Ii • (Oi (Oi X / • (j)i = N i (4.21)
4.3.5 Temperature initialization and rescaling
The initial kinetic energy o f an MD simulation is set by sampling the particles’ velocities from the Maxwell-Boltzmann distribution at the required temperature. The effective temperature o f the system is then calculated from the ensemble average o f its kinetic energy. However, the initial configuration is usually far from equilibrium and has a high potential energy. As the simulation proceeds, much o f the potential energy is converted to kinetic energy, thus raising the temperature o f the system.
It is therefore necessary to scale the kinetic energy o f the particles during the equilibration period. This is typically achieved by multiplying the linear and angular velocities at periodic intervals by a factor of:
s = (4,36)
where g is the number o f degrees o f freedom, T is the required temperature and is the instantaneous value o f the kinetic energy. By repeatedly setting the ‘instantaneous’ temperature to the correct value, the kinetic energy is made to approach its required value.
An MD simulation w ith scaling does not generate equilibrium particle trajectories, so the scaling must be switched o ff before any calculations o f thermodynamic averages are performed. The simulation can then proceed along the A, V,E ensemble, where the total number o f particles, the simulation cell volume and the total energy is held constant, or along an A ,F,T ensemble where the temperature, rather than the total energy, is fixed. The latter is achieved by thermostat methods:
the Nosé-Hoover method^^’^^ which couples the system to a heat bath, and the Gaussian thermostat^^ which replaces the Newton-Euler equations by variants in which the kinetic energy is conserved. The N^VJE ensemble was used in the simulations o f liquid ammonia and lithium-ammonia solutions, which are presented in Chapter 6.
4.3.6 Structural and dynamical properties from MD simulations
The MOLDY code enables configurations o f the system to be saved at periodic intervals. Particular information such as particle positions and molecule quaternions may then be extracted from these binary dump files, in order that time-averaged radial distribution functions and the mean square displacements o f different particle types may be calculated.
4.3.6.1 Radial distribution functions
The radial distribution function (RDF) is one o f the most important structural quantities which characterize a liquid system. In the case o f a molecular system, the partial RDF for atoms a and p is defined as:^^
where p is the molecular number density, V is the volume o f the simulation cell, N is the total number o f particles, r is the vector between the centre o f mass o f molecules i and j , and ria,rjp are the vectors jfrom the centres o f mass o f molecules i j to the atomic sites or,/? respectively. The angled brackets denote a spherical average in addition to the usual configurational average.
In the simulation, the RDF is evaluated
where fihis{b) is the accumulated number o f atom pairs per bin, b is the number o f the histogram bin, 5r is the bin width (hence r = 6& ) and r is the number o f timesteps over which the atom pair distances have been counted.
4.3.6.2 Mean square displacements and diffusion coefficients
The mean square displacement vs. time o f a particular species can be calculated from the particle positions extracted fi-om the binary dump files. For a species o f N particles, the mean square displacement is calculated via:
2 1 2
<1 r{t) - r(0) p > = — 2 ^ 2 ^ k „ (< + io ) - ^ (<o ) I (4.39) ^ «=1 tr,
where (t) is the position o f particle n at time t, and Nt is the total number o f timesteps over which the sum is performed for a particular t. The diffusion coefficient o f that species can then be calculated firom the gradient o f a plot o f the mean square displacement against time, using the Einstein relation:
{ \ r { t ) - r { Q ) Ÿ ) = 6Dt (4.40)
The radial distribution functions and diffusion coefficients extracted firom the MD simulations serve as a basis for comparison with experimental data, as detailed in Chapter 6. The experimental data can also be used as a stringent test for the potential functions used in the MD simulations.
4.4 References
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