The molecular dynamics method212,213can be classified into two main general forms: one for systems at equilibrium, and another one for systems away from equilibrium. Equilibrium molecular dynamics is typically applied to an isolated system containing a fixed number of molecules N, in a fixed volume V and, because of the isolated feature of the system, the total energyEis also constant.
Here the total energyEis the sum of the molecular kinetic and potential energies.
Then, the variablesN,V andEdetermine the thermodynamic state.
An NVE-molecular dynamics, the atom positions rN are obtained by solving Newton’s equations of motion:
Fi(t) =m¨ri(t) =−∂U(rN)
∂ri
(2.38) WhereFiis the force onicaused by theN−1other molecules, the dots indicate total time derivatives, andmis the molecular mass. The previous equation relates the force with the intermolecular potential energy. Integration once yields the atomic momenta, and twice gives the atomic positions.
Integration for several thousand times produces individual atomic trajectories from which time averagehAican be computed to set macroscopic properties.
hAi= lim
At equilibrium this average cannot depend on the initial timet0. Since positions and momenta are obtained, the time average represents both static properties, such as thermodynamics and dynamics properties, such as transport coefficients.
This is based on theergodic hypothesis, which makes the assumption that the average obtained by following a small number of particles over a long time is equivalent to averaging over a large number of particles for a short time. This
2.8 - Ab Initio Molecular Dynamics (AIMD) 63 implies that a time average over a single particle is equivalent to an average of a large number of particles at any given time snapshot.
The motion’s equation is usually calculated with theVerlet algorithm, which gives accurate approximation of the atomic trajectories if the∆tis small. Another way to approach the dynamical problem is by means of theLagrangian, where the equation of motion for each coordinate can be expressed in terms of:
d The atomic motions inMicrocanonicalenvironment are related to the temperature, which means that in a system in equilibrium at temperatureT, the velocities follow a Maxwell-Boltzmann distribution. In the NVE-type simulations, the temperature and pressure will fluctuate. The total energy can be calculated from the positions and velocities, as follows. The temperature of the system is proportional to the average kinetic energy:
hEKi= 1
2(3Natoms−Nconstraints)kT (2.42) Since the kinetic energy is the difference between the total energy and the potential energy,Ekwill vary significantly, and the temperature will be calculated as an average value with an associated fluctuation.
On the other hand, it is possible also to generate NVT or NPT ensembles by MD techniques by modifying the velocities or positions in each time step. The instant value of the temperature is given by the average of the kinetic energy, as indicated above. If this is different from the desired temperature, all velocities can be scaled by a factor of(Tdesired/Tactual)1/2in each time step to achieve the desired temperature. The system may be coupled to aheat bath, which gradually adds or removes energy to the system with a suitable time constant, this procedure is calledthermostat. The kinetic energy of the system is again modified by scaling the velocities, but the rate of heat transfer is controlled by a coupling parameterτ.
dT dt = 1
τ(Tdesired−Tactual) (2.43)
64 Chapter 2 - Theoretical Background The most commonheat bath, considered as a truecanonical ensemble, is the so-called Nosé-Hoover method, where the heat bath is considered an integral part of the system and assigned dynamic variables, which are evolved on an equal footing with the other variables.
The pressure can similarly be held constant by coupling to a pressure bath.
Instead of changing the velocities of the particles, the volume of the system is changed by scaling all coordinates according to the following equation.
dP dt =1
τ(Pdesired−Pactual) (2.44) Thesebarostatmethods are again widely used in molecular dynamics, but do not produce strictly correct ensembles. The pressure may alternatively be maintained by a Nosé-Hoover approach in order to produce correct ensembles.
The idea proposed by Nosé was to reduce the effect of an external system, acting as a heat reservoir, to an additional degree of freedom. This heat reservoir controls temperature or pressure of the given system, where it fluctuates around a target value. The thermal interaction between the heat reservoir and the system results in the exchange of the kinetic term between them.
Molecular dynamics (MD) simulations generate trajectories in phase-space by treating the nuclei classically and integrating Newton’s or Hamilton’s equations of motion numerically. Conventionally, the forces in the system are derived from a potential energy function which is ideally a good approximation to the true potential energy of the system. It is often the case, though, that an accurate potential function is not available, especially when it is not even clear physically what the form of that potential function should be; metal clusters are a good example of such a case. In order to perform MD simulations for such system, we need an alternate means of calculating internuclear forces. In ab initio molecular dynamics (AIMD) the potential energy of the system is calculated using quantum mechanics. The main drawback of AIMD is its computational cost.