Molecular dynamics harmonic oscillator

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. 

Semiclassical techniques like the instanton approach [211] can be applied to tunneling splittings. Finally, one can exploit the close correspondence between the classical and the quantum treatment of a harmonic oscillator and treat the nuclear dynamics classically. From the classical trajectories, correlation functions can be extracted and transformed into spectra. The particular charm of this method rests in the option to carry out the dynamics on the fly, using Born Oppenheimer or fictitious Car Parrinello dynamics [212]. Furthermore, multiple minima on the hypersurface can be treated together as they are accessed by thermal excitation. This makes these methods particularly useful for liquid state or other thermally excited system simulations. Nevertheless, molecular dynamics and Monte Carlo simulations can also provide insights into cold gas-phase cluster formation [213], if a reliable force field is available [189]. 

The attenuation of die dipole of the repeat unit owing to thermal oscillations was modeled by treating the dipole moment as a simple harmonic oscillator tied to the motion of the repeat unit and characterized by the excitation of a single lattice mode, the mode, which describes the in-phase rotation of the repeat unit as a whole about the chain axis. This mode was shown to capture accurately the oscillatory dynamics of the net dipole moment itself, by comparison with short molecular dynamics simulations. The average amplitude is determined from the frequency of this single mode, which comes directly out of the CLD calculation ... 

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... 

Molecular dynamics (MD) simulations, the most suitable theoretical tool for the investigation of internal motions, can be used to explore both equilibrium properties and time-dependent phenomena. Based on both experimental and theoretical observations two models for the internal motion of proteins have been suggested. Within the framework of the first model internal motions arise from harmonic or quasi-harmonic vibrations that occur in a single multidimensional well on the potential energy surface [4,5,6,7]. The second model assumes that motions are a superposition of oscillations within a well and... 

In Eq. (4-31), the first three terms describe a simple damped harmonic oscillator the first term is due to molecular accelerations, the second is due to viscous drag, and the third is due to the restoring force. Qq is the oscillator frequency, which is of order 10 sec", and p is a viscous damping coefficient. The crucial term producing the dynamic glass transition is, of course, the fourth term, which has the form of a memory integral, in which molecular motions produce a delayed response. The kernel m(t — t ) is determined self-consistently by the time-dependent structure. One simple choice relating m(s) to the structure is ... 

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. 

Depending on the character of the molecular motions, one can distinguish several physical situations. In most cases, the molecules are trapped in relatively deep potential wells. Then, they perform small translational and orientational oscillations around well-defined equilibrium positions and orientations. Such motions are reasonably well described by the harmonic approximation. The collective vibrational excitations of the crystal, which are considered as a set of harmonic oscillators, are called phonons. Those phonons that represent pure angular oscillations, or libra-tions, are called librons. For some properties it turns out to be necessary to look at the effects of anharmonicities. Anharmonic corrections to the harmonic model can be made by perturbation theory or by the self-consistent phonon method. These methods, which are summarized in Section III under the name quasi-harmonic theories, can be considered to be the standard tools in lattice dynamics calculations, in addition to the harmonic model. They are only applicable in the case of fairly small amplitude motions. Only the simple harmonic approximation is widely used the calculation of anharmonic corrections is often hard in practice. For detailed descriptions of these methods, we refer the reader to the books and reviews by Maradudin et al. (1968, 1971, 1974), Cochran and Cowley (1967), Barron and Klein (1974), Birman (1974), Wallace (1972), and Cali-fano et al. (1981). 

Supplementing this equation with an additional set of solvent oscillators one can incorporate a solvent environment. Notice that this does not necessarily imply harmonic solvent motions. In fact the full anharmonicity of the solvent can be accounted for in the context of linear response theory [39] where the interaction is described in terms of an effective harmonic oscillator bath. This allows calculation of relaxation rates from classical molecular dynamics simulations of the force fn(x) exerted by the solvent on the relevant system. This approach has found appli-... 

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