Molecular Dynamics at Constant Temperature and Pressure

1
Molecular Dynamics at Constant Temperature and
Pressure

• Section 6.7 in M.M.

2
Introduction

• Molecular mechanics simulations usually
  sample the microcanonical (constant NVE) ensemble
  .
• What if we are interested in some other
  ensemble such as the canonical (constant NVT) or
  isothermal-isobaric (constant NPT)?
• What if we are studying a system that gets
  too hot?
• Do a Monte Carlo calculation instead (canonical).
• Modify the molecular mechanics.

3
Scaling/Constraining Methods
Temperature is

• Simplest way to modify T
• Temperature depends on velocities so correct
  the velocities every step to give desired
  temperature.
• Multiply by
• Where TW is the temperature that you want and
  T(t) is the temperature at time t. Simple, but
  crude and may inhibit equilibration.

4
More Sophisticated S/C

• Encourage the temperature in the direction you
  want by coupling it to a heat bath. Have

• is the coupling parameter. If tdt the simpler
  form of scaling is recovered. Neither method
  samples the canonical ensemble.

5
More Sophisticated S/C

• Redefine equations of motion.
• Choose x such that
• To minimize the difference with Newtonian
  trajectories take
• This samples
• configurational part
• of canonical ensemble
• Note that it prevents changes in T but does not
  change it to a desired value

6
Example, S/C

• 10 atoms in a cell interacting via a
  Lennard-Jones Potential.

Simulate using leap-frog algorithm (6.3.1)
7
Example, S/C

• Simulate for a while

8
Example, S/C

• Obtain these properties (10,000,000 steps between
  65,0000
• and 650,001 not shown).

9
Example, S/C
Same system but with scaling to a temperature of
about 300.
10
Example, S/C
Same system but with scaling to a temperature of
about 300.
11
Stochastic Collisions

• Influence the system temperature by reassigning
  the velocity of a random particle (a
  collision). An element of Monte-Carlo.
• The new velocity is from the Maxwell-Boltzmann
  distribution corresponding to the desired TW.
• Between collisions sample a micro-canonical
  ensemble. It can be shown that overall the
  canonical ensemble is sampled.
• Collision frequency is important.
• Can also reassign some or all particle velocities.

12
Extended Systems

• Have a thermal reservoir coupled to the system.
• The reservoir has its own degree of freedom s and
  its own thermal inertia parameter Q.
• Energy is conserved in the total system and the
  micro-canonical ensemble of the total system is
  sampled.
• Two flavours
• 1.Nosé type
• 2.Hoover type

13
Nosé Method

• The extra degree of freedom s which scales
  the real velocities and time step
• s has its own kinetic and potential energies
  (f is the number of degrees of freedom)
• It can be shown that the partition function
  of this system is

14
Nosé Method

• Also,
• 
  
• for a given property A.
• Note that the total momentum and total
  angular momentum deviate from canonical by
  O(1/N).
• Q measures coupling between reservoir and
  system.It should not be too high (slow flow) or
  too low (oscillations).

15
Example of Nosé Method

• System made up of 108 argon atoms.
• S. Nosé Mol. Phys. 52, 255 (1984).

16
Hoover Method

• Start with the Nosé method and redefine the
  time variable
• Thus eliminate s from equations of motion
• Samples a canonical ensemble and is more
  gentle than straight scaling.

17
An Aside Ab Initio Molecular Dynamics (9.13.2-3)

• In the Car-Parinello method (e.g. PAW) Molecular
  Dynamics is performed using forces derived from
  QM.
• The nuclear and electronic degrees of freedom are
  relaxed simultaneously.
• When doing dynamics the electronic part must not
  heat up too much.
• Couple electronic and nuclear motions each to
  their own Nosé-Hoover thermostat.

18
Constant Pressure

• Pressure
• Can maintain constant pressure by changing
  V, the volume of the box. Long range corrections
  are important. Here f represents the forces.
• Volume changes can be large
• Gas in 20Å square box (volume 8000 Å3) has
  DVRMS18,100 Å3. Use a bigger box.

19
Scaling/Constraining

• Can encourage the pressure in the direction
  desired by scaling box size by c
• Can redefine equations of motion
• Where c is pretty ugly. Get

20
Extended Systems

• Have the system coupled to a piston .
• The piston has its own degree of freedom V and
  its own mass Q.
• Energy is conserved in the total system and the
  micro-canonical ensemble of the total system is
  sampled.

21
Anderson Method

• Variables are scaled.
• The piston has its own kinetic and potential
  energies.
• It can be shown that the time average of the
  trajectories derived equal the isoenthalpic-isobar
  ic ensemble average to O(N-2).

22
Extended Systems

• Again, the size of Q is important to avoid
  oscillations/slow exploration of phase space.
• Changing the box shape is a special case of this.
  Not so useful for liquids but good for solids.

23
Stochastic Methods

• None yet developed.

24
Constant Temperature and Pressure

• The isobaric-isothermal (constant NPT)
  ensemble is often of interest. Achieved by
  combing methods already described, e.g.
• Couple system with a piston then maintain
  temperature by the stochastic method including
  collisions with the piston.
• Redefine equations of motion to constrain T and P
  
• Here (cx) equals the previous
  definition of x and c is slightly less ugly than
  before.
• 3. Hoovers formulation

25
What Method to Use?

• Scaling is simple and easy and in the simplest
  case requires no parameters. Convergence may be a
  problem and do not sample cononical/isobaric/isoba
  ric-isothermal ensemble. Good for equilibration.
• Constraints a little more complicated but also
  require no parameters. Only keep T/P unchanged.
• Stochastic approach is more stable than scaling
  but method is no longer deterministic.
• Extended systems methods more complicated and
  require parameters. Nosé-Hoover thermostats
  enable the true canonical ensemble to be sampled.

26
Summary

• One may want to constrain/choose temperature
  and/or pressure in a molecular dynamics
  simulation for a number of reasons.
• The temperature can be fixed by a) scaling the
  velocities (partially or completely) or simply
  redefining the equations of motion so that T does
  not change, b) changing some or all of the
  velocities of the particles to a randomly
  selected member of the Maxwell- Boltzmann
  distribution of the desired T, c) couple the
  system to a heat bath
• Analagous methods exist to chose/maintain a
  constant pressure.
• Combinations of methods can be used to simulate a
  system at constant temperature and pressure.