Download Path-Integral Molecular Dynamics at Thermal Equilibrium

Path-Integral Molecular
Dynamics at Thermal
Equilibrium
Tom Markland
Department of Chemistry
Stanford University
[email protected]
Recap
• Yesterday Nicolas introduced molecular dynamics and Monte Carlo ideas. Many
of these concepts will be used today.
 These lectures will concentrate on the additional considerations needed
when performing a PIMD simulation which are not present in a classical
simulation.
• Florent and Joe then introduced the idea of path integrals and motivated the
ideas with some applications.
 Today we will talk more about the “nuts and bolts” that go into performing
PIMD simulations.
 The objective is to introduce ideas on how to initialize, evaluate forces,
thermostat and evolve PIMD systems.
 This will bring us into contact with lots of area upcoming in this workshop
such as colored noise, reducing the number of beads and CMD/RPMD.
Derivation
The quantum mechanical partition function is given by,
Insert n-1 complete sets of position Eigenstates:
For each matrix element:
Perform integral
over momentum
Putting it all together gives:
This form can be sampled by Monte Carlo method. But for MD we need momenta!
Introducing Momenta
Consider our result for the matrix element
To introduce momenta we use the
standard Gaussian integral:
Which allows us to write,
Using this identity we can introduce momentum associated with bead k.
Note that this is just the reverse of what is done in classical statistical mechanics when one
wants to sample the configuration integral by Monte Carlo:
Configuration
Integral
Introducing Momenta
With this insertion the quantum mechanical partition function becomes
We now define a ring polymer Hamiltonian,
So the partition function can be written as,
Note: For a particle
moving in a 1 dimensional
potential the path integral
isomorphism yields a
classical problem in ndimensions.
Path Integral Molecular Dynamics trajectories are created by integrating Hamilton’s classical
equations of motion,
Path Integral Isomorphism
Path integral molecular dynamics gives the
exact static properties of a quantum
mechanical system.
• Classical system at temperature T
• Path integral MD system
Classical Hamiltonian Harmonic springs between
for each “bead”
neighboring “beads”
• Each bead evolves at temperature nT
• Mapping converges to give the exact
quantum mechanical result as n is
increased
• n determined by how “quantum” the
problem is
Path integral molecular dynamics is simply an extended form of classical mechanics
Introducing Momenta
More general form of Path Integral Molecular Dynamics Hamiltonian
For completeness we note that nowhere in our insertion of the momenta was it required that the
mass assigned to each bead be the physical mass. Hence we can write a more general PIMD
Hamiltonian:
Mass matrix
These masses can be altered to achieve
efficient sampling.
Note: This mass cannot be changed!
Vector of bead momenta
With this arbitrary choice of mass matrix the partition function is given by,
Any reasonable choice of the mass matrix (real-symmetric and positive definite) will give the
same equilibrium averages. Only the dynamics will be changed.
The choice of mass matrix is one of the main differences between the CMD
and RPMD models of quantum dynamics (see later in Workshop).
Properties of the Ring Polymer
Some terminology and properties of the Ring Polymer
• Radius of gyration – the spread in imaginary time. For a free particle the root mean square
radius of gyration is:
De Broglie Wavelength of the particle.
• Bead to bead
distance. For a free
particle the average is:
• Centroid: The centre
of the polymer.
Note: distance between
beads decreases as number
of beads increases.
The overall object is referred to as an Imaginary Time Path or a Ring Polymer.
Properties of the Ring Polymer
• Normal modes – the free ring polymer potential can be diagonalized analytically.
Beads – Cartesian Coordinates
Beads (Cartesian Coordinates)
7 beads
Fourier
Transform
Normal modes
Normal mode Coordinates
7 normal modes
Doubly
degenerate
pairs of
normal
modes.
Zero frequency
normal mode is
the centroid.
Initializing the Simulation
Basic simulation scheme
Initial Structure
From known
crystal structure or
starting from
lattice and
equilibrating.
Get energy/forces
Pair/Empirical/
Ab initio
Evolve system:
Monte Carlo
Molecular Dynamics
Repeat until
average of
observables are
converged.
