The Aims of Molecular Dynamics: Simulation of molecular dynamics allows us to obtain accurate information about the relationships between the bulk properties of matter and the underlying interactions among the constituent atoms or molecules in the liquid, solid or gaseous state. It provide a direct route from the microscopic details of the system (the masses of atoms, the interactions between them, molecular geometry, etc.) to the macroscopic properties of experimental interest (the equation of state, transport coefficients, structural order parameters, and so on). Conceptual basis: Molecular dynamics predicts atomic trajectories by direct integration of the equations of motion – Newton’s second law for classical particles – with appropriate specification of an inter-atomic potential and suitable initial and boundary conditions. Molecular dynamics simulation consists of the numerical, step-by-step, solution of the classical equations of motion, which for a simple atomic system may be written as:
Interaction forces: Interactions are specified by giving an expression for the potential energy of the interaction, so the force can be written as a gradient of the potential. Non-bonded Interactions:
Non-bonded interactions model flexible interactions between particle pairs. Two wellknown non-bonded interaction potentials are the Coulomb potential and the LennardJones potential.
Here qi and qj are the charges of particles i and j. The Coulomb force on particle i due to j is given by:
The Lennard-Jones potential is given by:
Where ∈ is a constant determining the depth of the potential well, and where Ďƒ determines the diameter of the particle.
This term models a strong repelling force at very short distances.
This term models an attractive force at very long interaction range.
The force of the Lennard-Jones interaction exerted on particle i i by particle j j is given by:
In addition, the non-bonded interactions between atoms are traditionally split into 1-body, 2-body, 3-body . . . terms:
Here u(r) term indicate the externally applied potential field or the effects of the container Walls. But this is neglected in case of fully periodic simulation of bulk systems.
Bonded interactions: Bonded interactions model rather strong chemical bonds, and are not created or broken during a simulation. The three most widely used bonded interactions are the covalent interaction, the bond-angle interaction, and the dihedral interaction. Covalent interaction:
This interaction may be thought of as a very stiff linear spring between i and j . The spring has a neutral length b0 and a spring constant Kb. i
j
Bond-angle interaction (Three particles Interaction):
With
This interaction may be thought of as a torsion spring between the lines i,j and j,k The spring has a neutral angle θ 0 and a spring constant K θ . i; j
Dihedral-angle interaction(Four body interaction):
Where δ and Φ 0 are constants.
Periodic Boundary Conditions:
Whether or not thebe is surrounded by a containing wal,molecules on the surface will experience quite different forces from molecules in the bulk.This surface effect can be overcome by implementing periodic boundary conditions.
The Molecular Dynamic Algorithm: To start the simulation we must assign initial positions, velocities and accelerations to all the particles in the system. The integration of the equations of motion is carried out using the Velocity Verlet integration algorithm,Verlet algorithm and Leap-frog algorithm. The Velocity Verlet Algorithm: There is no compromise on precision.
The Verlet Algorithm:
By the Taylor expansion of the coordinate of a particle we can write:
By adding these two equation we will get
The Verlet algorithm uses positions and accelerations at time t and the positions from time t-δt to calculate new positions at time t+δt. The Verlet algorithm uses no explicit velocities. The advantages of the Verlet algorithm are, i) it is straightforward, and ii) the storage requirements are modest . The disadvantage is that the algorithm is of moderate precision.
• • •
The algorithm should conserve energy and momentum. It should be computationally efficient. It should permit a long time step for integration.
STATISTICAL MECHANICS
Molecular dynamics simulations generate information at the microscopic level, including atomic positions and velocities. The conversion of this microscopic information to macroscopic observables such as pressure, energy, heat capacities, etc., requires statistical mechanics. The connection between microscopic simulations and macroscopic properties is made via statistical mechanics which provides the rigorous mathematical expressions that relate macroscopic properties to the distribution and motion of the atoms and molecules of the N-body system; molecular dynamics simulations provide the means to solve the equation of motion of the particles and evaluate these mathematical formulas.
