into two major parts. We start with calculations of trajectories using approximate solution of atomically detailed equations (approach B). We then proceed to derive the equations for the conditional probability from which a rate constant can be extracted. We end with a simple numerical example of trajectory optimization. More complex problems are (and will be) discussed elsewhere [7]. 

It remains to be seen, if the approximation using large time steps is reasonable. We shall show later the effect of the approximation on the power spectrum of the trajectory. More specifically, we shall demonstrate that large time steps filter out high frequency motions. 

A saddle point approximation to the above integral provides the definition for optimal trajectories. The computations of most probable trajectories were discussed at length [1]. We consider the optimization of a discrete version of the action. 

In the derivation we used the exact expansion for X t), but an approximate expression for the last two integrals, in which we approximate the potential derivative by a constant at Xq- The optimization of the action S with respect to all the Fourier coefficients, shows that the action is optimal when all the d are zero. These coefficients correspond to frequencies larger than if/At. Therefore, the optimal solution does not contain contributions from these modes. Elimination of the fast modes from a trajectory, which are thought to be less relevant to the long time scale behavior of a dynamical system, has been the goal of numerous previous studies. 

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