Calculate properties
energy, pressure,
structure etc.
10
Initial Momenta
Sampling Initial Momenta
Momentum part is of form,
Gaussian Distribution
Hence we can sample the momentum of each bead from a Gaussian distribution with:
• Like classical sampling except at a
temperature n times higher.
• Allow beads to explore more regions of
space than a classical particle.
Initial Positions
Assume we know where to place our particles in a classical simulation e.g. we have run a
classical simulation previously or have crystal structure etc.
How do we initialize the bead positions?
Two options
1.) Start all beads at same position and equilibrate
As polymers expand under NVE conditions this
will cause the system temperature to drop
dramatically so strong thermostatting is needed.
2.) Sample from the free ring polymer distribution
This gives a spread consistent with free particle distribution. In practice this will be slightly too
extended.
How to do this?
For free particle the only potential is the harmonic spring terms between beads. In normal
mode representation it can be written as a sum of n independent terms,
Gaussian with
average 0 and:
Initializing the System
Summary
N = number of particles
n = number of beads
• To initialize path integral molecular dynamics simulations requires specification
of 3Nn positions and 3Nn momenta compared to only 3N for each in classical
simulations.
• The 3N momenta can be sampled from the Maxwell Boltzmann distribution.
• For the positions it is helpful to think of the ring polymer centroid as the
“classical” coordinate which can be initialized using the 3N classical positions
(not exact but often a decent starting guess).
Centroid
• With this choice of centroid positions the
remaining 3N(n-1) positions can be either sampled
from the free ring polymer position distribution
(generally an overestimate) or with all beads
started at the centroid (a massive underestimate).
Both the methods to initialize the positions are approximate so equilibration is
needed (with a thermostat) before properties can be extracted.
Evaluating the forces
Basic simulation scheme
Initial Structure
From known
crystal structure or
starting from
lattice and
equilibrating.
Get energy/forces
Pair/Empirical/
Ab initio
Evolve system:
Monte Carlo
Molecular Dynamics
Repeat until
average of
observables are
converged.
Calculate properties
energy, pressure,
structure etc.
14
Evaluating the forces
Multidimensional generalization
For a system of N distinguishable particles,
The PIMD Hamiltonian is,
Sum over particles
Potential energy: could be from empirical potential or ab
Bead index
initio calculations (Born Oppenheimer approximation).
Particle index
Sum over beads
and the partition function is,
Properties can be extracted from (see later),
Where,
Evaluating the forces
Interactions between ring polymers representing different particles
Particle i
Particle j
Evaluating the forces
Since each bead only interacts with its corresponding copies the number of force calculations
needed is linear in the number of beads required, n.
But recall:
How large does n need to be for convergence?
Frequency
Take the simplest possible case – 1D Harmonic Oscillator:
Potential energy for path integral is:
In a harmonic potential the normal mode frequencies are shifted to
Path integral solution as n∞ and
T 0 gives correct zero-point energy
of
.
Free particle normal mode frequency
Evaluating the forces
Using the normal mode representation the energy of the harmonic oscillator can be shown to be,
with
The exact solution is trivially shown to be,
Hence the fractional error as the number of beads, n, is increases is,
Plot Result:
Energy:
Convergence
is order:
Commonly used criteria
for convergence,
Error % for
Energy = 10%
Heat Capacity = 100%
Heat Capacity:
Evaluating the forces
A way to view the convergence criterion
Harmonic oscillator
levels spacing
Thermal energy of bead
• n increases the effective temperature (nT) until each bead is classical.
Can we reduce the force evaluations needed?
Spectrum of liquid water :
High frequency intra-molecular motion (stretches and bends)
Low frequency inter-molecular motion
Clearly wasteful to evaluate the intermolecular forces 32 times!
