Classical Oscillator

The nonnal-mode hannonic oscillator classical Hamiltonian is... 

In general, if a particle is bound (E < 0) it will oscillate (classically) between some limits r = a, and r = b. For example, in an elliptic orbit of a hydrogen atom, the radius oscillates periodically between inner and outer limits. Only for a circular orbit is there no oscillation. Among the eigenvalues which have the same n, the one with lowest l has the largest amplitude in the vicinity of the nucleus. 

The field Eo that surrounds an oscillating classical electric dipole po is shown in Fig. 3a,... 

Figure Al.6.1. Gaussian wavepacket in a hannonic oscillator. Note tliat the average position and momentum change according to the classical equations of motion (adapted from [6]).

The classical mechanical RRKM k(E) takes a very simple fonn, if the internal degrees of freedom for the reactant and transition state are assumed to be hamionic oscillators. The classical sum of states for s harmonic oscillators is [16]... 

Only in the high-energy limit does classical statistical mechanics give accurate values for the sum and density of states tenns in equation (A3.12.15) [3,14]. Thus, to detennine an accurate RRKM lc(E) for the general case, quantum statistical mechanics must be used. Since it is difficult to make anliannonic corrections, both the molecule and transition state are often assumed to be a collection of hannonic oscillators for calculating the... 

When looking at the snapshots in Figure A3.13.6 we see that the position of maximal probability oscillates back and forth along the stretching coordinate between the walls at = -20 and +25 pm, with an approximate period of 12 fs, which corresponds to the classical oscillation period r = 1 / v of a pendulum with... 

The reaction involving chlorite and iodide ions in the presence of malonic acid, the CIMA reaction, is another that supports oscillatory behaviour in a batch system (the chlorite-iodide reaction being a classic clock system the CIMA system also shows reaction-diffusion wave behaviour similar to the BZ reaction, see section A3.14.4). The initial reactants, chlorite and iodide are rapidly consumed, producing CIO2 and I2 which subsequently play the role of reactants . If the system is assembled from these species initially, we have the CDIMA reaction. The chemistry of this oscillator is driven by the following overall processes, with the empirical rate laws as given ... 

In order to illustrate some of the basic aspects of the nonlinear optical response of materials, we first discuss the anliannonic oscillator model. This treatment may be viewed as the extension of the classical Lorentz model of the response of an atom or molecule to include nonlinear effects. In such models, the medium is treated as a collection of electrons bound about ion cores. Under the influence of the electric field associated with an optical wave, the ion cores move in the direction of the applied field, while the electrons are displaced in the opposite direction. These motions induce an oscillating dipole moment, which then couples back to the radiation fields. Since the ions are significantly more massive than the electrons, their motion is of secondary importance for optical frequencies and is neglected. 

While the Lorentz model only allows for a restoring force that is linear in the displacement of an electron from its equilibrium position, the anliannonic oscillator model includes the more general case of a force that varies in a nonlinear fashion with displacement. This is relevant when tire displacement of the electron becomes significant under strong drivmg fields, the regime of nonlinear optics. Treating this problem in one dimension, we may write an appropriate classical equation of motion for the displacement, v, of the electron from equilibrium as... 

Finally, semi-classical approaches to non-adiabatic dynamics have also been fomuilated and siiccessfLilly applied [167. 181]. In an especially transparent version of these approaches [167], one employs a mathematical trick which converts the non-adiabatic surfaces to a set of coupled oscillators the number of oscillators is the same as the number of electronic states. This mediod is also quite accurate, except drat the number of required trajectories grows with time, as in any semi-classical approach. 

In this paper, we discuss semi-implicit/implicit integration methods for highly oscillatory Hamiltonian systems. Such systems arise, for example, in molecular dynamics [1] and in the finite dimensional truncation of Hamiltonian partial differential equations. Classical discretization methods, such as the Verlet method [19], require step-sizes k smaller than the period e of the fast oscillations. Then these methods find pointwise accurate approximate solutions. But the time-step restriction implies an enormous computational burden. Furthermore, in many cases the high-frequency responses are of little or no interest. Consequently, various researchers have considered the use of scini-implicit/implicit methods, e.g. [6, 11, 9, 16, 18, 12, 13, 8, 17, 3]. 

The limit equation governing limj -,o qc can be motivated by referring to the quantum adiabatic theorem which originates from work of Born and FOCK [4, 20] The classical position g influences the Hamiltonian very slowly compared to the time scale of oscillations of in fact, infinitely slowly in the limit e — 0. Thus, in analogy to the quantum adiabatic theorem, one would expect that the population of the energy levels remain invariant during the evolution ... 

Hence, as the second class of techniques, we discuss adaptive methods for accurate short-term integration (Sec. 4). For this class, it is the major requirement that the discretization allows for the stepsize to adapt to the classical motion and the coupling between the classical and the quantum mechanical subsystem. This means, that we are interested in discretization schemes which avoid stepsize restrictions due to the fast oscillations in the quantum part. We can meet this requirement by applying techniques recently developed for evaluating matrix exponentials iteratively [12]. This approach yields an adaptive Verlet-based exponential integrator for QCMD. 

Fig. 1. Total energy (in kj/mol) versus time (in fs) for different integrators for a collinear collision of a classical particle with a harmonic quantum oscillator (for details see [2]). Dashed line Nonsymplectic scheme. Dotted Symplectic integrator of first order. Solid PICKABACK (symplectic, second order).

The splitting of the quantum propagator negatively effects the efficiency of the scheme especially if m/M is small, i.e., if the quantum oscillation are much faster than the classical motion and the number n of substeps is becoming inefficiently large. 

Abstract. We present novel time integration schemes for Newtonian dynamics whose fastest oscillations are nearly harmonic, for constrained Newtonian dynamics including the Car-Parrinello equations of ab initio molecular dynamics, and for mixed quantum-classical molecular dynamics. The methods attain favorable properties by using matrix-function vector products which are computed via Lanczos method. This permits to take longer time steps than in standard integrators. 

In this section we consider the classical equations of motion of particles in cases where the highest-frequency oscillations are nearly harmonic The positions y t) = j/i (t) evolve according to the second-order system of differential equations... 

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