Parts I-IV built the equilibrium side of the path-integral picture: PIMD and PIMC both sample the finite-\(P\) quantum Boltzmann ring polymer. RPMD starts from that same ensemble but uses the ring-polymer Hamiltonian as an approximate real-time dynamics for Kubo-transformed correlation functions. This note makes that transition explicit and checks it on the harmonic oscillator, where the \(q\)-\(q\) result is known analytically.

From Sampling to Dynamics

PIMD trajectories are sampling trajectories. Their fictitious time is not the physical quantum time. The configurational target is

\[ \rho_P(q) \propto \exp\left[ -\beta_P \sum_{j=1}^{P} \left\{ \frac{1}{2}m\omega_P^2(q_{j+1}-q_j)^2+V(q_j) \right\} \right], \qquad \omega_P=\frac{P}{\beta\hbar}. \]

Part IV sampled this distribution directly with Metropolis PIMC. RPMD keeps the same ring-polymer potential but adds canonical momenta and then runs microcanonical dynamics under

\[ H_P(q,p)= \sum_{j=1}^{P} \left[ \frac{p_j^2}{2m} +\frac{1}{2}m\omega_P^2(q_{j+1}-q_j)^2 +V(q_j) \right], \]

with phase-space weight \(e^{-\beta_P H_P}\). The momenta are not sampled by PIMC; after the PIMC path is accepted, each bead momentum is drawn from the Maxwell factor \(p_j\sim N(0,m/\beta_P)\).

RPMD Correlation Estimator

The target real-time object is the Kubo-transformed correlation

\[ C_{AB}^{K}(t)= \frac{1}{\beta Z} \int_0^\beta d\lambda\, \operatorname{Tr} \left[ e^{-(\beta-\lambda)\hat H} \hat A e^{-\lambda\hat H} e^{i\hat Ht/\hbar} \hat B e^{-i\hat Ht/\hbar} \right]. \]

RPMD replaces that quantum real-time evolution by classical evolution of the ring-polymer Hamiltonian:

\[ C_{AB}^{\mathrm{RPMD}}(t)= \frac{1}{Z_P} \int dq\,dp\, e^{-\beta_P H_P(q,p)} A_P(q)\,B_P(q_t), \]

where \(q_t\) is the RPMD trajectory initialized at \((q,p)\). For a position autocorrelation, the natural linear estimator is the centroid

\[ q_c=\frac{1}{P}\sum_{j=1}^{P}q_j, \qquad C_{qq}^{\mathrm{RPMD}}(t)= \left\langle q_c(0)q_c(t)\right\rangle_{H_P}. \]

Limits and Failure Modes

  • For static equilibrium averages, the ring-polymer ensemble is exact as \(P\to\infty\); RPMD is not needed for that part.
  • For \(P=1\), RPMD reduces to ordinary classical molecular dynamics on the physical potential.
  • For a harmonic oscillator with linear position operators, RPMD reproduces the exact Kubo-transformed \(q\)-\(q\) correlation.
  • At short times, RPMD has the correct leading Kubo time derivatives for many position-dependent observables.
  • For anharmonic systems, nonlinear operators, tunneling splittings, coherence, and phase-sensitive quantum interference, RPMD is an approximation rather than an exact quantum dynamics.
  • The internal ring-polymer modes are artificial. They can create resonance artifacts, which is why Part III discussed free-step stability and why TRPMD later damps internal modes for spectra.

SHO Analytic Target

For \(V(q)=m\omega_0^2q^2/2\), the ring-polymer normal-mode frequencies are

\[ \Omega_k=\sqrt{\omega_0^2+\omega_k^2}, \qquad \omega_k=2\omega_P\sin\left(\frac{k\pi}{P}\right). \]

The centroid mode has \(\Omega_0=\omega_0\), so the RPMD centroid follows the same sinusoidal motion as a classical harmonic oscillator. The exact Kubo-transformed position autocorrelation is

\[ C_{qq}^{K}(t)= \frac{1}{\beta m\omega_0^2} \cos(\omega_0t). \]

This is not the same as the ordinary equal-time quantum variance \(\langle q^2\rangle=(\hbar/2m\omega_0)\coth(\beta\hbar\omega_0/2)\). For the parameters below, the Kubo \(C_{qq}^{K}(0)\) target is \(1/32=0.03125\), while the bead-coordinate quantum variance is about \(0.125\).