Evaluating the forces
Ring Polymer Contraction
Normal modes (11)
(keep n’=5 lowest)
Contracted
ring polymer
(n’=5 beads)
Full ring polymer
(n=11 beads)
Fourier
Transform
a.) Markland and Manolopoulos, JCP (2008)
Inverse
Transform
b.) Markland and Manolopoulos, CPL (2008)
Evaluating the forces
Ring Polymer Contraction: Water at 300K
Method
ΔVintra / %
ΔVinter / %
ΔT / %
ΔD / %
Beads for
Ewald
1+3
0.03
1.7
1.5
9.0
3+5
0.03
0.6
0.6
1.2
Beads
for LJ
5+7
0.00
0.3
0.3
0.8
7 bead
31
0.3
7.6
1.3
In all RPC runs the bends and stretches were evaluated on all 32 beads
• Vintra and Vinter are the intra and intermolecular contributions to the potential energy
• T is the virial estimate of quantum contribution to kinetic energy and is a direct probe
of the error in the approximation of the forces :
• D is the RPMD diffusion coefficient calculated from the velocity auto-correlation function
Evaluating the forces
Ring Polymer Contraction
• We can improve efficiency further by also
splitting the electrostatic interactions.
σ
σ/Å
ΔVintra / %
ΔVinter / %
ΔT / %
ΔD / %
3
0
0.4
0.5
0.5
4
0
0.02
0.1
0.8
5
0
0.05
0.05
0.3
Evaluating the forces
Ring Polymer Contraction
•
Classical effort can be achieved in the limit
of large system size where the Ewald sum
dominates.
•
For typical system sizes only 2-3 times the
cost of a classical simulation.
Force-force correlation of
imaginary time path:
fintra
finter
Inter force weakly
correlated along
polymer.
Note scale
Inter force highly
correlated along
polymer.
Evaluating the forces: Other Efficient Approaches
Treat some particles classically e.g. only one “important” H atom quantized.
• Mixed quantum/classical.
e.g. Mol. Phys. 103, 203,
JCP 134, 074112
Converge using less beads:
• Higher order path integrals discretization (e.g. J. Phys. Soc. Jpn. 53, 3765 ).
• Colored noise thermostats (e.g. JCP 134, 084104).
Evaluating the forces: Other Efficient Approaches
Summary
• For PIMD of distinguishable particles the forces are evaluated between
corresponding beads on each ring polymer.
• Standard PIMD converges as 1/n2 where n is the number of beads.
• A rough guess of the number of beads needed can be obtained using:
• Some properties converge much faster than other with increasing number of beads
e.g. radial distribution function converges fast but heat capacity very slowly.
• Method exists to greatly reduce the cost of path integral simulations.
Evolving the Polymers
Basic simulation scheme
Initial Structure
From known
crystal structure or
starting from
lattice and
equilibrating.
Get energy/forces
Pair/Empirical/
Ab initio
Evolve system:
Monte Carlo
Molecular Dynamics
Repeat until
average of
observables are
converged.
Calculate properties
energy, pressure,
structure etc.
26
Evolving the Polymers
Frequencies in a PIMD simulation
For a ring polymer in a harmonic potential of frequency ω the
normal mode frequencies are:
Normal mode frequency for free ring polymer.
Hence for a polymer of n beads the highest frequency is (l=n/2),
• The highest frequency present in the system increases as more beads are used.
If we use a typical convergence criteria:
This gives a rough value for the maximum frequency in a PIMD
simulation of a potential with frequency ω.
: where “a” determines
how accurate the
calculation is and is
typically ~1
Evolving the Polymers
Frequencies in a PIMD simulation
Largest ring
polymer frequency
Frequency of the
physical system.
• For typical convergence (a=1) the highest frequency in a PIMD simulation is
than would be present in a classical simulation of that potential.
larger
• The time step must be small enough to accurately integrate the highest frequency present
and so for a naïve implementation the time-step required will be
smaller.
Hence for a simulation of liquid water the highest frequency is
~8000cm-1 rather than 3500cm-1 classically.
Methods to allow bigger time-steps:
1.) Scale the normal mode masses so they all oscillate at the same frequency.
2.) Multiple time-scale molecular dynamics: use smaller time-steps for bead forces.
3.) Use an integrator where the free ring polymer is evolved exactly.
Integrators
How do we develop integrators?
• Consider the classical Hamiltonian,
• Perform Trotter factorization of the propagator.
