Kepler PredictorâCorrector Algorithm - Yale Chemistry - Yale ...

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Journal of Chemical Theory and Computation construction of pseudopotentials based on physical insights, such as the screening effect of electrons in the conduction band of metals. 6 In ionic solids (e.g., alkali halides, oxide insulators, or semiconductors), however, there are no electrons in the conduction band that could offer screening of Coulombic interactions. Therefore, alternative methods are required. Changes in the potential usually alter the underlying dynamics of the systems yielding artificial effects that disappear only when the pseudopotentials become more and more similar to the true potentials. In that limit, however, the numerical problems due to large gradients usually reappear. An approach that avoids changing the singular potential has been developed for gravitational systems, implementing a change of variables that regularizes the dynamics (e.g., the Kustaanheimo Stiefel (KS) regularization and related methods 12,17 21 ) and applies standard numerical integration for the new variables rather than for the original coordinates. This requires transformation of the time variable and coordinates that depend on the interparticle distance for the pair of particles experiencing a close encounter. Great care must be taken to keep track of transformations for multiple close encounters and to match up time-steps so that the interaction with other particles present in the simulation is properly accounted for, while not sacrificing efficiency. 20,21 As a consequence of these complications, the method is, to our knowledge, implemented in only a few stellar codes. 22 In this paper, we introduce a simple Kepler predictor corrector (KPC) algorithm where close-encounter collisions are integrated analytically by solving the Kepler two-body problem without altering (or smoothing) the Coulomb potential and updating coordinates based on the residual potential due to particles not participating in the close encounter. A simple well-known example for such an approach is the lightly damped harmonic oscillator, where the frequency of the resulting oscillation is approximately the same as that of the underlying undamped oscillator and only the amplitude may be viewed as modulated. 23 In molecular dynamics, the analytic solution of the harmonic oscillator has been employed by splitting linear molecule Hamiltonians into a harmonic and anharmonic part and treating high frequency components of the molecular vibrations analytically, while treating low frequency components numerically. 24 29 In biological systems, a large speedup was achieved by treating water molecules as rigid and using analytic solutions for their motion. 30 For the 1/r problem, an update scheme based on a single momentum shift has been proposed previously for the purpose of simulating ion collisions. 31 We propose a method that corrects the predicted scattering trajectories with an additional term rigorously derived from numerical integrators (e.g., velocity Verlet or Gear Predictor Corrector algorithms), accounting for the regular influence of other particles (or external fields) in the system. The integration method as a whole is time-reversal symmetric, does not require any time-variable transformation, and can be applied to the general multibody dynamics problem with close encounters, as in electron proton scattering processes, particle antiparticle dynamics, and charged gas dynamics. Algorithmically, the complete potential is separated into two parts, including an integrable two-body term that is dominant at close encounters and a correction due to interactions with all other particles (or potentials) in the system. With an integrable two-body problem (as is the case of Coulombic 1/r or van der Waals 1/r 6 potentials), 32 the analytic solution of close encounters is employed, and the effect due to interactions with all other particles is introduced by augmenting the analytic solution with ARTICLE Figure 1. KPC trajectory (red line) of a particle scattered by a singularity V s (r), obtained by solving the Kepler problem (dashed line), taking into account the effect of a weakly varying residual potential V r (r) (bold green lines) as described in the text. additional terms that stem from the numerical integrator (e.g., the velocity Verlet method 33,34 ). The resulting KPC method allows for integration time-steps on the same order as used for regular potentials and hence reduces the numerical effort for problems with close encounters. It is applicable for any integrable singular pair potential, 32 including the family of integrable spherically symmetric singular pair potentials that are the focus of this paper. The paper is organized as follows. Section 2 introduces the KPC method as applied to modeling the dynamics of a particle evolving on a potential with multiple singularities and its generalization to multibody dynamics. Section 3 describes its implementation for semiclassical simulations of electron proton scattering in the Wigner-transform time-dependent picture. Section 4 presents concluding remarks in perspective of the KPC limitations and applicability as a general method. 2. KEPLER PREDICTOR CORRECTOR ALGORITHM 2.1. Single Particle Colliding with Multiple Singularities. Consider the general case of a single particle moving in a stationary potential with multiple singularities, as shown in Figure 1. The total potential V acting on the particle is composed of the spherically symmetric potential V s with the singularity nearest to the particle, and the residual potential V r : VðxÞ ¼V s ðrÞ þV r ðxÞ As an example, we consider the semiclassical trajectory of a fast electron undergoing multiple scattering through proton plasma, under the Born Oppenheimer approximation, where V(x) is defined as the Coulomb potential due to the closest proton V s (x) plus the sum of Coulomb potentials V r (x) due to the other protons in the plasma. During a close encounter with the singularity V s (r), the electron scattering force is dominated by this closest proton, and accurate numerical integration faces several difficulties, including the following: 1. the large absolute value of potential energy, 2. the very large norm of the potential gradient and higher derivatives, and hence, 3. the very large acceleration of particles and curvature of trajectories, 4. the requirement of very small time-steps of standard integrators (such as the Verlet methods and the Nordsieck Gear Predictor Corrector methods 33,34 ). ð1Þ 25 dx.doi.org/10.1021/ct200452h |J. Chem. Theory Comput. 2012, 8, 24–35
