An Overview of Molecular Simulation - N. Allan

An overview of the current state of molecular simulation with examples of special interest taken from the literature.

Introduction to force fields - 
 

Statistical Mechanics 1 
In this lecture we will begin with an important question: why bother with statistical thermodynamics? We will progress to basic statistical quantities and concepts such as averages, fluctuations and correlations and how to use them in practice to calculate the physical properties of systems. This will lead us to the determination of the true statistical error for system properties obtained by simulation. We will apply these ideas to commonly calculated properties such as diffusion, radial distribution functions and velocity autocorrelation, while also examining the physical meaning of these properties. We will conclude with a look at distribution functions: how they arise and what they mean.

Statistical Mechanics 2 
In the second lecture we shall look at more theoretical aspects of statistical mechanics. Beginning with the Lagrange and Hamiltonian description of classical mechanics we shall progress to the idea of phase space and the concept of a probability distribution function. This will be followed by basic applications (and associated mathematical manipulations) of the distribution function to obtain various physical properties of a system. We will examine the common ensembles (NVE, NVT and NPT) and discuss their application and interrelation. Finally we shall look at time dependence, beginning with the Liouville Equation and its connection with other time dependent equations. We shall conclude with the fluctuation-dissipation theorem.

Monte Carlo 1 - 
Basics: The system. Random sampling. Importance sampling. Detailed balance. Metropolis algorithm in the canonical ensemble. Isothermal-isobaric ensemble. Grand-canonical ensemble. Which ensemble?

Monte Carlo 2 - 
Practicalities: Finite-size effects. Random number generators. Tuning the acceptance rate. Equilibration. Configurational temperature. Ergodicity and free-energy barriers. Measuring ensemble averages. Examples (showing ensemble independence for the Lennard-Jones fluid)

Monte Carlo 3 -
(Free) Energy Barriers: Quasi non-ergodicity. Vapour-liquid phase transition as an example. Removing the interface by Gibbs ensemble MC. Free-energy barrier in the grand-canonical ensemble. Multicanonical preweighting. Histogram reweighting. Parallel tempering

Molecular Dynamics 1 - 
Molecular dynamics: the basic methodology. Integration algorithms and their derivation. Static properties: thermodynamics and structure. Dynamic properties: correlation functions and collective properties

Molecular dynamics 2 - 
Practical aspects of molecular dynamics - Verlet neighbour list, link cell algorithm. Calculating pressure: the virial theorem and the thermodynamic method. Estimating statistical errors: the blocking method. Symplectic algorithms and the Tuckerman-Berne-Martyna approach. Extended systems: canonical (NVT) and isothermal-isobaric (NPT) ensembles.

Molecular dynamics 3 
Rigid Bodies, SHAKE, RATTLE.

Free energy methods 1 - J. Anwar
Free energy, chemical potential & thermodynamics. Applications. Essential statistical mechanics. Ensemble averages, probability distributions & simulations. Free energy, the challenge. Particle insertion & removal. Energy density distributions. The perturbation method.

Free energy methods 2 - J. Anwar
Review essential statistical mechanics. Thermodynamic integration. Potential of mean force calculations. Umbrella sampling. Absolute free energies. Free energy of liquids.Free energy of solids.

Optimization Methods - 
The energy landscape, geometrical optimisation and saddle points. Minimisation methods (steepest descent, conjugate gradient, genetic algorithm). Saddle-points (transition state theory, harmonic theory, nudged elastic band, dimer method).

Long timescale methods - 
Long timescales simulations - the problems. Transition state theory and kinetic Monte Carlo. Temperature accelerated hyperdynamics. Metadynamics.

 

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