Total Liouville
operator
Liouville operator
for Hamiltonian HV
Liouville operator
for Hamiltonian HT
Operator
Operation on particle i
This choice of splitting gives the Velocity Verlet Scheme
Integrators
What the Trotter factorization of the propagator says is that we need to be able to
split our total Hamiltonian into a sum of Hamiltonians where each one can be
solved exactly.
Full Hamiltonian Not
solvable for arbitrary
choice of potential, V.
Evolving a free particle.
Exactly solvable:
Evolving a stationary (dq/dt=0)
particle. Exactly solvable:
Integrators
A PIMD Velocity Verlet Scheme
For classical velocity Verlet integrator:
Evolve momentum by half timestep at constant position.
Move free particle through
full time-step Δt. Particles
have moved so update forces.
Evolve momentum using new
forces by half time-step at
constant position.
For PIMD velocity Verlet integrator:
Evolve bead momenta by half
time-step at constant position.
Move free ring polymer
through full time-step Δt.
Particles have moved so update
forces.
Evolve momentum using new
forces by half time-step at
constant position.
Evolving the Polymers
This scheme is equivalent to splitting the Hamiltonian as:
Ring polymer
Hamiltonian
Free ring polymer Hamiltonian
The factorization of the propagator is therefore,
Half time-step
evolution under
potential.
Potential Energy arising from the physical
forces (the potential due to the PI springs
terms are in the other Hamiltonian)
Full time-step
Half time-step
evolution of free evolution under
ring-polymer.
potential.
For the scheme to work we need to be able to solve the evolution
under each sub-Hamiltonian exactly.
Evolving the Polymers
Evolution under
potential:
Just like for classical simulation can be easily solved to give:
Evolution of free
ring polymer:
In Cartesian coordinates evolving the free ring polymer is not straightforward.
• Transform to normal modes:
The Hamiltonian of a set of uncoupled harmonic oscillators!
Textbook harmonic
oscillator solution:
Evolving the Polymers
In summary the overall evolution is:
Kick from potential (not including
ring polymer harmonic potential)
Sum over beads
Normal
mode
momentum
Transform from Cartesian
coordinates to normal modes.
Evolve free ring
polymer normal
modes.
Sum over normal modes
Bead
momentum
Orthogonal
transformati
on matrix
Transform back from normal
modes to Cartesian
coordinates.
Kick from potential (not including
ring polymer harmonic potential)
The transformation to normal modes can be done using a fast Fourier transform for efficiency.
Evolving the Polymers
Notes on this integration scheme:
1.) The free ring polymer is evolved exactly so for a free particle will give the exact for result for
any choice of time-step.
2.) The analytic free ring polymer evolution avoids the need to use small time-steps to integrate
the very fast ring polymer internal frequencies. This allows time-steps to be used which are
similar to those possible in the corresponding classical simulation.
3.) However, despite the exact evolution of the free ring polymer we have still performed a
Trotter factorization of the propagator so despite the exact free ring polymer evolution the ring
polymer normal mode frequencies can still contribute to resonance problems which limit the
maximum time-step.
For PIMD velocity Verlet integrator:
Evolve bead momenta by half
time-step at constant position.
Move free ring polymer
through full time-step Δt.
Particles have moved so update
forces.
Evolve momentum using new
forces by half time-step at
constant position.
Staging
We have just seen how using the normal modes of the ring polymer to diagonalize the
potential allows us to make efficient integrators:
Uncoupled harmonic oscillators
NM Transform
Bead coordinates and momenta
Normal mode coordinates and momenta
• Staging variables(a.) provide another way to diagonalize the potential.
Staging coordinate
Bead coordinate
Ring polymer spring term becomes,
Staging
transform
a.) Tuckerman, Berne, Martyna and Klein JCP, 99, 2796 (1993)
Evolving the Polymers
Summary
• The large range of frequencies in PIMD simulations causes simple integration of
the equations of motion to be inefficient.
• This can be alleviated by using integrators which evolve the free ring polymer
exactly.