Journal of Chemical Theory and Computation The KPC algorithm addresses these challenges by first predicting the coordinates and momenta due to the collision with the closest singularity V s (r) and then correcting the resulting coordinates and momenta according the residual term, as follows. Formally, x(t) and p(t) are obtained according to the velocity 33 35 Verlet method: xðt þ ΔtÞ ¼xðtÞ þ Δt m pðtÞ Δt 2 2m ∇Vj xðtÞ ð2Þ Δt pðt þ ΔtÞ ¼pðtÞ 2 ð∇Vj xðtÞ þ ∇Vj xðtþΔtÞ Þ ð3Þ where m = m e is the mass of the particle, and the time increment Δt is assumed to be sufficiently short to ensure energy conservation. However, to address the numerical challenge of the close encounter, we decompose the total force 3V into the contribution due to the nearest singularity 3V s and the contributions due to smaller residual forces 3V r and write suggestively: xðt þ ΔtÞ ¼xðtÞ þ Δt m pðtÞ Δt 2 2m ð∇V s j xðtÞ þ ∇V r j xðtÞ Þ pðt þ ΔtÞ ¼pðtÞ Δt 2 ð∇V r j xðtÞ þ ∇V s j xðtÞ þ ∇V s j xðtþΔtÞ þ ∇V r j xðtþΔtÞ Þ and introducing auxiliary coordinates x r ðtÞ ¼xðtÞ p r ðtÞ ¼pðtÞ ð4Þ ð5Þ ð6Þ Δt 2 ∇V r j xðtÞ ð7Þ x s ðt þ ΔtÞ ¼x r ðtÞ þ Δt m pr ðtÞ Δt 2 2m ∇V s j xr ðtÞ ¼ xðtÞ þ Δt m pðtÞ Δt 2 2m ð∇V s j xðtÞ þ ∇V r j xðtÞ Þ ð8Þ ð9Þ p s ðt þ ΔtÞ ¼p r Δt ðtÞ 2 ð∇V s j xr ðtÞ þ ∇V s j xs ðtþΔtÞ Þ ð10Þ We obtain by comparison with eqs 4 and 5: xðt þ ΔtÞ ¼x s ðt þ ΔtÞ ð11Þ pðt þ ΔtÞ ¼p s Δt ðt þ ΔtÞ 2 ∇V r j xðtþΔtÞ ð12Þ The auxiliary variables were constructed such that eqs 8 and 10 become velocity Verlet equations for only the singular potential V s (r), so they provide a practical way of predicting coordinates and momenta at time t + Δt, as solely determined by the twobody collision with the closest proton. During a close encounter, however, such equations become numerically stiff, and they are replaced by analytic solutions of the corresponding two-body Kepler initial value problem with initial coordinates (x r (t),p r (t)) ARTICLE [three-body collisions are usually prevented by Coulombic repulsion under cool plasma conditions] as described in section 2.3. [The two-body problem is integrable for a variety of spherically symmetric interaction potentials, including the Coulomb r 1 potential and van der Waals r 6 type potentials. 32 In particular, the two-body problem for the 1/r potential is known as the “Kepler problem,” since the elliptical trajectories obey Kepler’s laws of planetary motion.] Having obtained x s (t + Δt) and p s (t + Δt), we obtain x(t + Δt) and p(t + Δt) by correcting the predicted coordinates and momenta according to eq 12. It turns out that x r (t) and x s (t + Δt) are not, as indicated in Figure 1, distinct from x(t) and x(t + Δt), but their corresponding momenta are. The resulting KPC algorithm thus allows for the integration of close-encounter collisions beyond the capabilities of the standard velocity Verlet algorithm (i.e., eqs 4 and 5), as follows: A. Determine p r (t) by a momentum shift according to eq 6. B. Obtain (x s (t + Δt), p s (t + Δt)) by solving the two-body Kepler problem. The analytic solution S K is a set that contains position, momentum, and, in principle, all higher derivatives of the position at the final time t + Δt: ðx s ðt þ ΔtÞ, p s ðt þ ΔtÞÞ ⊂ S K ðx r ðtÞ, p r ðtÞÞ ð13Þ C. From p s (t + Δt), determine p(t + Δt) according to eq 12. Maximum efficiency is achieved when the analytic task is restricted to small regions surrounding the singularity closest to the scattering particle, since the solution of the Kepler problem is more involved than a velocity Verlet integration step. This is typically ensured by defining a cutoff distance r min from the singularity center, below which the KPC method is implemented. In the case under consideration, the potential is stationary, and the cutoff distance can be set to a constant value much smaller than half the minimal distance between singularities, r min , 1/2 min{r ij }. It should be small enough that the residual potential V r is always much more slowly varying than the close-encounter potential V s , guaranteeing that the singular potential dominates the dynamics and the influence of the residual potential leads only to small corrections. To make the method fully time-reversal symmetric, the cutoff criterion has to be made time-reversal symmetric as well, in the following way. The KPC method is used when the initial coordinate is inside the cutoff radius from a scattering center. If the coordinate after the time step is outside the cutoff radius, the result is discarded, and a velocity Verlet step is made instead. In contrast to ref 31 where a single momentum shift was applied, our symmetrized approach uses two momentum shifts per time step. However, the potential gradient needs only be evaluated once per time step. 2.2. Multibody Molecular Dynamics. The generalization of the KPC algorithm, introduced in section 2.1, to multibody molecular dynamics is straightforward. Let (x(t), ξ(t)) and (p(t), π(t)) be the position and momentum vectors of the system at time t. Without a loss of generality, let (x(t), p(t)) be the six-dimensional phase-space vector describing the relative motion of two particles undergoing a close encounter and m be their reduced mass, while their center-of-mass coordinate, along with the coordinates of all other particles (some of which may also be in a close encounter), is contained in the phase-space vector (ξ(t), π(t)). While the interparticle potential V(x(t), ξ(t)) includes all of the interactions, only the dynamics of the relative coordinate 26 dx.doi.org/10.1021/ct200452h |J. Chem. Theory Comput. 2012, 8, 24–35
Page 1: ARTICLE pubs.acs.org/JCTC Kepler Pr
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