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Molecular Dynamics

Molecular dynamics revolutionized the field of realistic computer simulation of complex molecular systems and processes, including chemical reactions , by unifying molecular dynamics and electronic structure theory. Molecular dynamics operates in the continuum in contrast to lattice based methods such as cellular automata, which are spatially discrete.

Molecular dynamics methods treating the atoms and the molecules as classical particles have developed over the last three decades, in to an important discipline to obtain information about thermodynamics, structure and dynamical properties in condensed matter from pure simple liquids to studies of complex bio molecular systems in solution.

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Molecular Dynamics Simulation

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The development of computer simulations of particle models has been closely connected with the development of computers. The first article about simulation in molecular dynamics was written by Alder and Wainwright in 1957.

Molecular dynamics simulations with constant pressure or constant temperature were described in the beginning of the eighties. Also, more complex potentials with many body interactions were introduced quite early into simulation.

The assumptions hold in transition state theory simulations for the simulations results to be valid. They are as follows.
  1. A particle that has passed through the dividing surface at the transition state will always continue on to the product state. This assumption can be relaxed by the inclusion of a recrossing factor with a value less than one.
  2. The "reactant" state is in thermal equilibrium. That is, the system spends enough energy in the potential energy basin defining the "reactant" state to come to a thermal equilibrium before crossing the transition state dividing surface.
  3. The "reactant" state is in thermal equilibrium with the state of the system at the transition state. That is, the system obtains a Boltzmann distribution of velocities at the transition state.
  4. Classical mechanics applies to the system. Recently this assumption has also been relaxed due to advances in semi-classical quantum mechanics.

Main classification of simulation technique is molecular dynamics (MD) and Monte Carlo (MC).

Molecular dynamics (MD) is a computer simulation of the physical movements of atoms and molecules. In this process, atoms and molecules interact for a given time period, providing a view of the motion of the atoms.

The Molecular systems are made up of a vast number of particles, hence it is impossible to discover and study the properties of such systems analytically. Computer simulations are done to understand the properties of assembly of molecules in terms of their structure and the microscopic interactions between them.

Md is a process by which the atomic trajectories of a system of N particles is generated by numerical integration of Newton's equation (motion). This is done for a specific potential existing between the atoms with certain initial condition and boundary condition.

The level of interaction between molecules is guessed, and an exact prediction of bulk properties is done. The accuracy of predictions is subject to the limitations imposed by our computer budget. Simulations of the experiments (difficult to be carried out experimentally in labs) are done.

MD simulation consists of the numerical solution of the classical equations of motion, which may be stated as

F = ma

F = $\frac{dU}{dr}$

For this purpose we need to calculate the forces acting on the atoms, which are derived from potential energy U. In a M.D. simulation the forces between particles depend on particle positions and not on the velocities.

By calculating force, the acceleration of an individual atom in the system can be determined easily. Integration of the motion equations then gives a trajectory that describes the v, r and a of the particles varying with time.
The average value of properties can be determined using the trajectory. This method is deterministic, i.e., once the position and velocities of each atom are known, the state of the system can be predicted at any time.

Main Principle

The main principle of M.D. simulation given the system state S(t0)( the position r and velocity v of every particle (atom) in the system at time t0) subsequent states S(t0+ $\Delta$ t),S(t0+ 2$\Delta$t), are calculated using Newton’s law F= ma. For accurate results small time steps $\Delta$t have to be used. To calculate S(t0+ (n+1)$\Delta$t) from S(t0+n$\Delta$t), Fi(t0+n$\Delta$t) is calculated for every particle i.

[Here Fi(t0+ n$\Delta$t) : the sum of the forces on i as exerted by the other particles of the system at time t0+ n$\Delta$t]. For every particle i the force Fi(t0+ n$\Delta$t) is then integrated to get the new velocity vi(t0+ n$\Delta$t). Using this velocity, for every particle i the new position ri(t0+ (n + 1)$\Delta$t) can be calculated.

Design constraints

Design of a molecular dynamics simulation is constrained by the availability of computational power. Simulation size (n=number of particles), time step and total time duration are selected, so that the calculations can be finished within a reasonable period of time. The time allocated to the simulations should be in close harmony to the actual time of completion of the natural process, i.e., for the conclusions to be valid the time span should match the kinetics of the natural process. To obtain these simulations, several days to years are needed. Parallel algorithms could be used for load distribution among the CPU's.