• To evolve the free ring polymer exactly one must transform to the normal modes
of the ring polymer or staging coordinates.
• Exact evolution of the free ring polymer does NOT prevent resonance problems
so the highest ring polymer frequencies do decrease the largest time-step which can
be used in some cases.
Coffee Break
Simulation
Basic simulation scheme
Initial Structure
From known
crystal structure or
starting from
lattice and
equilibrating.
Get energy/forces
Pair/Empirical/
Ab initio
Evolve system:
Monte Carlo
Molecular Dynamics
Repeat until
average of
observables are
converged.
Thermostat!
Calculate properties
energy, pressure,
structure etc.
39
Thermostats
Difficulties with path integral thermostatting:
• Highly non ergodic(a.).
• Thermostatting essential to obtain correct averages.
• The classical replicas are linked by high frequency harmonic springs.
• Much larger frequency range present than in the equivalent classical system:
e.g. for a simulation of liquid water the
highest PIMD frequency is ~8000cm-1
rather than 3500cm-1 classically.
(a.) Thirumalai and Berne, Ann. Rev. Phys. Chem. 37,401 (1986)
Thermostats
Thermostats
Local (Massive) : couple to
individual degrees of freedom
Global : couple to total kinetic
energy of a set of particles.
Stochastic
Deterministic
Stochastic
Deterministic
Andersen
(BGK)
Massive Nose
Hoover
Global
Stochastic
Rescaling
Nose Hoover
Langevin
(White Noise)
Coloured
Noise (GLE)
Berendsen
Andersen Thermostat
Andersen Thermostat
A safe choice to implement in a PIMD code since there is
little that can go wrong.
Approach:
1.) Evolve system under NVE conditions.
2.) Every time-step resample the momentum of the system
with a probability P from the Maxwell distribution (see
initializing momentum section).
3.) The probability controls the rate at which the velocity
decorrelates:
Time-step
Velocity
correlation time
for free particle.
Between these resampling “collisions” the motion is NVE
so the dynamics can be use to obtain RPMD
approximations to time correlation functions.
Thermostats
For a harmonic oscillator it can be shown that for a harmonic oscillator of frequency, ω the
most efficient choice of thermostat parameter is:
Resampling
probability for
Andersen
Frequency of
oscillator
Langevin friction coefficient:
“Most-efficient” here means the lowest
total Hamiltonian correlation time:
Hence for a given choice of the Andersen thermostat resampling probability or Langevin
thermostat friction only one frequency is optimally thermostatted and only a narrow range of
frequencies are efficiently thermostatted.
PIMD systems have a wide range of frequencies. So how can we create a thermostat
which thermostats them all efficiently?
Exploit out knowledge of the frequencies present in the system!
We know the frequencies for a free ring polymer:
(a.) Ceriotti, Parrinello, Markland and Manolopoulos JCP 133 124104 (2010)
Thermostats
Since in the normal mode representation we know the free ring polymer frequencies we can
use the criteria:
to select a friction for each which maximizes the efficiency.
Fourier
Transform
Connect a white noise Langevin
thermostat tuned to target
normal mode frequency:
Centroid mode oscillates at
physical frequency of system, ω.
Pick either global or massive
thermostat.
A Nose Hoover scheme like this stochastic one exists(a.)  see later in talk.
(a.) Tuckerman, Berne, Martyna and Klein, JCP, 99, 2796 (1993)
Thermostats
Integration scheme for Path Integral Langevin Equation (PILE) Thermostat
Thermostat evolution
NVE evolution (see earlier)
The thermostat evolution step is:
Thermostat evolution
The friction on the centroid is chose as for
a classical simulation of the same system.
Transform to normal mode
momentum.
Apply Langevin thermostat
with optimized friction for
that normal mode.
Transform to Cartesian
momentum.
Conserved Quantity
• NVE dynamics conserved the total PIMD Hamiltonian (i.e. the Energy).
• Langevin dynamics does not conserve the Hamiltonian since the system can give or take
energy from the bath.
• However if one keeps an account of the kinetic energy change each time the thermostat
operation is performed one can define a “conserved” quantity for stochastic dynamics (a.).
