Matter Modeling Asked on December 4, 2021

In similar spirit to recent questions on Quantum Monte Carlo, *ab-initio* Molecular Dynamics, types of SCF, and others, I would like to ask:

**What are the types of Quantum Molecular Dynamics (QMD)?**

As I have recently learned, Quantum Molecular Dynamics is different to AIMD and MD through the fact that in QMD, the *nuclei* are treated quantum mechanically.

I would like to ask here, if we can summarize the types of QMD in a few paragraphs:

- PIMD (Feynman Path Integral Molecular Dynamics)
- CMD (Centroid Molecular Dynamics)
- RPMD (Ring Polymer Molecular Dynamics) [link to answer]
- *Feel free to let me know of other methods or include them as an answer!

Also, it would be great if we limit one QMD method per answer (and per person).

**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:

$$ {small textrm{tr}left(e^{-beta H}right) approx frac{1}{left( 2pi hbar right)^{3Nn}}int !!!! int cdots int e^{-frac{beta}{n} H_nleft(mathbf{p}_1,ldots,mathbf{p}_{Nn},mathbf{q}_1,ldots,mathbf{q}_{Nn}right)} textrm{d}^3mathbf{p}_1ldots textrm{d}^3mathbf{p}_n textrm{d}^3mathbf{q}_1ldots textrm{d}^3mathbf{q}_ntag{5}, } $$

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 $nrightarrow infty$, 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_nleft(mathbf{p}_1(t),ldots,mathbf{p}_{Nn}(t),mathbf{q}_1(t),ldots,mathbf{q}_{Nn}(t)right)}hat{A}_nleft( 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}^3mathbf{p}_1(0)ldots textrm{d}^3mathbf{p}_n(0) textrm{d}^3mathbf{q}_1(0)ldots textrm{d}^3mathbf{q}_n(0)}{int !!! int cdots int e^{-frac{beta}{n} H_nleft(mathbf{p}_1,ldots,mathbf{p}_{Nn},mathbf{q}_1,ldots,mathbf{q}_{Nn}right)} textrm{d}^3mathbf{p}_1ldots textrm{d}^3mathbf{p}_n textrm{d}^3mathbf{q}_1ldots textrm{d}^3mathbf{q}_n }} tag{6}, end{align}

where $hat{A}_nleft( mathbf{q}_1(t),ldots,mathbf{q}_{Nn}(t)right)$ involves for each of the $N$ atoms, an average over all $n$ of its beads at time $t$:

$$ {small hat{A}_nleft( mathbf{q}_1(t),ldots,mathbf{q}_{Nn}(t)right) equiv frac{1}{n}sum_{j=1}^n hat{A}left( mathbf{q}_{j}(t),mathbf{q}_{n+j}(t),mathbf{q}_{2n+j}(t),ldots ,mathbf{q}_{(N-1)n+j}(t) right).tag{7} } $$

Unlike PIMD which calculates a static property exactly in the limit of $nrightarrow infty$, RPMD *approximates* a function of time (the auto-correlation function) even in the limit as $nrightarrow infty$. However the approximation is exact (in the limit where $nrightarrow infty$) 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.

Answered by Nike Dattani on December 4, 2021

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