Thethermo of a system is usually defined by a small set of parameters, for example, the temperature, T, the pressure, P, and the number of particles, N. Other thermodynamic properties may be derived from the equations of state and other fundamental thermodynamic equations. The microscopic state of a system is defined by the atomic positions, q, and momenta, p; these can also be considered as coordinates in a multidimensional space called phase space. For a system of N particles, this space has 6N dimensions. A single point in phase space, denoted by Γ, describes the state of the system. An ensemble is a collection of all possible systems which have different microscopic states but have an identical macroscopic or thermodynamic state.
The ensemble average is given by
Where
is the observable of interest and it is expressed as a function of the momenta, p, and the positions, r, of the system. The integration is over all possible variables of r and p.
The probability density of the ensemble is given by:
Where H is the Hamiltonian, T is the temperature, kB is Boltzmann’s constant and Q is the partition function. Partition function: In statistical mechanics the partition function Q is an important quantity that encodes the statistical properties of a system in thermodynamic equilibrium. It is a function of temperature and other parameters. Most of the aggregate thermodynamic variables of the system, such as the total energy, free energy, entropy, and pressure, can be expressed in terms of the partition function or its derivatives. There are actually several different types of partition functions, each corresponding to different types of statistical ensemble.Q is like to be in the form of
Another way, as done in an MD simulation, is to determine a time average of A, which is expressed as:
The Ergodic hypothesis states
Ensemble average = Time average
Now we have to define all the ensembles possible for a system and which one should be taken and all the thermodynamic connections with all the ensemble possible.
Principal ensembles of statistical thermodynamics: Different macroscopic environmental constraints lead to different types of ensembles, with particular statistical characteristics. The following are the most important: Microcanonical ensemble or NVE ensemble is an ensemble of systems, each of which is required to have the same total energy (i.e. thermally isolated). Canonical ensemble or NVT ensemble is an ensemble of systems, each of which can share its energy with a large heat reservoir or heat bath. The system is allowed to exchange energy with the reservoir, and the heat capacity of the reservoir is assumed to be so large as to maintain a fixed temperature for the coupled system. Grand canonical ensemble is an ensemble of systems, each of which is again in thermal contact with a reservoir. But now in addition to energy, there is also exchange of particles. The temperature is still assumed to be fixed.
Microcanonical ensemble: it is a theoretical tool used to describe the thermodynamic properties of an isolated system. It is also called the NVE ensemble because it describes a system with a fixed number of particles ("N"), a fixed volume ("V"), and a fixed energy ("E"). With N,V,E constant ,the system has fluctuating P and T.At each time step, the temperature is calculated from the kinetic energy as :
Τ=
1 ∑ mv(i)2 3ΝΚ i
Similarly the virial theorem is used for calculating pressure. Ρ=
ΝΚΤ 1 − ∑ ∨ 3∨ i
∑ j >i
r (ij )
∂U (ij ) ∂r (ij )
The key assumption of a microcanonical ensemble is that all accessible microstates are equally probable. Therefore the density function on the relevant region of phase space is constant, say it is 1 everywhere Let us call â„Ś(E) the number of micro-states corresponding to this value of the system's energy. The macroscopic state of maximal entropy for the system is the one in which all micro-states are equally likely to occur, with probability 1 / â„Ś(E), during the system's fluctuation.
Where S is the system entropy, and kB is Boltzmann's constant
Canonical ensemble:
In canonical ensemble N, V and T are fixed. Invoking the concept of the canonical ensemble, it is possible to derive the probability Pi that a macroscopic system in thermal equilibrium with its environment, will be in a given microstate with energy Ei according to the Boltzmann distribution:
Where
The temperature T arises from the fact that the system is in thermal equilibrium with its environment. The probabilities of the various microstates must add to one, and the normalization factor in the denominator is the canonical partition function:
Thermodynamic Connection: The partition function can be used to find the expected (average) value of any microscopic property of the system, which can then be related to macroscopic variables. For instance, the expected value of the microscopic energy E is interpreted as the microscopic definition of the
thermodynamic variable internal energy U, and can be obtained by taking the derivative of the partition function with respect to the temperature.
