RPMD (Ring Polymer Molecular Dynamics)
Introduction (correlation functions and Kubo transforms):
For a time-dependent quantum operator $\hat{A}(t)$ the auto-correlation function of the operator at time $t$ versus at time 0 can be written as:
$$
c_{AA}(t) \equiv \frac{\textrm{tr}\left( e^{-\beta H}\hat{A}(0)\hat{A}(t) \right)}{\textrm{tr}\left({e^{-\beta H}}\right)} \tag{1}.
$$
Applying the Kubo transformation to this, gives the Kubo-transformed auto-correlation function:
$$
\tilde{c}_{AA}(t) \equiv \frac{\int_0^\beta \textrm{tr}\left( e^{-(\beta - \lambda) H}\hat{A}(0) e^{-\lambda H }\hat{A}(t) \right)\textrm{d}\lambda}{\beta\, \textrm{tr}\left({e^{-\beta H}}\right)} \tag{2}.
$$
Remember from high school the chemical reaction rate $k$. It can be written in terms of the above two expressions! Reaction rates depend on temperature so we'll write $k(T)$:
\begin{align}
k(t) &=\frac{1}{Q_r(T)}\int_0^\infty c_{ff}(t)\textrm{d}t \tag{3}\\
&=\frac{1}{Q_r(T)}\int_0^\infty \tilde{c}_{ff}(t)\textrm{d}t. \tag{4}
\end{align}
Review of PIMD (Path Integral MD):
If we have $N$ atoms, and we treat each of them as an $n$-bead ring of artificial atoms (beads) we can make the approximation:
$$
{\small
\textrm{tr}\left(e^{-\beta H}\right) \approx \frac{1}{\left( 2\pi \hbar \right)^{3Nn}}\int \!\!\!\! \int \cdots \int e^{-\frac{\beta}{n} H_n\left(\mathbf{p}_1,\ldots,\mathbf{p}_{Nn},\mathbf{q}_1,\ldots,\mathbf{q}_{Nn}\right)} \textrm{d}^3\mathbf{p}_1\ldots \textrm{d}^3\mathbf{p}_n \textrm{d}^3\mathbf{q}_1\ldots \textrm{d}^3\mathbf{q}_n\tag{5},
}
$$
where $H_n$ is the Hamiltonian of the $N \times n$ beads representing $N$ atoms and the $n$ beads connected by harmonic spring potentials representing each of the $N$ atoms.
This approximation becomes exact when $n\rightarrow \infty$, which would mean we have an infinite-dimensional integral (known as a Feynman integral or "path integral", in this case actually a double-Feynman-integral or Feynman double-integral since there's two entirely different "path" sets over which a Feynman integral is being done).
The "RPMD" approximation:
It might sound bizarre, because really PIMD is MD on "ring polymers" (ring polymer just being another name for the set of beads representing each atom), so PIMD could be called RPMD. However when people use the term "RPMD" they are referring to this approximation:
\begin{align}{\tiny
\!\!\!\!\!\!\!\!\tilde{c}_{AA}(t) \approx \frac{\int \!\!\! \int \cdots \int e^{-\frac{\beta}{n} H_n\left(\mathbf{p}_1(t),\ldots,\mathbf{p}_{Nn}(t),\mathbf{q}_1(t),\ldots,\mathbf{q}_{Nn}(t)\right)}\hat{A}_n\left( \mathbf{q}_1(0),\ldots,\mathbf{q}_{Nn}(0)\right) \hat{A}_n \left( \mathbf{q}_1(t),\ldots,\mathbf{q}_{Nn}(t) \right) \textrm{d}^3\mathbf{p}_1(0)\ldots \textrm{d}^3\mathbf{p}_n(0) \textrm{d}^3\mathbf{q}_1(0)\ldots \textrm{d}^3\mathbf{q}_n(0)}{\int \!\!\! \int \cdots \int e^{-\frac{\beta}{n} H_n\left(\mathbf{p}_1,\ldots,\mathbf{p}_{Nn},\mathbf{q}_1,\ldots,\mathbf{q}_{Nn}\right)} \textrm{d}^3\mathbf{p}_1\ldots \textrm{d}^3\mathbf{p}_n \textrm{d}^3\mathbf{q}_1\ldots \textrm{d}^3\mathbf{q}_n }} \tag{6},
\end{align}
where $\hat{A}_n\left( \mathbf{q}_1(t),\ldots,\mathbf{q}_{Nn}(t)\right)$ involves for each of the $N$ atoms, an average over all $n$ of its beads at time $t$:
$$
{\small
\hat{A}_n\left( \mathbf{q}_1(t),\ldots,\mathbf{q}_{Nn}(t)\right) \equiv \frac{1}{n}\sum_{j=1}^n \hat{A}\left(
\mathbf{q}_{j}(t),\mathbf{q}_{n+j}(t),\mathbf{q}_{2n+j}(t),\ldots ,\mathbf{q}_{(N-1)n+j}(t) \right).\tag{7}
}
$$
Unlike PIMD which calculates a static property exactly in the limit of $n\rightarrow \infty$, RPMD approximates a function of time (the auto-correlation function) even in the limit as $n\rightarrow \infty$. However the approximation is exact (in the limit where $n\rightarrow \infty$) in some limits:
- infinitely high temperature,
- $t \approx 0$, (short-time limit),
- harmonic limit (where the $N$ atoms interact via harmonic potentials) if $A$ is linear,
- $\hat{A}=1$ (the identity operator).
Pros:
- It is relatively cheap to calculate (compared to exact real-time quantum dynamics). It costs only a bit more than doing classical MD several ($n$) times,
- it allows one to incorporate the effects of tunneling and zero-point-energy effects in an MD calculation.
Cons:
- "coherence" between paths in the Feynman integral is not taken into account,
- the formulation above does not give attention to how non-adiabatic effects would be treated. Several proposals have bee made for how to treat non-adiabatic effects, but no single proposal has stood out as a "gold standard" as far as I know.
- it's at least $n$ times more expensive than doing classical MD, so it's applicability is limited to cases where quantum mechanical effects of nuclei are important enough for it to be worth the extra cost.