For example for a standard classical Hamilonian of form:
The conserved quantity would be,
Energy change due
to thermostat.
• This provides a useful way to check the
time-step chosen is reasonable.
(a.) Bussi and Parrinello, Phys. Rev. E. 75, 056707 (2007)
Thermostats
Nosé-Hoover Chain Thermostats
•
Nosé-Hoover chains provide another efficient approach to
thermostatting classical and PIMD systems.
•
The idea is to connect each degree of freedom in the system to
additional degrees of freedom which can exchange energy with the
system.
•
The important difference with NH thermostats and Langevin is that
the chain is explicitly evolved under its own equations of motion.
•
NH chains generally provide less disturbance to the system’s natural
dynamics than white noise Langevin thermostats.
•
However a NH chain of length=1 does not produce the correct
canonical distribution. Hence one typically uses a chain of length 4.
•
This adds a lot of extra degrees of freedom to the system e.g. 256
water molecules with 32 beads for each and 4 chains:
4 x 32 x 256 x 3 = 98,304 extra degrees of freedom
4 chains
Degrees of freedom in water system
Thermostats
Integration scheme for Nose Hoover Path Integral Thermostat
NHC Thermostat evolution
NVE evolution (see earlier)
NHC Thermostat evolution
• As for PILE scheme one can select the
masses of the Nose Hoover chains for
optimal sampling efficiency of the free
ring polymer normal modes.
• NH chains are therefore connect to the
normal modes of the ring polymer.
Equations for the thermostat evolution once in the normal mode representation can be
found in: (a.) Martyna, Klein, Tuckerman JCP, 99, 2796 (1993)
(b.) Statistical Mechanics: Theory and Molecular Simulation, Tuckerman
(c.) Ceriotti, Parrinello, Markland and Manolopoulos JCP 133 124104 (2010)
Finally we note that :
• The PIMD system and chains have a conserved Hamiltonian.
• To conserve this Hamiltonian requires accurate integration of both the PIMD system and
the Nose Hoover Chain degrees of freedom.
(a.) Martyna, Klein, Tuckerman JCP, 99, 2796 (1993)
Thermostats
Colored Noise Thermostats (Generalized Langevin Equation = GLE)
•
Recent work2 has shown how colored noise can be used to thermostat molecular dynamics
simulations.
•
The colored noise thermostat is implemented by adding an additional set of stochastically
evolved “s” momenta to the system.
•
Colored noise gives the flexibility to allow the thermostat to couple in different ways to
different frequencies determined by the memory function:
Different colored noise profiles allow for many applications ranging from approximating nuclear
quantum effects to probing the vibrational spectrum of solids.
(a.) M. Ceriotti, G. Bussi, M. Parrinello. J. Chem. Theory Comput. 6, 1170 (2010)
Thermostats
Colored Noise Thermostats
Application to thermostat path integral simulations.
• The GLE (colored noise) thermostat can effectively thermostat a wide range of frequencies so
it is no necessary to apply it in the normal mode representation.
• Here GLE is being used to thermostat a full PIMD simulation. GLE can also be used to
accelerate the convergence of PIMD simulations with number of beads used (see later talk).
(a.) Ceriotti, Parrinello, Markland and Manolopoulos JCP 133 124104 (2010)
Global Thermostats
Local Thermostat
e.g. White Noise Langevin,
Massive Nose Hoover
Global Thermostat
e.g. Global Nose Hoover, Stochastic
Velocity Rescaling
Couple to individual particle momenta
Couple to total system kinetic energy
Global thermostat engender a smaller difference in the particle dynamics
Global Thermostats
Stochastic Velocity Rescaling
Scales all momenta in system by α:
Scaling coefficient simply depends on system’s total kinetic energy and the strength of coupling.
• Reduces to local Langevin when Nf=1
• Diffusion is much less affected for a
strongly coupled global thermostat.
• Global thermostats can act as good
thermostats for the centroid.
Global
Langevin dynamics disrupts diffusion a high coupling strengths.
