The Feynman Path Integral
For a single particle in potential $V(x)$, the canonical partition function is expressed as a functional integral over all paths $x(\tau)$ in imaginary time $\tau \in [0, \hbar\beta]$:
\[Z = \oint \mathcal{D}x(\tau)\,\exp\!\left(-\frac{1}{\hbar}\int_0^{\hbar\beta}\left[\frac{m\dot{x}^2}{2}+V(x(\tau))\right]d\tau\right)\]where $\beta = 1/k_BT$. Discretising imaginary time into $P$ slices of width $\varepsilon = \hbar\beta/P$ maps the quantum path integral onto a classical ring polymer with beads ${x_1,\ldots,x_P}$:
\[Z \approx \left(\frac{mP}{2\pi\hbar^2\beta}\right)^{P/2}\int\prod_{k=1}^P dx_k\,\exp\!\left(-\beta V_{RP}\right)\]Ring Polymer Effective Potential
The ring polymer potential couples neighbouring beads harmonically and applies the physical potential to each bead:
\[V_{RP} = \sum_{k=1}^{P}\left[\frac{m\omega_P^2}{2}(x_k-x_{k+1})^2 + \frac{1}{P}V(x_k)\right]\]with $\omega_P = P/(\hbar\beta)$ the harmonic spring frequency and periodic boundary conditions $x_{P+1} = x_1$. In the $P\to\infty$ limit, the classical ring polymer exactly reproduces the quantum partition function.
Ring Polymer MD and Tunnelling
RPMD evolves the ring polymer beads under classical equations of motion using fictitious momenta $p_k$, yielding real-time correlation functions and rate constants. The centroid $\bar{x} = P^{-1}\sum_k x_k$ corresponds to the physical particle position. Quantum tunnelling manifests through configurations where the ring polymer spans classically forbidden regions.
| Method | Nuclear Quantum Effects | Cost vs Classical MD |
|---|---|---|
| Classical MD | None | $1\times$ |
| PIMD (static) | ZPE, tunnelling (equilibrium) | $P\times$ |
| RPMD | ZPE, tunnelling (dynamics) | $P\times$ |
| CMD | ZPE, approximate tunnelling | $P\times$ |
A typical production calculation uses $P = 32$–$64$ beads per atom, making PIMD roughly two orders of magnitude more expensive than classical MD but essential for hydrogen-containing systems at temperatures below 500 K.