So now U will be:
The entropy can be calculated by:
which implies that
Where F is the free energy of the system. So we can write:
Having microscopic expressions for the basic thermodynamic potentials U (internal energy), S (entropy) and F (free energy) is sufficient to derive expressions for other thermodynamic quantities. Helmholtz free energy: (NVT) Internal energy:
Pressure:
Entropy:
Gibbs free energy: (NPT)
Enthalpy:
Constant volume heat capacity:
Constant pressure heat capacity:
Chemical potential:
Grand canonical ensemble: In grand canonical ensemble V, T and chemical potential are fixed. If the system under study is an open system, (matter can be exchanged), but particle number is not conserved, we would have to introduce chemical potentials, Âľj, j = 1,...,n and replace the canonical partition function with the grand canonical partition function:
Where Nij is the number of jth species particles in the ith configuration.
Grand potential:
Internal energy:
Particle number:
Entropy:
Helmholtz free energy:
The partition function itself is the product between pressure
Properties of "good" ensembles:
Representativeness:
and volume, divided by
The chosen probability measure on the phase space should be a Gibbs state of the ensemble, i.e. it should be invariant under time evolution. Ergodicity : The ergodicity requirement is that the ensemble average coincide with the time average. A sufficient condition for ergodicity is that the time evolution of the system is a mixing.
Limitations of molecular dynamics:
There are also significant limitations to MD. The two most important ones are: (a) Need for sufficiently realistic interatomic potential functions U: This depends on what is known fundamentally about the chemical binding of the system under study. Progress is being made in quantum and solid-state chemistry, and condensed-matter physics; these advances will make MD more and more useful in understanding and predicting the properties and behavior of physical systems. (b) Computational capabilities constraints: No computers will ever be big enough and fast enough. On the other hand, things will keep on improving as far as we can tell. Current limits on how big and how long are a billion atoms and about a microsecond in brute force simulation.
STEP BY GUIDE TO MOLECULAR DYNAMIC SIMULATION 1. We read in the parameters that specify the conditions of the run (e.g., initial temperature, number of particles, density, time step). 2. We initialize the system (i.e., we select initial positions and velocities). 3. We compute the forces on all particles. 4. We integrate Newton’s equations of motion. This step and the previous one make up the core of the simulation. They are repeated until we have computed the time evolution of the system for the desired length of time. 5. After completion of the central loop, we compute and print the averages of measured quantities, and stop.
STEPS: INITIALIZATION: The particle positions should be chosen compatiblewith the structure that we are aiming to simulate. This initial velocity distribution is Maxwellian neither in shape nor even in width. Subsequently, we shift all velocities, such that the total
momentum is zero and we scale the resulting velocities to adjust the mean kinetic energy to the desired value. We know that, in thermal equilibrium, the following relation should hold:
where vÎą is the Îą component of the velocity of a given particle. We can use this relation to define an instantaneous temperature at time t T(t):
The Force Calculation: If a given pair of particles is close enough to interact, we must computethe force between these particles, and the contribution to the potential energy.Suppose that we wish to compute the x-component of the force
Integrating the Equations of Motion: Now that we have computed all forces between the particles, we can integrate Newton’s equations of motion. Algorithms have been designed to do this like Verlet algorithm Velocity-verlet etc as described above in the different algorithm part.
THERMODYNAMIC PROPERTIES:After all these steps,we have to calculate all the thermodynamic properties from the ensemble averages but we hav to take care of one thing is that what are the intial parameters we use for the specific ensemble.
All the thermodynamic connections are described above for each ensemble.