Ring polymer molecular dynamics (RPMD)

Ring polymer molecular dynamics is a quantum dynamics methods that attempts to approximate Kubo-transformed real-time correlation functions ([15]).

The idea is to exploit the classical isomorphism that maps a quantum particle onto the extended phasespace of a classical ring polymer. It can be shown that the quantum partition function for a system can be manipulated such that it resembles the classical partition function of a system containing many replicas of the original particle joined to together with harmonic springs in a ring. In the limit of infinite beads or replicas in the ring polymer, the isomorphism becomes exact and it is possible to evaluate quantum expectation values by evaluating ensemble averages for the classical ring polymer system. This is referred to as the field of imaginary-time path integrals and the techniques used are Path Integral Monte Carlo (PIMC) and Path Integral Molecular Dynamics (PIMD) depending on whether molecular dynamics or Monte Carlo methods are used to explore the phasespace ([16]).

RPMD was proposed as a heuristic extension of imaginary-time path integrals to evaluate real-time dynamical quantities. To perform RPMD, it is necessary to solve Hamilton's equations for the ring polymer Hamiltonian:

\[H = \sum_\alpha^N \frac{1}{2} \mathbf{P}_\alpha^T \mathbf{M} \mathbf{P}_\alpha + \frac{1}{2} \omega_N^2 (\mathbf{R}_\alpha - \mathbf{R}_{\alpha+1})^T \mathbf{M} (\mathbf{R}_\alpha - \mathbf{R}_{\alpha+1}) + V(\mathbf{R}_\alpha)\]

where the ring polymer spring constant $\omega_N = 1 / \hbar\beta_N$ and $\beta_N = \beta / N$.

When the initial distribution is taken as the thermal ring polymer distribution and this Hamiltonian is used to generate configurations at later times, the correlation functions obtained can be used to approximate real-time quantum correlation functions.

Example

Let us perform some simple adiabatic ring polymer dynamics to get a feel for what the ring polymer dynamics looks like. We set up a 2D system for one hydrogen atom by giving the Free model 2 degrees of freedom and specify that the ring polymer should have 50 beads.

using NQCDynamics
using Unitful

atoms = Atoms([:H])
sim = RingPolymerSimulation(atoms, Free(2), 50; temperature=100u"K")

RingPolymerSimulation{Classical}:

  Atoms{Float64}([:H], [1], [1837.4715941070515])

  Free(2)
  with 50 beads.

We initialise the simulation with zero velocity and a random distribution for the ring polymer bead positions. For a real RPMD simulation you will use the thermal ring polymer distribution obtained from a PIMC or Langevin simulation but here for simplicity we use a normally distributed configuration.

u = DynamicsVariables(sim, zeros(size(sim)), randn(size(sim)))

Now we can run the simulation, for which we use the time interval 0.0 to 500.0 and a time step of dt = 2.5:

dt = 2.5
traj = run_dynamics(sim, (0.0, 500.0), u; output=OutputPosition, dt=dt)

[ Info: Performing 1 trajectory.
[ Info: Finished after 8.326674563 seconds.

We can visualise this ring polymer trajectory with a 2D scatter plot that shows how the ring polymer evolves in time. Here, we have joined the adjacent beads together with lines, with the end and start beads joined with a different color. This animation shows the cyclic nature of the ring polymer, and how every bead is connected to its two neighbours.

using Plots

rs = traj[:OutputPosition]

timestamps = 1:length(traj[:OutputPosition])
@gif for i in timestamps
    xs = rs[i][1,1,:]
    ys = rs[i][2,1,:]
    close_loop_x = [rs[i][1,1,end], rs[i][1,1,begin]]
    close_loop_y = [rs[i][2,1,end], rs[i][2,1,begin]]

    plot(
        xlims=(-3, 3),
        ylims=(-3, 3),
        legend=false
    )

    plot!(xs, ys, color=:black)
    scatter!(xs, ys)
    plot!(close_loop_x, close_loop_y)
end

Since this package is focused on nonadiabatic dynamics, you won't see much adiabatic RPMD elsewhere in the documentation, but it's useful to understand how the original adiabatic version works before moving on to the nonadiabatic extensions.