(a.) Bussi and Parrinello, Comput. Phys. Commun. 179 26 (2008) (b.) Bussi, Donadio and Parrinello, JCP 126, 014101 (2009)
Relative Efficiency: Liquid Water
Liquid water at 300K
Correlation time of property A
Decreasing the correlation time of a property means
that for a given simulation length the statistical
uncertainty can be reduced.
is the correlation time of the virial kinetic energy.
is the correlation time of the potential energy.
WNLE is the White Noise Langevin Equation
GLE is the colored noise thermostat.
PILE-L is the White Noise Langevin Equation
which is targeted on each normal mode using the
free particle frequency.
NHC-L is the targeted Nose Hoover scheme.
G indicates global coupling to the centroid.
(a.) Ceriotti, Parrinello, Markland and Manolopoulos JCP 133 124104 (2010)
Strength of thermostat coupling
Relative Efficiency: Liquid Water
Liquid water at 300K
How much are the dynamics changed?
is the correlation time of the square dipole
moment of the simulation box.
is the center-of-mass diffusion coefficient
of a water molecule.
WNLE is the White Noise Langevin Equation
GLE is the colored noise thermostat.
PILE-L is the White Noise Langevin Equation
which is targeted on each normal mode using the
free particle frequency.
NHC-L is the targeted Nose Hoover scheme.
G indicates global coupling to the centroid.
Stronger
thermostat
coupling
Relative Efficiency: H in Palladium
H in Palladium at 350K
H in Pd in much more harmonic and
inhomogeneous than water so
provides a further test.
is the correlation time of the radius of
gyration of the H atom.
is the correlation time of the virial
kinetic energy of the H atom.
WNLE is the White Noise Langevin Equation
GLE is the colored noise thermostat.
PILE-L is the White Noise Langevin Equation
which is targeted on each normal mode using the
free particle frequency.
NHC-L is the targeted Nose Hoover scheme.
G indicates global coupling to the centroid.
Absolute value to
avoid correlation
being cancelled by
anticorrelation.
Relative Efficiency: H in Palladium
H in Palladium at 350K
is the correlation time of the H atom
potential energy.
is the diffusion coefficient of a H atom.
WNLE is the White Noise Langevin Equation
GLE is the colored noise thermostat.
PILE-L is the White Noise Langevin Equation
which is targeted on each normal mode using the
free particle frequency.
NHC-L is the targeted Nose Hoover scheme.
G indicates global coupling to the centroid.
Initializing dynamics trajectories
RPMD and CMD are intended to provide approximations to quantum dynamics and not to
efficiently sample the quantum phase space.
Hence it is most efficient to launch RPMD and CMD trajectories using this scheme:
1.) Run a trajectory using an efficient thermostat scheme (blue)
2.) Pick configurations and momenta from the thermostatted trajectory and launch RPMD
or CMD trajectories from them (red lines).
3.) Ideally the separation in time between each should be determined by the correlation
time of the properties in the system.
(a.) Perez, Muser and Tuckerman JCP, 130, 184105 (2009)
PIMD Thermostats
Summary
• PIMD trajectories (particularly those with large numbers of beads) are non-ergodic
and so thermostats are needed to obtain correct equilibrium averages.
• The large range of frequencies in PIMD makes it challenging to simultaneously
thermostat all modes with optimal efficiency.
• By exploiting knowledge of the free ring polymer frequencies one can create
targeted Nose Hoover or Langevin schemes to thermostat path integrals.
• These targeted schemes do not give a indication of what to use on the centroid but
this can be estimated from efficient values used in classical simulations.
• Colored noise (GLE) thermostats can be give efficient sampling without the need
to transform to the normal modes.
• Dynamics trajectories should be launched from configurations taken from a PIMD
trajectory using an efficient thermostat.
Simulation
Basic simulation scheme
Initial Structure
From known
crystal structure or
starting from
lattice and
equilibrating.
Get energy/forces
Pair/Empirical/
Ab initio
Evolve system:
Monte Carlo
Molecular Dynamics
Repeat until
average of
observables are
converged.
Calculate properties
energy, pressure,
structure etc.
59
Any questions?
Thanks for listening.
Questions?