Workflow

  1. Use the same oscillator scale as the previous PIMD notes: \(m=\hbar=1\), \(\beta=2\), \(\omega_0=4\), and \(P=32\).
  2. Run six independent local Metropolis PIMC chains and retain 48,000 post-burn-in ring-polymer paths.
  3. For each retained path, sample bead momenta from \(p_j\sim N(0,m/\beta_P)\).
  4. Transform each RPMD initial condition to ring-polymer normal modes.
  5. Propagate the harmonic RPMD modes exactly and evaluate \(\langle q_c(0)q_c(t)\rangle\) from the trajectory ensemble.
  6. Compare the numerical curve with \(C_{qq}^{K}(t)=\cos(4t)/32\).

Result

The PIMC initial ensemble gives \(\langle q_c^2\rangle=0.03104\pm0.00073\), consistent with the exact Kubo \(C_{qq}(0)=0.03125\). After attaching RPMD momenta and propagating the trajectories, the full time series has RMS error \(1.75\times10^{-4}\) and maximum absolute error \(2.45\times10^{-4}\) over \(0\le t\le6\).

PIMC-initialized RPMD harmonic oscillator q-q correlation benchmark
PIMC-initialized RPMD benchmark for the harmonic oscillator with \(m=\hbar=1\), \(\beta=2\), \(\omega_0=4\), and \(P=32\). Left: horizontal axis is the centroid coordinate \(q_c=P^{-1}\sum_jq_j\); vertical axis is probability density. The blue histogram is the PIMC centroid distribution from 48,000 retained paths, and the black curve is the exact Gaussian centroid marginal with variance \(1/(\beta m\omega_0^2)=0.03125\). Right: horizontal axis is RPMD time; vertical axis is \(C_{qq}(t)=\langle q_c(0)q_c(t)\rangle\). The black solid curve is the exact Kubo result \(\cos(4t)/32\), and the red dashed curve is the PIMC-initialized RPMD estimate. Their near overlap is expected because harmonic RPMD is exact for this linear correlation.

Code used in this note

  • rpmd_sho_correlation.py runs the PIMC sampler, samples RPMD momenta, propagates harmonic normal modes, writes the figure, and saves the numerical tables.
  • rpmd-sho-correlation.csv stores the time grid, RPMD estimate, exact Kubo value, and pointwise error.
  • rpmd-sho-summary.csv stores the PIMC acceptance rate, centroid variance, \(C(0)\), RMS error, and maximum absolute error.

References

  • Craig and Manolopoulos, J. Chem. Phys. 121, 3368 (2004).
  • Habershon, Manolopoulos, Markland, and Miller, Annu. Rev. Phys. Chem. 64, 387 (2013).
  • Tuckerman, Statistical Mechanics: Theory and Molecular Simulation.

PIMD series map

  1. Part I: Ring-Polymer Basics - path-integral isomorphism, normal modes, estimators, and the SHO convergence test.
  2. Part II: NVT Thermostats and GLE - PILE, GLE, PIGLET-style covariance shaping, and sampling diagnostics.
  3. Part III: Cayley Stability - why exact free ring-polymer propagation can be unstable and how the Cayley step fixes it.
  4. Part IV: Path-Integral Monte Carlo - local Metropolis bead moves, constrained-PIMC context, and an SHO observable mean.
  5. Part V: From PIMD to RPMD - PIMC-initialized RPMD trajectories and an exact SHO \(q\)-\(q\) Kubo benchmark.
  6. Part VI: Normal-Mode Free Propagation - normal-mode and bead-coordinate free spring propagation, stability limits, and subcycling costs.