Lennard-Jones oscillator

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... 

Fig. 3.2 We compare the symplectic Euler, Verlet and Yoshida schemes in application to a Lennard-Jones oscillator. The plot shows the absolute deviation in the computed Hamiltonian (left) as a function of time, for each scheme at a fixed timestep h = 0.005. Moreover we simulate the system using different stepsizes and compute for each stepsize the mtiximum deviation in the Hamiltonian (right), comparing the results with guide lines associated to various powers of the step size. The scheme used as the base method for the Yoshida composition methods is denoted in the parenthesis (either position or velocity Verlet)...

Figure 3 Cumulative values of U)/kB (dashed line) and Cv/ks (solid line) for a Monte Carlo simulation of the one-dimensional Lennard-Jones oscillator (see text) at T = 1 K. Note that although Cy typically takes longer to equilibrate than does (U), there is no significant difference between the two for this simple system at this very low temperature.

Table 1 Convergence of Approximate 95% Confidence Level Error Estimates of a Metropolis Monte Carlo Estimate of (U) and Cy with Respect to the Number of Blocks Used for the One-Dimensional Lennard-Jones Oscillator at T = 1...

Table 2 Average Potential Energy, Heat Capacity, and Fractional 95% Confidence Level Statistical Errors for the One-Dimensional Lennard-Jones Oscillator ...

Figure 4 U) jk as a function of temperature (diamonds) for the one-dimensional Lennard-Jones oscillator (see text). The Equipartition Theorem requires that the slope approach the harmonic limit (— ) as the temperature approaches zero (solid line).

Fig. 5.1 A schematic projection of the 3n dimensional (per molecule) potential energy surface for intermolecular interaction. Lennard-Jones potential energy is plotted against molecule-molecule separation in one plane, the shifts in the position of the minimum and the curvature of an internal molecular vibration in the other. The heavy upper curve, a, represents the gas-gas pair interaction, the lower heavy curve, p, measures condensation. The lighter parabolic curves show the internal vibration in the dilute gas, the gas dimer, and the condensed phase. For the CH symmetric stretch of methane (3143.7 cm-1) at 300 K, RT corresponds to 8% of the oscillator zpe, and 210% of the LJ well depth for the gas-gas dimer (Van Hook, W. A., Rebelo, L. P. N. and Wolfsberg, M. /. Phys. Chem. A 105, 9284 (2001))...

Thus we again assume a Lennard-Jones form, where now the well depth and range parameters depend on the solute s internal vibrational coordinates. Without loss of generality we can define these coordinates so that q = Q = 0 corresponds to the minimum in the intramolecular potential. The solute-solvent potential in Hb above is actually then

(r, 0, 0), where clearly e = e(0, 0) and a = cr(0, 0). The oscillator-bath interaction term is... 

Exact calculations of AE have been carried out by Kelley and Wolfsberg [19] for colinear collisions between an atom and a diatomic molecule. The oscillator potential was considered to be both harmonic and Morse-type, and the interaction between the colliding pair was taken both as an exponential repulsion and as a Lennard-Jones 6 12 potential. Two important conclusions were reached First, when the initial energy of the oscillator increases, the total energy transferred from translation to vibration, AE, decreases. Second, the effect of using a Morse-oscillator potential in place of the harmonic oscillator was generally to decrease AE, often by more than a factor of 10. 

Recently, detailed molecular pictures of the interfacial structure on the time and distance scales of the ion-crossing event, as well as of ion transfer dynamics, have been provided by Benjamin s molecular dynamics computer simulations [71, 75, 128, 136]. The system studied [71, 75, 136] included 343 water molecules and 108 1,2-dichloroethane molecules, which were separately equilibrated in two liquid slabs, and then brought into contact to form a box about 4 nm long and of cross-section 2.17 nmx2.17 nm. In a previous study [128], the dynamics of ion transfer were studied in a system including 256 polar and 256 nonpolar diatomic molecules. Solvent-solvent and ion-solvent interactions were described with standard potential functions, comprising coulombic and Lennard-Jones 6-12 pairwise potentials for electrostatic and nonbonded interactions, respectively. While in the first study [128] the intramolecular bond vibration of both polar and nonpolar solvent molecules was modeled as a harmonic oscillator, the next studies [71,75,136] used a more advanced model [137] for water and a four-atom model, with a united atom for each of two... 

Because of their importance to nucleation kinetics, there have been a number of attempts to calculate free energies of formation of clusters theoretically. The most important approaches for the current discussion are harmonic models, " Monte Carlo studies, and molecular dynamics calcula-tions. In the harmonic model the cluster is assumed to be composed of constituent atoms with harmonic intermolecular forces. The most recent calculations, which use the harmonic model, have taken the geometries of the clusters to be those determined by the minimum in the two-body additive Lennard-Jones potential surface. The oscillator frequencies have been obtained by diagonalizing the Lennard-Jones force constant matrix. In the harmonic model the translational and rotational modes of the clusters are treated classically, and the vibrational modes are treated quantum mechanically. The harmonic models work best at low temjjeratures where anharmonic-ity effects are least important and the system is dominated by a single structure. 

Figure 13. The accumulated action, as a function of time, for a typical saddle-crossing trajectory of the three-particle Lennard-Jones cluster. The right panel is a close-up view of the central part of the longer trajectory on the left, with arrows indicating the individual oscillations. [Reprinted with permission from R. J. Hinde and R. S. Berry, J. Chem. Phys. 99, 2942 (1993). Copyright 1993, American Institute of Physics.]...

Figure 5 Uniaxial stress versus volume for an overdriven [111] direction shock simulation in a perfect Lennard Jones crystal. The gray line is the Rayleigh line, or constraint line provided by the volume equation of motion Eq. (16). The black line is the actual path of the simulation. The volume begins the simulation at V/V = 1 and subsequently undergoes elastic oscillations around V/V =0.85. As the amplitude of these oscillations decays with time, the simulation trajectory approaches the Rayleigh line. After the oscillations have decayed away, plastic deformation and further compression occur. During this slower plastic wave, the simulation trajectory closely follows the Rayleigh line, ensuring the correct sequence thermodynamic states are sampled.

Figure 6. Depicted is the time-dependence of the volume for three simulations of 2.2 km/sec shock waves in the [110] direction of a perfect 1400 atom Lennard-Jones crystal. Each simulation was performed with a different mass-like parameter Q in Eq. (9), given here in reduced Leimard-Jones units. If Q is chosen too large (top panel), long-lived oscillations can result. If Q is chosen too small (bottom panel) large amplitude oscillations that do not decay with time can result. An optimal value of Q results in fast decay of volume oscillations.

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