Molecular equation of motion

Molecular dynamics calculations are more time-consuming than Monte Carlo calculations. This is because energy derivatives must be computed and used to solve the equations of motion. Molecular dynamics simulations are capable of yielding all the same properties as are obtained from Monte Carlo calculations. The advantage of molecular dynamics is that it is capable of modeling time-dependent properties, which can not be computed with Monte Carlo simulations. This is how diffusion coefficients must be computed. It is also possible to use shearing boundaries in order to obtain a viscosity. Molec-... 

By integrating the Newtonian Equations of Motion, Molecular Dynamics simulations are able to describe the behavior of particles in a certain system within the observed period of time. The interaction of the atoms is described by the potential energy fxmction of the given force field [e.g. Amber (Cornell et al, 1995), CHARMM (Brooks et al., 1983), GROMOS (Scott et al., 1999), OPLS (Jorgensen Rives, 1988)]. Nowadays, there is an ongoing effort to ameliorate these parameters in a need for models being as less artificial as possible. 

Two simulation methods—Monte Carlo and molecular dynamics—allow calculation of the density profile and pressure difference of Eq. III-44 across the vapor-liquid interface [64, 65]. In the former method, the initial system consists of N molecules in assumed positions. An intermolecule potential function is chosen, such as the Lennard-Jones potential, and the positions are randomly varied until the energy of the system is at a minimum. The resulting configuration is taken to be the equilibrium one. In the molecular dynamics approach, the N molecules are given initial positions and velocities and the equations of motion are solved to follow the ensuing collisions until the set shows constant time-average thermodynamic properties. Both methods are computer intensive yet widely used. 

The alternative simulation approaches are based on molecular dynamics calculations. This is conceptually simpler that the Monte Carlo method the equations of motion are solved for a system of A molecules, and periodic boundary conditions are again imposed. This method pennits both the equilibrium and transport properties of the system to be evaluated, essentially by numerically solvmg the equations of motion... 

Molecular dynamics consists of the brute-force solution of Newton s equations of motion. It is necessary to encode in the program the potential energy and force law of interaction between molecules the equations of motion are solved numerically, by finite difference techniques. The system evolution corresponds closely to what happens in real life and allows us to calculate dynamical properties, as well as thennodynamic and structural fiinctions. For a range of molecular models, packaged routines are available, either connnercially or tlirough the academic conmuinity. 

Ryckaert J-P, Ciccotti G and Berendsen H J C 1977 Numerical integration of the Cartesian equations of motion of a system with constraints molecular dynamics of n-alkanes J. Comput. Phys. 23 327-41... 

Molecular dynamics tracks tire temporal evolution of a microscopic model system tlirough numerical integration of tire equations of motion for tire degrees of freedom considered. The main asset of molecular dynamics is tliat it provides directly a wealtli of detailed infonnation on dynamical processes. 

In this minimal END approximation, the electronic basis functions are centered on the average nuclear positions, which are dynamical variables. In the limit of classical nuclei, these are conventional basis functions used in moleculai electronic structure theoiy, and they follow the dynamically changing nuclear positions. As can be seen from the equations of motion discussed above the evolution of the nuclear positions and momenta is governed by Newton-like equations with Hellman-Feynman forces, while the electronic dynamical variables are complex molecular orbital coefficients that follow equations that look like those of the time-dependent Hartree-Fock (TDHF) approximation [24]. The coupling terms in the dynamical metric are the well-known nonadiabatic terms due to the fact that the basis moves with the dynamically changing nuclear positions. 

Molecular dynamics simulations ([McCammon and Harvey 1987]) propagate an atomistic system by iteratively solving Newton s equation of motion for each atomic particle. Due to computational constraints, simulations can only be extended to a typical time scale of 1 ns currently, and conformational transitions such as protein domains movements are unlikely to be observed. 

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 ... 

In order to solve the classical equations of motion numerically, and, thus, to t)btain the motion of all atoms the forces acting on every atom have to be computed at each integration step. The forces are derived from an energy function which defines the molecular model [1, 2, 3]. Besides other important contributions (which we shall not discuss here) this function contains the Coulomb sum... 