Another factor that impacts total CPU time required by a simulation is the size of the time-step of integration. This is the time period between subsequent evaluations of the potential. The time step should be small enough to avoid discretization errors (i.e. smaller than the fastest vibration frequency in the system). Typical time-steps are in the order of 10-15s. This value may be extended by using algorithms, e.g., SHAKE, which fixes the vibrations of the fastest atoms (for e.g. hydrogen) into place.

Input files needed for MD simulation :

  • A starting structure
  • Details of the structural connections and atom types
  • A force field for the atom types
  • An input script for the MD program

Molecular Dynamics Simulation Methods

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An attractive feature of molecular dynamic simulation is that its techniques are simple and it imposes fewer assumptions than other numerical methods. In classical molecular dynamic simulation the only material property to be specified beforehand is the intermolecular potential function.

In other cases in which quantum effects are important, such as in electron transport or photon absorption empirical potential functions may be unavailable. Therefore, non classical or quantum molecular dynamics such as the ab-initio method is developed.

Classical molecular dynamic simulation

Classical molecular dynamic simulation methods can be divided into equilibrium molecular dynamics and non equilibrium molecular dynamics. Most of the work on molecular dynamic was based on equilibrium molecular dynamics and non equilibrium molecular dynamics has not been treated thoroughly.

However, because it can handle calculation of transport coefficient easily, non equilibrium molecular dynamics will receive increasing attention in the future.

Non classical molecular dynamic simulation

Any phenomena that involves electrons or photons cannot be treated easily by classical molecular dynamics simulation unless empirical equations can be obtained to describe dynamics of these particles. Usually interactions among electrons and photons have the quantum mechanical nature and must be obtained by solving the Schrodinger equation.

However limited by computational capacity quantum molecular dynamics can simulate a system with only about 100 molecules. Attracted by the fact that classical molecular dynamics can treat a relatively large number of molecules researchers attempted to combine quantum molecular dynamics and classical molecular dynamics in one simulation and such a technique has been successfully applied in simulating the chemical reactions.

Finite difference methods

The essential idea here is that the integration is broken down into many small levels, each separated by a fixed time Δ t. The total force on each particle in the configuration at a time t is calculated as the vector sum of its interaction with other particles. From the force we can determine the ‘a’ of the particles, which are then combined with the positions and velocities at a time (t + Δt). The force is assumed to be constant during the time step. The force on the particles is then calculated for their new positions, leading to new positions and velocities at a time (t + Δt).

Predictor-corrector integration methods

These methods form a general family of integration algorithms from which one can select a scheme that is correct to a given order.

  1. New positions , velocities , accelerations and higher-order terms are predicted
  2. The forces are evaluated at a the new positions to give acceleration a (t + t)
  3. The difference between the predicted and calculated acceleration is then used to correct the positions, velocities etc.

Molecular Dynamics Simulation Uses

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The following list includes somewhat random and far from complete assortment of ways in which md simulation is used.
  1. Fundamental studies: equilibrium, tests of molecular chaos, kinetic theory, diffusion, transport properties, size dependence, tests of models and potential functions.
  2. Phase transitions: first and second order, phase coexistence, order parameters, critical phenomena.
  3. Collective behavior: decay of space and time correlation functions, coupling of translational and rotational motion, vibration, spectroscopic measurements, orientational order, dielectric properties.
  4. Complex fluids: structure and dynamics of glasses, molecular liquids, pure water and aqueous solutions, liquid crystals, ionic liquids, fluid interfaces, films and monolayers.
  5. Polymers: chains, rings and branched molecules, equilibrium conformation, relaxation and transport processes.
  6. Solids: defect formation and migration, fracture, grain boundaries, structural transformations, radiation damage, elastic and plastic mechanical properties, friction, shock waves, molecular crystals, epitaxial growth.
  7. Bio-molecules: structure and dynamics of proteins, protein folding, micelles, membranes, docking of molecules.
  8. Fluid dynamics: laminar flow, boundary layers, rheology of non-Newtonian fluids, unstable flow.
  9. M.D. simulations are used to simulate experiments in physics and chemistry. It has been used to study simple gases, polymers, crystals, proteins, liquids, proteins in liquids etc.
  10. M.D. has been used to calculate mechanical properties like compressibility and tensile strength for polymers.
  11. M.D. is used to calculate the free energy of a reaction for drug design.
  12. M.D. is used to refined the data related to inter-atomic distances.
  13. MD is used to treat the collision cascades in the heat spike regime (the effects that energetic neutron and ion irradiation have on solids solid surfaces)
  14. M.D supports simulation of experiments under extreme conditions like high pressure and temperature.
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