Given this effective potential, it is possible to define a constant temperature molecular dynamics algorithm such that the trajectory samples the distribution Pg(r ). The equation of motion then takes on a simple and suggestive form... 

Extending time scales of Molecular Dynamics simulations is therefore one of the prime challenges of computational biophysics and attracted considerable attention [2-5]. Most efforts focus on improving algorithms for solving the initial value differential equations, which are in many cases, the Newton s equations of motion. 

As is well known. Molecular Dynamics is used to simulate the motions in many-body systems. In a typical MD simulation one first starts with an initial state of an N particle system F = xi,..., Xf,pi,..., pf) where / = 3N is the number of degrees of freedom in the system. After sampling the initial state one numerically solves Hamilton s equations of motion ... 

Among the main theoretical methods of investigation of the dynamic properties of macromolecules are molecular dynamics (MD) simulations and harmonic analysis. MD simulation is a technique in which the classical equation of motion for all atoms of a molecule is integrated over a finite period of time. Harmonic analysis is a direct way of analyzing vibrational motions. Harmonicity of the potential function is a basic assumption in the normal mode approximation used in harmonic analysis. This is known to be inadequate in the case of biological macromolecules, such as proteins, because anharmonic effects, which MD has shown to be important in protein motion, are neglected [1, 2, 3]. 

Molecular dynamics (MD) studies the time evolution of N interacting particles via the solution of classical Newton s equations of motion. 

Molecular dynamics conceptually involves two phases, namely, the force calculations and the numerical integration of the equations of motion. In the first phase, force interactions among particles based on the negative gradient of the potential energy function U,... 

A molecular dynamics simulation samples the phase space of a molecule (defined by the position of the atoms and their velocities) by integrating Newton s equations of motion. Because MD accounts for thermal motion, the molecules simulated may possess enough thermal energy to overcome potential barriers, which makes the technique suitable in principle for conformational analysis of especially large molecules. In the case of small molecules, other techniques such as systematic, random. Genetic Algorithm-based, or Monte Carlo searches may be better suited for effectively sampling conformational space. 

There are many algorithms for integrating the equations of motion using finite difference methods, several of which are commonly used in molecular dynamics calculations. All algorithms assume that the positions and dynamic properties (velocities, accelerations, etc.) can be approximated as Taylor series expansions ... 

A key feature of the Car-Parrinello proposal was the use of molecular dynamics a simulated annealing to search for the values of the basis set coefficients that minimise I electronic energy. In this sense, their approach provides an alternative to the traditioi matrix diagonalisation methods. In the Car-Parrinello scheme, equations of motion ... 

Mesoscale simulations model a material as a collection of units, called beads. Each bead might represent a substructure, molecule, monomer, micelle, micro-crystalline domain, solid particle, or an arbitrary region of a fluid. Multiple beads might be connected, typically by a harmonic potential, in order to model a polymer. A simulation is then conducted in which there is an interaction potential between beads and sometimes dynamical equations of motion. This is very hard to do with extremely large molecular dynamics calculations because they would have to be very accurate to correctly reflect the small free energy differences between microstates. There are algorithms for determining an appropriate bead size from molecular dynamics and Monte Carlo simulations. 

To generate characteristic velocities and bring a molecular system to equilibrium at the simulation temperature, atoms are allowed to interact with each other through the equations of motion. Eor isothermal simulations, a temperature bath scales velocities to drive the system towards the simulation temperature. Scaling occurs at each step of a simulation, according to equation 28. 

Langevin dynamics simulates the effect of molecular collisions and the resulting dissipation of energy that occur in real solvents, without explicitly including solvent molecules. This is accomplished by adding a random force (to model the effect of collisions) and a frictional force (to model dissipative losses) to each atom at each time step. Mathematically, this is expressed by the Langevin equation of motion (compare to Equation (22) in the previous chapter) ... 

A classical molecular dynamics trajectory is simply a set of atoms with initial conditions consisting of the 3N Cartesian coordinates of N atoms A(X, Y, Z ) and the 3N Cartesian velocities (v a VyA v a) evolving according to Newton s equation of motion ... 

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